Holographic partition functions and phases for higher genus Riemann surfaces

TL;DR

The paper advances holographic calculations of 2D CFT partition functions on higher genus boundaries by numerically solving the Liouville equation for the boundary metric within Schottky uniformisation, allowing precise evaluation of the Takhtajan–Zograf action for bulk handlebody saddles. By focusing on genus two in a two‑dimensional moduli subspace with reflection and enhanced symmetries, it maps the phase diagram among connected and disconnected handlebodies and a non‑handlebody, finding that handlebodies always dominate over the non‑handlebody phase. The work provides both analytic pinching limit checks and detailed numerical results for horizon lengths and phase boundaries, revealing symmetry‑controlled transitions and instances of spontaneous symmetry breaking. These results illuminate the entanglement structure and phase competition in multi‑boundary AdS/CFT setups and set the stage for higher‑genus extensions with potential insights into Rényi entropies and replica symmetry in holographic theories.

Abstract

We describe a numerical method to compute the action of Euclidean saddlepoints for the partition function of a two-dimensional holographic CFT on a Riemann surface of arbitrary genus, with constant curvature metric. We explicitly evaluate the action for the saddles for genus two and map out the phase structure of dominant bulk saddles in a two-dimensional subspace of the moduli space. We discuss spontaneous breaking of discrete symmetries, and show that the handlebody bulk saddles always dominate over certain non-handlebody solutions.

Figures (9)

• Figure 1: The choice of fundamental domains $D$ we use for the relevant Schottky groups $G$, corresponding in the three boundary wormhole to the connected phase (left) and partially connected (right), and a cartoon of the surface we consider with various relevant cycles marked. The generators for the Schottky group for the left domain identify the two green circles on the right with one another, and similarly the two on the left; for the right domain they identify the concentric orange circles, and the yellow circles. The disconnected phase uses the left domain, but swaps the interpretations of A- and B-cycles. The three involutions which generate the symmetries of the surface can be visualised as reflections in the three coordinate axes. Reflection in the horizontal plane fixes the blue and yellow A-cycles, and is interpreted as the time-reversal for the three boundary wormhole, implemented as reflection in the horizontal axis for the left domain, and the vertical axis for the right. Reflection in the plane of the page, fixing green and orange B-cycles, is implemented as inversion in the unit circle for both domains. Finally, the reflection in the vertical plane fixing the purple 'waist', interpreted as time reversal for the torus wormhole, is represented by reflection in vertical and horizontal axes for left and right domains respectively. The hyperelliptic involution is the half rotation around the horizontal axis, fixing the six points where A- and B-cycles intersect.
• Figure 2: The maximally symmetric real genus 2 Riemann surface. The left plot shows a tessellation of the complex $x$-plane respecting the symmetries of the curve $y^2=x^6-1$. The symmetries acting on the $x$-plane include the obvious reflections and rotations, as well as inversion in the unit circle and the hyperelliptic involution $y\mapsto -y$ exchanging sheets. The red points are the sixth roots of unity, where the plane is ramified. Each triangle has a vertex at either $0$ or $\infty$ on the $x$-plane, where 12 triangles meet, so the angle is $\frac{\pi}{6}$, and a vertex at $e^{(2k+1)i\pi/6}$ for some integer $k$, where 4 triangles meet with right angles (the blue points in the figure), so the angle is $\pi/2$. Finally, there is a vertex at a sixth root of unity (the red points), which is a ramification point for the hyperelliptic curve, so there are in fact 8 triangles meeting at this point, four on each sheet, with angles $\pi/4$. The right diagram shows a fundamental domain of the hyperbolic plane for the surface, which can be represented as a quotient of $\mathbb{H}^2$. The surface is given by an appropriate identification of edges of this domain.
• Figure 3: Examples of the meshes used for the two types of Schottky group, in this case at the triple point where all phases meet.
• Figure 4: Plots of the solutions of the Liouville equation for each of the phases, at the triple point where all phases meet.
• Figure 5: Log-Log plot of error in the Gauss-Bonnet theorem: $1-\frac{A}{4\pi}$, as a function of the number of elements $N$. The linear fit gives a power $N^{-3.08}$.