Finite-g Strings

TL;DR

This paper advances understanding of string theory on curved backgrounds by applying integrability and algebro-geometric methods to derive finite-gap solutions and semiclassical quantization on RxS^3, serving as a model for AdS_5xS^5.

Contribution

It constructs finite-gap solutions for string theory on RxS^3 using integrability, and performs a semiclassical quantization, providing insights applicable to AdS_5xS^5.

Findings

Finite-gap solutions are obtained via algebraic curves.

Semiclassical quantization matches gauge theory expectations.

The algebraic curve discretizes naturally in the quantum regime.

Abstract

In view of one day proving the AdS/CFT correspondence, a deeper understanding of string theory on certain curved backgrounds such as AdS_5xS^5 is required. In this dissertation we make a step in this direction by focusing on RxS^3. It was discovered in recent years that string theory on AdS_5xS^5 admits a Lax formulation. However, the complete statement of integrability requires not only the existence of a Lax formulation, but also that the resulting integrals of motion are in pairwise involution. This idea is central to the first part of this thesis. Exploiting this integrability we apply algebro-geometric methods to string theory on RxS^3 and obtain the general finite-gap solution. The construction is based on an invariant algebraic curve previously found in the AdS_5xS^5 case. However, encoding the dynamics of the solution requires specification of additional marked points. By restricting the symplectic structure of the string to this algebro-geometric data we derive the action-angle variables of the system. We then perform a first-principle semiclassical quantisation of string theory on RxS^3 as a toy model for strings on AdS_5xS^5. The result is exactly what one expects from the dual gauge theory perspective, namely the underlying algebraic curve discretises in a natural way. We also derive a general formula for the fluctuation energies around the generic finite-gap solution. The ideas used can be generalised to AdS_5xS^5.