arXiv subjects

E. Frenkel

Publications and source records attributed to E. Frenkel.

13 recordsLinked to original sources

arXiv2009

Gaudin models with irregular singularities

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from hep-th/9402022 to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P^1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.

math.QA↗

arXiv2009

Soliton equations, vertex operators, and simple singularities

We prove the equivalence of two hierarchies of soliton equations associated to a simply-laced finite Dynkin diagram. The first was defined by Kac and Wakimoto using the principal realization of the basic representations of the corresponding affine Kac-Moody algebra. The second was defined in arXiv:math/0307176 using the Frobenius structure on the local ring of the corresponding simple singularity. We also obtain a deformation of the principal realization of the basic representation over the space of miniversal deformations of the corresponding singularity. As a by-product, we compute the operator product expansions of pairs of vertex operators defined in terms of Picard-Lefschetz periods for more general singularities. Thus, we establish a surprising link between twisted vertex operators and deformation theory of singularities.

math.QA↗

arXiv2008

Instantons beyond topological theory II

The present paper is the second part of our project in which we describe quantum field theories with instantons in a novel way by using the "infinite radius limit" (rather than the limit of free field theory) as the starting point. The theory dramatically simplifies in this limit, because the correlation functions of all, not only topological (or BPS), observables may be computed explicitly in terms of integrals over finite-dimensional moduli spaces of instanton configurations. In Part I (arXiv:hep-th/0610149) we discussed in detail the one-dimensional (that is, quantum mechanical) models of this type. Here we analyze the supersymmetric two-dimensional sigma models and four-dimensional Yang--Mills theory, using the one-dimensional models as a prototype. We go beyond the topological (or BPS) sectors of these models and consider them as full-fledged quantum field theories. We study in detail the space of states and find that the Hamiltonian is not diagonalizable, but has Jordan blocks. This leads to the appearance of logarithms in the correlation functions. We find that our theories are in fact logarithmic conformal field theories (theories of this type are of interest in condensed matter physics). We define jet-evaluation observables and consider in detail their correlation functions. They are given by integrals over the moduli spaces of holomorphic maps, which generalize the Gromov--Witten invariants. These integrals generally diverge and require regularization, leading to an intricate logarithmic mixing of the operators of the sigma model. A similar structure arises in the four-dimensional Yang--Mills theory as well.

hep-th↗

arXiv2007

Notes on instantons in topological field theory and beyond

This is a brief summary of our studies of quantum field theories in a special limit in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the corresponding models as full-fledged quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kahler manifolds the space of states exhibits holomorphic factorization. In particular, in dimensions two and four our theories are logarithmic conformal field theories.

hep-th↗

arXiv2006

Instantons beyond topological theory I

Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kahler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.

hep-th↗

arXiv2006

Geometric realizations of Wakimoto modules at the critical level

We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we describe explicitly geometric realizations of the Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group.

math.RT↗

arXiv2004

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

Let ${\mathfrak g}$ be a simple Lie algebra. For a level $κ$ (thought of as a symmetric ${\mathfrak g}$-invariant form of ${\mathfrak g}$), let $\hat{\mathfrak g}_κ$ be the corresponding affine Kac-Moody algebra. Let $Gr_G$ be the affine Grassmannian of ${\mathfrak g}$, and let $D_κ(Gr_G)-mod$ be the category of $κ$-twisted right D-modules on $Gr_G$. By taking global sections of a D-module, we obtain a functor $Γ:D_κ(Gr_G)-mod\to {\mathfrak g}_κ-mod$. It is known that this functor is exact and faithful when $κ$ is negative or irrational. In this paper, we show that the functor $Γ$ is exact and faithful also when $κ$ is the critical level.

math.AG↗

arXiv2001

On the geometric Langlands conjecture

Let X be a smooth, complete, geometrically connected curve over a field of characteristic p. The geometric Langlands conjecture states that to each irreducible rank n local system E on X one can attach a perverse sheaf on the moduli stack of rank n bundles on X (irreducible on each connected component), which is a Hecke eigensheaf with respect to E. In this paper we derive the geometric Langlands conjecture from a certain vanishing conjecture. Furthermore, using recent results of Lafforgue, we prove this vanishing conjecture, and hence the geometric Langlands conjecture, in the case when the ground field is finite.

math.AG↗

arXiv2000

Whittaker Patterns in the Geometry of Moduli Spaces of Bundles on Curves

Let G be a split connected reductive group over a finite field F_q, and N its maximal unipotent subgroup. V. Drinfeld has introduced a remarkable partial compactification of the moduli stack of N-bundles on a smooth projective curve X over F_q. In this paper we study Drinfeld's moduli space and a certain category of perverse sheaves on it. The definition of this category is motivated by the study of the Whittaker functions on the group G(K), where K=F_q((t)). We prove that our category is semi-simple, and that irreducible objects of this category are "clean", i.e., they are extenstions by 0 of local systems supported on the strata. As an application of these results, we obtain a purely geometric proof of the Casselman-Shalika formula for the Whittaker functions.

math.AG↗

arXiv1997

Geometric Realization of Whittaker Functions and the Langlands Conjecture

We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group GL_n(A) associated to an irreducible l-adic local system of rank n on an algebraic curve X over a finite field. The existence of such a function is predicted by the Langlands conjecture. The first construction, which was proposed by Shalika and Piatetski-Shapiro following Weil and Jacquet-Langlands (n=2), is based on considering the Whittaker function. The second construction, which was proposed recently by Laumon following Drinfeld (n=2) and Deligne (n=1), is geometric: the automorphic function is obtained via Grothendieck's ``faisceaux-fonctions'' correspondence from a complex of sheaves on an algebraic stack. Our proof of their equivalence is based on a local result about the spherical Hecke algebra, which we prove for an arbitrary reductive group. We also discuss a geometric interpretation of this result.

alg-geom↗

arXiv1997

Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras I. The case of Virasoro algebra

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL(2). The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL(2), and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in q-alg/9505025. We also consider a discrete analogue of this Poisson algebra. In the second part (q-alg/9702016) the construction is generalized to the case of an arbitrary semisimple Lie algebra.

q-alg↗

arXiv1994

W_{1+\infty} and W(gl_N) with central charge N

We study representations of the central extension of the Lie algebra of differential operators on the circle, the W-infinity algebra. We obtain complete and specialized character formulas for a large class of representations, which we call primitive; these include all quasi-finite irreducible unitary representations. We show that any primitive representation with central charge N has a canonical structure of an irreducible representation of the W-algebra W(gl_N) with the same central charge and that all irreducible representations of W(gl_N) with central charge N arise in this way. We also establish a duality between "integral" modules of W(gl_N) and finite-dimensional irreducible modules of gl_N, and conjecture their fusion rules.

hep-th↗

arXiv1993

Two dimensional black-hole as a topological coset model of c=1 string theory

We show that a special superconformal coset (with $\hat c =3$) is equivalent to $c=1$ matter coupled to two dimensional gravity. This identification allows a direct computation of the correlation functions of the $c=1$ non-critical string to all genus, and at nonzero cosmological constant, directly from the continuum approach. The results agree with those of the matrix model. Moreover we connect our coset with a twisted version of a Euclidean two dimensional black hole, in which the ghost and matter systems are mixed.

hep-th↗