# Parabolic Whittaker Functions and Topological Field Theories I - High Energy Physics - Theory

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Abstract: First, we define a generalization of the standard quantum Toda chain inspiredby a construction of quantum cohomology of partial flags spaces GL\ell+1-P, Pa parabolic subgroup. Common eigenfunctions of the parabolic quantum Todachains are generalized Whittaker functions given by matrix elements ofinfinite-dimensional representations of gl\ell+1. For maximal parabolicsubgroups i.e. for P such that GL\ell+1-P=\mathbb{P}^{\ell} we constructtwo different representations of the corresponding parabolic Whittakerfunctions as correlation functions in topological quantum field theories on atwo-dimensional disk. In one case the parabolic Whittaker function is given bya correlation function in a type A equivariant topological sigma model with thetarget space \mathbb{P}^{\ell}. In the other case the same Whittaker functionappears as a correlation function in a type B equivariant topologicalLandau-Ginzburg model related with the type A model by mirror symmetry. Thisnote is a continuation of our project of establishing a relation betweentwo-dimensional topological field theories and more generally topologicalstring theories and Archimedean \infty-adic geometry. From this perspectivethe existence of two, mirror dual, topological field theory representations ofthe parabolic Whittaker functions provide a quantum field theory realization ofthe local Archimedean Langlands duality for Whittaker functions. Theestablished relation between the Archimedean Langlands duality and mirrorsymmetry in two-dimensional topological quantum field theories should beconsidered as a main result of this note.

Author: ** Anton Gerasimov, Dimitri Lebedev, Sergey Oblezin**

Source: https://arxiv.org/