Probabilistic Schwarzian Field Theory

Abstract

In this thesis we use methods of stochastic analysis to study a quantum field theory called Schwarzian Theory, which emerged in the studies of two dimensional quantum gravity, black holes, and AdS/CFT correspondence. It is believed that Schwarzian Theory is a holographic dual of the Jackiw--Teitelboim gravity in the two dimensional disc. It is also expected to arise in the low energy limit of the Sachdev--Ye--Kitaev random matrix model. In Chapter 2 we rigorously construct a finite measure on $\Diff^1(\T)/\SL(2, \R)$, which corresponds to the Schwarzian Theory and compute its partition function (i.e. total mass). Here $\SL(2, \R)$ is the group of conformal isomorphisms of a unit disc restricted to its boundary. Our construction is based on the appropriate reparametrisation of the Brownian bridge measure, and the partition function calculation is done via an application of a suitable analogue of Girsanov formula. The obtained result for the partition function agrees with the formula derived in physics literature using a formal application of the Duistermaat–Heckman theorem on the infinite dimensional symplectic space $\Diff^1(\T)/\SL(2, \R)$. The content of Chapter~\ref{chapter_def} is based on the joint work with Roland Bauerschmidt and Peter Wildemann. In Chapter 3 we rigorously compute a natural class of correlation functions of cross-ratio observables. These correlation functions were originally formally derived in physics by taking a $c\to \infty$ limit of two dimensional Liouville CFT and, in particular, a limit of the DOZZ formula. Moreover, we prove that the computed correlation functions determine the measure uniquely, which further confirms that the measure we have constructed corresponds to the Schwarzian Theory studied in the physics literature. In addition, we show that using these observables we can make sense of and compute the stress-energy tensor correlation functions. We also prove that the stress-energy tensor correlation functions calculated this way agree with values obtained by differentiation of the partition function. In Chapter 4 we prove a large deviations principle for the probabilistic Schwarzian Theory at low temperatures. We demonstrate that the good rate function is equal to the action of the theory and find its minimisers. In addition, we define an analogue of the H"{o}lder condition on the functional space $\Diff^1(\T)/\SL(2, \R)$ in terms of cross-ratio observables, characterise them in terms of the usual H"{o}lder property for continuous functions, and deduce the corresponding compact embedding theorem. We also show that the Schwarzian measure concentrates on functions satisfying the defined condition.