Orbifolds by 2-groups and decomposition

Abstract

In this paper we study three-dimensional orbifolds by 2-groups with a trivially-acting one-form symmetry group BK. These orbifolds have a global two-form symmetry, and so one expects that they decompose into (are equivalent to) a disjoint union of other three-dimensional theories, which we demonstrate. These theories can be interpreted as sigma models on 2-gerbes, whose formal structures reflect properties of the orbifold construction.

Decomposition, Condensation Defects, and Fusion

“… For one, the theory must admit a local operator algebra. Chern‐Simons theories in three dimensions, unless noneffectively gauged as in [43, 48], do not admit such an algebra, and so do not decompose. Even if there is a local operator algebra, we also emphasize that we only speak of decomposition in unitary cases. For example, the topological subsector of the A model with target $${\mathbb {P}}^n$$ formally may be equivalent to a disjoint union; however, the full quantum field theory with that target does not decompose.…”

“…Decomposition in three‐dimensional orbifolds by 2‐groups was discussed in [48]. Specifically, that work discussed three‐dimensional orbifolds $$[X/\tilde{\Gamma }]$$ where $$\tilde{\Gamma }$$ is a two‐group extension of an ordinary finite group G by a trivially‐acting one‐form group $$BK$$: $$\begin{equation} 1 \: \longrightarrow \: BK \: \longrightarrow \: \tilde{\Gamma } \: \longrightarrow \: G \: \longrightarrow \: 1.$$…”

“…If the three‐manifold $$Y = T^3$$, then the partition function of $$[X/\tilde{\Gamma }]$$ is [48, equ'n (4.18)] $$\begin{eqnarray} Z_{T^3}{\left([X/\tilde{\Gamma } ] \right)} & = & \frac{ | H^0(T^3, K) | }{ | H^1(T^3, K) | } \frac{1}{|H^0(T^3,G)|} \nonumber \\ &&\times\,\sum _{z_1, z_2, z_3 \in K} {\sum _{g_1, g_2, g_3 \in G}}^{\!\!\!\!\!\prime } \: Z(g_1, g_2, g_3), \nonumber \\ & = & \frac{1}{|K|^2 |G|} \sum _{z_1, z_2, z_3 \in K} {\sum _{g_1, g_2, g_3 \in G}}^{\!\!\!\!\!\prime }\: Z(g_1, g_2, g_3), \end{eqnarray}$$where the prime (′) indicates that the sum over elements of G is restricted to those su...…”

“…For $$Y = T^3$$, $$\begin{equation} \epsilon _{Y=T^3}(g_1, g_2, g_3) \: = \: \frac{ \omega (g_1, g_2, g_3) }{ \omega (g_1, g_3, g_2) } \frac{ \omega (g_3, g_1, g_2) }{ \omega (g_3, g_2, g_1) } \frac{ \omega (g_2, g_3, g_1) }{ \omega (g_2, g_1, g_3) }, \end{equation}$$where $$\omega \in H^3(G,K)$$ is the class of the extension (). Projecting to G bundles of that form is equivalent to working with a sum over universes, and it was argued in [48] that $$\begin{equation} Z_{T^3}{\left([X/\tilde{\Gamma } ] \right)} \: = \: \sum _{\rho \in \hat{K}} Z_{T^3}{\left([X/G]_{\epsilon _{\rho })} \right)}, \end{equation}$$reflecting the decomposition ().…”

“…Let us check this in some simple examples. First, from [48, section 4.3], consider the case $$G={\mathbb {Z}}_2 = K$$, so that $$\begin{equation} 1 \: \longrightarrow \: B {\mathbb {Z}}_2 \: \longrightarrow \: \tilde{\Gamma } \: \longrightarrow \: {\mathbb {Z}}_2 \: \longrightarrow \: 1. \end{equation}$$As noted in [48, section 4.3], there is a nontrivial extension of this form, which we take.…”