Nilmanifolds and their associated non-local fields

For a three dimensional nilmanifold together with a three form on it, we build a module $\mathcal{H}$ of an affine Kac–Moody vertex algebras. Then, we associate logarithmic fields to the module $\mathcal{H}$ and we study their singularities. We also present 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 trilogarithm singularities whenever $kl \neq 0$.

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Published 2 September 2020