${\mathcal{C}\times\mathcal{C}}$${\mathcal{C}}$${\mathcal{E}\times\mathcal{E}}$${\mathcal{E}}$$\scriptstyle{\otimes_{\mathcal{C}}}$$\scriptstyle{Q\times
Q}$$\scriptstyle{Q}$$\scriptstyle{\otimes_{\mathcal{E}}}$$\scriptstyle{j}$

without further requirements. These functors, as we describe below, are
intimately related to the notion of anomalies. In particular, we will see that
one can have discrete torsion twists without a fiber functor.

The important observation for discrete torsion is that given a quasi-monoidal
functor $(Q,j)$ one may also twist it by a 2-cocycle $\omega$ to obtain
another quasi-monoidal functor $(Q,Q(\omega)\circ j)$, which as one can
readily verify is monoidal if and only if $(Q,j)$ itself is monoidal. This
allows to extend the notion of discrete torsion twists and actions to
categories of representations of quasi-Hopf algebras, for instance. We now
illustrate this concretely.

A first example is provided by considering quasi-monoidal functors in the
context of Section 5.2. For that purpose, consider the category
$\text{Vec}_{G}^{\alpha}$ of $G$-graded vector spaces whose associator is now
defined by a nontrivial cohomology class $0\neq[\omega]\in
H^{3}(G,\mathbb{C}^{\times})$. It is well known that this fusion category does
not admit a fiber functor, equivalently an action on Vec, and for this reason
one calls the category $\text{Vec}_{G}^{\alpha}$ anomalous [29, Section 2.3].
However, it is still a fusion category and as such it has a well-defined lazy
cohomology group, since it is an intrinsic notion. Indeed, its discrete
torsion twist group $H^{2}_{\ell}(\text{Vec}_{G}^{\alpha})$ is the same as for
the category $\text{Vec}_{G}$, that is, $H^{2}(G,U(1))$. These lazy
2-cocycles, in the same fashion described in (4.32), can then twist any
gaugeable algebra in $\text{Vec}_{G}^{\alpha}$ such as the algebras
corresponding to the subgroups $H\leq G$ for which $\alpha|_{H}$ is
cohomologically trivial.

Now, a lazy 2-cocycle can no longer give a fiber functor, since there are no
fiber functors on which it can act. However, $\text{Vec}_{G}^{\alpha}$ does
admit a quasi-fiber functor [26, Example 5.1.3]. This can consist, for
example, of the forgetful functor of underlying categories that ignores the
$G$-grading, along with the identity natural isomorphism

$\displaystyle Q:\text{Vec}_{G}^{\alpha}\to\text{Vec},$ (5.44) $\displaystyle
j_{X,Y}:Q(X)\otimes Q(Y)\xrightarrow{\text{id}}Q(X\otimes Y).$ (5.45)

The anomaly is then observed in Vec as the non-associativity of the images of
the objects of $\text{Vec}_{G}^{\alpha}$ in Vec, equivalently their actions on
Vec. More in detail, $(Q,j)$ defines a diagram for all triples of objects

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{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(5.46)

where $\Phi_{X,Y,Z}$ is generally a nontrivial morphism. Concentrating on
triples of simple objects $(U_{g},U_{h},U_{z})$ gives morphisms
$\Phi_{g,h,k}:\mathbb{C}\to\mathbb{C}$, which is simply the value of the
3-cocycle associator $\alpha(g,h,k)$. The reader should compare this to the
case of fiber functors (5.3) to Vec, in which case
$\Phi_{X,Y,Z}=a_{F(X),F(Y),F(Z)}=\text{id}_{F(X)\otimes F(Y)\otimes F(Z)}$.

Having a quasi-fiber functor at our disposal, one can proceed and twist by a
lazy 2-cocycle, which returns another quasi-fiber functor. This may be
understood as discrete torsion twists of anomalous actions on Vec.

We can also consider, as in Section 5.2, relaxing the condition on the action
functor from a monoidal functor to a quasi-monoidal functor $(Q,j)$ for
actions of gerbes, which we understand as anomalous actions. As in the case
above, an anomalous action gives rise to diagrams

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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-12.93365pt}{-49.08195pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{Q(a_{X,Y,Z})}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (5.47)

where the collection of such $\Phi_{X,Y,Z}$ are now potentially non-trivial
morphisms of hermitian vector bundles with connection

$\Phi_{X,Y,Z}:Q(X)\otimes Q(Y)\otimes Q(Z)\xrightarrow[]{\cong}Q(X)\otimes
Q(Y)\otimes Q(Z),$ (5.48)

where we omit the brackets since $\text{HVBdl}_{\nabla}(M)$ is strictly
associative. The collection of isomorphisms (5.48) characterizes the anomaly
of the $\mathcal{C}$-action.

In particular, for $\mathcal{C}=\text{Vec}_{G}^{\alpha}$ for $0\neq[\alpha]\in
H^{3}(G,U(1))$, we can see that an anomalous action on a gerbe is described,
apart from the cochains coming from the natural isomorphisms $j_{X,Y}$, by
Čech 3-cocycles

$(k^{g_{2},g_{3},g_{4}}_{\alpha})(k_{\alpha}^{g_{1},g_{2}g_{3},g_{4}})(k_{\alpha}^{g_{1},g_{2},g_{3}})=(k_{\alpha}^{g_{1}g_{2},g_{3},g_{4}})(k_{\alpha}^{g_{1},g_{2},g_{3}g_{4}}).$
(5.49)

If it furthermore happens that this action factors (5.39, 5.40) through the
trivial finite-dimensional Hilbert bundles $M\times H$, hence factors through
a quasi-fiber functor to $\text{Hilb}_{\rm f.d.}$, then these are genuine
group $3$-cocycles as the ones classifying the ’t Hooft anomaly of some
$G$-symmetry.

Just as before, these anomalous actions described by quasi-monoidal functors
can be acted on by the intrinsic discrete torsion twists of $\mathcal{C}$ to
yield other anomalous actions. This shows that the intrinsic formulation of
discrete torsion twists seamlessly allows for significant generalizations of
discrete torsion action.

##  6 Conclusions

In this article we have highlighted that the notion of discrete notion of
group-like symmetries actually admits two different but complementary
generalizations to non-invertible symmetries: the set of discrete torsion
choices, and the cohomology group of discrete torsion twists. These two
notions are complementary in the sense that the former always admits a
consistent twisting action by the latter, meaning that discrete torsion
choices not only form a set but a group-module. We showed that the discrete
torsion twists of a fusion category $\mathcal{C}$ are coherently described by
its lazy cohomology group $H^{2}_{\ell}(\mathcal{C})$. We explicitly described
how lazy 2-cocycles can consistently twist the gaugeable algebras
$\mathcal{A}$ inside $\mathcal{C}$ to produce new gaugeable algebras, and that
up to Morita equivalence the twist only depends on the cohomology class of the
lazy 2-cocycle. We gave explicit expressions for the resulting partition
functions, and showed they are consistent with modular invariance and multi-
loop factorization. We also described how lazy 2-cocycles can twist monoidal
functors from $\mathcal{C}$, where similarly only lazy 2-cocycles with
nontrivial cohomology classes can give rise to monoidal inequivalent twists.
In particular, we studied the case of twists of fiber functors to clarify the
distinction between discrete torsion twists, and discrete torsion choices,
described by equivalence classes of fiber functors. We also studied how
discrete torsion twists relate to non-invertible gauge actions on $B$-fields,
and recovered known results in the special case of group-like actions. The
perspective of separating between choices of structure and twists of such
structures thus allows a direct application of discrete torsion twisting
actions to a wide variety of settings, including actions on Vec (fiber
functors), but also explicit actions on gerbes and even their anomalous
variants.

##  7 Acknowledgements

We would like to thank D. Robbins, E. Sharpe, T. Vandermeulen, and X. Yu for
useful discussions.

##  Appendix A Lazy cohomology

For the reader’s convenience, we gather the definitions relevant to the
characterization of discrete torsion twists as elements of the lazy cohomology
group.

Let $\mathcal{C}$ be a fusion category with its tensor product functor denoted
as $\otimes$. A lazy 2-cocycle is a natural isomorphism

$\omega:\otimes\Rightarrow\otimes,$

satisfying the conditions

$\displaystyle\omega_{X,1}=\omega_{1,X}=\text{id}_{X};$
$\displaystyle\omega_{X,Y\otimes
Z}\circ(\text{id}_{X}\otimes\omega_{Y,Z})=\omega_{X\otimes
Y,Z}\circ(\omega_{X,Y}\otimes\text{id}_{Z}).$

The collection of lazy 2-cocycles form a group denoted as
$Z^{2}(\mathcal{C})$.

A lazy 2-coboundary is a lazy 2-cocycle whose components are of the form

$(d\phi)_{X,Y}:=\phi^{-1}_{X\otimes Y}\circ(\phi_{X}\otimes\phi_{Y}),$

where $\phi$ is an automorphism of $\mathcal{C}$

$\phi:\text{id}_{\mathcal{C}}\to\text{id}_{\mathcal{C}}.$ (A.1)

The lazy cohomology $H^{2}_{\ell}(\mathcal{C})$ of $\mathcal{C}$ is the
quotient of lazy 2-cocycles $Z^{2}(\mathcal{C})$ by its central subgroup
$B^{2}(\mathcal{C})$ of lazy 2-coboundaries

$H^{2}_{\ell}(\mathcal{C}):=Z^{2}(\mathcal{C})/B^{2}(\mathcal{C}).$

The cohomology classes in $H^{2}_{\ell}(\mathcal{C})$ is what we identify as
discrete torsion twists.

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