2010/02/12 by Anton Gerasimov, Gerasimov, Anton, Dimitri Lebedev +3
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th #math.AG
paper · pdf · doi:10.48550/arxiv.1002.2622
Section 1 is extended and Appendices are added, 23 pages
arxiv created 2010/06/13 · arxiv updated 2010/06/15
First, we define a generalization of the standard quantum Toda chain inspired by a construction of quantum cohomology of partial flags spaces GL(ℓ+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(ℓ+1). For maximal parabolic subgroups (i.e. for P such that GL(ℓ+1)/P=ℙℓ) 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 ℙℓ. 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 (∞-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.