Parabolic Whittaker Functions and Topological Field Theories I

TL;DR

This paper introduces a generalization of quantum Toda chains linked to quantum cohomology, constructs two topological field theory representations of parabolic Whittaker functions, and relates these to Archimedean Langlands duality and mirror symmetry.

Contribution

It establishes a novel connection between parabolic Whittaker functions, topological quantum field theories, and Archimedean Langlands duality, providing dual representations via mirror symmetry.

Findings

Two topological field theory representations of parabolic Whittaker functions are constructed.

The relation between Langlands duality and mirror symmetry in 2D topological theories is demonstrated.

The work links quantum integrable systems with geometric and field-theoretic dualities.

Abstract

First, we define a generalization of the standard quantum Toda chain inspired by a construction of quantum cohomology of partial flags spaces GL(\ell+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(\ell+1). For maximal parabolic subgroups (i.e. for P such that GL(\ell+1)/P=\mathbb{P}^{\ell}) 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 \mathbb{P}^{\ell}. 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 two-dimensional topological field theories (and more generally topological string theories) and Archimedean (\infty-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.