Classifying boundary conditions in JT gravity: from energy-branes to $α$-branes

Abstract

We classify the possible boundary conditions in JT gravity and discuss their exact quantization. Each boundary condition that we study will reveal new features in JT gravity related to its matrix integral interpretation, its factorization properties and ensemble averaging interpretation, the definition of the theory at finite cutoff, its relation to the physics of near-extremal black holes and, finally, its role as a two-dimensional model of cosmology.

Figures (6)

• Figure 1: Summary of the boundary conditions considered in this paper. On the left column, one can see the boundary conditions for fixed dilaton $\phi$ (D) on the boundary, whereas the right column is for fixed $K$, i.e., Neumann boundary conditions on the dilaton (N). This is indicated by the first letter of the two letters in brackets. The rows are analogous, but then for the boundary metric $g_{uu}$ and is specified in the second letter in brackets.
• Figure 2: Figure showing the different curves of constant $k$ in the Poincaré plane and disk. The sign of $k$ is dependent on the direction of the normal vector perpendicular to the boundary; for instance, for $|k|>1$, $k>1$ is the boundary of the compact space (the disk), and $k<-1$ is the boundary of the non-compact space (the outside of the disk). In this paper, we will focus on $k>0$ to simplify our higher genus analysis.
• Figure 3: Figure showing a region of a trumpet which ends on a single geodesic boundary $G_0$ of length $b$. The red curves represent geodesics on the Poincaré disk and the two geodesics $G_1$ and $G_2$ which are dotted are identified ($G_1 = G_2$). In order to be able to identify such two geodesics we require them to be perpendicular to the closed geodesic containing the segment of length $b$. To construct boundaries with constant $K$ in the same homotopy class as the geodesic boundary with length $b$ we construct hypercycles (shown in green) which, by definition, intersect the two geodesics $G_1$ and $G_2$ at equal distances. Note that because the hypercycles need to pass through the points where $G_0$ intersects the boundary, the hypercycles can never be fully contained on the Poincaré disk and always have $k < 1$.
• Figure 4: Summary of the classical Euclidean geometries for fixed $K$ and $g_{uu}$ boundary conditions. For $k>1$ we can have a smooth bulk geometry, the disk. For $k<1$ the geometry is cylindrical and we have drawn (in slightly darker green) another fixed $K$ and $g_{uu}$ boundary in the middle and cutting the geometry there gives the trumpet geometry.
• Figure 5: Left: replica geometry for the microcanonical thermofield double state. Right: replica geometry for the fixed area state.