Orbifold groupoids

TL;DR

This work develops the orbifold groupoid framework to classify and compose orbifold operations in 2d bosonic and fermionic QFTs by embedding them into 3d Dijkgraaf–Witten theories. It shows that discrete gauging, discrete torsion, and dualities are encoded as boundary conditions and interfaces in a 3d bulk, with emergent symmetries and EM dualities organizing the web of theories. The paper extends the construction to fermionic theories via spin structures and supercohomology, introduces spin-preserving interfaces, and analyzes generalized symmetries through the 3d–2d boundary correspondence. It concludes with applications to current–current deformations of WZW models and outlines open questions about fermionic boundary data and Lagrangian algebras in 3d TFTs.

Abstract

We review the properties of orbifold operations on two-dimensional quantum field theories, either bosonic or fermionic, and describe the "Orbifold groupoids" which control the composition of orbifold operations. Three-dimensional TQFT's of Dijkgraaf-Witten type will play an important role in the analysis. We briefly discuss the extension to generalized symmetries and applications to constrain RG flows.

Figures (15)

• Figure 1: We take a 2d theory with $G$ symmetry and anomaly $\mu_3$ and use it to produce a boundary condition for the dynamical 3d DW theory with gauge group $G$ and action $\mu_3$. The anomaly of the 2d theory is cancelled by anomaly inflow from the bulk 3d theory. This picture can also be understood in terms of a boundary state as depicted in Equation \ref{['eqn:boundaryState']}.
• Figure 2: If we take the boundary condition $B[T]$ for the 3d theory we may produce $T$ by compactifying $B[T]$ with Dirichlet boundary conditions on $M\times[0,1]$. See also Appendix \ref{['appendix:Interfaces']}.
• Figure 3: In the folding trick, we replace the setup with a gauge theory $B$ on the left of the interface and gauge theory $A$ on the right of the interface, by a product theory $A\times\bar{B}$ with a corresponding boundary condition.
• Figure 4: Coupling to the 3d bulk literally decouples a theory $T$ from its topological manipulations. If $T$ corresponds to some boundary condition $B[T]$ (in blue), and some topological manipulation corresponds to the interface $I_{\nu_2}$ (in yellow), we may produce the theory with the topological manipulation included (in green), by colliding the boundary $B[T]$ with $I_{\nu_2}$.
• Figure 5: For any theory with non-anomalous finite Abelian $A$ symmetry, we obtain a new theory with $\hat{A} \cong A$ symmetry by gauging all of $A$. These correspond to the Dirichlet and "entirely-Neumann" boundary conditions for the associated $\mathop{\mathrm{\mathrm{DW}}}\nolimits[A]$. In the case of a non-anomalous $A=\mathop{\mathrm{\mathbb{Z}}}\nolimits_p$ ($p$ prime) this is the complete orbifold groupoid (suppressing multi-edges and edges from a vertex to itself), as there are only two bosonic irreducible topological boundary conditions.