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Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory

Anton Kapustin, Natalia Saulina

TL;DR

This work builds a bridge between 3d TFT surface operators and the consistent gluing of chiral and anti-chiral sectors in 2d RCFTs, interpreting such gluings through Morita-equivalence classes of symmetric Frobenius algebras in the bulk-line category. It provides a 3d geometric and categorical framework for the Fuchs–Runkel–Schweigert construction and shows how boundary conditions induce commutative Frobenius algebras via a 3d boundary-bulk map, illustrated concretely in abelian U(1) Chern-Simons theory. The results unify 3d topological defect data with 2d RCFT data, offering a general mechanism to classify RCFT gluings and boundary conditions from 3d perspective, with potential extensions to nonabelian theories and spin TQFTs.

Abstract

We study surface operators in 3d Topological Field Theory and their relations with 2d Rational Conformal Field Theory. We show that a surface operator gives rise to a consistent gluing of chiral and anti-chiral sectors in the 2d RCFT. The algebraic properties of the resulting 2d RCFT, such as the classification of symmetry-preserving boundary conditions, are expressed in terms of properties of the surface operator. We show that to every surface operator one may attach a Morita-equivalence class of symmetric Frobenius algebras in the ribbon category of bulk line operators. This provides a simple interpretation of the results of Fuchs, Runkel and Schweigert on the construction of 2d RCFTs from Frobenius algebras. We also show that every topological boundary condition in a 3d TFT gives rise to a commutative Frobenius algebra in the category of bulk line operators. We illustrate these general considerations by studying in detail surface operators in abelian Chern-Simons theory.

Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory

TL;DR

This work builds a bridge between 3d TFT surface operators and the consistent gluing of chiral and anti-chiral sectors in 2d RCFTs, interpreting such gluings through Morita-equivalence classes of symmetric Frobenius algebras in the bulk-line category. It provides a 3d geometric and categorical framework for the Fuchs–Runkel–Schweigert construction and shows how boundary conditions induce commutative Frobenius algebras via a 3d boundary-bulk map, illustrated concretely in abelian U(1) Chern-Simons theory. The results unify 3d topological defect data with 2d RCFT data, offering a general mechanism to classify RCFT gluings and boundary conditions from 3d perspective, with potential extensions to nonabelian theories and spin TQFTs.

Abstract

We study surface operators in 3d Topological Field Theory and their relations with 2d Rational Conformal Field Theory. We show that a surface operator gives rise to a consistent gluing of chiral and anti-chiral sectors in the 2d RCFT. The algebraic properties of the resulting 2d RCFT, such as the classification of symmetry-preserving boundary conditions, are expressed in terms of properties of the surface operator. We show that to every surface operator one may attach a Morita-equivalence class of symmetric Frobenius algebras in the ribbon category of bulk line operators. This provides a simple interpretation of the results of Fuchs, Runkel and Schweigert on the construction of 2d RCFTs from Frobenius algebras. We also show that every topological boundary condition in a 3d TFT gives rise to a commutative Frobenius algebra in the category of bulk line operators. We illustrate these general considerations by studying in detail surface operators in abelian Chern-Simons theory.

Paper Structure

This paper contains 10 sections, 1 theorem, 36 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Generalities
  3. Surface operators and consistent gluings
  4. From surface operators to algebra-objects
  5. Example: $U(1)$ current algebra
  6. Surface operators in $U(1)$ Chern-Simons theory
  7. Algebra-objects corresponding to surface operators
  8. Fusion of surface operators
  9. Boundary-bulk map in 3d TFT
  10. Concluding remarks

Key Result

Proposition 1

(A. Tsymbalyuk) Let $N$ be a natural number, $v$ be a divisor of $N$, and $R_v$ be a relation (in fact, a subgroup) in ${\mathbb Z}_{2N}\times{\mathbb Z}_{2N}$ given by Then for any two divisors $v,v'$ we have where

Figures (15)

  • Figure 1: The space of RCFT primaries labeled by objects $W,W'$ is the space of states of the 3d TFT on $S^2$ with insertions of line operators at the poles and the surface operator at the equator.
  • Figure 2: 3d TFT on a ball with an insertion of a surface operator (shaded region) on the equatorial plane. Chiral and anti-chiral sectors of the RCFT live on northern and southern hemispheres respectively. The surface operator terminates on a line operator.
  • Figure 3: Boundary conditions are labeled by line operators on which the surface operator can terminate. Given two such line operators $\lambda,\lambda'$, the space of boundary-changing operators in the vacuum representation of the chiral algebra is the space of states of the 3d TFT on $S^2$ with insertions of a surface operator (thick line) along a meridian semicircle and line operators $\lambda$ and $\lambda'$ at the poles.
  • Figure 4: A strip of a surface operator $S$ bounded from the left by a line operator $\lambda$ and from the right by its orientation-reversal.
  • Figure 5: The product.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Proposition
  • proof