Quantum corrections to finite-gap solutions for Yang-Mills-Nahm equations via zeta-function technique

TL;DR

This paper computes quantum corrections to Yang-Mills-Nahm models using zeta-function techniques, expressing results via elliptic and hyperelliptic integrals, and linking different mathematical representations for further analysis.

Contribution

It introduces a novel approach to evaluate one-loop quantum corrections in Nahm models using zeta functions and connects Hermit equation solutions with Riemann theta functions.

Findings

Explicit expression for quantum corrections via hyperelliptic integrals.

Linking Hermit equation solutions with Riemann theta functions.

Enhanced understanding of elliptic potential in Yang-Mills-Nahm models.

Abstract

One-dimensional Yang-Mills-Nahm models are considered from algebrogeometric points of view. A quasiclassical quantization of the models based on path integral and its zeta function representation in terms of a Green function diagonal for a heat equation with an elliptic potential is considered. The Green function diagonal and, hence, zeta function and its derivative are expressed via solutions of Hermit equation and, alternatively, by means of Its-Matveev formalism in terms of Riemann teta-functions. For the Nahm model, which field is represented via elliptic (lemniscate) integral by construction, one-loop quantum corrections to action are evaluated as the zeta function derivative in zero point in terms of a hyperelliptic integral. The alternative expression should help to link the representations and continue investigation of the Yang-Mills-Nahm models. Keywords: Nahm model, one-loop quantum corrections, zeta function, elliptic potential, hyperelliptic integral, Its-Matveev formula. MSC numbers: 81Q30, 35J10, 35K08, 81T13.