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# Communications in Number Theory and Physics

## Volume 5 (2011)

### Number 1

### Parabolic Whittaker functions and topological field theories I

Pages: 135 – 201

DOI: http://dx.doi.org/10.4310/CNTP.2011.v5.n1.a4

#### Authors

#### Abstract

First, we define a generalization of the standard quantum Toda chaininspired by a construction ofquantum cohomology of partial flags spaces $GL_{\ell+1}/P$, $P$a parabolic subgroup. Common eigenfunctions of theparabolic quantum Toda chains are generalized Whittaker functionsgiven by matrix elements of infinite-dimen\-sional representations of$\mathfrak{gl}_{\ell+1}$. For maximal parabolic subgroups (i.e., for$P$ such that $GL_{\ell+1}/P=\IP^{\ell}$) we construct two different representationsof the corresponding parabolic Whittaker functions as correlation functionsin topological quantum field theories on a two-dimensional disk. Inone case the parabolic Whittaker function is given by a correlation functionin a type-$A$ equivariant topological sigma model with the targetspace $\IP^{\ell}$. In the other case, the same Whittaker functionappears as a correlation function in a type-$B$ equivarianttopological Landau–Ginzburg model related with the type-$A$ modelby mirror symmetry. This note is a continuation of ourproject of establishing a relation between two-dimensionaltopological field theories (and more generally topological string theories)and Archimedean ($\infty$-adic) geometry.From this perspective the existence oftwo, mirror dual, topological field theoryrepresentations of the parabolic Whittaker functionsprovide a quantum field theory realization of thelocal Archimedean Langlands duality for Whittaker functions.The established relation between the Archimedean Langlands dualityand mirror symmetry in two-dimensional topological quantum fieldtheories should be considered as a main result of this note.