Group Actions on fusion categories

!!! warning This section is highly experimental and may yield unexpected results. Use with caution.

Let $\mathcal C$ be a fusion category and $G$ finite group. A group action of $G$ on $\mathcal C$ is given by a monoidal functor

\[T \colon \mathrm{Cat}(G) \to \mathrm{Aut}_{\otimes}(\mathcal C),~~~g \mapsto T_g \colon \mathcal C \to \mathcal C,\]

i.e. each group element is mapped to an autoequivalence of $\mathcal C$ and for each pair of group elements $g,h$ we have a monoiodal natural transformation

\[\sigma_{g,h} \colon T_g\circ T_h \to T_{g,h}.\]

We follow the construction of a group action with the example of the Ising category.

I = ising_category()
G = cyclic_group(2)
aut = autoequivalences(I)

As a first step we computed the autoequivalences. Note that this method at the moment is supported only for some categories not all. A general way to compute autoequivalences are the inner autoequivalences given by $V \mapsto X \otimes V \otimes X^\ast$ for some invertible $X$.

inner_autoequivalences(C::Category)

action_by_inner_automorphisms(C::Category)

Next we figure out which autoequivalence is non-trivial and define the tensor action.

a,b = aut    

if length(monoidal_natural_transformations(a,identity_as_monoidal_functor(I))) == 0
    a,b = b,a
end

# the monoidal structure is given by a Dict 
monoidal_str = Dict(
    (1,1) => id(a),
    (1,2) => id(b),
    (2,1) => id(b),
    (2,2) => monoidal_natural_transformations(b∘b, a)[1] # the nontrivial monoidal structure on the identity functor
)

T = gtensor_action(I, elements(G), [a,b], monoidal_str)

Equivariantization

Let $\mathcal C$ be a monoidal category with an action $T$ by a group $G$. An equivariant object is a tuple $(X,u)$ such that $X$is an object and a family of isomorphisms $u_g \colon T_g(X) \to X$ compatible with the action. The equivariant objects form a category $\mathcal C^G$ called equivariantization of $\mathcal C$.

Induction

There is a canonical forgetful functor $F \colon \mathcal C^G \to \mathcal C$. This forgetful functor admits a left adjoint $I_G$ given by

\[I_g(X) = \bigoplus\limits_{g \in G} T_g(X)\]

and structure maps

\[u_g \colon \sum\limits_{h} \iota_h \circ (\sigma_{g,h})_X \circ T_g(p_h)\;. \]

The induction is implemented for fusion categories by the method equivariant_induction.

equivariant_induction(X::Object, T::GTensorAction)

Computation

We can compute the equivariantization of a fusion category with a given $G$-action explicitly. We follow the example of the Ising category with its non-trivial $\mathbb Z_2$-action from earlier. The resulting category of type Equivariantization allows for all the operations available for fusion categories, including computation of $F$-symbols.

E = equivariantization(I)
simples(E)

$G$-Crossed Extensions

Given a fusion category $\mathcal C$ and $G$-action $T$we can define the $G$-crossed product $\mathcal C \ltimes G$ of $\mathcal C$ and $G$, see [1, 4.15.5]. This category has the same objects as $\mathcal C \boxtimes \mathrm{Vec}_G$ but with alternative tensor product

\[(X\boxtimes g) \otimes (Y \boxtimes h) := (X \otimes T_g(Y)) \boxtimes gh\]

and the associativity is given by

\[(X \otimes T_g(Y)) \otimes T_{gh}(Z) \xrightarrow{a_{X,T_(Y),T_{gh}(Z)}} X \otimes (T_g(Y) \otimes T_{gh(Z)}) \xrightarrow{\mathrm{id}_X \otimes \left(\mathrm{id}_{T_g(Y)} \otimes \left(\sigma_{g,h}\right)_{Z}\right)} \cdots \\ \cdots \to X \otimes (T_g(Y) \otimes T_g(T_h(Z))) \xrightarrow{\mathrm{id}_X \otimes \mu_{Y,T_h(Z)}} X \otimes (T_g(Y \otimes T_h(Z)))\]