Abstract
We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry group . Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry group . After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry group . Use the electric-magnetic duality to pass to the Langlands dual Lie group . Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra 𝔤. Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groups .
1. Introduction
A quantization procedure can be described as a covariant functor from the category of classical Hamiltonian systems to the category of quantum systems: The main rule of a quantization procedure is that when the Planck constant approaches the system does become the starting classical system, that is, the classical limit.
There are some well-known rules of quantization, namely, Weyl quantization, related with the canonical representation of the commutation relations; pseudodifferential operators quantization, regarding the functions of position and momentum variables as symbols of some pseudodifferential operators; geometric quantization, thinking of the symplectic gradients of functions as vector fields acting on sections following some connection, and so forth.
Let us concentrate on the geometric quantization and explain in some more detail.
1.1. Geometric Quantization
In the general model, a Hamiltonian system is modeled as some symplectic manifold with a flat action of some Lie group of symmetry ; see [1, 2] for more details.
If is a function on the symplectic manifold, its symplectic gradient is the vector field such that Conversely, every element of the Lie algebra of symmetry provides a vector field . The condition is that the potential always exists and that it can be lifted to a homomorphism of corresponding Lie algebras
(1.4)On an arbitrary symplectic manifold, there exist the so-called Darboux coordinate systems at some neighborhood of every point. A global system of such separations of variables is the so-called polarization. In more general context, it is given by some -invariant integrable tangent distribution of the complexified tangent bundle. If in each coordinate chart one uses the pseudodifferential quantization through the oscillating integral Fourier and then glue the results between the chart, one meets some obstacle, which is the so-called Arnold-Maslov index classes.
It is given also by a maximal commutative subalgebra of functions, with respect to the Poisson brackets and one defines a momentum map from and arbitrary heomegeneous strictly Hamiltonian symplectic manifolds to the coadjoint orbits of the universal covering of Lie group or its central extension
in the dual space of the Lie algebra .
The sheaf of sections of a vector bundle vanishing along the direction provides the so-called quantum vector bundle and the space of square integrable sections depending only on the space direction is the Hilbert space of quantum states.
Geometric Quantization of Hamiltonian systems is given by the rule of assigning to each real- (or complex-) valued function an autoadjoint (or normal, resp.) operator such that
Application to group representations: can be considered as a function on an arbitrary coadjoint -orbit in the vector space dual to .
1.2. Fields Theories
The general conception of (physical) fields is physical systems (movements, forces, interactions, etc.) located in some parametrized region of space-time, for example, the null-dimensional fields are the same as particles, and the one-dimensional fields are the fields in the quantum mechanics. We refer the reader to [3] for discussion of the cases of one-dimensional and -dimensional fields.
In field theory one defines the partition function as where is the Wiener measure of the space of paths from a point to another one, where is the action which is the integral of the Lagrangian. The general field equation is obtained from the variation principle.
The sigma model for the general field theory is started with reduction to reduce the 4-dimensional Minkowski space to the product of a possibly noncompact Riemann surface and a compact Riemann surface . The Klein reduction requires to compactify and have some effective theory on .
Therefore one needs to consider the sigma-model on with target space or the moduli space of semistable Higgs bundles on , that is, the holomorphic vector bundles over the hyperKähler manifold , with self-dual connection . They can be obtained from the principal bundle , a finite-dimensional -module , and . The operator of taking partial trace related with the representation give the space of values of partial traces and has a pair of fibrations
(1.11)The fibers of these two fibrations are pairwise-dual tori. There are 3 complex structures , , on the corresponding spaces. The moduli space of semistable -bundles with holomorphic connection on endowed with the complex structure is denoted by . The -branes on are the same as these objects .
The manifold is endowed with the complex structure and become a symplectic manifold with respect to the symplectic structure .
By the mirror symmetry transformation, the -branes become the so called -branes, those are the Lagrangian submanifolds on .
1.3. Quantization of Fields
In the general scheme there are two models of -branes: the model with 't Hooft line operators and model with Wilson loop operators, in one side and the -modules on the stack of vector -bundles on with Hecke operators, on another side. With reduction the moduli stack is reduced to the moduli stack of -bundles on with connection , the curvature of which satisfies the Bogomolnyi system of equations This system means that the curvature , where . We have the general picture as
(1.13)Our tash is to realize the flesh from -branes to -modules which is equivalent to the Geometric Langlands Correspondence, through the mirror symmetry.
Going from the model of branes to Hecke eigensheaves of modules can be considered as a quantization procedure of fields, using the Fukaya category or mutidimensional Fedosov deformation quantization, or the . Tsygan deformation quantization. The most deficulties are related with the complicated category or analytic transformation in Tsygan approach. From the model to module Hecke eigensheaves can be considered as the second quantization procedure of fields, related much more with algebraic geometry. Our approach is related with ideas of geometric quantization.
The new quantization procedure we proposed consists of the following steps.
(i)Starting from a connection associated with the -bundle, use the Electric-Magnetic GNO Duality to obtain bundles with connection for the dual groups . (ii)Use the Kaluza-Klein Reduction to reduce the model to the case over complex curve extending the symmetry group . (iii)From a connection construct the correspoding representation . (iv)Construct the corresponding Momentum Maps. (v)Use the Orbit Method to obtain the representations of Lie group . (vi)Use ADHM construction and the Hitchin Fibration Construction to have some holomorphic bundle on . (vii)Use the positive energy representations of Virasoro algebras to obtain representations of loop algebras (Fock space construction).Together compose all the steps, we have the same automorphic representations those appeared in GLC. As the main result of this paper we have the following.
Theorem 1.1 (quantization procedure for fields). The obtained automorphic representations are exactly the automorphic representations of from the Geometric Langlands Correspondence.
In our method, beside the other things, the new idea involved the orbit method to provide a quantization procedure. The rest of this paper is devoted to prove this in exposing the corresponding theories in a suitable form.
1.4. Structure of the Paper
We describe in more detail the conception of quantization in the case of particle physiscs in Section 1. In Section 2 we discuss the electric-magnetic duality. In Section 3, we start this job by considering the embedding of the conplexified Minkowski space into the twistor space . Section 4 is devoted to the construction of representations starting by reduction and finished by the final construction of representations in Fock spaces. Section 5 is to show the corresponding construction by the Geometric Langlands Correspondence.
2. Electric-Magnetic GNO Duality
Let us discuss first about the Langlands duality or electric-magnetic GNO duality.
Theorem 2.1. Let be a -connection of the target space . Then there exists a unique dual connection on the Langlands dual -bundle on the same base .
Proof. This theorem is a direct consequence of the electric-magnetic dualty.
3. Kaluza-Klein Model
Following physical ideas, the only-nontrival quantum field theories that are believed exist have dimension and the most standard ones have . We can pass to different quantum field theories from each other by the operation of so called Kaluza-Klein Reduction. It means that we can consider the case when the target space is decomposed into a Cartesian product where the action may be very large for the field that are not constant over and therefore the correlation functions are localized along the fields that are constant along . The mirror symmetry theory says that one needs only to concentrate in the case of dimension or .
Following Kapustin and Witten [4] we reduce the theory to the case of complex curve .
Theorem 3.1. The connection on is uniquely defined by a representation of the fundamental groups .
Proof. Over the universal covering , there is a unique trivial connection. Passing on the manifold we have some fixed representation of the fundamential group.
4. General Momentum Maps
4.1. Momentum Maps
Theorem 4.1. There is local diffeomorphisms mapping the -orbits in the principal bundle total space and the coadjoint -orbits of in the space dual to the Lie algebra of .
Proof. One looks at each point of the manifold as some functional on the space of functions. This dual gives us the necessary momentum map; see [1] or [2] for more detail.
The general scheme of model consists of the following.
(i)Choose a Langangian submanifold (ii)Choose the ground state space (iii)Identify the vertex algebra. (iv)Define the Verma modules over the vertex algebra. (v)The resulting modules can be considered as some induced modules.4.2. Polarization and Ground States
Theorem 4.2. Suppose that is an coadjoint orbit of Lie group of symmetry in the dual space of its Lie algebra, is a polarization at then the Lie algebra is acting on the de Rham cohomology with coefficients in the sheaf of partially holomorphic and partially invariant sections of the vector bundle as differential operators with coefficients in the Lie algebra.
Proof. Following the Orbit method theory [1, 2] we have a representation of the Lie algebra by the differential operators in the space of partially invariant and partially holomorphic sections of the induced vector bundle. Therefore by the universal property of the enveloping algebra we have a corresponding homomorphism of associative algebras
4.3. Fock Space Construction
Theorem 4.3. The Fock space construction gives a realization of the weight modules of the Virassoro algebras on the Fock space as subspaces of the tensor product of of standard action obtained from the sigma models.
Proof. Taking polarizations of the coadjoint orbits of the Lie group one obtains the natural action of the Lie agebra on the representation space obtained from the orbit method. Taking the tensor product of these action, one have the corresponding Verma modules.
The rest of the paper is devoted to the corresponding Kapustin-Witten theory of Geometric Langlands Correspondence for the brane model.
5. Branes and GLC
The main ideas of the Geometric Langlands Correspondence is now summerized and compaired with our construction in order to show that the same automorphic representations are obtained by the both methods. The Geometric Langlands Correspondence [4] can be formulated as follows.
(i)One begins with the Langlands dual group and a semistable homomorphism . The space of such homomorphisms is the Hitchin moduli space . (ii)If is irreducible the corresponding point is a smooth point and a zerobrane supported at is an electric eigenbrane in the sigma model of the target space . (iii)Applying the duality to this electric eigenbrane will give a magnetic eigenbrane in the sigma-model of the target space , whose support is a fiber of the Hitchin fibration, endowed with a Chan-Paton bundle of rank 1. (iv)The main claim of the Langlands correspondence is that a homomorphism is associated in a natural way to a sheaf on , that is, a Hecke eigensheaf and also a holonomic -module.What follows is an entry into details.
5.1. Reduction to a Theory on Curve
Consider the four-manifold , where is a compact Riemann surface of genus greater than one, is either a complete but noncompact two-manifold such as , or a second compact Riemann surface.
To find an effective physics on , we must find the configuration on that minimize or nearly minimize the action in Euclidean signature or the energy in the Lorentz signature. In either case the four-dimensional twisted supersymmetric gauge theory reduces on to a sigma-model of maps , where is the moduli space of the solutions on of Hitchin's equation with gauge group.
The minimum is obtained if It is equivalent to the following system: where is the genus of the surface .
5.2. 't Hooft Operators and Operator Product Expansion
5.2.1. Wilson Loop and Line Operators
Let be the -bundle, associated with a representation . Let be a connection with curvature , a scalar field with values in the Lie algebra , , and , an oriented loop. The Wilson operator is defined as the trace in the representation of the holonomy that is, If is a line with endpoints and at infinity, we can define as a matrix of parallel transport (of the connection or , from the fiber at the point , taken in the representation , to the point .
The dual of a Wilson operator for and is a 't Hooft operator for the dual group and the corresponding representation .
To define a Wilson loop operator associated with a loop , must be oriented. The 't Hooft operator is labelled by the representations of the L-group and instead requires an orientation of the normal bundle to . A small neighborhood of can be identified with . Once is oriented we can ask for the orientation ds of and the orientation are compatible in the sense along The line operators that preserve the topological symmetry at rational values of are called mixed Wilson-'t Hooft operators.
Combined Wilson-'t Hooft operators: Abelian Case
To the action of the gauge field, add a term
, where is gauge invariant that has singularity at , but is smooth near ,
with being the generator of the topological symmetry , because is gauge invariant ,
This means that we can restore the topological symmetry if we include a Wilson operator as an additional factor in the path integral. The expression is gauge invariant if and only if an integer; that is, must be rational number Rational transformation
Combined Wilson-'t Hooft Operators: Nonabelian Case
Let be a homomorphism, , where is the singular gauge field with Dirac singularity along the curve . The required conditions are
5.2.2. 't Hooft Operators and Hecke Operators
The main result is to show that the 't Hooft operators correspond to the Hecke operators of the Geometric Langlands Correspondence. More precisely, the line operators provide an algebra isomorphic to the affine Hecke algebra.
Consider for some 3-manifold . The 't Hooft operators are supported on 1-manifold of the form , , two-dimensional -model with target : a supersymmetric classical field ; the first approximation: being an interval; the space of supersymmetric states: the cohomology of the space of constant maps to that obey the boundary conditions.
Reduction to Two Dimensions
We have , , with being a Riemann surface, an interval, at whose ends one takes the boundary conditions. Then the BPS equations are
In the term of complex connection , curvarture :
5.2.3. 't Hooft Operators and Eigenbranes
The Case of Group
A 't Hooft operator at the point is classified by an integer . The operator is defined by saying that near the point , the curvature has the singular behaviour
where is the distance from the point to the nearby point . This implies that if is a small enclosing the point , then
and the 't Hooft operator is acting by twisting with and
Nonabelian Case of
The singular part of the curvature near is diagonal
Near , the Bogomolny equation reduces to equation in some maximal torus of . The corresponding vector bundle near splits up to a sum of the line bundles . The effect of the 't Hooft operator on is
Therefore we have some action of the 't Hooft operators what is the same as the action of the Hecke operators on bunldes for . This proves that the eigensheaves of these 't Hooft operators are the same as Hecke eigensheaves. Remark that .
5.3. The Extended Bogomolny Equation
We consider only the supersymmetric time-independent and time-reversal invariant. Then the BPS equations are reducing to the ordinary Bogomolny equations, and the 't Hooft operators reduce to the usual geometric Hecke operators.
On the four-manifold we write the Higgs field as , where is the projection.
The gauge fields , where is a 3-dimensional connection with curvature .
The time-independent BPS equation for is where is the exterior derivative, the Hodge operator , and the operator .
Because of the time independence one deduces that . Choose a local coordinated . and . Because the metric on is , then the extended Bogomolny equations are The first equation means that reatricted on is a holomorphic section of , is the holommorphic bundle over defined by the operator The pair is a Higgs bundle or Hecke modification for any .
6. Proof of the Main Theorem
First we remark that following the previous results, under mirror symmetry, the category of -branes invariant under the 't Hooft operators is equivalent to the corresponding category of -branes invariant under the Wilson loop operators. And the both categories are equivalent to the category of -modules invariant under the Hecke operators. More precisely, the second category is equivalent to the third category under the Geometric Langlands correspondence and the first to the third under the Fukaya category.
Next, it is known that the Fukaya category has the analytic version as the Batalin-Vilkovski and B. Tsygan quantization following the deformation quantization formula where and the multi-index notation is used.
The main difficulty is that the formulas of the deformation quantization are formal and cannot be convergent.
Lemma 6.1. The first-order jet of the deformation quantization is equivalent to geometric quantization.
Proof. The first-order component is which is the geometric quantization formula.
Lemma 6.2. The th component of of the series is the continued th component of the universal enveloping algebra.
Proof. The linear differential operators are defined following the universal property of the universal enveloping algebra and therefore we have the identical components.
Lemma 6.3. The th component of of the series is linear operator from the tensor product of finite-dimensional Hilbert space into the Hilbert space
Proof. It is clear from the formula for deformation.
Lemma 6.4. The formal series are elements of the tensor product of Hilbert space .
Proof. The exponential series formula provides the convergence.
Now the proof of the main theorem is achieved.
Acknowledgment
The work was supported in part by NAFOSTED of Vietnam under the Project no. 101.01.24.09.