Quantum Field Theory II

Integrable system

From Wikipedia, the free encyclopedia




[ltr]In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.
In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical systems, there is the notion of Liouville integrability. More generally, in differentiable dynamical systems integrability relates to the existence of foliations by invariant submanifolds within the phase space. Each of these notions involves an application of the idea of foliations, but they do not coincide. There are also notions ofcomplete integrability, or exact solvability in the setting of quantum systems and statistical mechanical models. Integrability can often be traced back to the algebraic geometry of differential operators.[/ltr]

• 1 Frobenius integrability (overdetermined differential systems)
  
• 2 General dynamical systems
  
• 3 Hamiltonian systems and Liouville integrability
  
• 4 Action-angle variables
  
• 5 The Hamilton–Jacobi approach
  
• 6 Solitons and inverse spectral methods
  
• 7 Quantum integrable systems
  
• 8 Exactly solvable models
  
• 9 List of some well-known classical integrable systems
  
• 10 Notes
  
• 11 References
  
• 12 External links
  

[ltr]

Frobenius integrability (overdetermined differential systems)[edit]
A differential system is said to be completely integrable in the Frobeniussense if the space on which it is defined has a foliation by maximal integral manifolds. The Frobenius theorem states that a system is completely integrable if and only if it generates an ideal that is closed under exterior differentiation. (See the article on integrability conditions for differential systems for a detailed discussion of foliations by maximal integral manifolds.)
General dynamical systems[edit]
In the context of differentiable dynamical systems, the notion ofintegrability refers to the existence of invariant, regular foliations; i.e., ones whose leaves are embedded submanifolds of the smallest possible dimension that are invariant under the flow. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation. This concept has a refinement in the case ofHamiltonian systems, known as complete integrability in the sense ofLiouville (see below), which is what is most frequently referred to in this context.
An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describeevolution equations that either are systems of differential equations orfinite difference equations.
The distinction between integrable and nonintegrable dynamical systems thus has the qualitative implication of regular motion vs. chaotic motionand hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in exact form.
Hamiltonian systems and Liouville integrability[edit]
In the special setting of Hamiltonian systems, we have the notion of integrability in the Liouville sense. Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated to the invariants of the foliation span the tangent distribution. Another way to state this is that there exists a maximal set of Poisson commuting invariants (i.e., functions on the phase space whose Poisson brackets with the Hamiltonian of the system, and with each other, vanish).
In finite dimensions, if the phase space is symplectic (i.e., the center of the Poisson algebra consists only of constants), then it must have even dimension , and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is . The leaves of the foliation are totally isotropic with respect to the symplectic form and such a maximal isotropic foliation is called Lagrangian. All autonomousHamiltonian systems (i.e. those for which the Hamiltonian and Poisson brackets are not explicitly time dependent) have at least one invariant; namely, the Hamiltonian itself, whose value along the flow is the energy. If the energy level sets are compact, the leaves of the Lagrangian foliation are tori, and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical -form are called the actionvariables, and the resulting canonical coordinates are called action-angle variables (see below).
There is also a distinction between complete integrability, in theLiouville sense, and partial integrability, as well as a notion ofsuperintegrability and maximal superintegrability. Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. When the number of independent Poisson commuting invariants is less than maximal (but, in the case of autonomous systems, more than one), we say the system is partially integrable. When there exist further functionally independent invariants, beyond the maximal number that can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is less than n, we say the system is superintegrable. If there is a regular foliation with one-dimensional leaves (curves), this is called maximally superintegrable.
Action-angle variables[edit]
When a finite-dimensional Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are tori. There then exist, as mentioned above, special sets of canonical coordinates on thephase space known as action-angle variables, such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the torus. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.
The Hamilton–Jacobi approach[edit]
In canonical transformation theory, there is the Hamilton–Jacobi method, in which solutions to Hamilton's equations are sought by first finding acomplete solution of the associated Hamilton–Jacobi equation. In classical terminology, this is described as determining a transformation to a canonical set of coordinates consisting of completely ignorable variables; i.e., those in which there is no dependence of the Hamiltonian on a complete set of canonical "position" coordinates, and hence the corresponding canonically conjugate momenta are all conserved quantities. In the case of compact energy level sets, this is the first step towards determining the action-angle variables. In the general theory of partial differential equations of Hamilton–Jacobi type, a complete solution (i.e. one that depends on n independent constants of integration, where n is the dimension of the configuration space), exists in very general cases, but only in the local sense. Therefore the existence of a complete solution of the Hamilton–Jacobi equation is by no means a characterization of complete integrability in the Liouville sense. Most cases that can be "explicitly integrated" involve a complete separation of variables, in which the separation constants provide the complete set of integration constants that are required. Only when these constants can be reinterpreted, within the full phase space setting, as the values of a complete set of Poisson commuting functions restricted to the leaves of a Lagrangian foliation, can the system be regarded as completely integrable in the Liouville sense.
Solitons and inverse spectral methods[edit]
A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that solitons, which are strongly stable, localized solutions of partial differential equations like the Korteweg–de Vries equation (which describes 1-dimensional non-dissipative fluid dynamics in shallow basins), could be understood by viewing these equations as infinite-dimensional integrable Hamiltonian systems. Their study leads to a very fruitful approach for "integrating" such systems, theinverse scattering transform and more general inverse spectral methods (often reducible to Riemann–Hilbert problems), which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations.
The basic idea of this method is to introduce a linear operator that is determined by the position in phase space and which evolves under the dynamics of the system in question in such a way that its "spectrum" (in a suitably generalized sense) is invariant under the evolution. This provides, in certain cases, enough invariants, or "integrals of motion" to make the system completely integrable. In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of Liouville integrability. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to completely ignorable coordinates, in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.
Quantum integrable systems[edit]
There is also a notion of quantum integrable systems. In the quantum setting, functions on phase space must be replaced by self-adjoint operators on a Hilbert space, and the notion of Poisson commuting functions replaced by commuting operators.
To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are one-body reducible. A quantum system is said to be integrable if the dynamics are two-body irreducible. The Yang-Baxter equation is a consequence of this reducibility and leads to trace identities which provide an infinite set of conserved quantities. All of these ideas are incorporated into the Quantum inverse scattering method where the algebraic Bethe Ansatz can be used to obtain explicit solutions. Examples of quantum integrable models are the Lieb-Liniger Model, theHubbard model and several variations on the Heisenberg model. ^[1]
Exactly solvable models[edit]
In physics, completely integrable systems, especially in the infinite-dimensional setting, are often referred to as exactly solvable models. This obscures the distinction between integrability in the Hamiltonian sense, and the more general dynamical systems sense.
There are also exactly solvable models in statistical mechanics, which are more closely related to quantum integrable systems than classical ones. Two closely related methods: the Bethe ansatz approach, in its modern sense, based on the Yang–Baxter equations and the quantum inverse scattering method provide quantum analogs of the inverse spectral methods. These are equally important in the study of solvable models in statistical mechanics.
An imprecise notion of "exact solvability" as meaning: "The solutions can be expressed explicitly in terms of some previously known functions" is also sometimes used, as though this were an intrinsic property of the system itself, rather than the purely calculational feature that we happen to have some "known" functions available, in terms of which the solutions may be expressed. This notion has no intrinsic meaning, since what is meant by "known" functions very often is defined precisely by the fact that they satisfy certain given equations, and the list of such "known functions" is constantly growing. Although such a characterization of "integrability" has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems.
List of some well-known classical integrable systems[edit]
1. Classical mechanical systems (finite-dimensional phase space):[/ltr]

• Harmonic oscillators in n dimensions
  
• Central force motion (exact solutions of classical central-force problems)
  
• Two center Newtonian gravitational motion
  
• Geodesic motion on ellipsoids
  
• Neumann oscillator
  
• Lagrange, Euler and Kovalevskaya tops
  
• Integrable Clebsch and Steklov systems in fluids
  
• Calogero–Moser–Sutherland model
  
• Swinging Atwood's Machine with certain choices of parameters
  

[ltr]
2. Integrable lattice models[/ltr]

• Toda lattice
  
• Ablowitz–Ladik lattice
  
• Volterra lattice
  

[ltr]
3. Integrable systems of PDEs in 1 + 1 dimension[/ltr]

• Korteweg–de Vries equation
  
• Sine–Gordon equation
  
• Nonlinear Schrödinger equation
  
• AKNS system
  
• Boussinesq equation (water waves)
  
• Nonlinear sigma models
  
• Classical Heisenberg ferromagnet model (spin chain)
  
• Classical Gaudin spin system (Garnier system)
  
• Landau–Lif***z equation (continuous spin field)
  
• Benjamin–Ono equation
  
• Dym equation
  
• Three-wave equation
  

[ltr]
4. Integrable PDEs in 2 + 1 dimensions[/ltr]

• Kadomtsev–Petviashvili equation
  
• Davey–Stewartson equation
  
• Ishimori equation
  

[ltr]
5. Other integrable systems of PDEs in higher dimensions[/ltr]

• Self-dual Yang–Mills equations
  

[ltr]
Notes[edit][/ltr]

1. Jump up^ V.E. Korepin, N. M. Bogoliubov, A. G. Izergin (1997). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press.ISBN 978-0-521-58646-7.
   

[ltr]
References[edit][/ltr]

• V. I. Arnold (1997). Mathematical Methods of Classical Mechanics, 2nd ed. Springer. ISBN 978-0-387-96890-2.
  
• M. Dunajski (2009). Solitons, Instantons and Twistors,. Oxford University Press. ISBN 978-0-19-857063-9.
  
• L. D. Faddeev, L. A. Takhtajan (1987). Hamiltonian Methods in the Theory of Solitons. Addison-Wesley. ISBN 978-0-387-15579-1.
  
• A. T. Fomenko, Symplectic Geometry. Methods and Applications.Gordon and Breach, 1988. Second edition 1995, ISBN 978-2-88124-901-3.
  
• A. T. Fomenko, A. V. Bolsinov Integrable Hamiltonian Systems: Geometry, Topology, Classification. Taylor and Francis, 2003, ISBN 978-0-415-29805-6.
  
• H. Goldstein (1980). Classical Mechanics, 2nd. ed. Addison-Wesley.ISBN 0-201-02918-9.
  
• J. Harnad, P. Winternitz, G. Sabidussi, eds. (2000). Integrable Systems: From Classical to Quantum. American Mathematical Society. ISBN 0-8218-2093-1.
  
• V. E. Korepin, N. M. Bogoliubov, A. G. Izergin (1997). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press. ISBN 978-0-521-58646-7.
  
• V. S. Afrajmovich, V. I. Arnold, Yu. S. Il'yashenko, L. P. Shil'nikov.Dynamical Systems V. Springer. ISBN 3-540-18173-3.
  
• Giuseppe Mussardo. Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics. Oxford University Press.ISBN 978-0-19-954758-6.
  

[ltr]
External links[edit][/ltr]

• Hazewinkel, Michiel, ed. (2001), "Integrable system", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
  

Conformal field theory

From Wikipedia, the free encyclopedia


[ltr]A conformal field theory (CFT) is a quantum field theory, also recognized as a model of statistical mechanics at a thermodynamic critical point, that is invariant under conformal transformations. Conformal field theory is often studied in two dimensions where there is an infinite-dimensional group of local conformal transformations, described by theholomorphic functions.
Conformal field theory has important applications in string theory,statistical mechanics, and condensed matter physics.

[/ltr]

1 Scale invariance vs. conformal invariance

2 Dimensional considerations



• 2.1 Two dimensions
  
• 2.2 More than two dimensions
  



3 Conformal symmetry

4 See also

5 References

6 Further reading

[size][ltr]

Scale invariance vs. conformal invariance[edit]
While it is possible for a quantum field theory to be scale invariant but not conformally-invariant, examples are rare.^[1] For this reason, the terms are often used interchangeably in the context of quantum field theory, even though the scale symmetry is larger.
In some particular cases it is possible to prove that scale invariance implies conformal invariance in a quantum field theory, for example in unitarycompact conformal field theories in two dimensions.
Dimensional considerations[edit]
Two dimensions[edit]
There are two versions of 2D CFT: 1) Euclidean, and 2) Lorentzian. The former applies to statistical mechanics, and the latter to quantum field theory. The two versions are related by a Wick rotation.
Two-dimensional CFTs are (in some way) invariant under an infinite-dimensional symmetry group. For example, consider a CFT on theRiemann sphere. It has the Möbius transformations as the conformal group, which is isomorphic to (the finite-dimensional) PSL(2,C). However, the infinitesimal conformal transformations form an infinite-dimensional algebra, called the Witt algebra and only the primary fields (or chiral fields) are invariant with respect to the full infinitesimal conformal group.
In most conformal field theories, a conformal anomaly, also known as aWeyl anomaly, arises in the quantum theory. This results in the appearance of a nontrivial central charge, and the Witt algebra is modified to become the Virasoro algebra.
In Euclidean CFT, we have a holomorphic and an antiholomorphic copy of the Virasoro algebra. In Lorentzian CFT, we have a left-moving and a right moving copy of the Virasoro algebra (spacetime is a cylinder, with space being a circle, and time a line).
This symmetry makes it possible to classify two-dimensional CFTs much more precisely than in higher dimensions. In particular, it is possible to relate the spectrum of primary operators in a theory to the value of thecentral charge, c. The Hilbert space of physical states is a unitary moduleof the Virasoro algebra corresponding to a fixed value of c. Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
A chiral field is a holomorphic field W(z) which transforms as

and

Similarly for an antichiral field. Δ is the conformal weight of the chiral fieldW.
Furthermore, it was shown by Alexander Zamolodchikov that there exists a function, C, which decreases monotonically under the renormalization group flow of a two-dimensional quantum field theory, and is equal to the central charge for a two-dimensional conformal field theory. This is known as the Zamolodchikov C-theorem, and tells us that renormalization group flow in two dimensions is irreversible.
Frequently, we are not just interested in the operators, but we are also interested in the vacuum state, or in statistical mechanics, the thermal state. Unless c=0, there can't possibly be any state which leaves the entire infinite dimensional conformal symmetry unbroken. The best we can come up with is a state which is invariant under L_-1, L₀, L₁, L_i, . This contains the Möbius subgroup. The rest of the conformal group is spontaneously broken.
Two-dimensional conformal field theories play an important role in statistical mechanics, where they describe critical points of many lattice models.
More than two dimensions[edit]
In d > 2 dimensions, the conformal group is isomorphic to SO(d+1, 1 ) in Euclidean signature, or SO(d, 2 ) in Minkowski space.
Higher-dimensional conformal field theories are prominent in the AdS/CFT correspondence, in which a gravitational theory in anti-de Sitter space(AdS) is equivalent to a conformal field theory on the AdS boundary. Notable examples are d=4 N = 4 supersymmetric Yang–Mills theory, which is dual to Type IIB string theory on AdS₅ x S⁵, and d=3 N=6 super-Chern–Simons theory, which is dual to M-theory on AdS₄ x S⁷. (The prefix "super" denotes supersymmetry, N denotes the degree of extended supersymmetry possessed by the theory, and d the number of space-time dimensions on the boundary.)
Conformal symmetry[edit]
Conformal symmetry is a symmetry under scale invariance and under the special conformal transformations having the following relations.

where  generates translations,  generates scaling transformations as a scalar and  generates the special conformal transformations as acovariant vector under Lorentz transformation.
See also[edit]
[/ltr][/size]

• Logarithmic conformal field theory
  
• AdS/CFT correspondence
  
• Operator product expansion
  
• Vertex operator algebra
  
• WZW model
  
• Critical point
  
• Boundary conformal field theory
  
• Primary field
  
• Superconformal algebra
  
• Conformal algebra
  

[size][ltr]
References[edit]
[/ltr][/size]

1. Jump up^ One physical example is the theory of elasticity in two and three dimensions (also known as the theory of a vector field without gauge invariance). See Riva V, Cardy J (2005). "Scale and conformal invariance in field theory: a physical counterexample". Phys. Lett. B 622: 339–342.arXiv:hep-th/0504197. Bibcode:2005PhLB..622..339R.doi:10.1016/j.physletb.2005.07.010.
   

[size][ltr]
Further reading[edit]
[/ltr][/size]

• Martin Schottenloher, A Mathematical Introduction to Conformal Field Theory, Springer-Verlag, Berlin, Heidelberg, 1997. ISBN 3-540-61753-1, 2nd edition 2008, ISBN 978-3-540-68625-5.
  
• Paul Ginsparg, Applied Conformal Field Theory. arXiv:hep-th/9108028.
  
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York, 1997. ISBN 0-387-94785-X.
  
• Conformal Field Theory page in String Theory Wiki lists books and reviews.
  

[th] [hide] v t e Statistical mechanics[/th][th]Statistical ensembles[/th][th]Statistical thermodynamics[/th][th]Partition functions[/th][th]Equations of state[/th][th]Entropy[/th][th]Particle statistics[/th][th]Statistical field theory[/th][th]Quantum statistical mechanics[/th][th]See also[/th] Microcanonical Canonical Grand canonical Isothermal–isobaric Isoenthalpic–isobaric Open Characteristic state functions Translational Vibrational Rotational Dieterici Van der Waals Ideal gas law Birch–Murnaghan Sackur–Tetrode equation Tsallis entropy Von Neumann entropy Maxwell–Boltzmann statistics Fermi–Dirac statistics Fermi–Dirac distribution Bose–Einstein statistics Bose–Einstein distribution Conformal field theory Osterwalder–Schrader axioms Gibbs canonical ensemble Density matrix Gibbs measure Partition function Weyl quantization Slater determinant Probability distribution Structureless particles

Affine Lie algebra

From Wikipedia, the free encyclopedia




[ltr]In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. It is a Kac–Moody algebra for which thegeneralized Cartan matrix is positive semi-definite and has corank 1. From purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite dimensional, semisimple Lie algebras is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.
Affine Lie algebras play an important role in string theory and conformal field theory due to the way they are constructed: starting from a simple Lie algebra , one considers the loop algebra, , formed by the -valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra  is obtained by adding one extra dimension to the loop algebra and modifying a commutator in a non-trivial way, which physicists call a quantum anomaly and mathematicians acentral extension. More generally, if σ is an automorphism of the simple Lie algebra  associated to an automorphism of its Dynkin diagram, thetwisted loop algebra  consists of -valued functions f on the real line which satisfy the twisted periodicity condition f(x+2π) = σ f(x). Their central extensions are precisely the twisted affine Lie algebras. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations transform amongst themselves under the modular group.[/ltr]

• 1 Affine Lie algebras from simple Lie algebras
  
  
  
  • 1.1 Definition
    
  • 1.2 Constructing the Dynkin diagrams
    
  • 1.3 Classifying the central extensions
    
  
  
  
• 2 Applications
  
• 3 References
  

[ltr]

Affine Lie algebras from simple Lie algebras[edit]
Definition[edit]
If  is a finite dimensional simple Lie algebra, the corresponding affine Lie algebra  is constructed as a central extension of the infinite-dimensional Lie algebra , with one-dimensional center  As a vector space,

where  is the complex vector space of Laurent polynomials in the indeterminate t. The Lie bracket is defined by the formula

for all  and , where  is the Lie bracket in the Lie algebra  and  is the Cartan-Killing form on 
The affine Lie algebra corresponding to a finite-dimensional semisimple Lie algebra is the direct sum of the affine Lie algebras corresponding to its simple summands. There is a distinguished derivation of the affine Lie algebra defined by

The corresponding affine Kac-Moody algebra is defined by adding an extra generator d satisfying [d,A] = δ(A) (a semidirect product).
Constructing the Dynkin diagrams[edit]
The Dynkin diagram of each affine Lie algebra consists of that of the corresponding simple Lie algebra plus an additional node, which corresponds to the addition of an imaginary root. Of course, such a node cannot be attached to the Dynkin diagram in just any location, but for each simple Lie algebra there exists a number of possible attachments equal to the cardinality of the group of outer automorphisms of the Lie algebra. In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called an untwisted affine Lie algebra. When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond to twisted affine Lie algebras.[/ltr]
Dynkin diagrams for affine Lie algebras

The set of extended (untwisted) affine Dynkin diagrams, with added nodes in green

"Twisted" affine forms are named with (2) or (3) superscripts. (k is the number of nodes in the graph)

[ltr]
Classifying the central extensions[edit]
The attachment of an extra node to the Dynkin diagram of the corresponding simple Lie algebra corresponds to the following construction. An affine Lie algebra can always be constructed as a central extension of the loop algebra of the corresponding simple Lie algebra. If one wishes to begin instead with a semisimple Lie algebra, then one needs to centrally extend by a number of elements equal to the number of simple components of the semisimple algebra. In physics, one often considers instead the direct sum of a semisimple algebra and an abelian algebra . In this case one also needs to add n further central elements for the nabelian generators.
The second integral cohomology of the loop group of the corresponding simple compact Lie group is isomorphic to the integers. Central extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified by a two-class known as the first Chern class of the fibration. Therefore the central extensions of an affine Lie group are classified by a single parameter kwhich is called the level in the physics literature, where it first appeared. Unitary highest weight representations of the affine compact groups only exist when k is a natural number. More generally, if one considers a semi-simple algebra, there is a central charge for each simple component.
Applications[edit]
They appear naturally in theoretical physics (for example, in conformal field theories such as the WZW model and coset models and even on the worldsheet of the heterotic string), geometry, and elsewhere in mathematics.
References[edit][/ltr]

• Di Francesco, P.; Mathieu, P.; Sénéchal, D. (1997), Conformal Field Theory, Springer-Verlag, ISBN 0-387-94785-X
  
• Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521-48412-X
  
• Goddard, Peter; Olive, David (1988), Kac-Moody and Virasoro algebras: A Reprint Volume for Physicists, Advanced Series in Mathematical Physics 3, World Scientific, ISBN 9971-5-0419-7
  
• Kac, Victor (1990), Infinite dimensional Lie algebras (3 ed.), Cambridge University Press, ISBN 0-521-46693-8
  
• Kohno, To***ake (1998), Conformal Field Theory and Topology, American Mathematical Society, ISBN 0-8218-2130-X
  
• Pressley, Andrew; Segal, Graeme (1986), Loop groups, Oxford University Press, ISBN 0-19-853535-X
  

Verma module

From Wikipedia, the free encyclopedia


[ltr]Verma modules, named after Daya-Nand Verma, are objects in therepresentation theory of Lie algebras, a branch of mathematics.
Verma modules can be used to prove that an irreducible highest weight module with highest weight  is finite-dimensional, if and only if the weight  is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds.

[/ltr]

1 Definition of Verma modules

2 Basic properties

3 Homomorphisms of Verma modules

4 Jordan–Hölder series

5 Bernstein–Gelfand–Gelfand resolution

6 See also

7 Notes

8 References

[size][ltr]

Definition of Verma modules[edit]
The definition relies on a stack of relatively dense notation. Let  be a field and denote the following:
[/ltr][/size]

• , a semisimple Lie algebra over , with universal enveloping algebra .
  
• , a Borel subalgebra of , with universal enveloping algebra .
  
• , a Cartan subalgebra of . We do not consider its universal enveloping algebra.
  
• , a fixed weight.
  

[size][ltr]
To define the Verma module, we begin by defining some other modules:
[/ltr][/size]

• , the one-dimensional -vector space (i.e. whose underlying set is  itself) together with a -module structure such that  acts as multiplication by  and the positive root spaces act trivially. As  is a left -module, it is consequently a left -module.
  
• Using the Poincaré–Birkhoff–Witt theorem, there is a natural right -module structure on  by right multiplication of a subalgebra.  is naturally a left -module, and together with this structure, it is a -bimodule.
  

[size][ltr]
Now we can define the Verma module (with respect to ) as

which is naturally a left -module (i.e. a representation of ). The Poincaré–Birkhoff–Witt theorem implies that the underlying vector space of is isomorphic to

where  is the Lie subalgebra generated by the negative root spaces of .
Basic properties[edit]
Verma modules, considered as -modules, are highest weight modules, i.e. they are generated by a highest weight vector. This highest weight vector is  (the first  is the unit in  and the second is the unit in the field , considered as the -module ) and it has weight .
Verma modules are weight modules, i.e.  is a direct sum of all itsweight spaces. Each weight space in  is finite-dimensional and the dimension of the -weight space  is the number of possibilities how to obtain  as a sum of positive roots (this is closely related to the so-called Kostant partition function).
Verma modules have a very important property: If  is any representation generated by a highest weight vector of weight , there is a surjective -homomorphism  That is, all representations with highest weight  that are generated by the highest weight vector (so called highest weight modules) are quotients of 
 contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight 
The Verma module  itself is irreducible if and only if none of the coordinates of  in the basis of fundamental weights is from the set .
The Verma module  is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight . In other word, there exist an element w of the Weyl group W such that

where  is the affine action of the Weyl group.
The Verma module  is called singular, if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight  so that is on the wall of the fundamental Weyl chamber (δ is the sum of allfundamental weights).
Homomorphisms of Verma modules[edit]
For any two weights  a non-trivial homomorphism

may exist only if  and  are linked with an affine action of the Weyl group of the Lie algebra . This follows easily from the Harish-Chandra theorem on infinitesimal central characters.
Each homomorphism of Verma modules is injective and the dimension

for any . So, there exists a nonzero  if and only if  isisomorphic to a (unique) submodule of .
The full classification of Verma module homomorphisms was done by Bernstein-Gelfand-Gelfand^[1] and Verma^[2] and can be summed up in the following statement:
[/ltr][/size]

There exists a nonzero homomorphism  if and only if there exists
a sequence of weights
such that  for some positive roots  (and  is the corresponding root reflection and  is the sum of all fundamental weights) and for each  is a natural number ( is the coroot associated to the root ).

[size][ltr]
If the Verma modules  and  are regular, then there exists a uniquedominant weight  and unique elements w, w′ of the Weyl group W such that
P
and

where  is the affine action of the Weyl group. If the weights are furtherintegral, then there exists a nonzero homomorphism

if and only if

in the Bruhat ordering of the Weyl group.
Jordan–Hölder series[edit]
Let

be a sequence of -modules so that the quotient B/A is irreducible withhighest weight μ. Then there exists a nonzero homomorphism .
An easy consequence of this is, that for any highest weight modules  such that

there exists a nonzero homomorphism .
Bernstein–Gelfand–Gelfand resolution[edit]
Let  be a finite-dimensional irreducible representation of the Lie algebra with highest weight λ. We know from the section about homomorphisms of Verma modules that there exists a homomorphism

if and only if

in the Bruhat ordering of the Weyl group. The following theorem describes a resolution of  in terms of Verma modules (it was proved byBernstein–Gelfand–Gelfand in 1975^[3]) :
[/ltr][/size]

There exists an exact sequence of -homomorphisms

where n is the length of the largest element of the Weyl group.

[size][ltr]
A similar resolution exists for generalized Verma modules as well. It is denoted shortly as the BGG resolution.
Recently, these resolutions were studied in special cases, because of their connections to invariant differential operators in a special type of Cartan geometry, the parabolic geometries. These are Cartan geometries modeled on the pair (G, P) where G is a Lie group and P a parabolic subgroup).^[4]
See also[edit]
[/ltr][/size]

• Generalized Verma module
  
• Weyl module
  

[size][ltr]
Notes[edit]
[/ltr][/size]

1. Jump up^ Bernstein I.N., Gelfand I.M., Gelfand S.I., Structure of Representations that are generated by vectors of highest weight, Functional. Anal. Appl. 5 (1971)
   
2. Jump up^ Verma N., Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968)
   
3. Jump up^ Bernstein I. N., Gelfand I. M., Gelfand S. I., Differential Operators on the Base Affine Space and a Study of g-Modules, Lie Groups and Their Representations, I. M. Gelfand, Ed., Adam Hilger, London, 1975.
   
4. Jump up^ For more information, see: Eastwood M., Variations on the de Rham complex, Notices Amer. Math. Soc, 1999 - ams.org. Calderbank D.M., Diemer T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, Arxiv preprint math.DG/0001158, 2000 - arxiv.org [1]. Cap A., Slovak J., Soucek V., Bernstein-Gelfand-Gelfand sequences, Arxiv preprint math.DG/0001164, 2000 - arxiv.org [2]
   

[size][ltr]
References[edit]
[/ltr][/size]

• Carter, R. (2005), Lie Algebras of Finite and Affine Type, Cambridge University Press, ISBN 0-521-85138-6.
  
• Knapp, A. W. (2002), Lie Groups Beyond an introduction (2nd ed.), Birkhäuser, p. 285, ISBN 978-0-8176-3926-6.
  
• Dixmier, J. (1977), Enveloping Algebras, Amsterdam, New York, Oxford: North-Holland, ISBN 0-444-11077-1.
  
• Humphreys, J. (1980), Introduction to Lie Algebras and Representation Theory, Springer Verlag, ISBN 3-540-90052-7.
  
• Rocha, Alvany (2001), "BGG resolution", in Hazewinkel, Michiel,Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
  
• Roggenkamp, K.; Stefanescu, M. (2002), Algebra - Representation Theory, Springer, ISBN 0-7923-7114-3.
  

[size][ltr]
This article incorporates material from Verma module on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.[/ltr]
[/size]

Operator product expansion

From Wikipedia, the free encyclopedia

This article does not cite any references or sources. Please help improve this articleby adding citations to reliable sources. Unsourced material may be challenged andremoved. (January 2009)

• 1 2D Euclidean quantum field theory
  
• 2 General
  
• 3 See also
  
• 4 External links
  

[ltr]

2D Euclidean quantum field theory[edit]
In quantum field theory, the operator product expansion (OPE) is aLaurent series expansion of two operators. A Laurent series is a generalization of the Taylor series in that finitely many powers of the inverse of the expansion variable(s) are added to the Taylor series: pole(s) of finite order(s) are added to the series.
Heuristically, in quantum field theory one is interested in the result of physical observables represented by operators. If one wants to know the result of *** two physical observations at two points  and , one can time order these operators in increasing time.
If one maps coordinates in a conformal manner, one is often interested in radial ordering. This is the analogue of time ordering where increasing time has been mapped to some increasing radius on the complex plane. One is also interested in normal ordering of creation operators.
A radial-ordered OPE can be written as a normal-ordered OPE minus the non-normal-ordered terms. The non-normal-ordered terms can often be written as a commutator, and these have useful simplifying identities. The radial ordering supplies the convergence of the expansion.
The result is a convergent expansion of the product of two operators in terms of some terms that have poles in the complex plane (the Laurent terms) and terms that are finite. This result represents the expansion of two operators at two different points as an expansion around just one point, where the poles represent where the two different points are the same point e.g.
.
Related to this is that an operator on the complex plane is in general written as a function of  and . These are referred to as the holomorphicand anti-holomorphic parts respectively, as they are continuous and differentiable except at the (finite number of) singularities. One should really call them meromorphic but holomorphic is common parlance. In general, the operator product expansion may not separate into holormorphic and anti-holomorphic parts, especially if there are terms in the expansion. However, derivatives of the OPE can often separate the expansion into holomorphic and anti-holomorphic expansions. This expression is also an OPE and in general is more useful.
General[edit]
In quantum field theory, the operator product expansion (OPE) is aconvergent expansion of the product of two fields at different points as a sum (possibly infinite) of local fields.
More precisely, if x and y are two different points, and A and B areoperator-valued fields, then there is an open neighborhood of y, O such that for all x in O/{y}

where the sum is over finitely or countably many terms, Cⁱ are operator-valued fields, c_i are analytic functions over O/{y} and the sum is convergent in the operator topology within O/{y}.
OPEs are most often used in conformal field theory.
The notation  is often used to denote that the difference G(x,y)-F(x,y) remains analytic at the points x=y. This is anequivalence relation.
See also[edit][/ltr]

• QCD sum rules
  

[ltr]
External links[edit][/ltr]

• The OPE at Scholarpedia
  

Product operator formalism

From Wikipedia, the free encyclopedia

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacksinline citations. Please improve this article by introducing more precise citations. (May 2012)

[size][ltr]
In NMR spectroscopy, the product operator formalism is a method used to determine the outcome of pulse sequences in a rigorous but straightforward way. With this method it is possible to predict how the bulkmagnetization evolves with time under the action of pulses applied in different directions. It is a net improvement from the semi-classical vector model which is not able to predict many of the results in NMR spectroscopy and is a simplification of the complete density matrix formalism.
In this model, for a single spin, four base operators exist: , ,  and  which represent respectively polarization (population difference between the two spin states), single quantum coherence (magnetization on the xy plane) and the unit operator. Many other, non-classical operators exist for coupled systems. Using this approach, the evolution of the magnetization under free precession is represented by  and corresponds to a rotation about the z-axis with a phase angle proportional to the chemical shift of the spin in question:

Pulses about the x and y axis can be represented by  and respectively; these allow to interconvert the magnetization between planes and ultimately to observe it at the end of a sequence. Since every spin will evolve differently depending on its shift, with this formalism it is possible to calculate exactly where the magnetization will end up and hence devise pulse sequences to measure the desired signal while excluding others.
The product operator formalism is particularly useful in describingexperiments in two-dimensions like COSY, HSQC and HMBC.
References[edit]
[/ltr][/size]

• James Keeler. "Understanding NMR Spectroscopy" (reprinted atUniversity of Cambridge). University of California, Irvine. Retrieved 2012-08-05.
  
• David Donne; David Gorenstein. "A Pictorial Representation of Product Operator Formalism". University of Texas. Retrieved 2012-08-05.
  

Vertex operator algebra

From Wikipedia, the free encyclopedia
  (Redirected from Vertex algebras)

[th]String theory[/th]

Fundamental objects[show]

Perturbative string theory[show]

Non-perturbative results[show]

Phenomenology[show]

Mathematics[show]

Related concepts[show]

Theorists[show]

History Glossary

v t e

[size][ltr]
In mathematics, avertex operator algebra (VOA) is an algebraic structure that plays an important role inconformal field theoryand string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such asmonstrous moonshineand the geometric Langlands correspondence.
The related notion ofvertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due toFrenkel. In the course of this construction, one employs a Fock space that admits an action of vertex operators attached to lattice vectors. Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method.
The notion of vertex operator algebra was introduced as a modification of the notion of vertex algebra, by Frenkel, Lepowsky, and Meurman in 1988, as part of their project to construct the moonshine module. They observed that many vertex algebras that appear in nature have a useful additional structure (an action of the Virasoro algebra), and satisfy a bounded-below property with respect to an energy operator. Motivated by this observation, they added the Virasoro action and bounded-below property as axioms.
We now have post-hoc motivation for these notions from physics, together with several interpretations of the axioms that were not initially known. Physically, the vertex operators arising from holomorphic field insertions at points (i.e., vertices) in two dimensional conformal field theory admitoperator product expansions when insertions collide, and these satisfy precisely the relations specified in the definition of vertex operator algebra. Indeed, the axioms of a vertex operator algebra are a formal algebraic interpretation of what physicists call chiral algebras, or "algebras of chiral symmetries", where these symmetries describe the Ward identities satisfied by a given conformal field theory, including conformal invariance. Other formulations of the vertex algebra axioms include Borcherds's later work on singular commutative rings, algebras over certain operads on curves introduced by Huang, Kriz, and others, and D-module-theoretic objects called chiral algebras introduced by Alexander Beilinson andVladimir Drinfeld. While related, these chiral algebras are not precisely the same as the objects with the same name that physicists use.
Important basic examples of vertex operator algebras include lattice VOAs (modeling lattice conformal field theories), VOAs given by representations of affine Kac–Moody algebras (from the WZW model), the Virasoro VOAs (i.e., VOAs corresponding to representations of the Virasoro algebra) and the moonshine module V^♮, which is distinguished by its monster symmetry. More sophisticated examples such as affine W-algebras and the Chiral de Rham complex on a complex manifold arise in geometric representation theory and mathematical physics.

[/ltr][/size]

1 Formal definition

2 Commutative vertex algebras

3 Basic properties

4 Example: The rank 1 free boson

5 Example: Virasoro vertex operator algebras

6 Example: WZW vacuum modules

7 Modules

8 Vertex operator algebra defined by an even lattice

9 Vertex operator superalgebras

10 Superconformal structures

11 Additional constructions

12 Additional Examples

13 Related algebraic structures

14 Alternative definitions

15 Notes

16 References

[size][ltr]

Formal definition[edit]
A vertex algebra is the following data:
[/ltr][/size]

• a vector space V, called the space of states. The underlying field is typically taken to be the complex numbers, although Borcherds's original formulation allowed for an arbitrary commutative ring.
  
• an identity element 1 ∈ V, sometimes written  or Ω to indicate a vacuum state.
  
• an endomorphism T: V → V, called "translation". (Borcherds's original formulation included a system of divided powers of T, because he did not assume the ground ring was divisible.)
  
• a linear multiplication map Y:V⊗V → V((z)), where V((z)) is the space of all formal Laurent series with coefficients in V. This structure is alternatively presented as an infinite collection of bilinear products u_nv, or as a left-multiplication map V → End(V)[[z^±1]], called the state-field correspondence. For each u∈V, the operator-valued formal distributionY(u,z) is called a vertex operator or a field (inserted at zero), and the coefficient of z^–n–1 is the operator u_n. The standard notation for the multiplication is
  

[size][ltr]
.
These data are required to satisfy the following axioms:
[/ltr][/size]

• (Identity) For any u ∈ V, Y(1,z)u = u = uz⁰ and Y(u,z)1 ∈ u + zV[[z]].
  

• (Translation) T(1) = 0, and for any u, v ∈ V,
  

[size][ltr]

[/ltr][/size]

• (Locality, or Jacobi identity, or Borcherds identity) For any u,v∈V, there exists a positive integer N such that (z–x)^N Y(u,z)Y(v,x) = (z–x)^NY(v,x)Y(u,z). This has several equivalent formulations in the literature, e.g., Frenkel-Lepowsky-Meurman introduced the Jacobi identity:
  

[size][ltr]

for all vectors u, v, and w, where we define the formal delta series by:

Borcherds initially used the following identity:

for any vectors u, v, and w, and integers m and n. He later gave a more expansive version that is equivalent but easier to use:

for any vectors u, v, and w, and integers m, n, and q.
Finally, there is a formal function version of locality: For any u, v, w ∈ V, there is an element

such that

are the corresponding expansions of  in

A vertex operator algebra is a vertex algebra equipped with aconformal element ω, such that the vertex operator Y(ω,z) is the weight 2 Virasoro field L(z):

and satisfies the following properties:
[/ltr][/size]

• [L_m, L_n] = (m – n)L_m+n + (δ_{m + n, 0}/12) (m³–m)c Id_V, where c is a constant called the central charge, or rank of V. In particular, the coefficients of this vertex operator endow V with an action of the Virasoro algebra with central charge c.
  
• L₀ acts semisimply on V with integer eigenvalues that are bounded below.
  
• Under the grading provided by the eigenvalues of L₀, the multiplication on V is homogeneous in the sense that if u and v are homogeneous, then u_n v is homogeneous of degree deg(u)+deg(v)–n–1.
  
• The identity 1 has degree 0, and the conformal element ω has degree 2.
  
• L_–1 = T.
  

[size][ltr]
A homomorphism of vertex algebras is a map of the underlying vector spaces that respects the additional identity, translation, and multiplication structure. Homomorphisms of vertex operator algebras have "weak" and "strong" forms, depending on whether they respect conformal vectors.
Commutative vertex algebras[edit]
A vertex algebra V is commutative if all vertex operators commute with each other. This is equivalent to the property that all products Y(u,z)v lie inV[[z]]. Given a commutative vertex algebra, the constant terms of multiplication endow the vector space with a commutative ring structure, and T is a derivation. Conversely, any commutative ring V with derivation Thas a canonical vertex algebra structure, where we set Y(u,z)v = u_–1v z⁰ =uv. If the derivation T vanishes, we may set ω = 0 to obtain a vertex operator algebra concentrated in degree zero.
Any finite-dimensional vertex algebra is commutative. In particular, even the smallest examples of noncommutative vertex algebras require significant introduction.
Basic properties[edit]
The translation operator T in a vertex algebra induces infinitesimal symmetries on the product structure, and satisfies the following properties:
[/ltr][/size]

• Y(u,z)1 = e^zTu
  
• Tu = u_–21, so T is determined by Y.
  
• Y(Tu,z) = d(Y(u,z))/dz
  
• e^xTY(u,z)e^–xT = Y(e^xTu,z) = Y(u,z+x)
  
• (skew-symmetry) Y(u,z)v = e^zTY(v,–z)u
  

[size][ltr]
For a vertex operator algebra, the other Virasoro operators satisfy similar properties:
[/ltr][/size]

• x^L₀Y(u,z)x^–L₀ = Y(x^L₀u,xz)
  
• e^xL₁Y(u,z)e^–xL₁ = Y(e^{x(1–xz)L₁}(1–xz)^–2L₀u,z(1–xz)⁻¹)
  
• (quasi-conformality)  for all m≥–1.
  

• (Associativity, or Cousin property): For any u, v, w ∈ V, the element
  

[size][ltr]

given in the definition also expands to Y(Y(u,z–x)v,x)w in V((x))((z–x)).
The associativity property of a vertex algebra follows from the fact that the commutator of Y(u,z) and Y(v,x) is annihilated by a finite power of z–x, i.e., one can expand it as a finite linear combination of derivatives of the formal delta function in (z–x), with coefficients in End(V).
Reconstruction: Let V be a vertex algebra, and let {J^a} be a set of vectors, with corresponding fields J^a(z) ∈ End(V)[[z^±1]]. If V is spanned by monomials in the positive weight coefficients of the fields (i.e., finite products of operators J^a_n applied to 1, where n is negative), then we may write the operator product of such a monomial as a normally ordered product of divided power derivatives of fields (here, normal ordering means polar terms on the left are moved to the right). Specifically,

More generally, if one is given a vector space V with an endomorphism Tand vector 1, and one assigns to a set of vectors J^a a set of fields J^a(z) ∈ End(V)[[z^±1]] that are mutually local, whose positive weight coefficients generate V, and that satisfy the identity and translation conditions, then the previous formula describes a vertex algebra structure.
Example: The rank 1 free boson[edit]
A basic example of a noncommutative vertex algebra is the rank 1 free boson, also called the Heisenberg vertex operator algebra. It is "generated" by a single vector b, in the sense that by applying the coefficients of the field b(z) = Y(b,z) to the vector 1, we obtain a spanning set. The underlying vector space is the infinite-variable polynomial ringC[x₁,x₂,...], where for positive n, the coefficient b_–n of Y(b,z) acts as multiplication by x_n, and b_n acts as n times the partial derivative in x_n. The action of b₀ is multiplication by zero, producing the "momentum zero" Fock representation V₀ of the Heisenberg Lie algebra (generated by b_n for integers n, with commutation relations [b_n,b_m]=n δ_n,–m), i.e., induced by the trivial representation of the subalgebra spanned by b_n, n ≥ 0.
The Fock space V₀ can be made into a vertex algebra by following reconstruction:

where :..: denotes normal ordering (i.e. moving all derivatives in x to the right). The vertex operators may also be written as a functional of a multivariable function f as:

if we understand that each term in the expansion of f is normal ordered.
The rank n free boson is given by taking an n-fold tensor product of the rank 1 free boson. For any vector b in n-dimensional space, one has a field b(z) whose coefficients are elements of the rank n Heisenberg algebra, whose commutation relations have an extra inner product term: [b_n,c_m]=n (b,c) δ_n,–m.
Example: Virasoro vertex operator algebras[edit]
Virasoro vertex operator algebras are important for two reasons: First, the conformal element in a vertex operator algebra canonically induces a homomorphism from a Virasoro vertex operator algebra, so they play a universal role in the theory. Second, they are intimately connected to the theory of unitary representations of the Virasoro algebra, and these play a major role in conformal field theory. In particular, the unitary Virasoro minimal models are simple quotients of these vertex algebras, and their tensor products provide a way to combinatorially construct more complicated vertex operator algebras.
The Virasoro vertex operator algebra is defined as an induced representation of the Virasoro algebra: If we choose a central charge c, there is a unique one-dimensional module for the subalgebra C[z]∂_z + K for which K acts by cId, and C[z]∂_z acts trivially, and the corresponding induced module is spanned by polynomials in L_–n = –z^–n–1∂_z as n ranges over integers greater than 1. The module then has partition function
.
This space has a vertex operator algebra structure, where the vertex operators are defined by:

and . The fact that the Virasoro field L(z) is local with respect to itself can be deduced from the formula for its self-commutator:

where c is central charge.
Given a vertex algebra homomorphism from a Virasoro vertex algebra of central charge c to any other vertex algebra, the vertex operator attached to the image of ω automatically satisfies the Virasoro relations, i.e., the image of ω is a conformal vector. Conversely, any conformal vector in a vertex algebra induces a distinguished vertex algebra homomorphism from some Virasoro vertex operator algebra.
The Virasoro vertex operator algebras are simple, except when c has the form 1–6(p–q)²/pq for coprime integers p,q strictly greater than 1 - this follows from Kac's determinant formula. In these exceptional cases, one has a unique maximal ideal, and the corresponding quotient is called a minimal model. When p = q+1, the vertex algebras are unitary representations of Virasoro, and their modules are known as discrete series representations. They play an important role in conformal field theory in part because they are unusually tractable, and for small p, they correspond to well-known statistical-mechanical systems at criticality, e.g., the Ising model, the tri-critical Ising model, the three-state Potts model, etc. By work of Weiqang Wang ^[1] concerning fusion rules, we have a full description of the tensor categories of unitary minimal models. For example, when c=1/2 (Ising), there three irreducible modules with lowestL₀-weight 0, 1/2, and 1/16, and its fusion ring is Z[x,y]/(x²–1, y²–x–1, xy–y).
Example: WZW vacuum modules[edit]
By replacing the Heisenberg Lie algebra with an untwisted affine Kac–Moody Lie algebra (i.e., the universal central extension of the algebra of polynomial loops on a finite-dimensional simple Lie algebra), one may construct the Vacuum representation in much the same way as the free boson vertex algebra is constructed. Concretely, pulling back the central extension

along the inclusion  yields a split extension, and the vacuum module is induced from the one-dimensional representation of the latter on which a central basis element acts by some chosen constant called the "level". Since central elements can be identified with invariant inner products on the finite type Lie algebra , one typically normalizes the level so that the Killing form has level twice the dual Coxeter number. Equivalently, level one gives the inner product for which the longest root has norm 2. This matches the loop group convention, where levels are discretized by third cohomology of simply connected compact Lie groups.
By choosing a basis J^a of the finite type Lie algebra, one may form a basis of the affine Lie algebra using J^a_n = J^a tⁿ together with a central elementK. By reconstruction, we can describe the vertex operators by normally ordered products of derivatives of the fields

When the level is non-critical, i.e., the inner product is not minus one half of the Killing form, the vacuum representation has a conformal element, given by the Sugawara construction.^[2] For any choice of dual bases J^a, J_awith respect to the level 1 inner product, the conformal element is

and yields a vertex operator algebra whose central charge is . At critical level, the conformal structure is destroyed, since the denominator is zero, but one may produce operatorsL_n for n ≥ –1 by taking a limit as k approaches criticality.
This construction can be altered to work for the rank 1 free boson. In fact, the Virasoro vectors form a one-parameter family ω_s = 1/2 x₁² + s x₂, endowing the resulting vertex operator algebras with central charge 1−12s². When s=0, we have the following formula for the graded dimension:

This is known as the generating function for partitions, and is also written as q^1/24 times the weight −1/2 modular form 1/η. The rank n free boson then has an n parameter family of Virasoro vectors, and when those parameters are zero, the character is q^n/24 times the weight −n/2 modular form η^–n.[/ltr][/size]

[ltr]Modules[size=13][edit]
Much like ordinary rings, vertex algebras admit a notion of module, or representation. Modules play an important role in conformal field theory, where they are often called sectors. A standard assumption in the physics literature is that the full Hilbert space of a conformal field theory decomposes into a sum of tensor products of left-moving and right-moving sectors:

That is, a conformal field theory has a vertex operator algebra of left-moving chiral symmetries, a vertex operator algebra of right-moving chiral symmetries, and the sectors moving in a given direction are modules for the corresponding vertex operator algebra.
Given a vertex algebra V with multiplication Y, a V-module is a vector space M equipped with an action Y^M: V ⊗ M → M((z)), satisfying the following conditions:
(Identity) Y^M(1,z) = Id_M(Associativity, or Jacobi identity) For any u, v ∈ V, w ∈ M, there is an element
such that Y^M(u,z)Y^M(v,x)w and Y^M(Y(u,z–x)v,x)w are the corresponding expansions of  in M((z))((x)) and M((x))((z–x)). Equivalently, the following "Jacobi identity" holds:

The modules of a vertex algebra form an abelian category. When working with vertex operator algebras, the previous definition is given the name "weak module", and V-modules are required to satisfy the additional condition that L₀ acts semisimply with finite-dimensional eigenspaces and eigenvalues bounded below in each coset of Z. Work of Huang, Lepowsky, Miyamoto, and Zhang has shown at various levels of generality that modules of a vertex operator algebra admit a fusion tensor product operation, and form a braided tensor category.
When the category of V-modules is semisimple with finitely many irreducible objects, the vertex operator algebra V is called rational. Rational vertex operator algebras satisfying an additional finiteness hypothesis (known as Zhu's C₂-cofiniteness condition) are known to be particularly well-behaved, and are called "regular". For example, Zhu's 1996 modular invariance theorem asserts that the characters of modules of a regular VOA form a vector-valued representation of SL₂(Z). In particular, if a VOA is holomorphic, i.e., its representation category is equivalent to that of vector spaces, then its partition function is SL₂(Z)-invariant up to a constant. Huang showed that the category of modules of a regular VOA is a modular tensor category, and its fusion rules satisfy theVerlinde formula.
To connect with our first example, the irreducible modules of the rank 1 free boson are given by Fock spaces V_λ with some fixed momentum λ, i.e., induced representations of the Heisenberg Lie algebra, where the elementb₀ acts by scalar multiplication by λ. The space can be written asC[x₁,x₂,...]v_λ, where v_λ is a distinguished ground-state vector. The module category is not semisimple, since one may induce a representation of the abelian Lie algebra where b₀ acts by a nontrivial Jordan block. For the rank n free boson, one has an irreducible module V_λ for each vector λ in complex n-dimensional space. Each vector b ∈ Cⁿ yields the operator b₀, and the Fock space V_λ is distinguished by the property that each such b₀acts as scalar multiplication by the inner product (b,λ).
Unlike ordinary rings, vertex algebras admit a notion of twisted module attached to an automorphism. For an automorphism σ of order N, the action has the form V ⊗ M → M((z^1/N)), with the following monodromy condition: if u ∈ V satisfies σ u = exp(2πik/N)u, then u_n = 0 unless nsatisfies n+k/N ∈ Z (there is some disagreement about signs among specialists). Geometrically, twisted modules can be attached to branch points on an algebraic curve with a ramified Galois cover. In the conformal field theory literature, twisted modules are called twisted sectors, and are intimately connected with string theory on orbifolds.
Vertex operator algebra defined by an even lattice[edit]
The lattice vertex algebra construction was the original motivation for defining vertex algebras. It is constructed by taking a sum of irreducible modules for the free boson corresponding to lattice vectors, and defining a multiplication operation by specifying intertwining operators between them. That is, if Λ is an even lattice, the lattice vertex algebra V_Λ decomposes into free bosonic modules as:

Lattice vertex algebras are canonically attached to double covers of even integral lattices, rather than the lattices themselves. While each such lattice has a unique lattice vertex algebra up to isomorphism, the vertex algebra construction is not functorial, because lattice automorphisms have an ambiguity in lifting.^[3]
The double covers in question are uniquely determined up to isomorphism by the following rule: elements have the form ±e_α for lattice vectors α ∈ Λ (i.e., there is a map to Λ sending e_α to α that forgets signs), and multiplication satisfies the relations e_αe_β = (–1)^(α,β)e_βe_α. Another way to describe this is that given an even lattice Λ, there is a unique (up to coboundary) normalised cocycle ε(α,β) with values ±1 such that (–1)^(α,β) = ε(α,β)ε(β,α), where the normalization condition is that ε(α,0) = ε(0,α) = 1 for all α ∈ Λ. This cocycle induces a central extension of Λ by a group of order 2, and we obtain a twisted group ring C_ε[Λ] with basis e_α (α ∈ Λ), and multiplication rule e_αe_β = ε(α,β)e_α+β - the cocycle condition on ε ensures associativity of the ring.^[4]
The vertex operator attached to lowest weight vector v_λ in the Fock spaceV_λ is

where z^λ is a shorthand for the linear map that takes any element of the α-Fock space V_α to the monomial z^{(λ, α)}. The vertex operators for other elements of the Fock space are then determined by reconstruction.
As in the case of the free boson, one has a choice of conformal vector, given by an element s of the vector space Λ ⊗ C, but the condition that the extra Fock spaces have integer L₀ eigenvalues constrains the choice of s: for an orthonormal basis x_i, the vector 1/2 x_i,1² + s₂ must satisfy (s,λ) ∈ Zfor all λ ∈ Λ, i.e., s lies in the dual lattice.
If the even lattice Λ is generated by its "root vectors" (those satisfying (α,α)=2), and any two root vectors are joined by a chain of root vectors with consecutive inner products non-zero then the vertex operator algebra is the unique simple quotient of the vacuum module of the affine Kac–Moody algebra of the corresponding simply laced simple Lie algebra at level one. This is known as the Frenkel–Kac (or Frenkel–Kac–Segal) construction, and is based on the earlier construction by Sergio Fubini andGabriele Veneziano of the tachyonic vertex operator in the dual resonance model. Among other features, the zero modes of the vertex operators corresponding to root vectors give a construction of the underlying simple Lie algebra, related to a presentation originally due to Jacques Tits. In particular, one obtains a construction of all ADE type Lie groups directly from their root lattices. and this is commonly considered the simplest way to construct the 248 dimensional group E₈.^[5][6]
Vertex operator superalgebras[edit]
By allowing the underlying vector space to be a superspace (i.e., a Z/2Z-graded vector space ) one can define a vertex superalgebra by the same data as a vertex algebra, with 1 in V₊ and T an even operator. The axioms are essentially the same, but one must incorporate suitable signs into the locality axiom, or one of the equivalent formulations. That is, if a and b are homogeneous, one comparesY(a,z)Y(b,w) with εY(b,w)Y(a,z), where ε is –1 if both a and b are odd and 1 otherwise. If in addition there is a Virasoro element ω in the even part ofV₂, and the usual grading restrictions are satisfied, then V is called avertex operator superalgebra.
One of the simplest examples is the vertex operator superalgebra generated by a single free fermion ψ. As a Virasoro representation, it has central charge 1/2, and decomposes as a direct sum of Ising modules of lowest weight 0 and 1/2. One may also describe it as a spin representation of the Clifford algebra on the quadratic space t^1/2C[t,t⁻¹](dt)^1/2 with residue pairing. The vertex operator superalgebra is holomorphic, in the sense that all modules are direct sums of itself, i.e., the module category is equivalent to the category of vector spaces.
The tensor square of the free fermion is called the free charged fermion, and by Boson-Fermion correspondence, it is isomorphic to the lattice vertex superalgebra attached to the odd lattice Z.^[7] This correspondence has been used by Date-Jimbo-Kashiwara-Miwa to construct solitonsolutions to the KP hierarchy of nonlinear PDEs.
Superconformal structures[edit]
The Virasoro algebra has some supersymmetric extensions that naturally appear in superconformal field theory and superstring theory. The N=1, 2, and 4 superconformal algebras are of particular importance.
Infinitesimal holomorphic superconformal transformations of a supercurve (with one even local coordinate z and N odd local coordinates θ₁,...,θ_N) are generated by the coefficients of a super-stress–energy tensorT(z,θ₁,...,θ_N).
When N=1, T has odd part given by a Virasoro field L(z), and even part given by a field

subject to commutation relations
[/ltr][/size]

• 
  
• 
  

[size][ltr]
By examining the symmetry of the operator products, one finds that there are two possibilities for the field G: the indices n are either all integers, yielding the Ramond algebra, or all half-integers, yielding the Neveu-Schwarz algebra. These algebras have unitary discrete series representations at central charge

and unitary representations for all c greater than 3/2, with lowest weight honly constrained by h≥ 0 for Neveu-Schwartz and h ≥ c/24 for Ramond.
An N=1 superconformal vector in a vertex operator algebra V of central charge c is an odd element τ ∈ V of weight 3/2, such that

G_-1/2τ = ω, and the coefficients of G(z) yield an action of the N=1 Neveu-Schwarz algebra at central charge c.
For N=2 supersymmetry, one obtains even fields L(z) and J(z), and odd fields G⁺(z) and G^-(z). The field J(z) generates an action of the Heisenberg algebras (described by physicists as a U(1) current). There are both Ramond and Neveu-Schwartz N=2 superconformal algebras, depending on whether the indexing on the G fields is integral or half-integral. However, the U(1) current gives rise to a one-parameter family of isomorphic superconformal algebras interpolating between Ramond and Neveu-Schwartz, and this deformation of structure is known as spectral flow. The unitary representations are given by discrete series with central charge c = 3-6/m for integers m at least 3, and a continuum of lowest weights for c > 3.
An N=2 superconformal structure on a vertex operator algebra is a pair of odd elements τ⁺, τ^- of weight 3/2, and an even element µ of weight 1 such that τ^± generate G^±(z), and µ generates J(z).
For N=3 and 4, unitary representations only have central charges in a discrete family, with c=3k/2 and 6k, respectively, as k ranges over positive integers.
Additional constructions[edit]
[/ltr][/size]

• Fixed point subalgebras: Given an action of a symmetry group on a vertex operator algebra, the subalgebra of fixed vectors is also a vertex operator algebra. In 2013, Miyamoto proved that two important finiteness properties, namely Zhu's condition C₂ and regularity, are preserved when taking fixed points under finite solvable group actions.
  

• Current extensions: Given a vertex operator algebra and some modules of integral conformal weight, one may under favorable circumstances describe a vertex operator algebra structure on the direct sum. Lattice vertex algebras are a standard example of this. Another family of examples are framed VOAs, which start with tensor products of Ising models, and add modules that correspond to suitably even codes.
  

• Orbifolds: Given a finite cyclic group acting on a holomorphic VOA, it is conjectured that one may construct a second holomorphic VOA by adjoining irreducible twisted modules and taking fixed points under an induced automorphism, as long as those twisted modules have suitable conformal weight. This is known to be true in special cases, e.g., groups of order at most 3 acting on lattice VOAs.
  

• The coset construction (due to Goddard, Kent, and Olive): Given a vertex operator algebra V of central charge c and a set S of vectors, one may define the commutant C(V,S) to be the subspace of vectors vstrictly commute with all fields coming from S, i.e., such that Y(s,z)v ∈ V[[z]] for all s ∈ S. This turns out to be a vertex subalgebra, with Y, T, and identity inherited from V. and if S is a VOA of central charge c_S, the commutant is a VOA of central charge c–c_S. For example, the embedding of SU(2) at level k+1 into the tensor product of two SU(2) algebras at levels k and 1 yields the Virasoro discrete series withp=k+2, q=k+3, and this was used to prove their existence in the 1980s. Again with SU(2), the embedding of level k+2 into the tensor product of level k and level 2 yields the N=1 superconformal discrete series.
  

• BRST reduction: For any degree 1 vector v satisfying v₀²=0, the cohomology of this operator has a graded vertex superalgebra structure. More generally, one may use any weight 1 field whose residue has square zero. The usual method is to tensor with fermions, as one then has a canonical differential. An important special case is quantum Drinfeld-Sokolov reduction applied to affine Kac–Moody algebras to obtain affine W-algebras as degree 0 cohomology. TheseW algebras also admit constructions as vertex subalgebras of free bosons given by kernels of screening operators.
  

[size][ltr]
Additional Examples[edit]
[/ltr][/size]

• The monster vertex algebra V^♮ (also called the "moonshine module"), the key to Borcherds's proof of the Monstrous moonshine conjectures, was constructed by Frenkel, Lepowsky, and Meurman in 1988. It is notable because its partition function is the modular invariant j–744, and its automorphism group is the largest sporadic simple group, known as the monster group. It is constructed by orbifolding the Leech lattice VOA by the order 2 automorphism induced by reflecting the Leech lattice in the origin. That is, one forms the direct sum of the Leech lattice VOA with the twisted module, and takes the fixed points under an induced involution. Frenkel, Lepowsky, and Meurman conjectured in 1988 that V^♮ is the unique holomorphic vertex operator algebra with central charge 24, and partition function j–744. This conjecture is still open.
  

• Chiral de Rham complex: Malikov, Schechtman, and Vaintrob showed that by a method of localization, one may canonically attach a bcβγ (boson-fermion superfield) system to a smooth complex manifold. This complex of sheaves has a distinguished differential, and the global cohomology is a vertex superalgebra. Ben-Zvi, Heluani, and Szczesny showed that a Riemannian metric on the manifold induces an N=1 superconformal structure, which is promoted to an N=2 structure if the metric is Kähler and Ricci-flat, and a hyperKähler structure induces anN=4 structure. Borisov and Libgober showed that one may obtain the two-variable elliptic genus of a compact complex manifold from the cohomology of Chiral de Rham - if the manifold is Calabi-Yau, then this genus is a weak Jacobi form.
  

[size][ltr]
Related algebraic structures[edit]
[/ltr][/size]

• If one considers only the singular part of the OPE in a vertex algebra, one arrives at the definition of a Lie conformal algebra. Since one is often only concerned with the singular part of the OPE, this makes Lie conformal algebras a natural object to study. There is a functor from vertex algebras to Lie conformal algebras that forgets the regular part of OPEs, and it has a left adjoint, called the "universal vertex algebra" functor. Vacuum modules of affine Kac–Moody algebras and Virasoro vertex algebras are universal vertex algebras, and in particular, they can be described very concisely once the background theory is developed.
  
• There are several generalizations of the notion of vertex algebra in the literature. Some mild generalizations involve a weakening of the locality axiom to allow monodromy, e.g., the abelian intertwining algebras of Dong and Lepowsky. One may view these roughly as vertex algebra objects in a braided tensor category of graded vector spaces, in much the same way that a vertex superalgebra is such an object in the category of super vector spaces. More complicated generalizations relate to q-deformations and representations of quantum groups, such as in work of Frenkel–Reshetikhin, Etingof–Kazhdan, and Li.
  
• Beilinson and Drinfeld introduced a sheaf-theoretic notion of chiral algebra that is closely related to the notion of vertex algebra, but is defined without using any visible power series. Given an algebraic curve X, a chiral algebra on X is a D_X-module A equipped with a multiplication operation  on X×X that satisfies an associativity condition. They also introduced an equivalent notion of factorization algebra that is a system of quasicoherent sheaves on all finite products of the curve, together with a compatibility condition involving pullbacks to the complement of various diagonals. Any translation-equivariant chiral algebra on the affine line can be identified with a vertex algebra by taking the fiber at a point, and there is a natural way to attach a chiral algebra on a smooth algebraic curve to any vertex operator algebra.
  

[size][ltr]
Alternative definitions[edit]
Another way to think of vertex algebras is as a generalisation of a Lie algebra with the addition of a continuous variable to both the generators of the algebra and the structure constant. (The structure constants define the algebra). So we would have:

which can also be expressed without the y dependence as:

which reveals the Vertex Multiplication as a group operation in which the group constants vary with their position on the conformal sheet.
Notes[edit]
[/ltr][/size]

1. Jump up^ Wang 1993
   
2. Jump up^ [1], The history of the Sugawara construction is complicated, with several attempts required to get the formula correct.
   
3. Jump up^ Borcherds 1986
   
4. Jump up^ Kac 1998
   
5. Jump up^ Kac 1998
   
6. Jump up^ Frenkel, Meurman & Lepowsky 1988
   
7. Jump up^ Kac 1998
   

[size][ltr]
References[edit]
[/ltr][/size]

• Borcherds, Richard (1986), "Vertex algebras, Kac-Moody algebras, and the Monster", Proc. Natl. Acad. Sci. USA. 83: 3068–3071,doi:10.1073/pnas.83.10.3068
  
• Frenkel, Igor; Lepowsky, James; Meurman, Arne (1988), Vertex operator algebras and the Monster, Pure and Applied Mathematics134, Academic Press, ISBN 0-12-267065-5
  
• Kac, Victor (1998), Vertex algebras for beginners, University Lecture Series 10 (2nd ed.), American Mathematical Society, ISBN 0-8218-1396-X
  
• Frenkel, Edward; Ben-Zvi, David (2001), Vertex algebras and Algebraic Curves, Mathematical Surveys and Monographs, American Mathematical Society, ISBN 0-8218-2894-0
  
• Wang, Weiqiang (1993), "Rationality of Virasoro vertex operator algebras", Duke Math. J. IMRN 71: 197–211
  
• Xu, Xiaoping (1998), Introduction to vertex operator superalgebras and their modules, Springer, ISBN 0792352424
  

Algebraic curve

From Wikipedia, the free encyclopedia
  (Redirected from Algebraic curves)



[ltr]In mathematics, an algebraic curve or plane algebraic curve is the setof points on the Euclidean plane whose coordinates are zeros of somepolynomial in two variables.
For example, the unit circle is an algebraic curve, being the set of zeros of the polynomial x² + y² − 1
Various technical considerations have led to consider that the complexzeros of a polynomial belong to the curve. Also, the notion of algebraic curve has been generalized to allow the coefficients of the defining polynomial and the coordinates of the points of the curve to belong to anyfield, leading to the following definition.
In algebraic geometry, a plane affine algebraic curve defined over a field k is the set of points of K² whose coordinates are zeros of somebivariate polynomial with coefficients in k, where K is some algebraically closed extension of k. The points of the curve with coordinates in k are thek-points of the curve and, all together, are the k part of the curve.
For example, (2,√−3) is a point of the curve defined by x² + y² − 1 = 0and the usual unit circle is the real part of this curve. The term "unit circle" may refer to all the complex points as well to only the real points, the exact meaning being usually clear from the context. The equationx² + y² + 1 = 0 defines an algebraic curve, whose real part is empty.
More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve. The simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and even algebraic curves that are defined independently to any embedding in an affine or projective space. This leads to the most general definition of an algebraic curve:
In algebraic geometry, an algebraic curve is an algebraic variety ofdimension one.[/ltr]

• 1 In Euclidean geometry
  
• 2 Plane projective curves
  
• 3 Remarkable points of a plane curve
  
  
  
  • 3.1 Intersection with a line
    
  • 3.2 Tangent at a point
    
  • 3.3 Asymptotes
    
  • 3.4 Singular points
    
  
  
  
• 4 Non plane algebraic curves
  
• 5 Algebraic function fields
  
• 6 Complex curves and real su***ces
  
  
  
  • 6.1 Compact Riemann su***ces
    
  
  
  
• 7 Singularities
  
  
  
  • 7.1 Classification of singularities
    
  
  
  
• 8 Examples of curves
  
  
  
  • 8.1 Rational curves
    
  • 8.2 Elliptic curves
    
  • 8.3 Curves of genus greater than one
    
  
  
  
• 9 See also
  
  
  
  • 9.1 Classical algebraic geometry
    
  • 9.2 Modern algebraic geometry
    
  • 9.3 Geometry of Riemann su***ces
    
  
  
  
• 10 References
  
• 11 Notes
  

[ltr]

In Euclidean geometry[edit]
An algebraic curve in the Euclidean plane is the set of the points whosecoordinates are the solutions of a bivariate polynomial equation p(x, y) = 0. This equation is often called the implicit equation of the curve, by opposition to the curves that are the graph of a function defining explicitlyy as a function of x.
Given a curve given by such an implicit equation, the first problems that occur is to determine the shape of the curve and to draw it. These problems are not as easy to solve as in the case of the graph of a function, for which y may easily be computed for various values of x. The fact that the defining equation is a polynomial implies that the curve has some structural properties that may help to solve these problems.
Every algebraic curve may be uniquely decomposed into a finite number of smooth monotone arcs (also called branches) connected by some points sometimes called "remarkable points". A smooth monotone arc is the graph of a smooth function which is defined and monotone on an open interval of the x-axis. In each direction, an arc is either unbounded (one talk of an infinite arc) or has an end point which is either a singular point (this will be defined below) or a point with a tangent parallel to one of the coordinate axes.
For example, for the Tschirnhausen cubic of the figure, there are two infinite arcs having the origin (0,0) as end point. This point is the only singular point of the curve. There are two arcs having this singular point as one end point and having a second end point with a horizontal tangent. Finally, there are two other arcs having these points with horizontal tangent as first end point and sharing the unique point with vertical tangent as second end point. On the other hand, the sinusoid is certainly not an algebraic curve, having an infinite number of monotone arcs.
To draw an algebraic curve, it is important to know the remarkable points and their tangents, the infinite branches and their asymptote (if any) and the way in which the arcs connect them. It is also useful to consider also the inflection points as remarkable points. When all this information is drawn on a paper sheet, the shape of the curve appears usually rather clearly. If not it suffices to add a few other points and their tangents to get a good description of the curve.
The methods for computing the remarkable points and their tangents are described below, after section Projective curves.
Plane projective curves[edit]
It is often desirable to consider curves in the projective space. An algebraic curve in the projective plane or plane projective curve is the set of the points in a projective plane whose projective coordinates are zeros of a homogeneous polynomial in three variables P(x, y, z).
Every affine algebraic curve of equation p(x, y) = 0 may be completed into the projective curve of equation  where

is the result of the homogenization of p. Conversely, if P(x, y, z) = 0 is the homogeneous equation of a projective curve, then P(x, y, 1) = 0 is the equation of an affine curve, which consists of the points of the projective curve whose third projective coordinate is not zero. These two operations are reciprocal one to the other, as  and, if p is defined by , then , as soon as the homogeneous polynomial P is not divisible by z.
For example, the projective curve of equation x² + y² − z² is the projective completion of the unit circle of equation x² + y² − 1 = 0.
This allows to consider that an affine curve and its projective completion are the same curve, or, more precisely that the affine curve is a part of the projective curve that is large enough to well define the "complete" curve. This point of view is commonly expressed by calling "points at infinity" of the affine curve the points (in finite number) of the projective completion that do not belong to the affine part.
Projective curves are frequently studied for themselves. They are also useful for the study of affine curves. For example, if p(x, y) is the polynomial defining an affine curve, beside the partial derivatives  and , it is useful to consider the derivative at infinity

For example, the equation of the tangent of the affine curve of equationp(x, y) = 0 at a point (a, b) is

Remarkable points of a plane curve[edit][/ltr]

This section requires expansion.(October 2012)

[ltr]
In this section, we consider a plane algebraic curve defined by a bivariate polynomial p(x, y) and its projective completion, defined by the homogenization  of p.
Intersection with a line[edit]
Knowing the points of intersection of a curve with a given line is frequently useful. The intersection with the axes of coordinates and the asymptotesare useful to draw the curve. Intersecting with a line parallel to the axes allows to find at least a point in each branch of the curve. If an efficientroot-finding algorithm is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to the y-axis and passing through each pixel on the x-axis.
If the polynomial defining the curve has degree d, any line cuts the curve in at most d points. Bézout's theorem asserts that this number is exactly d, if the points are searched in the projective plane over an algebraically closed field (for example the complex numbers), and counted with theirmultiplicity. The method of computation that follows proves again this theorem, in this simple case.
To compute the intersection of the curve defined by the polynomial p with the line of equation ax+by+c = 0, one solves in x (or in y if a = 0) the equation of the line. Substituting the result in p, one gets a univariate equation q(y) = 0 (or q(x) = 0, if the equation of the line has been solved iny), whose roots are one coordinate of the intersection points. The other coordinate is deduced from the equation of the line. The multiplicity of an intersection point is the multiplicity of the corresponding root. There is an intersection point at infinity, if the degree of q is lower than the degree ofp; the multiplicity of such an intersection point at infinity is the difference of the degrees of p and q.
Tangent at a point[edit]
The tangent at a point (a, b) of the curve is the line of equation , like for everydifferentiable curve defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric:

where  is the derivative at infinity. The equivalence of the two equations results from Euler's homogeneous function theorem applied to P.
If  the tangent is not defined and the point is a singular point.
This extends immediately to the projective case: The equation of the tangent of at the point of projective coordinates (a:b:c) of the projective curve of equation P(x, y, z) = 0 is

and the points of the curves that are singular are the points such that

(The condition P(a, b, c) = 0 is implied by these conditions, by Euler's homogeneous function theorem.)
Asymptotes[edit]
Every infinite branch of an algebraic curve corresponds to a point at infinity on the curve, that is a point of the projective completion of the curve that does not belongs to its affine part. The correspondingasymptote is the tangent of the curve at that point. The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case.
Let  be the decomposition of the polynomial defining the curve into its homogeneous parts, where p_i is the sum of the monomials of p of degree i. It follows that

and

A point at infinity of the curve is a zero of p of the form (a, b, 0). Equivalently, (a, b) is a zero of p_d. The fundamental theorem of algebraimplies that, over an algebraically closed field (typically, the field of complex numbers), p_d factors into a product of linear factors. Each factor defines a point at infinity on the curve: if bx − ay is such a factor, then it defines the point at infinity (a, b, 0). Over the reals, p_d factors into linear and quadratic factors. The irreducible quadratic factors define non-real points at infinity, and the real points are given by the linear factors. If (a, b, 0) is a point at infinity of the curve, one says that (a, b) is an asymptotic direction. Setting q = p_d the equation of the corresponding asymptote is

If  and  the asymptote is the line at infinity, and, in the real case, the curve has a branch that looks like a parabola. In this case one says that the curve has a parabolic branch. If

the curve has a singular point at infinity and may have several asymptotes. They may be computed by the method of computing the tangent cone of a singular point.
Singular points[edit]
The singular points of a curve of degree d defined by a polynomial p(x,y) of degree d are the solutions of the system of equations:

In characteristic zero, this system is equivalent with

where, with the notation of the preceding section,  The systems are equivalent because ofEuler's homogeneous function theorem. The latter system has the advantage of having its third polynomial of degree d-1 instead of d.
Similarly, for a projective curve defined by a homogeneous polynomialP(x,y,z) of degree d, the singular points have the solutions of the system

as homogeneous coordinates. (In positive characteristic, the equation  has to be added to the system.)
This implies that the number of singular points is finite as soon as p(x,y) orP(x,y,z) is square free. Bézout's theorem implies thus that the number of singular points is at most (d−1)², but this bound is not sharp because the system of equations is overdetermined. If reducible polynomials are allowed, the sharp bound is d(d−1)/2, this value being reached when the polynomial factors in linear factors, that is if the curve is the union of dlines. For irreducible curves and polynomials, the number of singular points is at most (d−1)(d−2)/2, because of the formula expressing the genus in term of the singularities (see below). The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below).
The equation of the tangents at a singular point are given by the nonzero homogeneous part of lowest degree in the Taylor series of the polynomial at the singular point. When one changes the coordinates to put the singular point at the origin, the equation of the tangents at the singular point is thus the nonzero homogeneous part of lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.
Non plane algebraic curves[edit]
An algebraic curve is an algebraic variety of dimension one. This implies that an affine curve in an affine space of dimension n is defined by, at least, n−1 polynomials in n variables. To define a curve, these polynomials must generate a prime ideal of Krull dimension 1. This condition is not easy to test in practice. Therefore the following way to represent non plane curves may be preferred.
Let  be n − 1 polynomials in two variables x₁ and x₂such that f is irreducible. The points in the affine space of dimension nsuch whose coordinates satisfy the equations and inequations

are all the points of an algebraic curve in which a finite number of points have been removed. This curve is defined by a system of generators of the ideal of the polynomials h such that it exists an integer k such belongs to the ideal generated by . This representation is a rational equivalence between the curve and the plane curve defined by f. Every algebraic curve may be represented in this way. However, a linear change of variables may be needed in order to make almost always injective the projection on the two first variables. When a change of variables is needed, almost every change is convenient, as soon as it is defined over an infinite field.
This representation allows to deduce easily any property of a non-plane algebraic curve, including its graphical representation, from the corresponding property of its plane projection.
For a curve defined by its implicit equations, above representation of the curve may easily deduced from a Gröbner basis for a block ordering such that the block of the smaller variables is (x₁, x₂). The polynomial f is the unique polynomial in the base that depends only of x₁ and x₂. The fractions g_i/g₀ are obtained by choosing, for i = 3, ..., n, a polynomial in the basis that is linear in x_i and depends only on x₁, x₂ and x_i. If these choices are not possible, this means either that the equations define an algebraic set that is not a variety, or that the variety is not of dimension one, or that one must change of coordinates. The latter case occurs when f exists and is unique, and, for i = 3, ..., n, there exist polynomials whose leading monomial depends only on x₁, x₂ and x_i.
Algebraic function fields[edit]
The study of algebraic curves can be reduced to the study of irreduciblealgebraic curves. Up to birational equivalence, these are categorically equivalent to algebraic function fields. An algebraic function field is a field of algebraic functions in one variable K defined over a given field F. This means there exists an element x of K which is transcendental over F, and such that K is a finite algebraic extension of F(x), which is the field of rational functions in the indeterminate x over F.
For example, consider the field C of complex numbers, over which we may define the field C(x) of rational functions in C. If y² = x³ − x − 1, then the field C(x, y) is an elliptic function field. The element x is not uniquely determined; the field can also be regarded, for instance, as an extension of C(y). The algebraic curve corresponding to the function field is simply the set of points (x, y) in C² satisfying y² = x³ − x − 1.
If the field F is not algebraically closed, the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, "curves" with no points on them. If the base field F is the field R of real numbers, then x² + y² = −1 defines an algebraic extension field of R(x), but the corresponding curve considered as a locus has no points in R. However, it does have points defined over the algebraic closure C of R.
Complex curves and real su***ces[edit]
A complex projective algebraic curve resides in n-dimensional complex projective space CPⁿ. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, andorientable. An algebraic curve likewise has topological dimension two; in other words, it is a su***ce. A nonsingular complex projective algebraic curve will then be a smooth orientable su***ce as a real manifold, embedded in a compact real manifold of dimension 2n which is CPⁿregarded as a real manifold.
The topological genus of this su***ce, that is the number of handles or donut holes, is equal to the genus of the algebraic curve that may be computed by algebraic means. In short, if one consider a plane projection of a non singular curve, that has degree d and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is (d − 1)(d − 2)/2 − k, where k is the number of these singularities.
Compact Riemann su***ces[edit]
The theory of compact Riemann su***ces consists in studying non-singular complex algebraic curves through the complex analytic structure induced on this real compact su***ce.
A Riemann su***ce is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions. It is compact if it is compact as a topological space.
There is a triple equivalence of categories between the category of smooth projective algebraic curves over the complex numbers (with rational mapsas morphisms), the category of compact Riemann su***ces, and the category of complex algebraic function fields, so that in studying these subjects we are in a sense studying the same thing. This allows complex analytic methods to be used in algebraic geometry, and algebraic-geometric methods in complex analysis, and field-theoretic methods to be used in both, which is characteristic of a much wider class of problems than simply curves and Riemann su***ces.
See also Algebraic geometry and analytic geometry, as more general theory.
Singularities[edit]
Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth or non-singular, or else singular. Givenn−1 homogeneous polynomials in n+1 variables, we may find the Jacobian matrix as the (n−1)×(n+1) matrix of the partial derivatives. If the rank of this matrix is n−1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remainsn−1 when the Jacobian matrix is evaluated at a point P on the curve, then the point is a smooth or regular point; otherwise it is a singular point. In particular, if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial equation f(x,y,z) = 0, then the singular points are precisely the points P where the rank of the 1×(n+1) matrix is zero, that is, where

Since f is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the field F, which in particular need not be the real or complex numbers. It should of course be recalled that (0,0,0) is not a point of the curve and hence not a singular point.
Similarly, for an affine algebraic curve defined by a single polynomial equation f(x,y) = 0, then the singular points are precisely the points P of the curve where the rank of the 1×n Jacobian matrix is zero, that is, where

The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing thegenus, which is a birational invariant. For this to work, we should consider the curve projectively and require F to be algebraically closed, so that all the singularities which belong to the curve are considered.
Classification of singularities[edit][/ltr]


[ltr]
Singular points include multiple points where the curve crosses over itself, and also various types of cusp, for example that shown by the curve with equation x³ = y² at (0,0).
A curve C has at most a finite number of singular points. If it has none, it can be called smooth ornon-singular. For this definition to be correct, we must use analgebraically closed field and a curve C in projective space (i.e.,complete in the sense of algebraic geometry). If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m(m−1)/2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of  where  is the local ring at P and  is its integral closure. See also Hartshorne, Algebraic Geometry, IV Ex. 1.8.
The Milnor number μ of the singularity is the degree of the mapping gradf(x,y)/|grad f(x,y)| on the small sphere of radius ε, in the sense of the topological degree of a continuous mapping, where grad f is the (complex) gradient vector field of f. It is related to δ and r by the Milnor-Jung formula,
μ = 2δ − r + 1.
Another singularity invariant of note is the multiplicity m, defined as the maximum integer such that the derivatives of f to all orders up to m vanish.
Computing the delta invariants of all of the singularities allows the genus gof the curve to be determined; if d is the degree, then

where the sum is taken over all singular points P of the complex projective plane curve. It is called the genus formula.
Singularities may be classified by the triple [m, δ, r], where m is the multiplicity, δ is the delta-invariant, and r is the branching number. In these terms, an ordinary cusp is a point with invariants [2,1,1] and an ordinary double point is a point with invariants [2,1,2]. An ordinary n-multiple point may be defined as one having invariants [n, n(n−1)/2, n].
Examples of curves[edit]
Rational curves[edit]
A rational curve, also called a unicursal curve, is any curve which isbirationally equivalent to a line, which we may take to be a projective line; accordingly, we may identify the function field of the curve with the field of rational functions in one indeterminate F(x). If F is algebraically closed, this is equivalent to a curve of genus zero; however, the field of all real algebraic functions defined on the real algebraic variety x²+y² = −1 is a field of genus zero which is not a rational function field.
Concretely, a rational curve of dimension n over F can be parameterized (except for isolated exceptional points) by means of n rational functions defined in terms of a single parameter t; by clearing denominators we can turn this into n+1 polynomial functions in projective space. An example would be the rational normal curve.
Any conic section defined over F with a rational point in F is a rational curve. It can be parameterized by drawing a line with slope t through the rational point, and intersection with the plane quadratic curve; this gives a polynomial with F-rational coefficients and one F-rational root, hence the other root is F-rational (i.e., belongs to F) also.[/ltr]


[ltr]
For example, consider the ellipsex² + xy + y² = 1, where (−1, 0) is a rational point. Drawing a line with slope t from (−1,0), y = t(x+1), substituting it in the equation of the ellipse, factoring, and solving for x, we obtain

We then have that the equation fory is

which defines a rational parameterization of the ellipse and hence shows the ellipse is a rational curve. All points of the ellipse are given, except for (−1,1), which corresponds to t = ∞; the entire curve is parameterized therefore by the real projective line.
Such a rational parameterization may be considered in the projective space by equating the first projective coordinates to the numerators of the parameterization and the last one to the common denominator. As the parameter is defined in a projective line, the polynomials in the parameter should be homogenized. For example, the projective parameterization of above ellipse is

Eliminating T and U between these equations we get again the projective equation of the ellipse

which may be easily obtained directly by homogenizing above equation.
Many of the curves on Wikipedia's list of curves are rational, and hence have similar rational parameterizations.
Elliptic curves[edit]
An elliptic curve may be defined as any curve of genus one with a rational point: a common model is a nonsingular cubic curve, which suffices to model any genus one curve. In this model the distinguished point is commonly taken to be an inflection point at infinity; this amounts to requiring that the curve can be written in Tate-Weierstrass form, which in its projective version is

Elliptic curves carry the structure of an abelian group with the distinguished point as the identity of the group law. In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions.
The intersection of two quadric su***ces is in general a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic, or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve a line, or two conics, or a conic and two lines, or four lines).
Curves of genus greater than one[edit]
Curves of genus greater than one differ markedly from both rational and elliptic curves. Such curves defined over the rational numbers, by Faltings' theorem, can have only a finite number of rational points, and they may be viewed as having a hyperbolic geometry structure. Examples are thehyperelliptic curves, the Klein quartic curve, and the Fermat curvexⁿ + yⁿ = zⁿ when n is greater than three.
See also[edit]
Classical algebraic geometry[edit][/ltr]

• Acnode
  
• Bézout's theorem
  
• Crunode
  
• Curve
  
• Curve sketching
  
• Jacobian variety
  
• Klein quartic
  
• List of curves
  
• Hilbert's sixteenth problem
  
• Cubic plane curve
  
• Hyperelliptic curve
  

[ltr]Modern algebraic geometry[size=13][edit]
[/ltr][/size]

• Birational geometry
  
• Conic section
  
• Elliptic curve
  
• Fractional ideal
  
• Function field of an algebraic variety
  
• Function field (scheme theory)
  
• Genus (mathematics)
  
• Riemann–Roch theorem for algebraic curves
  
• Quartic plane curve
  
• Rational normal curve
  
• Weber's theorem
  

[size][ltr]
Geometry of Riemann su***ces[edit]
[/ltr][/size]

• Riemann–Hurwitz formula
  
• Riemann–Roch theorem for Riemann su***ces
  
• Riemann su***ce
  

[size][ltr]
References[edit]
[/ltr][/size]

Wikimedia Commons has media related to Algebraic curves.

• Egbert Brieskorn and Horst Knörrer, Plane Algebraic Curves, John Stillwell, translator, Birkhäuser, 1986
  
• Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, American Mathematical Society, Mathematical Surveys Number VI, 1951
  
• Hershel M. Farkas and Irwin Kra, Riemann Su***ces, Springer, 1980
  
• W. Fulton, Algebraic Curves: an introduction to algebraic geometryavailable at ^[1]
  
• C.G. Gibson, Elementary Geometry of Algebraic Curves: An Undergraduate Introduction, Cambridge University Press, 1998.
  
• Phillip A. Griffiths, Introduction to Algebraic Curves, Kuniko Weltin, trans., American Mathematical Society, Translation of Mathematical Monographs volume 70, 1985 revision
  
• Robin Hartshorne, Algebraic Geometry, Springer, 1977
  
• Shigeru Iitaka, Algebraic Geometry: An Introduction to the Birational Geometry of Algebraic Varieties, Springer, 1982
  
• John Milnor, Singular Points of Complex Hypersu***ces, Princeton University Press, 1968
  
• George Salmon, Higher Plane Curves, Third Edition, G. E. Stechert & Co., 1934
  
• Jean-Pierre Serre, Algebraic Groups and Class Fields, Springer, 1988
  
• Claire Voisin LECTURES ON THE HODGE AND GROTHENDIECK–HODGE CONJECTURES;^[2] ANTICANONICAL DIVISORS AND CURVE CLASSES ON FANO MANIFOLDS;^[3] Voisin C. Hodge theory and complex algebraic geometry 1;^[4] Green's canonical syzygy conjecture for generic curves of odd genus;^[5] Green’s generic syzygy conjecture for curves of even genus lying on a K3 su***ce ^[6]
  
• Montserrat Teixidor i Bigas ON A CONJECTURE OF LANGE;^[7] Moduli spaces of vector bundles on reducible curves;^[8] Green’s Conjecture for the generic r-gonal curve of genus g ¸ 3r ¡ 7 ^[9]
  
• Ernst Kötter (1887). "Grundzüge einer rein geometrischen Theorie der algebraischen ebenen Kurven (Fundamentals of a purely geometrical theory of algebraic plane curves)". Transactions of the Royal Academy of Berlin. — gained the 1886 Academy prize^[10]
  

[size][ltr]
Notes[edit]
[/ltr][/size]

1. Jump up^ Algebraic Curves: an introduction to algebraic geometry
   
2. Jump up^ LECTURES ON THE HODGE AND GROTHENDIECK–HODGE CONJECTURES
   
3. Jump up^ ANTICANONICAL DIVISORS AND CURVE CLASSES ON FANO MANIFOLDS
   
4. Jump up^ Hodge theory and complex algebraic geometry 1
   
5. Jump up^ Green's canonical syzygy conjecture for generic curves of odd genus
   
6. Jump up^ Green’s generic syzygy conjecture for curves of even genus lying on a K3 su***ce
   
7. Jump up^ ON A CONJECTURE OF LANGE
   
8. Jump up^ Moduli spaces of vector bundles on reducible curves
   
9. Jump up^ Green’s Conjecture for the generic r-gonal curve of genus g ¸ 3r ¡ 7
   
10. Jump up^ Norman Fraser (Feb 1888). "Kötter's synthetic geometry of algebraic curves". Proceedings of the Edinburgh Mathematical Society 7: 46–61.Here: p.46
    

[th] [hide] v t e Topics in algebraic curves[/th][th]Rational curves[/th][th]Elliptic curves[/th][th]Higher genus[/th][th]Plane curves[/th][th]Riemann su***ces[/th][th]Constructions[/th][th]Structure of curves[/th] Five points determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic [th]Analytic theory[/th][th]Arithmetic theory[/th][th]Applications[/th] Elliptic function Elliptic integral Fundamental pair of periods Modular form Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm Elliptic curve cryptography Elliptic curve primality De Franchis theorem Faltings' theorem Hurwitz's automorphisms theorem Hurwitz su***ce Hyperelliptic curve AF+BG theorem Bézout's theorem Bitangent Cayley–Bacharach theorem Conic section Cramer's paradox Cubic plane curve Fermat curve Genus–degree formula Hilbert's sixteenth problem Nagata's conjecture on curves Plücker formula Quartic plane curve Real plane curve Belyi's theorem Bolza su***ce Dessin d'enfant Differential of the first kind Klein quartic Riemann's existence theorem Riemann–Roch theorem Teichmüller space Torelli theorem Dual curve Polar curve Smooth completion [th]Divisors on curves[/th][th]Moduli[/th][th]Morphisms[/th][th]Singularities[/th][th]Vector bundles[/th] Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem for algebraic curves Weierstrass point Weil reciprocity law ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem Acnode Crunode Cusp Delta invariant Tacnode Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

Seiberg-Witten map

From Wikipedia, the free encyclopedia




[ltr]The Seiberg-Witten map is a map used in gauge theory and string theoryintroduced by Nathan Seiberg and Edward Witten which relates non-commutative degress of freedom of a gauge theory to their commutative counterparts. It was argued by Seiberg and Witten that certain non-commutative gauge theories are equivalent to commutative ones and that there exists a map from a commutative gauge field to a non-commutative one, which is compatible with the gauge structure of each.
References[size=13][edit][/ltr][/size]

• Seiberg, Nathan; Witten, Edward (1999), String theory and noncommutative geometry
  

Noncommutative geometry

From Wikipedia, the free encyclopedia


[ltr]Not to be confused with Anabelian geometry.
Noncommutative geometry (NCG) is a branch of mathematicsconcerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which  does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.

[/ltr]

1 Motivation



• 1.1 Applications in mathematical physics
  
• 1.2 Motivation from ergodic theory
  



2 Noncommutative C*-algebras, von Neumann algebras

3 Noncommutative differentiable manifolds

4 Noncommutative affine and projective schemes

5 Invariants for noncommutative spaces

6 Examples of noncommutative spaces

7 See also

8 Notes

9 References

10 Further reading

11 External links

[size][ltr]

Motivation[edit]
The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics, spaces, which are geometric in nature, can be related to numerical functions on them. In general, such functions will form a commutative ring. For instance, one may take the ring C(X) of continuous complex-valued functions on atopological space X. In many cases (e.g., if X is a compact Hausdorff space), we can recover X from C(X), and therefore it makes some sense to say that X has commutative topology.
More specifically, in topology, compact Hausdorff topological spaces can be reconstructed from the Banach algebra of functions on the space (Gel'fand-Neimark). In commutative algebraic geometry, algebraic schemes are locally prime spectra of commutative unital rings (A. Grothendieck), and schemes can be reconstructed from the categories of quasicoherent sheaves of modules on them (P. Gabriel-A. Rosenberg). For Grothendieck topologies, the cohomological properties of a site are invariant of the corresponding category of sheaves of sets viewed abstractly as a topos (A. Grothendieck). In all these cases, a space is reconstructed from the algebra of functions or its categorified version—some category of sheaves on that space.
Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.
The dream of noncommutative geometry is to generalize this duality to the duality between
[/ltr][/size]

• noncommutative algebras, or sheaves of noncommutative algebras, or sheaf-like noncommutative algebraic or operator-algebraic structures
  

• and geometric entities of certain kind,
  

[size][ltr]
and interact between the algebraic and geometric description of those via this duality.
Regarding that the commutative rings correspond to usual affine schemes, and commutative C*-algebras to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization oftopological spaces, as "non-commutative spaces". For this reason, some talk about non-commutative topology, though the term also has other meanings.
Applications in mathematical physics[edit]
Some applications in particle physics are described on the entriesNoncommutative standard model and Noncommutative quantum field theory. Sudden rise in interest in noncommutative geometry in physics, follows after the speculations of its role in M-theory made in 1997.^[1]
Motivation from ergodic theory[edit]
Some of the theory developed by Alain Connes to handle noncommutative geometry at a technical level has roots in older attempts, in particular inergodic theory. The proposal of George Mackey to create a virtual subgroup theory, with respect to which ergodic group actions would become homogeneous spaces of an extended kind, has by now been subsumed.
Noncommutative C*-algebras, von Neumann algebras[edit]
(The formal duals of) non-commutative C*-algebras are often now called non-commutative spaces. This is by analogy with the Gelfand representation, which shows that commutative C*-algebras are dual tolocally compact Hausdorff spaces. In general, one can associate to any C*-algebra S a topological space Ŝ; see spectrum of a C*-algebra.
For the duality between σ-finite measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are callednon-commutative measure spaces.
Noncommutative differentiable manifolds[edit]
A smooth Riemannian manifold M is a topological space with a lot of extra structure. From its algebra of continuous functions C(M) we only recoverM topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple. It is constructed from a smooth vector bundleE over M, e.g. the exterior algebra bundle. The Hilbert space L²(M,E) of square integrable sections of E carries a representation of C(M) by multiplication operators, and we consider an unbounded operator D inL²(M,E) with compact resolvent (e.g. the signature operator), such that the commutators [D,f] are bounded whenever f is smooth. A recent deep theorem^[2] states that M as a Riemannian manifold can be recovered from this data.
This suggests that one might define a noncommutative Riemannian manifold as a spectral triple (A,H,D), consisting of a representation of a C*-algebra A on a Hilbert space H, together with an unbounded operator D onH, with compact resolvent, such that [D,a] is bounded for all a in some dense subalgebra of A. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.
Noncommutative affine and projective schemes[edit]
In analogy to the duality between affine schemes and commutative rings, we define a category of noncommutative affine schemes as the dual of the category of associative unital rings. There are certain analogues of Zariski topology in that context so that one can glue such affine schemes to more general objects.
There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a Serre's theorem on Proj. Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length; there is also analogous theorem for coherent sheaves when the algebra is Noetherian. This theorem is extended as a definition ofnoncommutative projective geometry by Michael Artin and J. J. Zhang,^[3] who add also some general ring-theoretic conditions (e.g. Artin-Schelter regularity).
Many properties of projective schemes extend to this context. For example, there exist an analog of the celebrated Serre duality for noncommutative projective schemes of Artin and Zhang.^[4]
A. L. Rosenberg has created a rather general relative concept ofnoncommutative quasicompact scheme (over a base category), abstracting the Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.^[5] There is also another interesting approach via localization theory, due to Fred Van Oystaeyen, Luc Willaert and Alain Verschoren, where the main concept is that of a schematic algebra.^[6]
Invariants for noncommutative spaces[edit]
Some of the motivating questions of the theory are concerned with extending known topological invariants to formal duals of noncommutative (operator) algebras and other replacements and candidates for noncommutative spaces. One of the main starting points of the Alain Connes' direction in noncommutative geometry is his discovery (and independently by Boris Tsygan) of a new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely the cyclic homology and its relations to the algebraic K-theory (primarily via Connes-Chern character map).
The theory of characteristic classes of smooth manifolds has been extended to spectral triples, employing the tools of operator K-theory andcyclic cohomology. Several generalizations of now classical index theoremsallow for effective extraction of numerical invariants from spectral triples. The fundamental characteristic class in cyclic cohomology, the JLO cocycle, generalizes the classical Chern character.
Examples of noncommutative spaces[edit]
[/ltr][/size]

• In Weyl quantization, the symplectic phase space of classical mechanics is deformed into a non-commutative phase space generated by the position and momentum operators.
  

• The standard model of particle physics is another example of a noncommutative geometry, cf noncommutative standard model.
  

• The noncommutative torus, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple. This class of examples has been studied intensively and still functions as a test case for more complicated situations.
  

• Snyder space^[7]
  

• Noncommutative algebras arising from foliations.
  

• Examples related to dynamical systems arising from number theory, such as the Gauss shift on continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.
  

[size][ltr]
See also[edit]
[/ltr][/size]

• Commutativity
  
• Phase space formulation
  
• Moyal product
  
• Fuzzy sphere
  
• Noncommutative algebraic geometry
  

[size][ltr]
Notes[edit]
[/ltr][/size]

1. Jump up^ Alain Connes, Michael R. Douglas, Albert Schwarz, Noncommutative geometry and matrix theory: compactification on tori. J. High Energy Phys. 1998, no. 2, Paper 3, 35 pp. doi, hep-th/9711162
   
2. Jump up^ Connes, Alain, On the spectral characterization of manifolds,arXiv:0810.2088v1
   
3. Jump up^ M. Artin, J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228--287, doi
   
4. Jump up^ Amnon Yekutieli, James J. Zhang, Serre duality for noncommutative projective schemes, Proc. Amer. Math. Soc. 125, n. 3, 1997, 697-707,pdf
   
5. Jump up^ A. L. Rosenberg, Noncommutative schemes, Compositio Math. 112 (1998) 93--125, doi; Underlying spaces of noncommutative schemes, preprint MPIM2003-111, dvi, ps; MSRI lecture Noncommutative schemes and spaces (Feb 2000): video
   
6. Jump up^ Freddy van Oystaeyen, Algebraic geometry for associative algebras,ISBN 0-8247-0424-X - New York: Dekker, 2000.- 287 p. - (Monographs and textbooks in pure and applied mathematics , 232); F. van Oystaeyen, L. Willaert, Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras, J. Pure Appl. Alg. 104 (1995), p. 109--122
   
7. Jump up^ H. S. Snyder, Quantized Space-Time, Phys. Rev. 71 (1947) 38
   

[size][ltr]
References[edit]
[/ltr][/size]

• Connes, Alain (1994), Non-commutative geometry, Boston, MA:Academic Press, ISBN 978-0-12-185860-5
  
• Connes, Alain; Marcolli, Matilde (2008), "A walk in the noncommutative garden", An invitation to noncommutative geometry, World Sci. Publ., Hackensack, NJ, pp. 1–128, arXiv:math/0601054, MR 2408150
  
• Connes, Alain; Marcolli, Matilde (2008), Noncommutative geometry, quantum fields and motives, American Mathematical Society Colloquium Publications 55, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4210-2, MR 2371808
  
• Gracia-Bondia, Jose M; Figueroa, Hector; Varilly, Joseph C (2000),Elements of Non-commutative geometry, Birkhauser, ISBN 978-0-8176-4124-5
  
• Landi, Giovanni (1997), An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics. New Series m: Monographs51, Berlin, New York: Springer-Verlag, arXiv:hep-th/9701078,ISBN 978-3-540-63509-3, MR 1482228
  
• Van Oystaeyen, Fred; Verschoren, Alain (1981), Non-commutative algebraic geometry, Lecture Notes in Mathematics 887, Springer-Verlag, ISBN 978-3-540-11153-5
  

[size][ltr]
Further reading[edit]
[/ltr][/size]

• Consani, Caterina; Connes, Alain, eds. (2011), Noncommutative geometry, arithmetic, and related topics. Proceedings of the 21st meeting of the Japan-U.S. Mathematics Institute (JAMI) held at Johns Hopkins University, Baltimore, MD, USA, March 23–26, 2009, Baltimore, MD: Johns Hopkins University Press, ISBN 1-4214-0352-8,Zbl 1245.00040
  

[size][ltr]
External links[edit]
[/ltr][/size]

• Introduction to Quantum Geometry by Micho Đurđevich
  
• Lectures on Noncommutative Geometry by Victor Ginzburg
  
• Very Basic Noncommutative Geometry by Masoud Khalkhali
  
• Lectures on Arithmetic Noncommutative Geometry by Matilde Marcolli
  
• Noncommutative Geometry for Pedestrians by J. Madore
  
• An informal introduction to the ideas and concepts of noncommutative geometry by Thierry Masson (an easier introduction that is still rather technical)
  
• Noncommutative geometry on arxiv.org
  
• MathOverflow, Theories of Noncommutative Geometry
  
• S. Mahanta, On some approaches towards non-commutative algebraic geometry, math.QA/0501166
  
• G. Sardanashvily, Lectures on Differential Geometry of Modules and Rings (Lambert Academic Publishing, Saarbrücken, 2012); arXiv: 0910.1515
  
• Noncommutative geometry and particle physics
  

Modular form

From Wikipedia, the free encyclopedia
  (Redirected from Modular forms)

[ltr]"Modular function" redirects here. A distinct use of this term appears in relation to Haar measure.
In mathematics, a modular form is a (complex) analytic function on theupper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs tocomplex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology and string theory.
A modular function is a modular form invariant with respect to the modular group but without the condition that f(z) be holomorphic at infinity. Instead, modular functions are meromorphic at infinity.
Modular form theory is a special case of the more general theory ofautomorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups.

[/ltr]

1 Modular forms for SL₂(Z)



• 1.1 Definition in terms of lattices or elliptic curves
  
• 1.2 Modular functions
  



2 Modular forms for more general groups



• 2.1 The Riemann su***ce G\H^∗
  
• 2.2 Definition
  
• 2.3 Consequences
  
• 2.4 Line bundles
  



3 Miscellaneous



• 3.1 Entire forms
  
• 3.2 Automorphic factors and other generalizations
  



4 Examples

5 Generalizations

6 History

7 Notes

8 References

[size][ltr]

Modular forms for SL₂(Z)[edit]
A modular form of weight k for the modular group

is a complex-valued function f on the upper half-planeH = {z ∈ C, Im(z) > 0}, satisfying the following three conditions: firstly, f is aholomorphic function on H. Secondly, for any z in H and any matrix in SL(2,Z) as above, the equation

is required to hold. Thirdly, f is required to be holomorphic as z → i∞. The latter condition is also phrased by saying that f is "holomorphic at the cusp", a terminology that is explained below. The weight k is typically a positive integer.
The second condition, with the matrices  and  reads

and

respectively. Since S and T generate the modular group SL(2,Z), the second condition above is equivalent to these two equations. Note that since
,
modular forms are periodic functions, with period 1, and thus have aFourier series.
Note that for odd k, only the zero function can satisfy the second condition.
Definition in terms of lattices or elliptic curves[edit]
A modular form can equivalently be defined as a function F from the set oflattices in C to the set of complex numbers which satisfies certain conditions:
(1) If we consider the lattice  generated by a constant α and a variable z, then F(Λ) is an analytic function of z.(2) If α is a non-zero complex number and αΛ is the lattice obtained by multiplying each element of Λ by α, then F(αΛ) = α^−kF(Λ) where k is a constant (typically a positive integer) called the weight of the form.(3) The absolute value of F(Λ) remains bounded above as long as the absolute value of the smallest non-zero element in Λ is bounded away from 0.
The key idea in proving the equivalence of the two definitions is that such a function F is determined, because of the first property, by its values on lattices of the form , where ω ∈ H.
Modular functions[edit]
When the weight k is zero, the only modular forms are constant functions, as can be shown. However, relaxing the requirement that f be holomorphic leads to the notion of modular functions. A function f : H → C is called modular iff it satisfies the following properties:
[/ltr][/size]

1. f is meromorphic in the open upper half-plane H.
   
2. For every matrix  in the modular group Γ, .
   
3. As pointed out above, the second condition implies that f is periodic, and therefore has a Fourier series. The third condition is that this series is of the form  It is often written in terms of  (the square of the nome), as:
   

[size][ltr]

This is also referred to as the q-expansion^[1] of f. The coefficients  are known as the Fourier coefficients of f, and the number m is called the order of the pole of f at i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-n coefficients are non-zero, so the q-expansion is bounded below, guaranteeing that it is meromorphic at q=0. ^[2]
Another way to phrase the definition of modular functions is to use elliptic curves: every lattice Λ determines an elliptic curve C/Λ over C; two lattices determine isomorphic elliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex number α. Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism classes of elliptic curves. For example, the j-invariant j(z) of an elliptic curve, regarded as a function on the set of all elliptic curves, is a modular function. More conceptually, modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves.
A modular form f that vanishes at q = 0 (equivalently, a₀ = 0, also paraphrased as z = i∞) is called a cusp form (Spitzenform in German). The smallest n such that a_n ≠ 0 is the order of the zero of f at i∞.
A modular unit is a modular function whose poles and zeroes are confined to the cusps.^[3]
Modular forms for more general groups[edit]
The functional equation, i.e., the behavior of f with respect to  can be relaxed by requiring it only for matrices in smaller groups.
The Riemann su***ce G\H^∗[edit]
Let G be a subgroup of SL(2,Z) that is of finite index. Such a group G actson H in the same way as SL(2,Z). The quotient topological space G\H can be shown to be a Hausdorff space. Typically it is not compact, but can be compactified by adding a finite number of points called cusps. These are points at the boundary of H, i.e., either in Q, the rationals, or ∞, such that there is a parabolic element of G (a matrix with trace ±2) fixing the point. Here, a matrix  sends ∞ to a/c. This yields a compact topological space G\H^∗. What is more, it can be endowed with the structure of aRiemann su***ce, which allows one to speak of holo- and meromorphic functions.
Important examples are, for any positive integer N, either one of thecongruence subgroups

and

For G = Γ₀(N) or Γ(N), the spaces G\H and G\H^∗ are denoted Y₀(N) andX₀(N) and Y(N), X(N), respectively.
The geometry of G\H^∗ can be understood by studying fundamental domains for G, i.e. subsets D ⊂ H such that D intersects each orbit of theG-action on H exactly once and such that the closure of D meets all orbits. For example, the genus of G\H^∗ can be computed.^[4]
Definition[edit]
A modular form for G of weight k is a function on H satisfying the above functional equation for all matrices in G, that is holomorphic on H and at all cusps of G. Again, modular forms that vanish at all cusps are called cusp forms for G. The C-vector spaces of modular and cusp forms of weight kare denoted M_k(G) and S_k(G), respectively. Similarly, a meromorphic function on G\H^∗ is called a modular function for G. In case G = Γ₀(N), they are also referred to as modular/cusp forms and functions of level N. For G = Γ(1) = SL₂(Z), this gives back the afore-mentioned definitions.
Consequences[edit]
The theory of Riemann su***ces can be applied to G\H^∗ to obtain further information about modular forms and functions. For example, the spacesM_k(G) and S_k(G) are finite-dimensional, and their dimensions can be computed thanks to the Riemann-Roch theorem in terms of the geometry of the G-action on H.^[5] For example,

where  denotes the floor function.
The modular functions constitute the field of functions of the Riemann su***ce, and hence form a field of transcendence degree one (over C). If a modular function f is not identically 0, then it can be shown that the number of zeroes of f is equal to the number of poles of f in the closure of thefundamental region R_Γ.It can be shown that the field of modular function of level N (N ≥ 1) is generated by the functions j(z) and j(Nz).^[6]
Line bundles[edit]
The situation can be profitably compared to that which arises in the search for functions on the projective space P(V): in that setting, one would ideally like functions F on the vector space V which are polynomial in the coordinates of v ≠ 0 in V and satisfy the equation F(cv) = F(v) for all non-zero c. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let F be the ratio of two homogeneous polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence onc, letting F(cv) = c^kF(v). The solutions are then the homogeneous polynomials of degree k. On the one hand, these form a finite dimensional vector space for each k, and on the other, if we let k vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space P(V).
One might ask, since the homogeneous polynomials are not really functions on P(V), what are they, geometrically speaking? The algebro-geometric answer is that they are sections of a sheaf (one could also say aline bundle in this case). The situation with modular forms is precisely analogous.
Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.
Miscellaneous[edit]
Entire forms[edit]
If f is holomorphic at the cusp (has no pole at q = 0), it is called an entire modular form.
If f is meromorphic but not holomorphic at the cusp, it is called a non-entire modular form. For example, the j-invariant is a non-entire modular form of weight 0, and has a simple pole at i∞.
Automorphic factors and other generalizations[edit]
Other common generalizations allow the weight k to not be an integer, and allow a multiplier  with  to appear in the transformation, so that

Functions of the form  are known as automorphic factors.
Functions such as the Dedekind eta function, a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors. Thus, for example, let χ be a Dirichlet character mod N. A modular form of weight k, level N (or level group ) with nebentypus χ is aholomorphic function f on the upper half-plane such that for any

and any z in the upper half-plane, we have

and f is holomorphic at all the cusps; when the form vanishes at all cusps, it is called a cusp form.
Examples[edit]
The simplest examples from this point of view are the Eisenstein series. For each even integer k > 2, we define E_k(Λ) to be the sum of λ^−k over all non-zero vectors λ of Λ:

The condition k > 2 is needed for convergence; for odd k there is cancellation between λ^−k and (−λ)^−k, so that such series are identically zero.
An even unimodular lattice L in Rⁿ is a lattice generated by n vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer. As a consequence of the Poisson summation formula, the theta function

is a modular form of weight n/2. It is not so easy to construct even unimodular lattices, but here is one way: Let n be an integer divisible by 8 and consider all vectors v in Rⁿ such that 2v has integer coordinates, either all even or all odd, and such that the sum of the coordinates of v is an even integer. We call this lattice L_n. When n = 8, this is the lattice generated by the roots in the root system called E₈. Because there is only one modular form of weight 8 up to scalar multiplication,

even though the lattices L₈×L₈ and L₁₆ are not similar. John Milnorobserved that the 16-dimensional tori obtained by dividing R¹⁶ by these two lattices are consequently examples of compact Riemannian manifoldswhich are isospectral but not isometric (see Hearing the shape of a drum.)
The Dedekind eta function is defined as

Then the modular discriminant Δ(z) = η(z)²⁴ is a modular form of weight 12. The presence of 24 can be connected to the Leech lattice, which has 24 dimensions. A celebrated conjecture of Ramanujan asserted that the q^pcoefficient for any prime p has absolute value ≤2p^11/2. This was settled byPierre Deligne as a result of his work on the Weil conjectures.
The second and third examples give some hint of the connection between modular forms and classical questions in number theory, such as representation of integers by quadratic forms and the partition function. The crucial conceptual link between modular forms and number theory are furnished by the theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory.
Generalizations[edit]
There are a number of other usages of the term modular function, apart from this classical one; for example, in the theory of Haar measures, it is a function Δ(g) determined by the conjugation action.
Maass forms are real-analytic eigenfunctions of the Laplacian but need not be holomorphic. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's mock theta functions. Groups which are not subgroups of SL(2,Z) can be considered.
Hilbert modular forms are functions in n variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field.
Siegel modular forms are associated to larger symplectic groups in the same way in which the forms we have discussed are associated to SL(2,R); in other words, they are related to abelian varieties in the same sense that our forms (which are sometimes called elliptic modular forms to emphasize the point) are related to elliptic curves.
Jacobi forms are a mixture of modular forms and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the Fourier coefficients of Siegel modular forms of genus two - but it is a relatively recent observation that the Jacobi forms have an arithmetic theory very analogous to the usual theory of modular forms.
Automorphic forms extend the notion of modular forms to general Lie groups.
History[edit]
The theory of modular forms was developed in three or four periods: first in connection with the theory of elliptic functions, in the first part of the nineteenth century; then by Felix Klein and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable); then by Erich Hecke from about 1925; and then in the 1960s, as the needs of number theory and the formulation of themodularity theorem in particular made it clear that modular forms are deeply implicated.
The term modular form, as a systematic description, is usually attributed to Hecke.
Notes[edit]
[/ltr][/size]

1. Jump up^ Elliptic and Modular Functions
   
   
2. Jump up^ A meromorphic function can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most a pole at q=0, not an essential singularity as exp(1/q) has.
   
   
3. Jump up^ Kubert, Daniel S.; Lang, Serge (1981), Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science] 244, Berlin, New York: Springer-Verlag, p. 24,ISBN 978-0-387-90517-4, Zbl 0492.12002, MR 648603
   
   
4. Jump up^ Gunning, Robert C. (1962), Lectures on modular forms, Annals of Mathematics Studies 48, Princeton University Press, p. 13
   
   
5. Jump up^ Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan11, Tokyo: Iwanami Shoten, Theorem 2.33, Proposition 2.26
   
   
6. Jump up^ Milne, James (2010), Modular Functions and Modular Forms, Theorem 6.1.
   
   

[size][ltr]
References[edit]
[/ltr][/size]

• Jean-Pierre Serre, A Course in Arithmetic. Graduate Texts in Mathematics 7, Springer-Verlag, New York, 1973. Chapter VII provides an elementary introduction to the theory of modular forms.
  
• Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0
  
• Goro Shimura, Introduction to the arithmetic theory of automorphic functions. Princeton University Press, Princeton, N.J., 1971. Provides a more advanced treatment.
  
• Stephen Gelbart, Automorphic forms on adele groups. Annals of Mathematics Studies 83, Princeton University Press, Princeton, N.J., 1975. Provides an introduction to modular forms from the point of view of representation theory.
  
• Robert A. Rankin, Modular forms and functions, (1977) Cambridge University Press, Cambridge. ISBN 0-521-21212-X
  
• Stein's notes on Ribet's course Modular Forms and Hecke Operators
  
• Erich Hecke, Mathematische Werke, Goettingen, Vandenhoeck & Ruprecht, 1970.
  
• N.P. Skoruppa, D. Zagier, Jacobi forms and a certain space of modular forms, Inventiones Mathematicae, 1988, Springer
  

Moduli space

From Wikipedia, the free encyclopedia
  (Redirected from Moduli spaces)

[ltr]In algebraic geometry, a moduli space is a geometric space (usually ascheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects.

[/ltr]

1 Motivation

2 Basic examples



• 2.1 Projective space and Grassmannians
  
• 2.2 Chow variety
  
• 2.3 Hilbert scheme
  



3 Definitions



• 3.1 Fine moduli spaces
  
• 3.2 Coarse moduli spaces
  
• 3.3 Moduli stacks
  



4 Further examples



• 4.1 Moduli of curves
  
• 4.2 Moduli of varieties
  
• 4.3 Moduli of vector bundles
  



5 Methods for constructing moduli spaces

6 In physics

7 References

8 External links

[size][ltr]

Motivation[edit]
Moduli spaces classify geometric objects. Given a class of thing (such as lines, su***ces, or elliptic curves), moduli spaces say which things are considered isomorphic ("the same for present purposes") and how the things can modulate (vary) throughout the family.
[/ltr][/size][size][ltr]
As a motivating example, consider how to describe the collection of lines in R² which intersect the origin. We want to assign a quantity, a modulus, to each line Lof this family that can uniquely identify it, for example a positive angle θ(L) with 0 ≤ θ < π radians, which will yield all lines in R² which intersect the origin. The set of lines L just constructed is known as P¹(R) and is called the real projective line.
We can also describe the collection of lines in R² that intersect the origin by means of a topological construction. That is, consider S¹ ⊂ R² and notice that to every point s ∈ S¹ that we can identify a line L(s) in the collection if the line intersects the origin and s. Yet, this map is two-to-one, so we want to identify s ~ −s to yield P¹(R) ≅ S¹/~ where the topology on this space is the quotient topology induced by the quotient map S¹ →P¹(R).
Thus, when we consider P¹(R) as a moduli space of lines that intersect the origin in R², we capture the ways in which the members of the family (lines in the case) can modulate by continuously varying 0 ≤ θ < π.
Basic examples[edit]
Projective space and Grassmannians[edit]
The real projective space Pⁿ is a moduli space which parametrizes the space of lines in Rⁿ⁺¹ which pass through the origin. Similarly, complex projective space is the space of all complex lines in Cⁿ⁺¹ passing through the origin.
More generally, the Grassmannian G(k, V) of a vector space V over a fieldF is the moduli space of all k-dimensional linear subspaces of V.
Chow variety[edit]
The Chow variety Chow(d,P³) is a projective algebraic variety which parametrizes degree d curves in P³. It is constructed as follows. Let C be a curve of degree d in P³, then consider all the lines in P³ that intersect the curve C. This is a degree d divisor D_C in G(2, 4) the Grassmannian of lines in P³. When C varies, by associating C to D_C, we obtain a parameter space of degree d curves as a subset of the space of degree ddivisors of the Grassmannian: Chow(d,P³).
Hilbert scheme[edit]
The Hilbert scheme Hilb(X) is a moduli scheme. Every closed point ofHilb(X) corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.
Definitions[edit]
There are several different related notions of things we could call moduli spaces. Each of these definitions formalizes a different notion of what it means for the points of a space M to represent geometric objects.
Fine moduli spaces[edit]
This is the standard concept. Heuristically, if we have a space M for which each point m∈ M corresponds to an algebro-geometric object U_m, then we can assemble these objects into a topological family U over M. (For example, the Grassmannian G(k, V) carries a rank k bundle whose fiber at any point [L] ∈ G(k, V) is simply the linear subspace L ⊂ V.) M is called abase space of the family U. We say that such a family is universal if any family of algebro-geometric objects T over any base space B is thepullback of U along a unique map B → M. A fine moduli space is a spaceM which is the base of a universal family.
More precisely, suppose that we have a functor F from schemes to sets, which assigns to a scheme B the set of all suitable families of objects with base B. A space M is a fine moduli space for the functor F if Mcorepresents F, i.e., there is a natural isomorphism τ : F → Hom(−, M), where Hom(−, M) is the functor of points. This implies that M carries a universal family; this family is the family on M corresponding to the identity map 1_M ∈ Hom(M, M).
Coarse moduli spaces[edit]
Fine moduli spaces are very useful, but they do not always exist and are frequently difficult to construct, so mathematicians sometimes use a weaker notion, the idea of a coarse moduli space. A space M is a coarse moduli space for the functor F if there exists a natural transformation τ : F→ Hom(−, M) and τ is universal among such natural transformations. More concretely, M is a coarse moduli space for F if any family T over a base Bgives rise to a map φ_T : B → M and any two objects V and W (regarded as families over a point) correspond to the same point of M if and only if Vand W are isomorphic. Thus, M is a space which has a point for every object that could appear in a family, and whose geometry reflects the ways objects can vary in families. Note, however, that a coarse moduli space does not necessarily carry any family of appropriate objects, let alone a universal one.
In other words, a fine moduli space includes both a base space M and universal family T → M, while a coarse moduli space only has the base space M.
Moduli stacks[edit]
It is frequently the case that interesting geometric objects come equipped with lots of natural automorphisms. This in particular makes the existence of a fine moduli space impossible (intuitively, the idea is that if L is some geometric object, the trivial family L × [0,1] can be made into a twisted family on the circle S¹ by identifying L × {0} with L × {1} via a nontrivial automorphism. Now if a fine moduli space X existed, the map S¹ → Xshould not be constant, but would have to be constant on any proper open set by triviality), one can still sometimes obtain a coarse moduli space. However, this approach is not ideal, as such spaces are not guaranteed to exist, are frequently singular when they do exist, and miss details about some non-trivial families of objects they classify.
A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any base B one can consider the category of families on B with only isomorphisms between families taken as morphisms. One then considers the fibred category which assigns to any space B the groupoid of families over B. The use of thesecategories fibred in groupoids to describe a moduli problem goes back to Grothendieck (1960/61). In general they cannot be represented by schemes or even algebraic spaces, but in many cases they have a natural structure of an algebraic stack.
Algebraic stacks and their use to analyse moduli problems appeared in Deligne-Mumford (1969) as a tool to prove the irreducibility of the (coarse)moduli space of curves of a given genus. The language of algebraic stacks essentially provides a systematic way to view the fibred category that constitutes the moduli problem as a "space", and the moduli stack of many moduli problems is better-behaved (such as smooth) than the corresponding coarse moduli space.
Further examples[edit]
Moduli of curves[edit]
For more details on this topic, see Moduli of algebraic curves.
The moduli stack  classifies families of smooth projective curves of genus g, together with their isomorphisms. When g > 1, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is stable if it has only a finite group of automorphisms. The resulting stack is denoted . Both moduli stacks carry universal families of curves. One can also define coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was invented. In fact, the idea of a moduli stack was invented by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object.
Both stacks above have dimension 3g−3; hence a stable nodal curve can be completely specified by choosing the values of 3g−3 parameters, wheng > 1. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence the dimension of  is
dim(space of genus zero curves) - dim(group of automorphisms) = 0 − dim(PGL(2)) = −3.
Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence the stack  has dimension 0. The coarse moduli spaces have dimension 3g-3 as the stacks when g > 1 because the curves with genus g > 1 have only a finite group as its automorphism i.e. dim(group of automorphisms) = 0. Eventually, in genus zero the coarse moduli space has dimension zero, and in genus one, it has dimension one.
One can also enrich the problem by considering the moduli stack of genusg nodal curves with n marked points. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli stacks of smooth (or stable) genus g curves with n-marked points are denoted  (or ), and have dimension 3g−3+n.
A case of particular interest is the moduli stack  of genus 1 curves with one marked point. This is the stack of elliptic curves, and is the natural home of the much studied modular forms, which are meromorphic sections of bundles on this stack.
Moduli of varieties[edit]
In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties. This is the problem underlying Siegel modular formtheory. See also Shimura variety.
Moduli of vector bundles[edit]
Another important moduli problem is to understand the geometry of (various substacks of) the moduli stack Vect_n(X) of rank n vector bundleson a fixed algebraic variety X. This stack has been most studied when X is one-dimensional, and especially when n equals one. In this case, the coarse moduli space is the Picard scheme, which like the moduli space of curves, was studied before stacks were invented. Finally, when the bundles have rank 1 and degree zero, the study of the coarse moduli space is the study of the Jacobian variety.
In applications to physics, the number of moduli of vector bundles and the closely related problem of the number of moduli of principal G-bundles has been found to be significant in gauge theory.^{[citation needed]}
Methods for constructing moduli spaces[edit]
The modern formulation of moduli problems and definition of moduli spaces in terms of the moduli functors (or more generally the categories fibred in groupoids), and spaces (almost) representing them, dates back to Grothendieck (1960/61), in which he described the general framework, approaches and main problems using Teichmüller spaces in complex analytical geometry as an example. The talks in particular describe the general method of constructing moduli spaces by first rigidifying the moduli problem under consideration.
More precisely, the existence of non-trivial automorphisms of the objects being classified makes it impossible to have a fine moduli space. However, it is often possible to consider a modified moduli problem of classifying the original objects together with additional data, chosen in such a way that the identity is the only automorphism respecting also the additional data. With a suitable choice of the rigidifying data, the modified moduli problem will have a (fine) moduli space T, often described as a subscheme of a suitable Hilbert scheme or Quot scheme. The rigidifying data is moreover chosen so that it corresponds to a principal bundle with an algebraic structure group G. Thus one can move back from the rigidified problem to the original by taking quotient by the action of G, and the problem of constructing the moduli space becomes that of finding a scheme (or more general space) that is (in a suitably strong sense) the quotient T/G of T by the action of G. The last problem in general does not admit a solution; however, it is addressed by the groundbreaking geometric invariant theory(GIT), developed by David Mumford in 1965, which shows that under suitable conditions the quotient indeed exists.
To see how this might work, consider the problem of parametrizing smooth curves of genus g > 2. A smooth curve together with a complete linear system of degree d > 2g is equivalent to a closed one dimensional subscheme of the projective space P^d−g. Consequently, the moduli space of smooth curves and linear systems (satisfying certain criteria) may be embedded in the Hilbert scheme of a sufficiently high-dimensional projective space. This locus H in the Hilbert scheme has an action of PGL(n) which mixes the elements of the linear system; consequently, the moduli space of smooth curves is then recovered as the quotient of H by the projective general linear group.
Another general approach is primarily associated with Michael Artin. Here the idea is to start with any object of the kind to be classified and study itsdeformation theory. This means first constructing infinitesimaldeformations, then appealing to prorepresentability theorems to put these together into an object over a formal base. Next an appeal toGrothendieck's formal existence theorem provides an object of the desired kind over a base which is a complete local ring. This object can be approximated via Artin's approximation theorem by an object defined over a finitely generated ring. The spectrum of this latter ring can then be viewed as giving a kind of coordinate chart on the desired moduli space. By gluing together enough of these charts, we can cover the space, but the map from our union of spectra to the moduli space will in general be many to one. We therefore define an equivalence relation on the former; essentially, two points are equivalent if the objects over each are isomorphic. This gives a scheme and an equivalence relation, which is enough to define an algebraic space (actually an algebraic stack if we are being careful) if not always a scheme.
In physics[edit]
For more details on this topic, see moduli (physics).
The term moduli space is sometimes used in physics to refer specifically to the moduli space of vacuum expectation values of a set of scalar fields, or to the moduli space of possible string backgrounds.
Moduli spaces also appear in physics in cohomological field theory, where one can use Feynman path integrals to compute the intersection numbersof various algebraic moduli spaces.
References[edit]

[/ltr][/size]

• Grothendieck, Alexander (1960/1961). "Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes.". Séminaire Henri Cartan 13 no. 1, Exposés No. 7 and 8 (Paris: Secrétariat Mathématique).
  

• Mumford, David, Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34 Springer-Verlag, Berlin-New York 1965 vi+145 pp MR 0214602
  

• Mumford, David; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. MR 1304906 ISBN 3-540-56963-4
  

• Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029,ISBN 978-3-03719-029-6, MR2284826
  

• Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, doi:10.4171/055,ISBN 978-3-03719-055-5, MR2524085
  

• Papadopoulos, Athanase, ed. (2012), Handbook of Teichmüller theory. Vol. III, IRMA Lectures in Mathematics and Theoretical Physics, 17, European Mathematical Society (EMS), Zürich, doi:10.4171/103,ISBN 978-3-03719-103-3.
  

• Deligne, Pierre; Mumford, David (1969). "The irreducibility of the space of curves of given genus". Publications Mathématiques de l'IHÉS(Paris) 36: 75–109. doi:10.1007/bf02684599.
  

• Harris, Joe; Morrison, Ian (1998). Moduli of Curves. Springer Verlag.ISBN 0-387-98429-1.
  

• Katz, Nicholas M; Mazur, Barry (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press. ISBN 0-691-08352-5.
  

• Faltings, Gerd; Chai, Ching-Li (1990). Degeneration of Abelian Varieties. Springer Verlag. ISBN 3-540-52015-5.
  

• Viehweg, Eckart (1995). Quasi-Projective Moduli for Polarized Manifolds. Springer Verlag. ISBN 3-540-59255-5.
  

• Simpson, Carlos (1994). "Moduli of representations of the fundamental group of a smooth projective variety I". Publications Mathématiques de l'IHÉS (Paris) 79: 47–129. doi:10.1007/bf02698887.
  

[size][ltr]
External links[edit]
[/ltr][/size]

• J. Lurie, Moduli Problems for Ring Spectra
  

Quantum cohomology

From Wikipedia, the free encyclopedia




[ltr]In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called smalland big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects its structure, as well.
While the cup product of ordinary cohomology describes how submanifolds of the manifold intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a "fuzzy", "quantum" way. More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product.
Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has important implications for enumerative geometry. It also connects to many ideas in mathematical physics andmirror symmetry. In particular, it is ring-isomorphic to Floer homology.
Throughout this article, X is a closed symplectic manifold with symplectic form ω.[/ltr]

• 1 Novikov ring
  
• 2 Small quantum cohomology
  
• 3 Geometric interpretation
  
• 4 Example
  
• 5 Properties of the small quantum cup product
  
• 6 Dubrovin connection
  
• 7 Big quantum cohomology
  
• 8 References
  

[ltr]

Novikov ring[edit]
See also: Novikov ring
Various choices of coefficient ring for the quantum cohomology of X are possible. Usually a ring is chosen that encodes information about the second homology of X. This allows the quantum cup product, defined below, to record information about pseudoholomorphic curves in X. For example, let

be the second homology modulo its torsion. Let R be any commutative ring with unit and Λ the ring of formal power series of the form

where[/ltr]

• the coefficients  come from R,
  
• the  are formal variables subject to the relation ,
  
• for every real number C, only finitely many A with ω(A) less than or equal to C have nonzero coefficients .
  

[ltr]
The variable  is considered to be of degree , where  is the first Chern class of the tangent bundle TX, regarded as a complex vector bundle by choosing any almost complex structure compatible with ω. Thus Λ is a graded ring, called the Novikov ring for ω. (Alternative definitions are common.)
Small quantum cohomology[edit]
Let

be the cohomology of X modulo torsion. Define the small quantum cohomology with coefficients in Λ to be

Its elements are finite sums of the form

The small quantum cohomology is a graded R-module with

The ordinary cohomology H*(X) embeds into QH*(X, Λ) via , and QH*(X, Λ) is generated as a Λ-module by H*(X).
For any two cohomology classes a, b in H*(X) of pure degree, and for anyA in , define (a∗b)_A to be the unique element of H*(X) such that

(The right-hand side is a genus-0, 3-point Gromov–Witten invariant.) Then define

This extends by linearity to a well-defined Λ-bilinear map

called the small quantum cup product.
Geometric interpretation[edit]
The only pseudoholomorphic curves in class A = 0 are constant maps, whose images are points. It follows that

in other words,

Thus the quantum cup product contains the ordinary cup product; it extends the ordinary cup product to nonzero classes A.
In general, the Poincaré dual of (a∗b)_A corresponds to the space of pseudoholomorphic curves of class A passing through the Poincaré duals of a and b. So while the ordinary cohomology considers a and b to intersect only when they meet at one or more points, the quantum cohomology records a nonzero intersection for a and b whenever they are connected by one or more pseudoholomorphic curves. The Novikov ring just provides a bookkeeping system large enough to record this intersection information for all classes A.
Example[edit]
Let X be the complex projective plane with its standard symplectic form (corresponding to the Fubini–Study metric) and complex structure. Let  be the Poincaré dual of a line L. Then

The only nonzero Gromov–Witten invariants are those of class A = 0 or A= L. It turns out that

and

where δ is the Kronecker delta. Therefore

In this case it is convenient to rename  as q and use the simpler coefficient ring Z[q]. This q is of degree . Then

Properties of the small quantum cup product[edit]
For a, b of pure degree,

and

The small quantum cup product is distributive and Λ-bilinear. The identity element  is also the identity element for small quantum cohomology.
The small quantum cup product is also associative. This is a consequence of the gluing law for Gromov–Witten invariants, a difficult technical result. It is tantamount to the fact that the Gromov–Witten potential (a generating function for the genus-0 Gromov–Witten invariants) satisfies a certain third-order differential equation known as the WDVV equation.
An intersection pairing

is defined by

(The subscripts 0 indicate the A = 0 coefficient.) This pairing satisfies the associativity property

Dubrovin connection[edit]
When the base ring R is C, one can view the evenly-graded part H of the vector space QH*(X, Λ) as a complex manifold. The small quantum cup product restricts to a well-defined, commutative product on H. Under mild assumptions, H with the intersection pairing  is then a Frobenius algebra.
The quantum cup product can be viewed as a connection on the tangent bundle TH, called the Dubrovin connection. Commutativity and associativity of the quantum cup product then correspond to zero-torsionand zero-curvature conditions on this connection.
Big quantum cohomology[edit]
There exists a neighborhood U of 0 ∈ H such that  and the Dubrovin connection give U the structure of a Frobenius manifold. Any a in Udefines a quantum cup product

by the formula

Collectively, these products on H are called the big quantum cohomology. All of the genus-0 Gromov–Witten invariants are recoverable from it; in general, the same is not true of the simpler small quantum cohomology.
Small quantum cohomology has only information of 3-point Gromov–Witten invariants, but the big quantum cohomology has of all (n ≧ 4) n-point Gromov–Witten invariants. To obtain enumerative geometrical information for some manifolds, we need to use big quantum cohomology. Small quantum cohomology would corresponds to 3-point correlation functions in physics while big quantum cohomology would corresponds to all of n-point correlation functions.
References[edit][/ltr]

• McDuff, Dusa & Salamon, Dietmar (2004). J-Holomorphic Curves and Symplectic Topology, American Mathematical Society colloquium publications. ISBN 0-8218-3485-1.
  

• Fulton, W; Pandharipande, R (1996). "Notes on stable maps and quantum cohomology". arXiv:alg-geom/9608011.
  

• Piunikhin, Sergey; Salamon, Dietmar & Schwarz, Matthias (1996). Symplectic Floer–Donaldson theory and quantum cohomology. In C. B. Thomas (Ed.), Contact and Symplectic Geometry, pp. 171–200. Cambridge University Press. ISBN 0-521-57086-7
  

Frobenius manifold

From Wikipedia, the free encyclopedia
  (Redirected from Frobenius manifolds)

[ltr]In the mathematical field of differential geometry, a Frobenius manifold is a flat Riemannian manifold with a certain compatible multiplicative structure on the tangent space. The concept generalizes the notion of Frobenius algebra to tangent bundles. They were introduced by Dubrovin.^[1]
Frobenius manifolds occur naturally in the subject of symplectic topology, more specifically quantum cohomology. The broadest definition is in the category of Riemannian supermanifolds. We will limit the discussion here to smooth (real) manifolds. A restriction to complex manifolds is also possible.

[/ltr]

1 Definition

2 Elementary properties

3 Examples

4 References

[size][ltr]

Definition[edit]
Let M be a smooth manifold. An affine flat structure on M is a sheaf T^f of vector spaces that pointwisely span TM the tangent bundle and the tangent bracket of pairs of its sections vanishes.
As a local example consider the coordinate vectorfields over a chart of M. A manifold admits an affine flat structure if one can glue together such vectorfields for a covering family of charts.
Let further be given a Riemannian metric g on M. It is compatible to the flat structure if g(X, Y) is locally constant for all flat vector fields X and Y.
A Riemannian manifold admits a compatible affine flat structure if and only if its curvature tensor vanishes everywhere.
A family of commutative products * on TM is equivalent to a section A ofS²(T^*M) ⊗ TM via

We require in addition the property

Therefore the composition g^#∘A is a symmetric 3-tensor.
This implies in particular that a linear Frobenius manifold (M, g, *) with constant product is a Frobenius algebra M.
Given (g, T^f, A), a local potential Φ is a local smooth function such that

for all flat vector fields X, Y, and Z.
A Frobenius manifold (M, g, *) is now a flat Riemannian manifold (M, g) with symmetric 3-tensor A that admits everywhere a local potential and is associative.
Elementary properties[edit]
The associativity of the product * is equivalent to the following quadraticPDE in the local potential Φ

where Einstein's sum convention is implied, Φ_,a denotes the partial derivative of the function Φ by the coordinate vectorfield ∂/∂x^a which are all assumed to be flat. g^ef are the coefficients of the inverse of the metric.
The equation is therefore called associativity equation or Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation.
Examples[edit]
Beside Frobenius algebras, examples arise from quantum cohomology. Namely, given a semipositive symplectic manifold (M, ω) then there exists an open neighborhood U of 0 in its even quantum cohomologyQH^even(M, ω) with Novikov ring over C such that the big quantum product *_a for a in U is analytic. Now U together with the intersection form g = <·,·> is a (complex) Frobenius manifold.
References[edit]
[/ltr][/size]

1. Jump up^ B. Dubrovin: Geometry of 2D topological field theories. In: Springer LNM, 1620 (1996), pp. 120–348.
   

[size][ltr]
2. Yu.I. Manin, S.A. Merkulov: Semisimple Frobenius (super)manifolds and quantum cohomology of P^r, Topol. Methods in Nonlinear Analysis 9 (1997), pp. 107–161[/ltr]
[/size]