Mirror symmetry, Langlands duality, and the Hitchin system

Hausel, Tamás; Thaddeus, Michael

doi:10.1007/s00222-003-0286-7

Cited by 217 publications

(350 citation statements)

References 44 publications

Supporting

Mentioning

342

Contrasting

Order By: Relevance

“…[25]). In particular construction of a mirror dual description of type-A topological sigma models associated with flag spaces G/B in terms of eigenfunctions of the quantum Toda chains associated with the dual Lie groups G ∨ [21,22].…”

Section: Introductionmentioning

confidence: 99%

Parabolic Whittaker functions and topological field theories I

Gerasimov

Лебедев

Oblezin

2011

Communications in Number Theory and Physics

View full text Add to dashboard Cite

First, we define a generalization of the standard quantum Toda chain inspired by a construction of quantum cohomology of partial flags spaces GL +1 /P , P a parabolic subgroup. Common eigenfunctions of the parabolic quantum Toda chains are generalized Whittaker functions given by matrix elements of infinite-dimensional representations of gl +1 . For maximal parabolic subgroups (i.e., for P such that GL +1 /P = P ) we construct two different representations of the corresponding parabolic Whittaker functions as correlation functions in topological quantum field theories on a two-dimensional disk. In one case the parabolic Whittaker function is given by a correlation function in a type-A equivariant topological sigma model with the target space P . In the other case, the same Whittaker function appears as a correlation function in a type-B equivariant topological Landau-Ginzburg model related with the type-A model by mirror symmetry. This note is a continuation of our project of establishing a relation between twodimensional topological field theories (and more generally topological string theories) and Archimedean (∞-adic) geometry. From this perspective the existence of two, mirror dual, topological field theory representations of the parabolic Whittaker functions provide a quantum field theory realization of the local Archimedean Langlands duality for Whittaker functions. The established relation between the Archimedean Langlands duality and mirror symmetry in two-dimensional topological quantum field theories should be considered as a main result of this note. IntroductionIn [17,18] we propose two-dimensional topological field theories as a proper framework for a description of the Archimedean completion of arithmetic schemes (∞-adic geometry according to [28]). In particular, we give a representation of local Archimedean L-factors (we include local epsilon-factor in the definition of the L-factors) in terms of two-dimensional topological field theories. It is well-known that local L-factors allow two types of 135 136 Anton Gerasimov, Dimitri Lebedev and Sergey Oblezin constructions -"arithmetic" construction based on representation theory of the Weil-Deligne group of the local field and "automorphic" construction relying on representation theory of reductive groups over local field (see, e.g., [1,6,7]). The equivalence of these constructions for various types of L-factors is a subject of the local Langlands duality. In an interpretation suggested in [17,18] the "arithmetic" construction of local Archimedean L-factors is naturally identified with a type-A topological field theory description [17] in terms of equivariant volumes of spaces of holomorphic maps of a disk into complex vector spaces. The "automorphic" construction of the same local L-factors is realized using a type-B topological field theory via periods of holomorphic forms [18]. The Archimedean Langlands duality between these two constructions of the local Archimedean L-factors appears as a mirror duality between underlying type-A and type-B topolo...

show abstract

Section: Introductionmentioning

confidence: 99%

Parabolic Whittaker functions and topological field theories I

Gerasimov

Лебедев

Oblezin

2011

Communications in Number Theory and Physics

View full text Add to dashboard Cite

show abstract

“…This in fact was first pointed out in Hausel & Thaddeus (2002), and used in Kapustin & Witten (2007) as a key ingredient in understanding the geometric Langlands correspondence.…”

Section: Mirror Symmetry and Hitchin's Equationsmentioning

confidence: 78%

“…In the present context, there is a very beautiful description of the dual fibration: it is, as first shown in Hausel & Thaddeus (2002), simply the Hitchin fibration of the dual group. Thus one considers M H ( L G, C), the moduli space of solutions of Hitchin's equation for the dual group L G. It turns out that the bases of the Hitchin fibrations for G and L G can be identified in a natural way.…”

Section: The Hitchin Fibrationmentioning

confidence: 85%

“…With a little bit of hindsight (the question was first addressed in Hausel & Thaddeus (2002), following earlier work by Bershadsky et al (1995), and Harvey et al (1995)), we may ask whether the B-model of Y( L G, C) may be mirror to the A-model of Y(G, C). Even once this question is asked, it is difficult to answer it without some additional structure.…”

Section: Mirror Symmetry and Hitchin's Equationsmentioning

confidence: 99%

“…There is, however, another way to look at the Hitchin fibration, as first described in Hausel & Thaddeus (2002). Let us go back to the A-model defined with the real symplectic structure ω.…”

Section: The Hitchin Fibrationmentioning

confidence: 99%

See 2 more Smart Citations

Mirror Symmetry, Hitchin's Equations, and Langlands Duality

Witten¹

2010

The Many Facets of Geometry

View full text Add to dashboard Cite

Abstract. Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold C. But understanding these statements is extremely difficult without picking a complex structure on C and using Hitchin's equations. We sketch the essential statements both for the "unramified" case that C is a compact oriented two-manifold without boundary, and the "ramified" case that one allows punctures. We also give a few indications of why a more precise description requires a starting point in four-dimensional gauge theory.

show abstract