Tropical Limits of Dirac Operators

Abstract

We explore the tropical analog of spinors by representing tropical geometries as foliated Riemann surfaces endowed with degenerate complex structures. We investigate tropical limits of the Laplace-Beltrami operator and explicitly construct its square root, which defines a tropical Dirac operator. We find that the tropical Clifford algebra is classified as a degenerate Clifford algebra with nilpotent generators. The nilpotent generator allows us to work with a new kind of representation that allows for Grassmann odd numbers, effectively supersymmetrizing the tropical spin bundle. We show through Dirac-Bergmann's quantization procedure, that the corresponding tropicalized quantum field theories enjoy a purely fermionic topological symmetry which can be expected to give a new class of path integral localization that we call tropical localization similar to the alternative localization method recently constructed by Choi and Takhtajan. We also discuss how the tropical Dirac operator, when twisted by gauge fields, obeys a tropical version of the Lichnerowicz identity, thereby demonstrating how some elements of Yang-Mills curvature should arise in the tropical limit.