Quantum Thetas on Noncommutative T^d with General Embeddings

TL;DR

This paper extends the construction of quantum theta functions on noncommutative tori to more general embeddings, revealing differences in holomorphicity and translation properties between vector space and lattice components.

Contribution

It generalizes Manin's quantum theta construction to include embeddings involving vector spaces, lattices, and tori, highlighting new properties of quantum translations.

Findings

Holomorphic theta vectors exist only in the vector space part of the embedding.

Quantum translations from lattice parts are non-additive.

Translations from vector space parts are additive.

Abstract

In this paper we construct quantum theta functions over noncommutative T^d with general embeddings. Manin has constructed quantum theta functions from the lattice embedding into vector space x finite group. We extend Manin's construction of quantum thetas to the case of general embedding of vector space x lattice x torus. It turns out that only for the vector space part of the embedding there exists the holomorphic theta vector, while for the lattice part there does not. Furthermore, the so-called quantum translations from embedding into the lattice part become non-additive, while those from the vector space part are additive.