Interface Correlators in Symmetric Product Orbifolds

TL;DR

This work develops a comprehensive framework for interfaces in symmetric product orbifolds by introducing generalized interface covering maps, a corresponding interface Riemann–Hurwitz formula, and an extended Pakman–Rastelli–Razamat diagrammatics. It extends the Lunin–Mathur construction to interfaces via boundary Liouville action and the method of images, enabling efficient computations of interface correlators and revealing a structured large-$N$ expansion that matches string perturbation theory to all orders. A novel interface grand-canonical ensemble is defined, with fugacities controlling $N_\pm$ and transmission $P$, producing a genus-expansion consistent with open/closed string amplitudes in AdS$_3$/CFT$_2$. Collectively, the results provide analytic tools for probing AdS$_2$ branes in NSNS AdS$_3$ backgrounds and lay groundwork for open-string holography in symmetric product orbifolds. The framework holds promise for SUSY extensions, thermal/holographic analyses, and deformations via partial permutations and interface twists.

Abstract

Symmetric product orbifolds provide a controlled environment to explore generic features of gauge theory and holography. The tractability of these theories lies in the complete characterisation of their gauge structure through holomorphic covering maps. In this paper, we introduce a novel class of generalised covering maps, which define a universal family of interfaces between symmetric product orbifolds. These interfaces coincide with the holographic interfaces that were recently proposed as duals to AdS$_2$ branes in pure NSNS AdS$_3$ backgrounds. The new covering-map description enables efficient evaluation of interface correlators via a generalisation of the Lunin-Mathur method. To organise these computations, we derive a generalised Riemann-Hurwitz formula for interface coverings and introduce novel diagrammatic rules that systematically classify these maps. The new framework allows us to define a concrete grand-canonical ensemble that has the correct properties to compute correlation functions dual to open string scattering amplitudes. Using the generalised Riemann-Hurwitz formula, we explicitly show that the correlators of the ensemble structurally match string perturbation theory to all orders in the string coupling.

Figures (16)

• Figure 1: Visualisation of coverings in a neighbourhood of interface ramification points. The covering map is the vertical projection down the symmetry axis of the spiral. The preimage of the interface is marked by horizontal black lines on the covering space.
• Figure 2: Lines on $\mathbb{S}^2$ whose preimage w.r.t. a covering map $\Gamma$ produce $\mathcal{D}_\varepsilon[\Gamma]$. Branch points in the upper and lower hemisphere are denoted by $w_+$ and $w_-$. Orientations are represented by half-arrows extending into the disk on whose boundary they are drawn.
• Figure 3: This figure shows half of the vertices of the PRR diagrams. Two more vertices can be generated by turning around all orientations and exchanging $+ \leftrightarrow -$.
• Figure 4: Reduction to standard PRR diagrams.
• Figure 5: Diagram corresponding to the covering map that computes the two-point function of twist $w$ fields.