Nilmanifolds and their associated non local fields

TL;DR

This paper constructs modules of affine Kac-Moody vertex algebras on six-dimensional nilmanifolds, explores associated logarithmic fields with singularities, and connects these mathematical structures to physical theories, especially in specific fibration cases.

Contribution

It introduces a novel module construction for affine Kac-Moody vertex algebras on nilmanifolds and analyzes their logarithmic fields and singularities, with a focus on physical motivations.

Findings

Logarithmic fields exhibit tri-logarithm singularities when certain parameters are non-zero.

Construction of modules on nilmanifolds extends vertex algebra theory to new geometric contexts.

Physical motivation links the mathematical structures to non-local fields in theoretical physics.

Abstract

For six dimensional nilmanifolds we build a module $\mathcal{H}$ of an affine Kac Moody vertex algebras. Then, we associate some logarithmic fields for the module $\mathcal{H}$ and we study their singularities. We also presented a physics motivation behind this construction. We study a particular case, we show that when the nilmanifold $N$ is a $k$ degree $S^1$--fibration over the two torus and a choice of $l \in \mathbb{Z} \simeq H^3(N, \mathbb{Z})$ the fields associated to the space $\mathcal{H}$ have tri-logarithm singularities whenever $kl \neq 0$.