Semi-stable vector bundles on elliptic curves and the associative Yang-Baxter equation

TL;DR

This paper constructs and explicitly computes solutions to the associative Yang-Baxter equation with spectral parameters, linking them to elliptic curves and vector bundles, advancing understanding in mathematical physics and algebraic geometry.

Contribution

It introduces a method to generate solutions of the AYBE from elliptic curves and invertible matrices, providing explicit examples and a new perspective on the equation's structure.

Findings

Solutions depend holomorphically on spectral parameters and matrices.

Explicit solutions are computed for specific cases.

Links between vector bundles on elliptic curves and AYBE solutions are established.

Abstract

In this paper we study unitary solutions of the associative Yang--Baxter equation (AYBE) with spectral parameters. We show that to each point $\tau$ from the upper half-plane and an invertible $n \times n$ matrix $B$ with complex coefficients one can attach a solution of AYBE with values in $Mat_{n \times n}(\CC) \otimes Mat_{n \times n}(\CC)$, depending holomorphically on $\tau$ and $B$. Moreover, we compute some of these solutions explicitly.