Communications in Number Theory and Physics

Volume 5 (2011)

Number 1

Parabolic Whittaker functions and topological field theories I

Pages: 135 – 201



Anton Gerasimov (Institute for Theoretical and Experimental Physics, Moscow, Russia)

Dimitri Lebedev (School of Mathematics, Trinity College, Dublin, Ireland)

Sergey Oblezin (Hamilton Mathematics Institute, Trinity College, Dublin, Ireland)


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.

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