Discrete Quantum Field Theories and the Intersection Form

TL;DR

This paper introduces a novel four-dimensional lattice model based on the mod-$p$ intersection form, demonstrating subdivision invariance and computing its partition function on complex manifolds.

Contribution

It develops a new gauge theory model using the intersection form on second cohomology, establishing subdivision invariance and explicit partition function calculations.

Findings

Partition function computed for P^{2} shows non-trivial topological features.

Model achieves subdivision invariance with quantized coupling constants.

Uses mod-orm intersection form to define gauge-invariant Boltzmann weights.

Abstract

It is shown that the standard mod-$p$ valued intersection form can be used to define Boltzmann weights of subdivision invariant lattice models with gauge group $Z_{p}$. In particular, we discuss a four dimensional model which is based upon the assignment of field variables to the $2$-simplices of the simplicial complex. The action is taken to be the intersection form defined on the second cohomology group of the complex, with coefficients in $Z_{p}$. Subdivision invariance of the theory follows when the coupling constant is quantized and the field configurations are restricted to those satisfying a mod-$p$ flatness condition. We present an explicit computation of the partition function for the manifold $\pm CP^{2}$, demonstrating non-triviality.