Given a double cover
of finite groupoids, we explicitly construct cochain-level twisted loop transgression maps,
and
, thereby associating
to a Jandl
–gerbe
on
a Jandl
–gerbe
on the quotient loop
groupoid of
and
an ordinary
–gerbe
on the unoriented
quotient loop groupoid of .
For
,
we prove that the character theory (resp. centre) of the category of Real
–twisted
–vector
bundles over
admits a natural interpretation in terms of flat sections of the
–vector bundle
associated to
(resp. the
Real
–vector bundle
associated to
).
We relate our results to Real versions of twisted Drinfeld doubles of finite groups and
fusion categories and to discrete torsion in orientifold string theory and
–theory.
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