Finite 2-group background cochains

Let $G$ be a finite group with identity $e$, let $A$ be a finite abelian group written additively, and let $\rho :G\to \mathrm{Aut}(A)$ be a left action, so ${\rho }_{gh}={\rho }_g{\rho }_h$ and ${\rho }_e={\mathrm{id}}_A$. A group $2$-cochain is a function $c:G^2\to A$; it is normalized when it vanishes if either input is $e$. Its coboundary is

$$({\delta }_{\rho }c)(g,h,k)={\rho }_{g}c(h,k)-c(gh,k)+c(g,hk)-c(g,h)\mathrm{.}$$

A normalized group $3$-cocycle is a function $\beta :G^3\to A$ that vanishes whenever any argument is $e$ and satisfies

$${\rho }_g\beta (h,k,l)-\beta (gh,k,l)+\beta (g,hk,l)-\beta (g,h,kl)+\beta (g,h,k)=0$$

for all $g,h,k,l\in G$. Two normalized $3$-cocycles represent the same class when their difference is ${\delta }_{\rho }c$ for a normalized $2$-cochain $c$; this is an equivalence relation by additivity of ${\delta }_{\rho }$. The class $[\beta ]$ is the Postnikov class in this skeletal model.

The corresponding skeletal categorical $2$-group has objects $g\in G$ and, for each $g$, one automorphism labelled by each $u\in A$; there are no morphisms between distinct objects. Composition of automorphisms adds their labels, tensor product is

$$(g,u)\otimes (h,v)=(gh,u+{\rho }_{g}v),$$

and the associator from $(gh)k$ to $g(hk)$ has label $\beta (g,h,k)$. The two pentagon path labels are $\beta (gh,k,l)+\beta (g,h,kl)$ and ${\rho }_g\beta (h,k,l)+\beta (g,hk,l)+\beta (g,h,k)$; equality of these labels is exactly the displayed $3$-cocycle equation.

A finite ordered simplicial complex $K$ is a finite vertex set with a total order and a collection of nonempty finite vertex subsets, called simplices, closed under taking nonempty subsets. Write each simplex in increasing order. A $q$-cochain with values in a local coefficient system assigns an element to each ordered $q$-simplex at its first vertex. A flat $G$-valued edge cochain is a family $a_{ij}\in G$ for each edge $i<j$ such that $a_{ij}{a}_{jk}=a_{ik}$ on every triangle. It defines the local coefficient system with fiber $A$ at each vertex and transport ${\rho }_{a_{ij}}$ from vertex $j$ to vertex $i$.

For each integer $q\ge 0$ and local-coefficient $q$-cochain $c$, define the twisted coboundary on each ordered $(q+1)$-simplex by

$$({\delta }_{a}c{)}_{i_0\dots i_{q+1}}={\rho }_{a_{i_0{i}_1}}{c}_{i_1\dots i_{q+1}}+\sum _{r=1}^{q+1}(-1{)}^r{c}_{i_0\dots \widehat{i_r}\dots i_{q+1}}\mathrm{.}$$

In particular, for an $A$-valued $2$-cochain $B$,

$$({\delta }_{a}B{)}_{ijkl}={\rho }_{a_{ij}}{B}_{jkl}-B_{ikl}+B_{ijl}-B_{ijk},(a^{\ast }\beta {)}_{ijkl}=\beta (a_{ij},a_{jk},a_{kl})\mathrm{.}$$

A finite $2$-group background on $K$ is a flat edge cochain $a$ and an $A$-valued $2$-cochain $B$ satisfying ${\delta }_{a}B=a^{\ast }\beta$ on every ordered tetrahedron. This fixes the cochain representative, local coefficient action, and sign convention.

A $1$-gauge transformation consists of vertex labels $g_i\in G$ and edge labels ${\lambda }_{ij}\in A$ at vertex $i$. Define

$$a_{ij}^g=g_i{a}_{ij}{g}_j^{-1},$$ $${\zeta }_{ijk}^g=-\beta (g_i,a_{ij},a_{jk})+\beta (a_{ij}^g,g_j,a_{jk})-\beta (a_{ij}^g,a_{jk}^g,g_k),$$ $$B_{ijk}^{g,\lambda }={\rho }_{g_i}{B}_{ijk}+{\zeta }_{ijk}^g+({\delta }_{a^g}\lambda {)}_{ijk},({\delta }_{a^g}\lambda {)}_{ijk}={\rho }_{a_{ij}^g}{\lambda }_{jk}-{\lambda }_{ik}+{\lambda }_{ij}\mathrm{.}$$

Changing the chosen representative to ${\beta }^{\mathrm{\prime }}=\beta +{\delta }_{\rho }c$ for a normalized $2$-cochain $c$ and simultaneously replacing $B$ by $B+a^{\ast }c$ changes the cochain presentation but preserves the background equation: on every tetrahedron, flatness gives

$$({\delta }_a(a^{\ast }c){)}_{ijkl}={\rho }_{a_{ij}}c(a_{jk},a_{kl})-c(a_{ik},a_{kl})+c(a_{ij},a_{jl})-c(a_{ij},a_{jk})=({\delta }_{\rho }c)(a_{ij},a_{jk},a_{kl})\mathrm{.}$$

This finite mathematical model makes no assertion that a physical theory realizes the symmetry or has an anomaly.

Benini, Córdova, and Hsin, §2.3, PDF pp. 18–19

Baez and Lauda, §8.3, PDF pp. 54–59

Schäfer-Nameki, §§2.5 and 2.8.1, printed pp. 23–33