Infinite dimensional groups and Riemann surface field theories
We show how to obtain positive energy representations of the group Script G sign of smooth maps from a union of circles to U(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case where N = 1. Then our main result sh...
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| Format: | Journal article |
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1996
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| Summary: | We show how to obtain positive energy representations of the group Script G sign of smooth maps from a union of circles to U(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case where N = 1. Then our main result shows that Mackey induction may be applied and yields representations of the connected component of the identity of Script G sign 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 in Script G sign 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 of Script G sign (which have a simple interpretation in terms of function theory on the Riemann 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 of Script G sign. |
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