KdV Equation for Theta Functions on Non-commutative Tori

TL;DR

This paper explores whether quantum theta functions on non-commutative tori satisfy the KdV equation, revealing promising results that suggest potential for defining integrable hierarchies in quantum geometry.

Contribution

It demonstrates that quantum theta functions on the 2-torus satisfy the KdV equation using a naive differentiation approach, opening avenues for quantum integrability research.

Findings

Quantum theta functions on 2-torus satisfy KdV equation.

Successful application of differentiation on quantum tori.

Potential for defining quantum integrable hierarchies.

Abstract

In the fields of non-commutative geometry and string theory, quantum tori appear in different mathematical and physical contexts. Therefore, quantized theta functions defined on quantum tori are also studied (Yu. I. Manin, A. Schwartz; note that a comparison between the two definitions of quantum theta is still an open problem). One important application of classical theta functions is in soliton theory. Certain soliton equations, including the KdV equation, have algebro-geometric solutions that are given by theta functions (we refer to F. Gesztesy and H. Holden), and as such belong to an "integrable hierarchy." While quantized integrability is a very active and complicated subject, in this work we take a different, naive approach. We conduct an experiment: using a definition of differentiation on quantum tori (M. Rieffel), we ask whether the quantum theta function satisfies non-linear PDE. The experiment is successful on the 2-torus and for the KdV equation. This opens the way to future investigations, such as the quest for a compatible hierarchy satisfied by quantum theta, and a consistent definition of complete integrability.