Notes on the Quantization of Tropological Yang Mills Theory

Abstract

We continue our investigation into anisotropic topological field theories which arise from a tropical limit of conventional isotropic topological field theories. We analyze both the TBF theory and the tropical analogue of 2D topological Yang-Mills theory (TrYM) through a direct path integral calculation which probes a deformed analytic torsion and also through canonical quantization. The explicit construction of the Hilbert space of TrYM theory demonstrates that the TrYM theory provides an example of a solvable field theory where anisotropy properties and topological invariance can simultaneously hold. We show that the partition function has an asymptotic limit, which verifies that the dimension of the moduli space of tropicalized flat connection on a Riemann surface of genus $g>1$ is precisely given by $(g-1) \operatorname{rank}(\mathfrak{g})$. We show that the interpretation of this result is that the random matrix model associated to the U$(N)$ TrYM is in fact a novel random matrix theory whose integration space is still the same space of hermitian matrices (similar to a GUE) however the Dyson index matches that of a GOE consistent with the usual intuition that tropicalization reduces down complex objects to their real counterparts.