Quantum Spectral Curve for the η-deformed AdS_5 × S^5 superstring

Abstract

Being able to solve an interacting quantum field theory exactly is by itself an exciting prospect, as having full control allows for the precise study of phenomena described by the theory. In the context of the AdS/CFT correspondence, which hypothesises a duality between certain string and gauge theories, planar N = 4 super Yang-Mills theory shows signs of great tractability due to integrability. The spectral problem of finding the scaling dimensions of local operators or equivalently of the string energies of string states of its AdS/CFT-dual superstring theory on the AdS_5 × S^5 background turned out to be very tractable. After over a decade of research it was found that the spectral problem can be solved very efficiently through a set of functional equations known as the quantum spectral curve. Although it allowed for a detailed study of the aforementioned theory, many open questions regarding the wider applicability and underlying mechanisms of the quantum spectral curve exist and are worth studying. In this thesis we discuss how one can derive the quantum spectral curve for the η- deformed AdS_5 × S^5 superstring, an integrable deformation of the AdS_5 × S^5 superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the AdS_5 × S^5 superstring, like the 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 η-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 Pμ 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 η-deformed string and its quantum spectral curve interpolate between the AdS_5 × S^5 superstring and a superstring on “mirror” AdS_5 × S^5 , reflecting a more general relationship between the spectral and thermodynamic data of the η-deformed string.