[Paper Review] Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory

Anton Kapustin, Natalia Saulina|arXiv (Cornell University)|Dec 4, 2010

Black Holes and Theoretical Physics4 citations

TL;DR

This paper establishes a correspondence between surface operators in 3d topological field theories (TFTs) and consistent gluing structures in 2d rational conformal field theories (RCFTs). It shows that surface operators correspond to Morita-equivalence classes of symmetric Frobenius algebras in the ribbon category of bulk line operators, providing a physical interpretation of the Fuchs-Runkel-Schweigert construction and extending the boundary-bulk map to 3d TFTs.

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.

Motivation & Objective

• To clarify the physical role of surface operators in 3d TFTs from the perspective of 2d RCFTs.
• To provide a physical interpretation of the Fuchs-Runkel-Schweigert (FRS) construction of 2d RCFTs from modular tensor categories.
• To generalize the boundary-bulk map in 2d TFT to 3d TFTs by associating commutative Frobenius algebras to topological boundary conditions.
• To analyze surface operators in abelian Chern-Simons theory as a concrete realization of the general framework.
• To explore the algebraic structure of surface operators via category-theoretic tools such as Morita equivalence and symmetric Frobenius algebras.

Proposed method

• Use the folding trick to reinterpret a 3d TFT on a cylinder as a 2d RCFT on the boundary, linking bulk line operators to chiral and anti-chiral sectors.
• Define surface operators as 2-morphisms in a 2-category of TFTs, with the invisible surface operator acting as the monoidal unit.
• Show that a surface operator without local operators induces a consistent gluing of chiral and anti-chiral sectors via a symmetric Frobenius algebra in the ribbon category of bulk line operators.
• Establish a one-to-one correspondence between surface operators and Morita-equivalence classes of symmetric Frobenius algebras in the category of bulk line operators.
• Derive the boundary-bulk map in 3d TFT by associating a commutative Frobenius algebra to any topological boundary condition in the 3d theory.
• Analyze the $U(1)_k \times U(1)_{-k}$ Chern-Simons theory to explicitly realize surface operators and their algebraic invariants, including D-type and E-type operators for small $k$.

Experimental results

Research questions

• RQ1 How do surface operators in 3d TFTs relate to the gluing of chiral and anti-chiral sectors in 2d RCFTs?
• RQ2 What algebraic structure in the category of bulk line operators corresponds to a given surface operator?
• RQ3 How does the Fuchs-Runkel-Schweigert construction of 2d RCFTs from Frobenius algebras emerge from 3d TFT considerations?
• RQ4 Can the boundary-bulk map in 2d TFT be generalized to 3d TFTs via topological boundary conditions?
• RQ5 What is the classification of surface operators in abelian Chern-Simons theories, and how do they relate to modular data and duality?

Key findings

• Surface operators in 3d TFTs correspond to Morita-equivalence classes of symmetric Frobenius algebras in the ribbon category of bulk line operators.
• The construction of 2d RCFTs via symmetric Frobenius algebras in the FRS program is physically realized as the insertion of surface operators in 3d TFT.
• Topological boundary conditions in 3d TFTs give rise to commutative Frobenius algebras in the category of bulk line operators, generalizing the 2d boundary-bulk map.
• In $U(1)_k \times U(1)_{-k}$ Chern-Simons theory, surface operators are classified by relations in $\mathbb{Z}_{2N} \times \mathbb{Z}_{2N}$, with D-type and E-type operators emerging for small $k$.
• The composition of surface operators is governed by a formula involving gcd and lcm of divisors, with the composition of $R_v$ and $R_{v'}$ yielding $g R_x$ where $g = \gcd(v,v',N/v,N/v')$.
• The algebraic structure of surface operators is fully captured by the category-theoretic data of symmetric Frobenius algebras, with the D-type operator defined by agreement of $SO(3)$ gauge fields on both sides.