Review of AdS/CFT Integrability, Chapter II.4: The Spectral Curve

TL;DR

This paper reviews the spectral curve approach for classical strings in AdS5xS5, highlighting how integrability leads to a powerful characterization of solutions and their semi-classical quantization.

Contribution

It provides a comprehensive review of the spectral curve construction and semi-classical quantization methods for classical strings in AdS5xS5, emphasizing the role of monodromies and quasi-momenta.

Findings

Spectral curve encodes classical string solutions in AdS5xS5.

Semi-classical quantization can be performed using quasi-momenta.

Circular string solution exemplifies the framework.

Abstract

We review the spectral curve for the classical string in AdS5xS5. Classical integrability of the AdS5xS5 string implies the existence of a flat connection, whose monodromies generate an infinite set of conserved charges. The spectral curve is constructed out of the quasi-momenta, which are eigenvalues of the monodromy matrix, and each finite-gap classical solution can be characterized in terms of such a curve. This provides a concise and powerful description of the classical solution space. In addition, semi-classical quantization of the string can be performed in terms of the quasi-momenta. We review the general frame-work of the semi-classical quantization in this context and exemplify it with the circular string solution which is supported on RxS3 in AdS5xS5.