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Journal Article

Deformation quantization and homological reduction of a lattice gauge model


Pflaum,  M. J.
Max Planck Institute for Mathematics, Max Planck Society;

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Pflaum, M. J., Rudolph, G., & Schmidt, M. (2021). Deformation quantization and homological reduction of a lattice gauge model. Communications in Mathematical Physics, 382(2), 1061-1109. doi:10.1007/s00220-020-03896-w.

Cite as: https://hdl.handle.net/21.11116/0000-0007-D9E1-6
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 $G = \operatorname{SU} (2)$ 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.