Dualities and Flows: Sine-Dilaton Gravity, DSSYK and TTbar-Deformed Chiral Yang–Mills

This dissertation explores the interplay between two-dimensional Quantum Gravity, Gauge Theories, and T-Tbar deformation, with the aim of clarifying the emergence of dualities and flows between theories in low-dimensional systems. The first part focuses on the double-scaled SYK model (DSSYK) and its proposed gravitational dual, Sine-Dilaton gravity. Building on the parallel duality between JT-gravity and SYK, we review how the two models are classically connected through the particle-on-SUq(1,1) formulation and the Poisson Sigma Model framework. A notable tension arises between the linear entropy expected from a candidate bulk dual and the non-monotonic entropy characteristic of DSSYK. It is clear that, to resolve the mismatch within Sine-Dilaton gravity or its dual formulations, additional prescriptions should emerge. In particular, a suitable gauge fixing of the SUq(1,1) particle leads to a tractable theory, referred to as q-Liouville, in which a positivity constraint, motivated by the holographic dictionary, enforces the correct entropy behavior. Our main result establishes that the equivalence with DSSYK persists up to one-loop order, both at the level of the partition function and of the two-point correlators. In the second part, after summarizing the theoretical background, we turn to a comprehensive study of the T-Tbar deformation of two-dimensional chiral U(N) Yang-Mills theory on the torus. By combining the exact solvability of T-Tbar-deformed theories with the rich mathematical structure of two-dimensional Yang-Mills theory, we develop a systematic approach circumventing the lack of geometric moduli necessary for direct application of the Cardy flow equation. Our central innovation is a sequential deformation procedure that leverages the Okuyama-Sakai formulation of chiral Yang-Mills as a local bosonic theory with controlled anti-holomorphic dependence. We demonstrate that this construction yields a well-defined deformation preserving fundamental dualities and structural properties. The T-Tbar flow manifests as a square-root differential operator acting on the modular parameter, whose action we analyze through perturbative techniques. We derive explicit results for the deformed partition function, establish a first-order deformed version of the holomorphic anomaly equation, and prove the persistence of boson-fermion duality under deformation.