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

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.

205 pages, lots of figures. PhD thesis, partially based on arXiv:1708.02894. Contains abstract in German. v2: updated references and fixed typos