Multi-Dimensional Sigma-Functions

TL;DR

This paper reviews the development of multi-dimensional sigma-functions, extending classical Abelian function theory to higher genera, with applications in mathematical physics and integrable systems, highlighting differences from theta-functions and recent advances since 1997.

Contribution

It provides an updated review of sigma-function theory beyond hyperelliptic curves, including new results and extensions to arbitrary algebraic curves since 1997.

Findings

Extended sigma-function theory to non-hyperelliptic curves.

Clarified differences between theta and sigma functions.

Summarized recent developments in the field since 1997.

Abstract

In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by K.Weierstrass and F.Klein was aimed to extend to higher genera Weierstrass theory of elliptic functions based on the Weierstrass $\sigma$-functions. Our development was motivated by the recent achievements of mathematical physics and theory of integrable systems that were based of the results of classical theory of multi-dimensional theta functions. Both theta and sigma-functions are integer and quasi-periodic functions, but worth to remark the fundamental difference between them. While theta-function are defined in the terms of the Riemann period matrix, the sigma-function can be constructed by coefficients of polynomial defining the curve. Note that the relation between periods and coefficients of polynomials defining the curve is transcendental. Since the publication of our 1997-review a lot of new results in this area appeared (see below the list of Recent References), that promoted us to submit this draft to ArXiv without waiting publication a well-prepared book. We complemented the review by the list of articles that were published after 1997 year to develop the theory of $\sigma$-functions presented here. Although the main body of this review is devoted to hyperelliptic functions the method can be extended to an arbitrary algebraic curve and new material that we added in the cases when the opposite is not stated does not suppose hyperellipticity of the curve considered.