The quantum Hilbert space of a chiral two-form in d = 5 + 1 dimensions

TL;DR

This paper analyzes the quantum Hilbert space structure of a two-form gauge field in six-dimensional spacetime, revealing its decomposition into chiral and anti-chiral sectors on a compact five-manifold.

Contribution

It demonstrates that the non-chiral two-form Hilbert space is a subspace of a tensor product of chiral and anti-chiral spaces, clarifying their relationship in quantum theory.

Findings

Hilbert space decomposes into chiral and anti-chiral parts

Observable operators split into chiral and anti-chiral contributions

Explicit structure of electric, magnetic, and holonomy operators

Abstract

We consider the quantum theory of a two-form gauge field on a space-time which is a direct product of time and a spatial manifold, taken to be a compact five-manifold with no torsion in its cohomology. We show that the Hilbert space of this non-chiral theory is a certain subspace of a tensor product of two spaces, that are naturally interpreted as the Hilbert spaces of a chiral and anti-chiral two-form theory respectively. We also study the observable operators in the non-chiral theory that correspond to the electric and magnetic field strengths, the Hamiltonian, and the exponentiated holonomy of the gauge-field around a spatial two-cycle. All these operators can be decomposed into contributions pertaining to the chiral and anti-chiral sectors of the theory.