arXiv subjects

Edward Frenkel

Publications and source records attributed to Edward Frenkel.

At least 37 records · Page 2Linked to original sources

arXiv2014

Baxter's Relations and Spectra of Quantum Integrable Models

Generalized Baxter's relations on the transfer-matrices (also known as Baxter's TQ relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the category O introduced by Jimbo and the second author in arXiv:1104.1891 involving infinite-dimensional representations constructed in arXiv:1104.1891, which we call here "prefundamental". We define the transfer-matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Combining these two results, we express the spectra of the transfer-matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture of Reshetikhin and the first author formulated in 1998 (arXiv:math/9810055). We also obtain generalized Bethe Ansatz equations for all untwisted quantum affine algebras.

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arXiv2013

Geometric Langlands Correspondence Near Opers

We extend the categorical geometric Langlands correspondence from the locus of opers in the stack of de Rham local systems on a smooth projective algebraic curve to the formal neighborhood of opers (for any semi-simple complex algebraic group).

math.AG↗

arXiv2012

Mathematics, Love, and Tattoos

Reflections on the film "Rites of Love and Math" (directed by Reine Graves and the author) and its "formula of love".

math.HO↗

arXiv2011

Geometrization of Trace Formulas

Following our joint work arXiv:1003.4578 with Robert Langlands, we make the first steps toward developing geometric methods for analyzing trace formulas in the case of the function field of a curve defined over a finite field. We also suggest a conjectural framework of geometric trace formulas for curves defined over the complex field, which exploits the categorical version of the geometric Langlands correspondence.

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arXiv2011

Langlands duality for representations of quantum groups

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an "interpolating quantum group" depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.

math.QA↗

arXiv2011

Langlands duality for finite-dimensional representations of quantum affine algebras

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of q-alg/9708006 and 0809.4453. We prove this duality for the Kirillov-Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct "interpolating (q,t)-characters" depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.

math.QA↗

arXiv2010

Formule des Traces et Fonctorialité: le Début d'un Programme

We outline an approach to proving functoriality of automorphic representations using trace formula. More specifically, we construct a family of integral operators on the space of automorphic forms whose eigenvalues are expressed in terms of the L-functions of automorphic representations and begin the analysis of their traces using the orbital side of the stable trace formula. We show that the most interesting part, corresponding to regular conjugacy classes, is nothing but a sum over a finite-dimensional vector space over the global field, which we call the Steinberg-Hitchin base. Therefore it may be analyzed using the Poisson summation formula. Our main result is that the leading term of the dual sum (the value at 0) is precisely the dominant term of the trace formula (the contribution of the trivial representation). This gives us hope that the full Poisson summation formula would reveal the patterns predicted by functoriality.

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arXiv2010

Any flat bundle on a punctured disc has an oper structure

We prove that any flat G-bundle, where G is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in arXiv:math/0508382. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine Weyl groups on the homology of these deformations generalizing representations constructed by Lusztig.

math.RT↗

arXiv2009

Quantization of soliton systems and Langlands duality

We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac--Moody algebras. Our experience with the Gaudin models associated to finite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an affine algebra should be encoded by affine opers associated to the Langlands dual affine algebra. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of the soliton systems. In particular, for the KdV system the corresponding affine opers may be expressed as Schroedinger operators with spectral parameter, and our predictions in this case match those recently made by Bazhanov, Lukyanov and Zamolodchikov. This suggests that this and other recently found examples of the correspondence between quantum integrals of motion and differential operators may be viewed as special cases of the Langlands duality.

math.QA↗

arXiv2009

D-modules on the affine flag variety and representations of affine Kac-Moody algebras

We study the connection between the category of modules over the affine Kac-Moody Lie algebra at the critical level, and the category of D-modules on the affine flag scheme G((t))/I, where I is the Iwahori subgroup. We prove a localization-type result, which establishes an equivalence between certain subcategories on both sides. We also establish an equivalence between a certain subcategory of Kac-Moody modules, and the category of quasi-coherent sheaves on the scheme of Miura opers for the Langlands dual group, thereby proving a conjecture of [FG2].

math.RT↗

arXiv2009

A rigid irregular connection on the projective line

In this paper we construct a connection on the trivial G-bundle on the projective line for any simple complex algebraic group G, which is regular outside of the points 0 and infinity, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point infinity, with slope 1/h, the reciprocal of the Coxeter number of G. This connection, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of l-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These l-adic representations, and their characteristic 0 counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in representations of G, and describe its differential Galois group as a subgroup of G.

math.AG↗

arXiv2009

Gauge Theory and Langlands Duality

The Langlands Program was launched in the late 60s with the goal of relating Galois representations and automorphic forms. In recent years a geometric version has been developed which leads to a mysterious duality between certain categories of sheaves on moduli spaces of (flat) bundles on algebraic curves. Three years ago, in a groundbreaking advance, Kapustin and Witten have linked the geometric Langlands correspondence to the S-duality of 4D supersymmetric gauge theories. This and subsequent works have already led to striking new insights into the geometric Langlands Program, which in particular involve the Homological Mirror Symmetry of the Hitchin moduli spaces of Higgs bundles on algebraic curves associated to two Langlands dual Lie groups.

math.RT↗

arXiv2009

On the endomorphisms of Weyl modules over affine Kac-Moody algebras at the critical level

We present an independent short proof of the main result of arXiv:0706.3725 that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. We derive this from the results of arXiv:0712.1183 about the shift of argument subalgebras.

math.QA↗

arXiv2008

Gerbal Representations of Double Loop Groups

A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac-Moody groups and Lie algebras). Loop groups embed into the group GL_\infty of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL_\infty. In a similar way, double loop groups embed into a group of automorphisms of C((t))((s)), denoted by GL_{\infty,\infty}, which has a non-trivial third cohomology. In this paper we explain how to realize a third cohomology class in representation theory of a group: it naturally arises when we consider representations on categories rather than vector spaces. We call them "gerbal representations." We then construct a gerbal representation of GL_{\infty,\infty} (and hence of double loop groups), realizing its non-trivial third cohomology class, on a category of modules over an infinite-dimensional Clifford algebra. This is a two-dimensional analogue of the fermionic Fock representations of the ordinary loop groups.

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arXiv2008

The q-characters of representations of quantum affine algebras and deformations of W-algebras

We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck ring of representations of a quantum affine algebra to a polynomial ring. We conjecture that the image of this homomorphism is equal to the intersection of certain "screening operators". We also discuss the connection between q-characters and Bethe Ansatz.

math.QA↗

arXiv2008

Geometric Endoscopy and Mirror Symmetry

The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T-duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations, in particular, in the stabilization of the trace formula. Our goal is to use the mirror symmetry of the Hitchin fibrations to expose the special role played by these local systems in the geometric theory. The study of the categories of A-branes on the dual Hitchin fibers allows us to uncover some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence. We then follow our predictions back to the classical theory of automorphic functions. This enables us to test and confirm them. The geometry we use is similar to that which is exploited in recent work by B.-C. Ngo, a fact which could be significant for understanding the trace formula.

math.AG↗

arXiv2008

Opers with irregular singularity and spectra of the shift of argument subalgebra

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singulairity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, we obtain the structure of a Gorenstein ring on any such module. This fact may have geometric significance related to the intersection cohomology of Schubert varieties in the affine Grassmannian.

math.QA↗

arXiv2007

Weyl modules and opers without monodromy

We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. This result may be used to test the local geometric Langlands correspondence proposed in our earlier work arXiv:math/0508382.

math.QA↗