Communications in Number Theory and Physics

Volume 5 (2011)

Number 1

Archimedean $L$-factors and topological field theories I

Pages: 57 – 100



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)


We propose a functional integral representation for Archimedean $L$-factors given by products of $\Gamma$-functions.The corresponding\break functional integral arises in the description of type-A equivariant topological linear sigma modelon a disk. The functional integral representation provides in particular an interpretation of\break the $\Gamma$-function asan equivariant symplectic volume of an infinite-\break dimensional space of holomorphic maps of the disk to $\IC$. Thisshould be considered as a mirror dual to the classical Euler integral representation of the $\Gamma$-function. We givean analogous functional integral representation of $q$-deformed $\Gamma$-functions using a three-dimensionalequivariant topological linear sigma model on a handlebody. In general, upon proper ultra violent completion, thetopological sigma model considered on a particular class of three-dimensional spaces with a compact Kähler targetspace provides a quantum field theory description of a $K$-theory version of Gromov–Witten invariants.

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