Infinite dimensional groups and Riemann surface field theories

Abstract

We show how to obtain positive energy representations of the groupG of smooth maps from a union of circles toU(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case whereN=1. Then our main result shows that Mackey induction may be applied and yields representations of the connected component of the identity ofG which have the form of a Fock representation of an infinite dimensional Heisenberg group tensored with a finite dimensional representation of a subgroup isomorphic to the first cohomology group of the surface obtained by capping the boundary circles with discs. We give geometric sufficient conditions for the correlation functions to be positive definite and derive explicit formulae for them and for the vacuum (or cyclic) vector. (This gives a geometric construction of correlation functions which had been obtained earlier using tau functions.) By choosing particular functions inG with non-zero winding numbers on the boundary we obtain analogues of vertex operators described by Segal in the genus zero case. These special elements ofG (which have a simple interpretation in terms of function theory on theRiemann surface) approximate fermion (or Clifford algebra) operators. They enable a rigorous derivation of a form of boson-fermion correspondence in the sense that we construct generators of a Clifford algebra from the unitaries representing these elements ofG.