Tropical Limits of Dirac Operators

TL;DR

The paper develops a tropical analogue of spin geometry by tropicalizing complex geometry through Maslov dequantization, representing tropical spaces as foliated Riemann surfaces with nilpotent Jordan structures. It constructs a tropical Laplacian $\Delta$ with a square root given by a degenerate Clifford algebra $Cl(0,2,1)$, yielding a tropical Dirac operator $\slashed{D}$ whose square reproduces the tropical Laplacian, and introduces a Grassmann-odd extension (via a Grassmann parameter $\epsilon$) to form a supersymmetric-complexified spinor representation. Twisted Dirac operators satisfy a tropical Lichnerowicz identity, revealing a tropical Yang-Mills curvature, and a Dirac-Bergmann quantization shows a purely fermionic topological symmetry that motivates tropical (and arctic) localization for path integrals, including a tropicalization-based computation of the Schwinger effect. The work proposes a framework for tropical/topological field theories on foliated geometries and suggests rich future directions, such as tropical supersymmetry, arctic dualities, and spectral invariants like the eta invariant in noncompact/tropical limits.

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.

Figures (1)

• Figure 1: The contour used for $\Gamma+$. Notice that as $\hbar$ goes to zero the horizontal real part no longer contributes to the integral, however, the integration cycle itself fully extends to the origin as expected. There is a similar contour for the backwards branch whose contribution to the final sum is equal.