Quantum corrections to static solutions of Nahm equation and Sin-Gordon models via generalized zeta-function

TL;DR

This paper calculates quantum corrections to static solutions of Nahm, Ginzburg-Landau, and Sin-Gordon models using generalized zeta-functions, Green functions, and elliptic integrals, providing a unified approach for various nonlinear field theories.

Contribution

It introduces a novel method for evaluating quantum corrections via generalized zeta-functions and Green functions, applicable to a broad class of nonlinear Klein-Gordon-Fock models.

Findings

Quantum corrections to mass are expressed through elliptic integrals.

A universal construction of Green function diagonals for finite-gap potentials is developed.

An alternative approach using Baker-Akhiezer functions is proposed.

Abstract

One-dimensional Yang-Mills Equations are considered from a point of view of a class of nonlinear Klein-Gordon-Fock models. The case of self-dual Nahm equations and non-self-dual models are discussed. A quasiclassical quantization of the models is performed by means of generalized zeta-function and its representation in terms of a Green function diagonal for a heat equation with the correspondent potential. It is used to evaluate the functional integral and quantum corrections to mass in the quasiclassical approximation. Quantum corrections to a few periodic (and kink) solutions of the Nahm as a particular case of the Ginzburg-Landau (phi-in-quadro) and and Sin-Gordon models are evaluated in arbitrary dimensions. The Green function diagonal for heat equation with a finite-gap potential is constructed by universal description via solutions of Hermit equation. An alternative approach based on Baker-Akhiezer functions for KP equation is proposed . The generalized zeta-function and its derivative at zero point as the quantum corrections to mass is expressed in terms of elliptic integrals.