Special Quantum Field Theories In Eight And Other Dimensions

TL;DR

This paper constructs nearly topological quantum field theories in various dimensions, focusing on 8D theories related to special holonomy manifolds, and explores their connections to supersymmetric and gauge theories.

Contribution

It introduces new 8D quantum field theories based on different holonomy groups and relates them to known theories like Donaldson invariants and Seiberg-Witten equations.

Findings

Constructed 8D theories for Calabi-Yau and Joyce manifolds.

Linked 8D theories to supersymmetric Yang-Mills and Seiberg-Witten equations.

Presented theories coupling gauge fields to Chern classes in 8D.

Abstract

We build nearly topological quantum field theories in various dimensions. We give special attention to the case of 8 dimensions for which we first consider theories depending only on Yang-Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in SO(8), namely SU(4) and Spin(7). The choice of SU(4) gives a quantum field theory for a Calabi-Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of Spin(7) defines another eight dimensional theory for a Joyce manifold which could be of relevance in M- and F-theories. Relations to the eight dimensional supersymmetric Yang-Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg-Witten equations. The latter are thus related to pure Yang-Mills self-duality equations in 8 dimensions as well as to the N=1, D=10 super Yang-Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions.