Quantum Thetas on Noncommutative T^4 from Embeddings into Lattice

TL;DR

This paper extends the construction of quantum theta functions on noncommutative T^4 by embedding into a combination of vector space and lattice, revealing partial holomorphicity and non-additive quantum translations.

Contribution

It generalizes Manin's lattice embedding construction to include vector space x lattice embeddings, analyzing the properties of the resulting quantum theta functions.

Findings

Holomorphic theta vector exists only over the vector space part.

Quantum theta functions satisfy the required properties despite partial holomorphicity.

Quantum translations from lattice embedding are non-additive, unlike those from vector space embedding.

Abstract

In this paper we investigate the theta vector and quantum theta function over noncommutative T^4 from the embedding of R x Z^2. Manin has constructed the quantum theta functions from the lattice embedding into vector space (x finite group). We extend Manin's construction of the quantum theta function to the embedding of vector space x lattice case. We find that the holomorphic theta vector exists only over the vector space part of the embedding, and over the lattice part we can only impose the condition for Schwartz function. The quantum theta function built on this partial theta vector satisfies the requirement of the quantum theta function. However, two subsequent quantum translations from the embedding into the lattice part are non-additive, contrary to the additivity of those from the vector space part.