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Edward Frenkel

Publications and source records attributed to Edward Frenkel.

At least 55 records · Page 3Linked to original sources
arXiv2007

Local Geometric Langlands Correspondence: the Spherical Case

A module over an affine Kac--Moody algebra g^ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical g^-modules of critical level. In this paper we prove that this category is equivalent to the category of quasi-coherent sheaves on the ind-scheme of opers on the punctured disc which are unramified as local systems. This result is a categorical version of the well-known description of spherical vectors in representations of groups over local non-archimedian fields. It may be viewed as a special case of the local geometric Langlands correspondence proposed in arXiv:math/0508382.

math.QA↗
arXiv2007

Fusion and convolution: applications to affine Kac-Moody algebras at the critical level

Let g be a semi-simple Lie algebra, and let g^ be the corresponding affine Kac-Moody algebra. Consider the category of g^-modules at the critical level, on which the action of the Iwahori subalgebra integrates to algebraic action of the Iwahori subgroup I. We study the action on this category of the convolution functors with the "central" sheaves on the affine flag scheme G((t))/I. We show that each object of our category is an "eigen-module" with respect to these functors. In order to prove this, we use the fusion product of modules over the affine Kac-Moody algebra.

math.RT↗
arXiv2006

Ramifications of the geometric Langlands Program

The global geometric Langlands correspondence relates Hecke eigensheaves on the moduli stack of G-bundles on a smooth projective algebraic curve X and holomorphic G'-bundles with connection on X, where G' is the Langlands dual group of G. This correspondence is relatively well understood when the connection has no singularities, i.e., is unramified. But what should happen if the connection has singularities, or ramification, at finitely many points of X? In this case we expect to attach to it a category of Hecke eigensheaves on a suitable moduli stack of G-bundles on X with parabolic (or level) structures at the ramification points. In this paper I review an approach to constructing these categories developed by D.Gaitsgory and myself. It uses representations of affine Kac-Moody algebras of critical level and localization functors from these representations to D-modules on the moduli spaces of bundles with parabolic (or level) structures. As explained in hep-th/0512172, these D-modules may be viewed as sheaves of conformal blocks naturally arising in the framework of Conformal Field Theory.

math.QA↗
arXiv2006

Local geometric Langlands correspondence and affine Kac-Moody algebras

By a local geometric Langlands correspondence for a complex reductive group G we understand a construction which assigns to a local system on the punctured disc for the Langlands dual group of G, a category equipped with an action of the formal loop group G((t)). We propose a conjectural description of these categories as categories of representations of the corresponding affine Kac-Moody algebra of critical level, and, in some cases, as categories of D-modules on the ind-schemes G((t))/K. We describe in detail these categories and interrelations between them and provide supporting evidence for our conjectures. In particular, we prove that a certain quotient category of representations of critical level is equivalent to the category of quasicoherent sheaves on a thickening of the scheme of nilpotent opers associated to the Langlands dual group.

math.RT↗
arXiv2005

Localization of g^-modules on the affine Grassmannian

We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I^0 equivariant categories, where I^0 is the radical of the Iwahori subgroup of G((t)). Our result may be viewed as an affine analogue of the equivalence of categories of g-modules and D-modules on the flag variety G/B, due to Beilinson-Bernstein and Brylinski-Kashiwara.

math.RT↗
arXiv2005

Mirror symmetry in two steps: A-I-B

We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A-model). The second theory is an intermediate model, which we call the I-model. The equivalence between the A-model and the I-model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T-duality. On the other hand, the I-model is closely related to the twisted Landau-Ginzburg model (the B-model) that is mirror dual to the A-model. Thus, the mirror symmetry is realized in two steps, via the I-model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I-model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.

hep-th↗
arXiv2005

Lectures on the Langlands Program and Conformal Field Theory

These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. We start with a motivated introduction to the Langlands Program, including its geometric reformulation, addressed primarily to physicists. I tried to make it as self-contained as possible, requiring very little mathematical background. Next, we describe the connections between the Langlands Program and two-dimensional conformal field theory that have been found in the last few years. These connections give us important insights into the physical implications of the Langlands duality.

hep-th↗
arXiv2005

Self-extensions of Verma modules and differential forms on opers

We compute the algebras of self-extensions of the vacuum module and the Verma modules over an affine Kac-Moody algebra g^ in suitable categories of Harish-Chandra modules. We show that at the critical level these algebras are isomorphic to the algebras of differential forms on various spaces of opers associated to the Langlands dual Lie algebra of g, whereas away from the critical level they become trivial. These results rely on and generalize the description of the corresponding algebras of endomorphisms obtained by Feigin and Frenkel and the description of the corresponding graded versions due to Fishel, Grojnowski and Teleman.

math.QA↗
arXiv2005

Opers on the projective line, flag manifolds and Bethe Ansatz

We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called "opers" on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation. The Bethe Ansatz is a procedure to construct explicitly the eigenvectors of the generalized Gaudin hamiltonians. We show that each solution of the Bethe Ansatz equations defines what we call a "Miura oper" on the projective line. Moreover, we show that the space of Miura opers is a union of copies of the flag manifold (of the dual group), one for each oper. This allows us to prove that all solutions of the Bethe Ansatz equations, corresponding to a fixed oper, are in one-to-one correspondence with the points of an open dense subset of the flag manifold. The Bethe Ansatz equations can be written for an arbitrary Kac-Moody algebra, and we prove an analogue of the last result in this more general setting. For the Lie algebras of types A,B,C similar results were obtained by other methods by I.Scherbak and A.Varchenko and by E.Mukhin and A.Varchenko.

math.QA↗
arXiv2005

Gaudin model and opers

This is a review of our previous works (some of them joint with B. Feigin and N. Reshetikhin) on the Gaudin model and opers. We define a commutative subalgebra in the tensor power of the universal enveloping algebra of a simple Lie algebra g. This algebra includes the hamiltonians of the Gaudin model, hence we call it the Gaudin algebra. It is constructed as a quotient of the center of the completed enveloping algebra of the affine Kac-Moody algebra g^ at the critical level. We identify the spectrum of the Gaudin algebra with the space of opers associated to the Langlands dual Lie algebra of g on the projective line with regular singularities at the marked points. Next, we recall the construction of the eigenvectors of the Gaudin algebra using the Wakimoto modules over g^ of critical level. The Wakimoto modules are naturally parameterized by Miura opers (or, equivalently, Cartan connections), and the action of the center on them is given by the Miura transformation. This allows us to relate solutions of the Bethe Ansatz equations to Miura opers and ultimately to the flag varieties associated to the Langlands dual Lie algebra of g.

math.QA↗
arXiv2003

Recent Advances in the Langlands Program

These are the notes for the lecture given by the author at the "Current Events" Special Session of the AMS meeting in Baltimore on January 17, 2003. Topics reviewed include the Langlands correspondence for GL(n) in the function field case and its proof by V.Drinfeld and L.Lafforgue, the geometric Langlands correspondence for GL(n) and its proof by D.Gaitsgory, K.Vilonen and the author, and the work of A.Beilinson and V.Drinfled on the quantization of the Hitchin system and the Langlands correspondence for an arbitrary semisimple algebraic group.

math.AG↗
arXiv2003

Twisted Modules over Vertex Algebras on Algebraic Curves

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let $V$ be a vertex algebra, $H$ a finite group of automorphisms of $V$, and $C$ an algebraic curve such that $H \subset \on{Aut}(C)$. We show that a suitable collection of twisted $V$--modules gives rise to a section of a certain sheaf on the quotient $X=C/H$. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.

math.AG↗
arXiv2003

Chiral de Rham Complex and Orbifolds

Suppose that a finite group $G$ acts on a smooth complex variety $X$. Then this action lifts to the Chiral de Rham Complex of $X$ and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for the Chiral de Rham Complex (and their cohomologies) as sheaves of twisted vertex algebra modules supported on the components of the fixed-point sets $X^{g}, g \in G$. Each twisted sector sheaf carries a BRST differential and is quasi-isomorphic to the de Rham complex of $X^{g}$. Putting the twisted sectors together with the vacuum sector and taking $G$--invariants, we recover the additive and graded structures of Chen-Ruan orbifold cohomology. Finally, we show that the orbifold elliptic genus is the partition function of the direct sum of the cohomologies of the twisted sectors.

math.AG↗
arXiv2003

Geometric Realization of the Segal--Sugawara Construction

We apply the technique of localization for vertex algebras to the Segal-Sugawara construction of an ``internal'' action of the Virasoro algebra on affine Kac-Moody algebras. The result is a lifting of twisted differential operators from the moduli of curves to the moduli of curves with bundles, with arbitrary decorations and complex twistings. This construction gives a uniform approach to a collection of phenomena describing the geometry of the moduli spaces of bundles over varying curves: the KZB equations and heat kernels on non-abelian theta functions, their critical level limit giving the quadratic parts of the Beilinson-Drinfeld quantization of the Hitchin system, and their infinite level limit giving a Hamiltonian description of the isomonodromy equations.

math.AG↗
arXiv2002

Spectral Curves, Opers and Integrable Systems

We establish a general link between integrable systems in algebraic geometry (expressed as Jacobian flows on spectral curves) and soliton equations (expressed as evolution equations on flat connections). Our main result is a natural isomorphism between a moduli space of spectral data and a moduli space of differential data, each equipped with an infinite collection of commuting flows. The spectral data are principal G-bundles on an algebraic curve, equipped with an abelian reduction near one point. The flows on the spectral side come from the action of a Heisenberg subgroup of the loop group. The differential data are flat connections known as opers. The flows on the differential side come from a generalized Drinfeld-Sokolov hierarchy. Our isomorphism between the two sides provides a geometric description of the entire phase space of the Drinfeld-Sokolov hierarchy. It extends the Krichever construction of special algebro-geometric solutions of the n-th KdV hierarchy, corresponding to G=SL(n). An interesting feature is the appearance of formal spectral curves, replacing the projective spectral curves of the classical approach. The geometry of these (usually singular) curves reflects the fine structure of loop groups, in particular the detailed classification of their Cartan subgroups. To each such curve corresponds a homogeneous space of the loop group and a soliton system. Moreover the flows of the system have interpretations in terms of Jacobians of formal curves.

math.AG↗
arXiv2002

Lectures on Wakimoto modules, opers and the center at the critical level

Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In these lectures we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc. These results were originally obtained by B. Feigin and the author.

math.QA↗
arXiv2002

The Hopf algebra $Rep U_q \hat{gl}_\infty$

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of $U_q \hat{gl}_N$ in the limit $N \to \infty$. The resulting Hopf algebra $Rep U_q \hat{gl}_\infty$ is a tensor product of its Hopf subalgebras $Rep_a U_q \hat{gl}_\infty$, $a\in\C^\times/q^{2\Z}$. When $q$ is generic (resp., $q^2$ is a primitive root of unity of order $l$), we construct an isomorphism between the Hopf algebra $Rep_a U_q \hat{gl}_\infty$ and the algebra of regular functions on the prounipotent proalgebraic group $SL_\infty^-$ (resp., $\tilde{GL}_l^-$). When $q$ is a root of unity, this isomorphism identifies the Hopf subalgebra of $Rep_a U_q \hat{gl}_\infty$ spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of $\tilde{GL}_l^-$. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver with $l$ vertices) on $Rep_a U_q \hat{gl}_infty$ and describe the span of the tensor products of the evaluation representations taken at fixed points as a module over this Hall algebra.

math.QA↗