Holomorphic factorization of correlation functions in (4k+2)-dimensional (2k)-form gauge theory

TL;DR

The paper establishes a holomorphic factorization for correlation functions of a free $(2k)$-form gauge field on a $(4k+2)$-dimensional manifold by splitting fluctuations into harmonic (classical) and non-harmonic (quantum) sectors. Using a complex structure on the space of $(2k+1)$-forms, the non-chiral correlator is shown to decompose into holomorphic and anti-holomorphic blocks, with holomorphicity corresponding to chiral $(2k)$-form correlators after Wick rotation. The classical sector is encoded by a lattice sum that reorganizes into Jacobi theta-functions with characteristics, while the quantum sector yields determinant factors; both parts factorize, up to non-holomorphic anomalous terms, into holomorphic and anti-holomorphic components. After discarding these anomalies (and under suitable regularizations), the full correlator becomes a finite sum of products of holomorphic and anti-holomorphic factors, generalizing chiral-form factorization and connecting to the physics of self-dual fields in higher dimensions.

Abstract

We consider a free (2 k)-form gauge-field on a Euclidean (4 k + 2)-manifold. The parameters needed to specify the action and the gauge-invariant observables take their values in spaces with natural complex structures. We show that the correlation functions can be written as a finite sum of terms, each of which is a product of a holomorphic and an anti-holomorphic factor. The holomorphic factors are naturally interpreted as correlation functions for a chiral (2 k)-form, i.e. a (2 k)-form with a self-dual (2 k + 1)-form field strength, after Wick rotation to a Minkowski signature.