CP7 Arbeitsgruppenseminar 2022/23

What: The CP7 Arbeitsgruppenseminar in the academic year of 2022/23 was about the so-called Schwarzian theory, which has enjoyed much interest from many areas in physics and mathematics.

At heart, the Schwarzian theory can be described as a one-dimensional QFT whose space of fields is Diff(S1) (modulo symmetries) and whose action func- tional is governed by the Schwarzian derivative.

From a mathematician’s point of view, the Schwarzian theory is the study of integrals over Virasoro coadjoint orbits. Its defining action functional is the moment map of a certain circle action on the orbit. The partition function can therefore formally be obtained by a Duistermaat-Heckman integral and more generally, the theory computes the asymptotics of Virasoro characters via formal Duistermaat-Heckman integrals.

From a physicist’s perspective, the Schwarzian theory can be derived in a variety of ways. For example, it describes the low energy limit of the SYK model, a quantum mechanical system of Majorana fermions with random interactions.

As an other example, it is holographically dual to JT gravity, a two-dimensional theory which approximates the near horizon geometry of (near) extremal black holes. Studying JT gravity on a disk, one is lead to the study of the Schwarzian theory at the boundary whose configuration space is Diff(S1)/SL(2,R) which is a model for the smooth part of the universal Teichmüller space.

Additionally, JT gravity can be described by a two-dimensional topological SL(2,R)-gauge theory which reduces to a free particle on SL(2,R) on the boundary. The Schwarzian theory is then obtained by choosing appropriate boundary conditions. These boundary conditions can be given by first-class constraints and the resulting gauge theory on the boundary can be interpreted in terms of a Hamiltonian reduction.

The CP7 Arbeitsseminar aims at the study of the ubiquitous appearance of the Schwarzian theory and of its various interpretations which seem to bond numerous areas of physics and mathematics further and further together.

For a rough overview of possible topics, please click here