Exact solution of two-dimensional Palatini Gauss-Bonnet theory on a strip

TL;DR

This paper provides an exact analysis of the boundary degrees of freedom in two-dimensional Palatini Gauss-Bonnet theory, revealing its relation to geodesics on SL(2,R) and discussing boundary symmetries and quantum aspects.

Contribution

It explicitly solves the boundary dynamics of 2D Palatini Gauss-Bonnet theory, connecting it to geodesic motion on SL(2,R) and exploring boundary symmetries and quantum implications.

Findings

The phase space is the cotangent bundle of SL(2,R) with a quadratic constraint.

For zero boundary Hamiltonian, the theory describes geodesics on SL(2,R).

Boundary symmetries include group translations acting on the interval ends.

Abstract

We analyze the two-dimensional Palatini Gauss-Bonnet theory on an infinite strip (product of a finite interval with the infinite line, corresponding to ``time"). The theory has only boundary degrees of freedom. Its phase space is the cotangent bundle to the group manifold of $SL(2,\mathbf{R})$, subject to a (first-class) constraint quadratic in the momenta. With the simplest choice of boundary Hamiltonian, namely $H = 0$, the theory is shown to describe geodesics on the group manifold of $SL(2,\mathbf{R})$, with a ``mass" determined by the Palatini Gauss-Bonnet coupling constant. Other choices of boundary Hamiltonians compatible with gauge invariance are also possible. The symmetry group contains (left and right) group translations on $SL(2,\mathbf{R})$. These are ``boundary symmetries" from the bulk point of view, one copy acting on one end of the interval, the other copy acting on the other end. Comments on the quantum theory are also given.