# A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

Authors: Do Ngoc Diep
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## 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 groupGL. 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 GL. After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry group GL. Use the electric-magnetic duality to pass to the Langlands dual Lie group G. Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra 𝔤=Lie(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 G.


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## Body

## 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:(1.1){Classicalsystems}→{Quantumsystems}e.g.,classicalobservables↦quantumobservablesi.e.,real(complex)valuedfunctions↦autoadjoint(normal)operators
The main rule of a quantization procedure is that when the Planck constant ℏ approaches 0 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 qi and momentum variables pi 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(M,ω) with a flat action ∇ of some Lie group of symmetry G; see [1, 2] for more details.Iff is a function on the symplectic manifold, its symplectic gradient is the vector field ξf such that(1.2)ı(ξf)ω+df≡0.
Conversely, every element X of the Lie algebra of symmetry 𝔤=LieG provides a vector field ξX(m)=d/dt|t=0exp(tX)m. The condition is that the potential f always exists
(1.3)ξX↦fX,
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(q1,…,qn;p1,…,pn) 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 G-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(1.5){f,g}=ω(ξf,ξg),
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 G or its central extension G̃(1.6)0→R→G̃→G→0
in the dual space of the Lie algebra 𝔤̃.The sheaf of sections of a vector bundle vanishing along the directionp1,…,pn provides the so-called quantum vector bundle and the space of square integrable sections depending only on the space direction q1,…,qn 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
(1.7)f↦f̂=ℏi∇ξf+f,
such that
(1.8)[f̂,ĝ]=iℏ{f,g}̂,1̂=Idℋ.Application to group representations:X∈𝔤:=LieG can be considered as a function on an arbitrary coadjoint G-orbit Ω in the vector space 𝔤* dual to 𝔤.


### 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 (1+1)-dimensional fields.In field theory one defines thepartition function as
(1.9)〈f(X)〉=∫f(X)e-S[X]DX∫e-S[X]DX,
where DX is the Wiener measure of the space of paths from a point to another one, where
(1.10)S[X]=∫t1t2L(X,Ẋ)dτ
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 spaceM4 to the product M4=C×Σ of a possibly noncompact Riemann surface C and a compact Riemann surface Σ. The Klein reduction requires to compactify C and have some effective theory on Σ.Therefore one needs to consider the sigma-model onΣ with target space ℳH(G) or ℳH(LG) the moduli space of semistable Higgs bundles (ℰ,Φ) on C, that is, the holomorphic vector bundles E↠C over the hyperKähler manifold C, with self-dual connection ∇. They can be obtained from the principal bundle GC↠C, a finite-dimensional GC-module E, and Φ∈H0(C,E×G𝔤). The operator of taking partial trace TrR related with the representation R give the space B 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 structuresI, J, K on the corresponding spaces. The moduli space ℳH(LG) of semistable LG-bundles with holomorphic connection ℰ=(E,∇) on C endowed with the complex structure J is denoted by Y(LG). The ℬ-branes on C are the same as these objects ℰ.The manifoldℳH(G) is endowed with the complex structure K and become a symplectic manifold with respect to the symplectic structure ωK.By the mirror symmetry transformation, theℬ-branes become the so called 𝒜-branes, those are the Lagrangian submanifolds on ℳH(G).


### 1.3. Quantization of Fields

In the general scheme there are two models ofD-branes: the 𝒜 model with 't Hooft line operators and ℬ model with Wilson loop operators, in one side and the 𝒟-modules on the stack BunG of vector G-bundles on X with Hecke operators, on another side. With reduction X=C×Σ the moduli stack BunG is reduced to the moduli stack of G-bundles on C with connection ∇, the curvature of which satisfies the Bogomolnyi system of equations
(1.12)FA-Φ⋀Φ=0,DAΦ-DA⋆Φ=0.
This system means that the curvature FÃ=0, where Ã:=A+iΦ. We have the general picture as(1.13)Our tash is to realize the flesh from𝒜-branes to𝒟-modules which is equivalent to the Geometric Langlands Correspondence, through the mirror symmetry.Going from the𝒜 model of D branes to Hecke eigensheaves of 𝒟 modules can be considered as a quantization procedure of fields, using the Fukaya category or mutidimensional Fedosov deformation quantization, or the B. Tsygan deformation quantization. The most deficulties are related with the complicated category or analytic transformation in Tsygan approach. From the ℬ model to 𝒟 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.Thenew quantization procedure we proposed consists of the following steps.(i)
Starting from a connectionL∇ associated with the LG-bundle, use the Electric-Magnetic GNO Duality to obtain bundles with connection ∇ for the dual groups G.(ii)
Use theKaluza-Klein Reduction to reduce the model to the case over complex curve C extending the symmetry group G.(iii)
From a connection∇ construct the correspoding representation σ:π1(C)→G.(iv)
Construct the correspondingMomentum Maps.(v)
Use theOrbit Method to obtain the representations of Lie group G.(vi)
UseADHM construction and the Hitchin Fibration Construction to have some holomorphic bundle on CP3.(vii)
Use thepositive energy representations of Virasoro algebras to obtain representations of loop algebras (Fock space construction).Together compose all the steps, we have the sameautomorphic 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 ofG 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 Section1. 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 MC into the twistor space T=C4. 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.


## 1.1. Geometric Quantization

In the general model, a Hamiltonian system is modeled as some symplectic manifold(M,ω) with a flat action ∇ of some Lie group of symmetry G; see [1, 2] for more details.Iff is a function on the symplectic manifold, its symplectic gradient is the vector field ξf such that(1.2)ı(ξf)ω+df≡0.
Conversely, every element X of the Lie algebra of symmetry 𝔤=LieG provides a vector field ξX(m)=d/dt|t=0exp(tX)m. The condition is that the potential f always exists
(1.3)ξX↦fX,
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(q1,…,qn;p1,…,pn) 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 G-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(1.5){f,g}=ω(ξf,ξg),
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 G or its central extension G̃(1.6)0→R→G̃→G→0
in the dual space of the Lie algebra 𝔤̃.The sheaf of sections of a vector bundle vanishing along the directionp1,…,pn provides the so-called quantum vector bundle and the space of square integrable sections depending only on the space direction q1,…,qn 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
(1.7)f↦f̂=ℏi∇ξf+f,
such that
(1.8)[f̂,ĝ]=iℏ{f,g}̂,1̂=Idℋ.Application to group representations:X∈𝔤:=LieG can be considered as a function on an arbitrary coadjoint G-orbit Ω in the vector space 𝔤* dual to 𝔤.


## 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 (1+1)-dimensional fields.In field theory one defines thepartition function as
(1.9)〈f(X)〉=∫f(X)e-S[X]DX∫e-S[X]DX,
where DX is the Wiener measure of the space of paths from a point to another one, where
(1.10)S[X]=∫t1t2L(X,Ẋ)dτ
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 spaceM4 to the product M4=C×Σ of a possibly noncompact Riemann surface C and a compact Riemann surface Σ. The Klein reduction requires to compactify C and have some effective theory on Σ.Therefore one needs to consider the sigma-model onΣ with target space ℳH(G) or ℳH(LG) the moduli space of semistable Higgs bundles (ℰ,Φ) on C, that is, the holomorphic vector bundles E↠C over the hyperKähler manifold C, with self-dual connection ∇. They can be obtained from the principal bundle GC↠C, a finite-dimensional GC-module E, and Φ∈H0(C,E×G𝔤). The operator of taking partial trace TrR related with the representation R give the space B 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 structuresI, J, K on the corresponding spaces. The moduli space ℳH(LG) of semistable LG-bundles with holomorphic connection ℰ=(E,∇) on C endowed with the complex structure J is denoted by Y(LG). The ℬ-branes on C are the same as these objects ℰ.The manifoldℳH(G) is endowed with the complex structure K and become a symplectic manifold with respect to the symplectic structure ωK.By the mirror symmetry transformation, theℬ-branes become the so called 𝒜-branes, those are the Lagrangian submanifolds on ℳH(G).


## 1.3. Quantization of Fields

In the general scheme there are two models ofD-branes: the 𝒜 model with 't Hooft line operators and ℬ model with Wilson loop operators, in one side and the 𝒟-modules on the stack BunG of vector G-bundles on X with Hecke operators, on another side. With reduction X=C×Σ the moduli stack BunG is reduced to the moduli stack of G-bundles on C with connection ∇, the curvature of which satisfies the Bogomolnyi system of equations
(1.12)FA-Φ⋀Φ=0,DAΦ-DA⋆Φ=0.
This system means that the curvature FÃ=0, where Ã:=A+iΦ. We have the general picture as(1.13)Our tash is to realize the flesh from𝒜-branes to𝒟-modules which is equivalent to the Geometric Langlands Correspondence, through the mirror symmetry.Going from the𝒜 model of D branes to Hecke eigensheaves of 𝒟 modules can be considered as a quantization procedure of fields, using the Fukaya category or mutidimensional Fedosov deformation quantization, or the B. Tsygan deformation quantization. The most deficulties are related with the complicated category or analytic transformation in Tsygan approach. From the ℬ model to 𝒟 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.Thenew quantization procedure we proposed consists of the following steps.(i)
Starting from a connectionL∇ associated with the LG-bundle, use the Electric-Magnetic GNO Duality to obtain bundles with connection ∇ for the dual groups G.(ii)
Use theKaluza-Klein Reduction to reduce the model to the case over complex curve C extending the symmetry group G.(iii)
From a connection∇ construct the correspoding representation σ:π1(C)→G.(iv)
Construct the correspondingMomentum Maps.(v)
Use theOrbit Method to obtain the representations of Lie group G.(vi)
UseADHM construction and the Hitchin Fibration Construction to have some holomorphic bundle on CP3.(vii)
Use thepositive energy representations of Virasoro algebras to obtain representations of loop algebras (Fock space construction).Together compose all the steps, we have the sameautomorphic 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 ofG 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 Section1. 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 MC into the twistor space T=C4. 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.
LetL∇ be a LG-connection of the target space X. Then there exists a unique dual connection on the Langlands dual G-bundle on the same base X.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 dimensiond≤6 and the most standard ones have d≤4. 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 M is decomposed into a Cartesian product
(3.1)M=N×K,
where the action may be very large for the field that are not constant over K and therefore the correlation functions are localized along the fields that are constant along K. The mirror symmetry theory says that one needs only to concentrate in the case of dimension 1+1 or 2.Following Kapustin and Witten [4] we reduce the theory to the case of complex curve C.Theorem 3.1.
The connection∇ on X=C is uniquely defined by a representation of the fundamental groups π1(C)→G.Proof.
Over the universal coveringC̃, there is a unique trivial connection. Passing on the manifold C 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 theG-orbits in the principal bundle total space P and the coadjoint G-orbits of G in the space 𝔤* dual to the Lie algebra 𝔤=Lie(G) of G.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 ofσ model consists of the following.(i)
Choose a Langangian submanifoldM.(ii)
Choose the ground state spaceℋ(0)=HDR(M).(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 thatΩ is an coadjoint orbit of Lie group of symmetry G in the dual space of its Lie algebra, (𝔭,F,H,M,ρ) is a polarization at F∈Ω, then the Lie algebra 𝔤 is acting on the de Rham cohomology ℋDR*(Γ(ℰρ,F(V))) with coefficients in the sheaf of partially holomorphic and partially invariant sections of the vector bundle ℰρ,F(V)=G×M,ρV 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 𝔤 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 U(𝔤) we have a corresponding homomorphism of associative algebras
(4.1)U(𝔤)→PSDO(ℰρ,F(V)).


### 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 groupG one obtains the natural action of the Lie agebra 𝔤 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.


## 4.1. Momentum Maps

Theorem 4.1.
There is local diffeomorphisms mapping theG-orbits in the principal bundle total space P and the coadjoint G-orbits of G in the space 𝔤* dual to the Lie algebra 𝔤=Lie(G) of G.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 ofσ model consists of the following.(i)
Choose a Langangian submanifoldM.(ii)
Choose the ground state spaceℋ(0)=HDR(M).(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 thatΩ is an coadjoint orbit of Lie group of symmetry G in the dual space of its Lie algebra, (𝔭,F,H,M,ρ) is a polarization at F∈Ω, then the Lie algebra 𝔤 is acting on the de Rham cohomology ℋDR*(Γ(ℰρ,F(V))) with coefficients in the sheaf of partially holomorphic and partially invariant sections of the vector bundle ℰρ,F(V)=G×M,ρV 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 𝔤 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 U(𝔤) we have a corresponding homomorphism of associative algebras
(4.1)U(𝔤)→PSDO(ℰρ,F(V)).


## 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 groupG one obtains the natural action of the Lie agebra 𝔤 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. TheGeometric Langlands Correspondence [4] can be formulated as follows.(i)
One begins with the Langlands dual groupLG and a semistable homomorphism θ:π1(C)→LGC. The space of such homomorphisms is the Hitchin moduli space ℳH(LG,C).(ii)
Ifθ is irreducible the corresponding point x(θ)∈ℳH(LG,C) is a smooth point and a zerobrane ℬx(θ) supported at x(θ) is an electric eigenbrane in the sigma model of the target space ℳH(LG,C).(iii)
Applying theS duality to this electric eigenbrane will give a magnetic eigenbrane in the sigma-model of the target space ℳH(G,C), 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 θ:π1(C)→LGC is associated in a natural way to a sheaf on ℳH(G,C), that is, a Hecke eigensheaf and also a holonomic 𝒟-module.What follows is an entry into details.


### 5.1. Reduction to a Theory on Curve

Consider the four-manifoldM=Σ×C, where C is a compact Riemann surface of genus greater than one, Σ is either a complete but noncompact two-manifold such as R2, or a second compact Riemann surface.To find an effective physics onΣ, we must find the configuration on M that minimize or nearly minimize the action in Euclidean signature or the energy in the Lorentz signature. In either case the four-dimensional twisted 𝒩=4 supersymmetric gauge theory reduces on Σ to a sigma-model of maps Φ:Σ→ℳH(G,C), where ℳH(G,C) is the moduli space of the solutions on C of Hitchin's equation with gauge group.The minimum is obtained if(5.1)ℱ=D*Φ=0.
It is equivalent to the following system:
(5.2)F-Φ⋀Φ=0,DΦ=D*Φ=0,dimℳH(G,C)=(2g-2)dimG,
where g is the genus of the surface C.


### 5.2. 't Hooft Operators and Operator Product Expansion

#### 5.2.1. Wilson Loop and Line Operators

LetE→M be the G-bundle, associated with a representation R. Let A be a connection with curvature F, Φ a scalar field with values in the Lie algebra 𝔤=LieG, 𝒜=A+iΦ, and 𝒜̅=A-iΦ, S an oriented loop. The Wilson operator is defined as the trace in the representation R of the holonomy
(5.3)W0(R,S)=TrRHolS(A),
that is,
(5.4)W0(R,S)=TrRPexp(-∮S𝒜):=TrRPexp(-∮S(A+iΦ)),W0(R,S)=TrRPexp(-∮S𝒜̅):=TrRPexp(-∮S(A-iΦ)).
If S is a line with endpoints p and q at infinity, we can define W(R,S) as a matrix of parallel transport (of the connection 𝒜 or 𝒜̅, from the fiber at the point p, taken in the representation R, to the point q.The dual of a Wilson operator forG and R is a 't Hooft operatorT(LR,S) for the dual group LG and the corresponding representation LR.To define a Wilson loop operator associated with a loopS⊂M, S must be oriented. The 't Hooft operator is labelled by the representations LR of the L-group LG and instead requires an orientation of the normal bundle to S. A small neighborhood of S can be identified with S×R3. Once M is oriented we can ask for the orientation ds of S and the orientation e3 are compatible in the sense e4=ds⋀e3 along S(5.5)e3=ds⋀dr⋀dVol=-dr⋀ds⋀dVol.
The line operators that preserve the topological symmetry at rational values of Ψ are called mixed Wilson-'t Hooft operators.Combined Wilson-'t Hooft operators: Abelian CaseG=U(1)
To the action of the gauge field, add a term(5.6)Iθ=iΨ4π∫MTrF⋀F,A=A0+Â, where A0 is gauge invariant that has singularity at S, but Â is smooth near S,
(5.7)Îθ=iimΨ2π∫MTrF⋀F̂
with δT being the generator of the topological symmetry δTA0=0, because A0 is gauge invariant δTÂ=δTA, δTF̂=d(δTÂ)(5.8)δTÎθ=iimΨ2π∫Md(F0⋀δTÂ)=imΨ2π∫∂VεF0⋀δTÂ,δIθ=mΨ∫SδTÂ,δT(exp(-Iθ)),δT(exp(mΨ∫SÂ)exp(-Iθ))=0.
This means that we can restore the topological symmetry if we include a Wilson operator exp(mΨ∫SÂ) as an additional factor in the path integral. The expression exp(-nΨ∫SÂ) is gauge invariant if and only if n=mΨ an integer; that is, Ψ must be rational number -n/m Rational transformation
(5.9)(m,n)↦(m,n)(d-b-ca).Combined Wilson-'t Hooft Operators: Nonabelian Case
Letρ:U(1)→G be a homomorphism, A=ρ(A0)+Â, where A0 is the singular U(1) gauge field with Dirac singularity along the curve S. The required conditions are
(5.10)[ρ(A0),Â]=0,δTIθ=Ψ∫STrρ(1)δTÂ,δT(exp(Ψ∫STrρ(1)Â)exp(-Iθ))=0.


#### 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.ConsiderM=R×W for some 3-manifold W. The 't Hooft operators are supported on 1-manifold of the form R×p, p∈W,two-dimensionalA-model with targetX: a supersymmetric classical field Φ:Σ→X; the first approximation: Σ=R×I,I being an interval; the space of supersymmetric states: the cohomology of the space of constant maps to X that obey the boundary conditions.Reduction to Two Dimensions
We haveM=R×W, W=I×C, with C being a Riemann surface, I an interval, at whose ends one takes the boundary conditions. Then the BPS equations are
(5.11)(F-Φ⋀Φ+tDΦ)+=0,(F-Φ⋀Φ-t-1DΦ)-=0,D*Φ=0.In the term of complex connection𝒜=A+iΦ, curvarture ℱ=d𝒜+𝒜⋀𝒜:
(5.12)ℱ+i⋆ℱ̅=0.


#### 5.2.3. 't Hooft Operators and Eigenbranes

The Case of GroupG=U(1)
A 't Hooft operatorT(m;p0) at the point p0 is classified by an integer m. The operator T(m) is defined by saying that near the point p=p0×y0, the curvature has the singular behaviour
(5.13)F~⋆d(im21|x-p|),
where |x-p| is the distance from the point p to the nearby point x∈R×C. This implies that if S is a small enclosing the point p, then
(5.14)∫Sc1(ℒ)=m,
and the 't Hooft operator is acting by twisting with 𝒪(p0)m and
(5.15)T(m,p0)=T(1;p0)m.Nonabelian Case ofSU(N)
The singular part of the curvature nearp is diagonal
(5.16)F~⋆d(i21|x-p|)(m1m2m3⋱mN).
Near p, the Bogomolny equation reduces to equation in some maximal torus T=U(1)N of U(N). The corresponding vector bundle ℒy near p0×y∈Cy splits up to a sum ℒ1⊕ℒ2⊕⋯⊕ℒN of the line bundles ℒi. The effect of the 't Hooft operator on ℒi is(5.17)ℒi↦ℒi⊗𝒪(p0)mi.
Therefore we have some action of the 't Hooft operators T(Lw) what is the same as the action of the Hecke operators on bunldes for SU(N). This proves that the eigensheaves of these 't Hooft operators are the same as Hecke eigensheaves. Remark that U(N)C=GL(N,C).


### 5.3. The Extended Bogomolny Equation

We consider only the supersymmetric time-independent and time-reversal invariant. Then the BPS equations are(5.18)(F-Φ⋀Φ+tDΦ)+=0,(F-Φ⋀Φ-t-1DΦ)-=0,D*Φ=0,
reducing to the ordinary Bogomolny equations, and the 't Hooft operators reduce to the usual geometric Hecke operators.On the four-manifoldM=R×W we write the Higgs field Φ as Φ0dx0+π*Φ̃, where π:M→W is the projection.The gauge fieldsA=A0dx0+Ã, where Ã is a 3-dimensional connection with curvature F̃.The time-independent BPS equation fort=1 is
(5.19)F̃-Φ̃⋀Φ̃=⋆(DΦ0-[A0,Φ̃]),*DΦ̃=[Φ0,Φ̃]+DA0,D*Φ̃+[A0,Φ0]=0,
where D is the exterior derivative, the Hodge operator ⋆, and the operator D*=⋆D⋆.Because of the time independence one deduces thatΦ1=A0=0. Choose a local coordinated z=x2+ix3.  y=x1 and Ãy=0.  Because the metric on W is ds2=h|dz|2+dy2, then the extended Bogomolny equations are
(5.20)Dz̅Φz=0,DyAz̅=-iDz̅Φ0,DyΦz=-i[Φz,Φ0],Fzz̅-[Φz,Φz̅]=ih2∂zΦ0.
The first equation means that φ=Φzdz reatricted on Cy={y}×C is a holomorphic section of End(E)⊗KC, E is the holommorphic bundle over Cy defined by the ∂̅ operator(5.21)D̅=dz̅(∂z̅+Az̅).
The pair (E,φ) is a Higgs bundle or Hecke modification for any y.


## 5.1. Reduction to a Theory on Curve

Consider the four-manifoldM=Σ×C, where C is a compact Riemann surface of genus greater than one, Σ is either a complete but noncompact two-manifold such as R2, or a second compact Riemann surface.To find an effective physics onΣ, we must find the configuration on M that minimize or nearly minimize the action in Euclidean signature or the energy in the Lorentz signature. In either case the four-dimensional twisted 𝒩=4 supersymmetric gauge theory reduces on Σ to a sigma-model of maps Φ:Σ→ℳH(G,C), where ℳH(G,C) is the moduli space of the solutions on C of Hitchin's equation with gauge group.The minimum is obtained if(5.1)ℱ=D*Φ=0.
It is equivalent to the following system:
(5.2)F-Φ⋀Φ=0,DΦ=D*Φ=0,dimℳH(G,C)=(2g-2)dimG,
where g is the genus of the surface C.


## 5.2. 't Hooft Operators and Operator Product Expansion

### 5.2.1. Wilson Loop and Line Operators

LetE→M be the G-bundle, associated with a representation R. Let A be a connection with curvature F, Φ a scalar field with values in the Lie algebra 𝔤=LieG, 𝒜=A+iΦ, and 𝒜̅=A-iΦ, S an oriented loop. The Wilson operator is defined as the trace in the representation R of the holonomy
(5.3)W0(R,S)=TrRHolS(A),
that is,
(5.4)W0(R,S)=TrRPexp(-∮S𝒜):=TrRPexp(-∮S(A+iΦ)),W0(R,S)=TrRPexp(-∮S𝒜̅):=TrRPexp(-∮S(A-iΦ)).
If S is a line with endpoints p and q at infinity, we can define W(R,S) as a matrix of parallel transport (of the connection 𝒜 or 𝒜̅, from the fiber at the point p, taken in the representation R, to the point q.The dual of a Wilson operator forG and R is a 't Hooft operatorT(LR,S) for the dual group LG and the corresponding representation LR.To define a Wilson loop operator associated with a loopS⊂M, S must be oriented. The 't Hooft operator is labelled by the representations LR of the L-group LG and instead requires an orientation of the normal bundle to S. A small neighborhood of S can be identified with S×R3. Once M is oriented we can ask for the orientation ds of S and the orientation e3 are compatible in the sense e4=ds⋀e3 along S(5.5)e3=ds⋀dr⋀dVol=-dr⋀ds⋀dVol.
The line operators that preserve the topological symmetry at rational values of Ψ are called mixed Wilson-'t Hooft operators.Combined Wilson-'t Hooft operators: Abelian CaseG=U(1)
To the action of the gauge field, add a term(5.6)Iθ=iΨ4π∫MTrF⋀F,A=A0+Â, where A0 is gauge invariant that has singularity at S, but Â is smooth near S,
(5.7)Îθ=iimΨ2π∫MTrF⋀F̂
with δT being the generator of the topological symmetry δTA0=0, because A0 is gauge invariant δTÂ=δTA, δTF̂=d(δTÂ)(5.8)δTÎθ=iimΨ2π∫Md(F0⋀δTÂ)=imΨ2π∫∂VεF0⋀δTÂ,δIθ=mΨ∫SδTÂ,δT(exp(-Iθ)),δT(exp(mΨ∫SÂ)exp(-Iθ))=0.
This means that we can restore the topological symmetry if we include a Wilson operator exp(mΨ∫SÂ) as an additional factor in the path integral. The expression exp(-nΨ∫SÂ) is gauge invariant if and only if n=mΨ an integer; that is, Ψ must be rational number -n/m Rational transformation
(5.9)(m,n)↦(m,n)(d-b-ca).Combined Wilson-'t Hooft Operators: Nonabelian Case
Letρ:U(1)→G be a homomorphism, A=ρ(A0)+Â, where A0 is the singular U(1) gauge field with Dirac singularity along the curve S. The required conditions are
(5.10)[ρ(A0),Â]=0,δTIθ=Ψ∫STrρ(1)δTÂ,δT(exp(Ψ∫STrρ(1)Â)exp(-Iθ))=0.


### 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.ConsiderM=R×W for some 3-manifold W. The 't Hooft operators are supported on 1-manifold of the form R×p, p∈W,two-dimensionalA-model with targetX: a supersymmetric classical field Φ:Σ→X; the first approximation: Σ=R×I,I being an interval; the space of supersymmetric states: the cohomology of the space of constant maps to X that obey the boundary conditions.Reduction to Two Dimensions
We haveM=R×W, W=I×C, with C being a Riemann surface, I an interval, at whose ends one takes the boundary conditions. Then the BPS equations are
(5.11)(F-Φ⋀Φ+tDΦ)+=0,(F-Φ⋀Φ-t-1DΦ)-=0,D*Φ=0.In the term of complex connection𝒜=A+iΦ, curvarture ℱ=d𝒜+𝒜⋀𝒜:
(5.12)ℱ+i⋆ℱ̅=0.


### 5.2.3. 't Hooft Operators and Eigenbranes

The Case of GroupG=U(1)
A 't Hooft operatorT(m;p0) at the point p0 is classified by an integer m. The operator T(m) is defined by saying that near the point p=p0×y0, the curvature has the singular behaviour
(5.13)F~⋆d(im21|x-p|),
where |x-p| is the distance from the point p to the nearby point x∈R×C. This implies that if S is a small enclosing the point p, then
(5.14)∫Sc1(ℒ)=m,
and the 't Hooft operator is acting by twisting with 𝒪(p0)m and
(5.15)T(m,p0)=T(1;p0)m.Nonabelian Case ofSU(N)
The singular part of the curvature nearp is diagonal
(5.16)F~⋆d(i21|x-p|)(m1m2m3⋱mN).
Near p, the Bogomolny equation reduces to equation in some maximal torus T=U(1)N of U(N). The corresponding vector bundle ℒy near p0×y∈Cy splits up to a sum ℒ1⊕ℒ2⊕⋯⊕ℒN of the line bundles ℒi. The effect of the 't Hooft operator on ℒi is(5.17)ℒi↦ℒi⊗𝒪(p0)mi.
Therefore we have some action of the 't Hooft operators T(Lw) what is the same as the action of the Hecke operators on bunldes for SU(N). This proves that the eigensheaves of these 't Hooft operators are the same as Hecke eigensheaves. Remark that U(N)C=GL(N,C).


## 5.2.1. Wilson Loop and Line Operators

LetE→M be the G-bundle, associated with a representation R. Let A be a connection with curvature F, Φ a scalar field with values in the Lie algebra 𝔤=LieG, 𝒜=A+iΦ, and 𝒜̅=A-iΦ, S an oriented loop. The Wilson operator is defined as the trace in the representation R of the holonomy
(5.3)W0(R,S)=TrRHolS(A),
that is,
(5.4)W0(R,S)=TrRPexp(-∮S𝒜):=TrRPexp(-∮S(A+iΦ)),W0(R,S)=TrRPexp(-∮S𝒜̅):=TrRPexp(-∮S(A-iΦ)).
If S is a line with endpoints p and q at infinity, we can define W(R,S) as a matrix of parallel transport (of the connection 𝒜 or 𝒜̅, from the fiber at the point p, taken in the representation R, to the point q.The dual of a Wilson operator forG and R is a 't Hooft operatorT(LR,S) for the dual group LG and the corresponding representation LR.To define a Wilson loop operator associated with a loopS⊂M, S must be oriented. The 't Hooft operator is labelled by the representations LR of the L-group LG and instead requires an orientation of the normal bundle to S. A small neighborhood of S can be identified with S×R3. Once M is oriented we can ask for the orientation ds of S and the orientation e3 are compatible in the sense e4=ds⋀e3 along S(5.5)e3=ds⋀dr⋀dVol=-dr⋀ds⋀dVol.
The line operators that preserve the topological symmetry at rational values of Ψ are called mixed Wilson-'t Hooft operators.Combined Wilson-'t Hooft operators: Abelian CaseG=U(1)
To the action of the gauge field, add a term(5.6)Iθ=iΨ4π∫MTrF⋀F,A=A0+Â, where A0 is gauge invariant that has singularity at S, but Â is smooth near S,
(5.7)Îθ=iimΨ2π∫MTrF⋀F̂
with δT being the generator of the topological symmetry δTA0=0, because A0 is gauge invariant δTÂ=δTA, δTF̂=d(δTÂ)(5.8)δTÎθ=iimΨ2π∫Md(F0⋀δTÂ)=imΨ2π∫∂VεF0⋀δTÂ,δIθ=mΨ∫SδTÂ,δT(exp(-Iθ)),δT(exp(mΨ∫SÂ)exp(-Iθ))=0.
This means that we can restore the topological symmetry if we include a Wilson operator exp(mΨ∫SÂ) as an additional factor in the path integral. The expression exp(-nΨ∫SÂ) is gauge invariant if and only if n=mΨ an integer; that is, Ψ must be rational number -n/m Rational transformation
(5.9)(m,n)↦(m,n)(d-b-ca).Combined Wilson-'t Hooft Operators: Nonabelian Case
Letρ:U(1)→G be a homomorphism, A=ρ(A0)+Â, where A0 is the singular U(1) gauge field with Dirac singularity along the curve S. The required conditions are
(5.10)[ρ(A0),Â]=0,δTIθ=Ψ∫STrρ(1)δTÂ,δT(exp(Ψ∫STrρ(1)Â)exp(-Iθ))=0.


## 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.ConsiderM=R×W for some 3-manifold W. The 't Hooft operators are supported on 1-manifold of the form R×p, p∈W,two-dimensionalA-model with targetX: a supersymmetric classical field Φ:Σ→X; the first approximation: Σ=R×I,I being an interval; the space of supersymmetric states: the cohomology of the space of constant maps to X that obey the boundary conditions.Reduction to Two Dimensions
We haveM=R×W, W=I×C, with C being a Riemann surface, I an interval, at whose ends one takes the boundary conditions. Then the BPS equations are
(5.11)(F-Φ⋀Φ+tDΦ)+=0,(F-Φ⋀Φ-t-1DΦ)-=0,D*Φ=0.In the term of complex connection𝒜=A+iΦ, curvarture ℱ=d𝒜+𝒜⋀𝒜:
(5.12)ℱ+i⋆ℱ̅=0.


## 5.2.3. 't Hooft Operators and Eigenbranes

The Case of GroupG=U(1)
A 't Hooft operatorT(m;p0) at the point p0 is classified by an integer m. The operator T(m) is defined by saying that near the point p=p0×y0, the curvature has the singular behaviour
(5.13)F~⋆d(im21|x-p|),
where |x-p| is the distance from the point p to the nearby point x∈R×C. This implies that if S is a small enclosing the point p, then
(5.14)∫Sc1(ℒ)=m,
and the 't Hooft operator is acting by twisting with 𝒪(p0)m and
(5.15)T(m,p0)=T(1;p0)m.Nonabelian Case ofSU(N)
The singular part of the curvature nearp is diagonal
(5.16)F~⋆d(i21|x-p|)(m1m2m3⋱mN).
Near p, the Bogomolny equation reduces to equation in some maximal torus T=U(1)N of U(N). The corresponding vector bundle ℒy near p0×y∈Cy splits up to a sum ℒ1⊕ℒ2⊕⋯⊕ℒN of the line bundles ℒi. The effect of the 't Hooft operator on ℒi is(5.17)ℒi↦ℒi⊗𝒪(p0)mi.
Therefore we have some action of the 't Hooft operators T(Lw) what is the same as the action of the Hecke operators on bunldes for SU(N). This proves that the eigensheaves of these 't Hooft operators are the same as Hecke eigensheaves. Remark that U(N)C=GL(N,C).


## 5.3. The Extended Bogomolny Equation

We consider only the supersymmetric time-independent and time-reversal invariant. Then the BPS equations are(5.18)(F-Φ⋀Φ+tDΦ)+=0,(F-Φ⋀Φ-t-1DΦ)-=0,D*Φ=0,
reducing to the ordinary Bogomolny equations, and the 't Hooft operators reduce to the usual geometric Hecke operators.On the four-manifoldM=R×W we write the Higgs field Φ as Φ0dx0+π*Φ̃, where π:M→W is the projection.The gauge fieldsA=A0dx0+Ã, where Ã is a 3-dimensional connection with curvature F̃.The time-independent BPS equation fort=1 is
(5.19)F̃-Φ̃⋀Φ̃=⋆(DΦ0-[A0,Φ̃]),*DΦ̃=[Φ0,Φ̃]+DA0,D*Φ̃+[A0,Φ0]=0,
where D is the exterior derivative, the Hodge operator ⋆, and the operator D*=⋆D⋆.Because of the time independence one deduces thatΦ1=A0=0. Choose a local coordinated z=x2+ix3.  y=x1 and Ãy=0.  Because the metric on W is ds2=h|dz|2+dy2, then the extended Bogomolny equations are
(5.20)Dz̅Φz=0,DyAz̅=-iDz̅Φ0,DyΦz=-i[Φz,Φ0],Fzz̅-[Φz,Φz̅]=ih2∂zΦ0.
The first equation means that φ=Φzdz reatricted on Cy={y}×C is a holomorphic section of End(E)⊗KC, E is the holommorphic bundle over Cy defined by the ∂̅ operator(5.21)D̅=dz̅(∂z̅+Az̅).
The pair (E,φ) is a Higgs bundle or Hecke modification for any y.


## 6. Proof of the Main Theorem

First we remark that following the previous results, under mirror symmetry, the category of𝒜-branes invariant under the 't Hooft operators is equivalent to the corresponding category of ℬ-branes invariant under the Wilson loop operators. And the both categories are equivalent to the category of 𝒟-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(6.1)f̂g=exp(ℏ-1ω(∂y,∂z))f(y)g(z)|y=z=∑i1,…,in=0∞1i1!⋯in!(ℏ-1)|(i1,…,in)|ωi1j1⋯ωinjn∂yi1⋯∂yinf(y)∂zj1⋯∂zjng(z)|y=z,
where |(i1,…,in)|=i1+⋯+in 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(6.2)f+ℏ-1ωij∂if∂j=f+ℏ-1∇ξf,
which is the geometric quantization formula.Lemma 6.2.
The(i1,…,in)th component of of the series is the continued (i,…,in)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(i1,…,in)th component of of the series is linear operator from the tensor product of finite-dimensional Hilbert space TxM⊗Ωxn(M)=TxM⊗Ωx1(M)⊗⋯⊗Ωx1(M) into the Hilbert space TxM.Proof.
It is clear from the formula for deformation.Lemma 6.4.
The formal series are elements of the tensor product of Hilbert spaceTxM⊗exp(Ωx1(M)).Proof.
The exponential series formula provides the convergence.Now the proof of the main theorem is achieved.



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*Source: 749631-2009-12-30.xml*