Exact computation of one-loop correction to energy of spinning folded string in AdS_5 x S^5

TL;DR

This paper computes the exact one-loop quantum correction to the energy of a folded spinning string in AdS_5 x S^5, using elliptic functions and Lamé operators, providing expansions for small and large spins.

Contribution

It introduces a method to exactly evaluate fluctuation determinants for the folded spinning string by transforming them into Lamé form, enabling explicit energy corrections.

Findings

Exact one-loop energy correction derived

Fluctuation operators expressed as Lamé equations

Small and large spin expansions obtained

Abstract

We consider the 1-loop correction to the energy of folded spinning string solution in the AdS_3 part of AdS_5 x S^5. The classical string solution is expressed in terms of elliptic functions so an explicit computation of the corresponding fluctuation determinants for generic values of the spin appears to be a non-trivial problem. We show how it can be solved exactly by using the static gauge expression for the string partition function (which we demonstrate to be equivalent to the conformal gauge one) and observing that all the corresponding second order fluctuation operators can be put into the standard (single-gap) Lam\'e form. We systematically derive the small spin and large spin expansions of the resulting expression for the string energy and comment on some of their applications.