Higher central charges and topological boundaries in 2+1-dimensional TQFTs

TL;DR

This work analyzes when 2+1d gapped TQFTs admit topological boundaries, extending beyond the familiar chiral central charge obstruction by introducing higher central charges and generalized gcd conditions. It shows that for Abelian TQFTs, a gapped boundary exists if and only if all extended higher central charges vanish, equivalently meaning the theory is an Abelian Dijkgraaf-Witten gauge theory; it also provides a complete obstruction set via Frobenius-Schur data and proves a prime-factor factorization reduction. In the non-Abelian case, the authors develop a geometric framework around Lagrangian algebras, gauging procedures, and the Turaev-Viro TQFT, establishing a one-to-one correspondence between gapped boundaries and Lagrangian algebras, and using Galois action to derive higher-central-charge obstructions. They further connect these 3d insights to 3-manifold invariants, lens-space partition functions, and the 3d–3d correspondence, yielding computable invariants and a unified view of obstructions beyond anomalies. The results provide practical criteria and conceptual unification for the existence of gapped boundaries in both Abelian and non-Abelian 2+1d TQFTs with potential implications for topological phases and the classification of boundary conditions.

Abstract

A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition for the existence of a topological boundary condition is that the chiral central charge $c_-$ has to vanish. In this paper, we consider conditions associated with "higher" central charges, which have been introduced recently in the math literature. In terms of these new obstructions, we identify necessary and sufficient conditions for the existence of a topological boundary in the case of bosonic, Abelian TQFTs, providing an alternative to the identification of a Lagrangian subgroup. Our proof relies on general aspects of gauging generalized global symmetries. For non-Abelian TQFTs, we give a geometric way of studying topological boundary conditions, and explain certain necessary conditions given again in terms of the higher central charges. Along the way, we find a curious duality in the partition functions of Abelian TQFTs, which begs for an explanation via the 3d-3d correspondence.

Figures (4)

• Figure 1: Anyon $\mathcal{A}$ from gapped boundary condition.
• Figure 2: The anyons $\overline{a}$ and $\mathcal{A}$ wrap an $S^1$ (not shown) and are located at points on $S^2$. Replacing the algebra anyon $\mathcal{A}$ with an empty tube turns the $S^2$ into a disk, and the partition function of this configuration gives the dimension of the Hilbert space on $D^2$ with the insertion of $a$.
• Figure 3: The Hopf link between $a$ and $\mathcal{A}$ can be evaluated to $(SZ)_{0a}$. On the other hand, replacing $\mathcal{A}$ by an empty tube gives a solid torus with $a$ wrapping the non-contractible cycle, which can be evaluated to the dimension of the Hilbert space on $D^2$ with an insertion of $a$, i.e. $Z_{0a}$.
• Figure 4: Replacing the $\mathcal{A}$ anyons with empty tubes makes it clear that fusion and braiding are trivial. This gives a geometric interpretation of the defining axioms of a Lagrangian algebra $\cal A$.

Theorems & Definitions (3)

• Theorem 2.1
• Corollary 2.2
• Theorem C.1