Diagrams for Symmetric Product Orbifolds

TL;DR

The paper develops a diagrammatic framework for symmetric product orbifolds in 2D CFTs, recasting twist-field correlators as finite sums over diagrams that correspond to branched coverings and revealing a gauge-theory–like genus expansion. It provides explicit diagrammatic (Feynman) rules, the Hurwitz-theoretic correspondence to coverings, and a concrete procedure to compute leading large-$N$ four-point functions, including a detailed plan for genus-zero planar coverings via Heun’s equation. For planar four-point functions, the authors derive a constructive method to obtain all covering maps, analyze monodromies and channel-crossings, and connect these coverings to dual worldsheet pictures, culminating in a localization conjecture that the covering data localizes the string-like moduli space. Altogether, the work strengthens the bridge between symmetric orbifold CFTs and their string duals, and lays out precise combinatorial and analytic tools for evaluating twist correlators and their large-$N$ behavior.

Abstract

We develop a diagrammatic language for symmetric product orbifolds of two-dimensional conformal field theories. Correlation functions of twist operators are written as sums of diagrams: each diagram corresponds to a branched covering map from a surface where the fields are single-valued to the base sphere where twist operators are inserted. This diagrammatic language facilitates the study of the large N limit and makes more transparent the analogy between symmetric product orbifolds and free non-abelian gauge theories. We give a general algorithm to calculate the leading large N contribution to four-point correlators of twist fields.

Figures (19)

• Figure 1: We illustrate the construction of the diagram for the term $(1\,2\,3)_a(1\,2)_b(2\,3)_c$. On the left we have the first step of the construction: we draw fatgraphs loops for each of the indices (active colors), marking the appropriate vertices (letters) on the outer side of the loops. On the right we glue the vertices to obtain the diagram.
• Figure 2: On the left, the diagram corresponding to $(132)_a (24)_b (34)_c (241)_d$. A red (solid) dot is drawn for clarity on the inside of each color (solid) loop and is labeled by a color index. Each vertex (letter) corresponds to a twist field: going around the vertex counterclockwise one reads off the color indices of the corresponding cyclic permutation. On the right, the (graph theoretic) dual diagram, obtained as usual by dualizing vertices into faces. Each loop in the dual graph corresponds to a twist field.
• Figure 3: The vertex corresponding to $\sigma_{(123)}(z_a, \bar{z}_a)$. The solid lines (color lines) are numbered counter-clockwise in the cyclic ordering $(123)$. The letter in the center labels the coordinate of the twist operator.
• Figure 4: Two examples of illegal diagrams. (To avoid cluttering of the Figures we draw the fatgraph propagators with a single line, and use red dots to denote the "color" (solid) sides of the propagators.) On the left the numbers of two types of loops do not coincide (three color loops and two non-color loops). On the right the partial orderings defined by the color loops are incompatible: color $1$ defines partial ordering on vertices $abc$, and color $2$ defines the inverse ordering $bac$.
• Figure 5: Connected diagrams contributing to $\langle \sigma_{[3]}(a) \sigma_{[2]}(b) \sigma_{[3]}(c) \sigma_{[2]}(d) \rangle$ when $|z_a|<|z_b|<|z_c|<|z_d|$.