Spectral curves are transcendental

H W Braden

doi:10.1007/s11005-020-01349-y

Abstract

Some arithmetic properties of spectral curves are discussed: the spectral curve, for example, of a charge nge 2 Euclidean BPS monopole is not defined over overline{mathbb {Q}} if smooth.

Highlights

A fundamental ingredient of the modern theory of integrable systems is a curve, the spectral curve, and the function theory of this curve enables the solution of the system
We will show that for an integrable system of interest the associated spectral curves are not defined over Q, the transcendental of the title
The transcendence of periods is familiar: this paper provides a number of new examples where this is relevant

Summary

Introduction

A fundamental ingredient of the modern theory of integrable systems is a curve, the spectral curve, and the function theory of this curve enables (via the Baker–Akhiezer function, for example) the solution of the system. We will show that for an integrable system of interest the associated spectral curves are not defined over Q, the transcendental of the title. This aspect is a manifestation of why it is so difficult to construct specific examples of some systems. We will focus on a particular integrable system and remark on other examples. Neither a detailed knowledge of this particular physical system nor the arcane lore of integrable systems will be needed to understand this paper.

Page 2 of 9

The monopole spectral curve and Hitchin’s constraints

Page 4 of 9

Page 6 of 9

Proof and discussion of the theorem

Examples