Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$ superstring

TL;DR

This thesis derives the quantum spectral curve for the $\, ext{eta}$-deformed AdS$_5 imes S^5$ superstring, revealing its integrable structure and interpolation between different string models, extending the understanding of quantum integrability in deformed AdS/CFT systems.

Contribution

It presents the first derivation of the quantum spectral curve for the $\, ext{eta}$-deformed superstring, connecting it to the undeformed case and exploring its spectral properties.

Findings

Constructed the quantum spectral curve from thermodynamic Bethe ansatz equations.

Reformulated the spectral problem as an analytic $Y$-system and $T$-system.

Showed the interpolation between the original and mirror superstring models.

Abstract

In this thesis we discuss how one can derive the quantum spectral curve for the $\eta$-deformed AdS$_5 \times S^5$ superstring, an integrable deformation of the AdS$_5 \times $S$^5$ superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the AdS$_5 \times $S$^5$ superstring, like the Heisenberg xxz spin chain is a trigonometric version of the xxx spin chain. Our derivation starts from the ground-state thermodynamic Bethe ansatz equations and discusses the construction of both the undeformed and the $\eta$-deformed quantum spectral curve. We reformulate it first as an analytic $Y$-system, and map this to an analytic $T$-system which upon suitable gauge fixing leads to a $\mathbf{P}\mu$ system, the quantum spectral curve. We then discuss constraints on the asymptotics of this system to single out particular excited states. At the spectral level the $\eta$-deformed string and its quantum spectral curve interpolate between the AdS$_5 \times $S$^5$ superstring and a superstring on "mirror" AdS$_5 \times $S$^5$, reflecting a more general relationship between the spectral and thermodynamic data of the $\eta$-deformed string. The thesis is set up such that it simultaneously reviews the development of the undeformed AdS$_5 \times $S$^5$ quantum spectral curve.