Deformation Quantization and Homological Reduction of a Lattice Gauge Model

Abstract

For a compact Lie group G we consider a lattice gauge model given by the G-Hamiltonian system which consists of the cotangent bundle of a power of G with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-Civita connection of the Killing metric on G. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for G = SU(2) the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the-in general singular-symplectically reduced phase space of the corresponding lattice gauge model carries a star product.

On a connection used in deformation quantization

“…Let us refer to this connection as the BNW lift of ∇. Recently, the BNW lift was used in the context of homological reduction, see [9].…”