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

TL;DR

The paper presents a canonical quantization of a chiral two-form in six dimensions by treating it as a subsector of the non-chiral two-form on a spacetime Y = ${\mathbb R} \times M$. It shows that the full Hilbert space ${\cal V}$ is a subspace of ${\cal V}_+ \otimes {\cal V}_-$, with a precise decomposition of observables $G$, $H$, ${\cal H}$, and the Wilson surface $W(\Sigma)$ into chiral and anti-chiral contributions, both in the harmonic and non-harmonic sectors. The harmonic sector yields quantized spectra described by $H^0=2\pi m^0$ and $*G^0=4\pi n^0$, with a detailed mapping to chiral coordinates $(k_+,k_-,a_+)$ and a factorized Wilson surface $W^0(\Sigma)=W^0_+(\Sigma)W^0_-(\Sigma)$. The non-harmonic sector is quantized via Laplacian eigenmodes, introducing oscillator algebras for the chiral and anti-chiral sectors and revealing a nontrivial linking-number phase in the Wilson surface algebra, which necessitates regularization to control divergences. Overall, the work provides a concrete framework for understanding holomorphic factorization and the coupling structure of chiral two-forms, with implications for the (2,0) theories and their connections to M-theory and string theory.

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.