# Frobenius–Schur indicators for twisted Real representation theory and two
dimensional unoriented topological field theory

Levi Gagnon-Ririe  Department of Mathematics and Statistics
Utah State University
Logan, Utah 84322
USA<EMAIL_ADDRESS>and  Matthew B. Young  Department of Mathematics
and Statistics
Utah State University
Logan, Utah 84322
USA<EMAIL_ADDRESS>
###### Abstract.

We construct a two dimensional unoriented open/closed topological field theory
from a finite graded group $\pi:\hat{G}\twoheadrightarrow\\{1,-1\\}$, a
$\pi$-twisted $2$-cocycle $\hat{\theta}$ on $B\hat{G}$ and a character
$\lambda:\hat{G}\rightarrow U(1)$. The underlying oriented theory is a twisted
Dijkgraaf–Witten theory. The construction is based in the
$(\hat{G},\hat{\theta},\lambda)$-twisted Real representation theory of
$\ker\pi$. In particular, twisted Real representations are boundary conditions
and the generalized Frobenius–Schur element is its crosscap state.

###### Key words and phrases:

Real representation theory. Topological field theory.

###### 2020 Mathematics Subject Classification:

Primary: 20C25; Secondary 81T45.

## Introduction

Associated to a finite group $G$ and a $U(1)$-valued $2$-cocycle $\theta$ on
its classifying space $BG$ is a two dimensional topological gauge theory known
as Dijkgraaf–Witten theory [DW90]. This is an oriented open/closed topological
quantum field theory (TFT) $\mathcal{Z}_{(G,\theta)}$ with boundary conditions
the category $\textup{\text{Rep}}^{\theta}(G)$ of finite dimensional
$\theta$-twisted complex representations of $G$ [Fre94, MS06]. In particular,
$\mathcal{Z}_{(G,\theta)}$ assigns a partition function to each compact
oriented $2$-manifold with boundary components labelled by twisted
representations. Open/closed TFT was introduced as a framework to axiomatize
the structure of topological D-branes in string theory [Laz01, KR04, MS06] and
has found a variety of applications in pure mathematics [Cos07, BCT09, Abo10].
The open/closed structure of Dijkgraaf–Witten theory plays an important role
in the descriptions of D-branes in orbifold string theory [DW90], generalized
symmetries in quantum field theory [Sha15, HLS21] and boundary degrees of
freedom in topological phases of matter [SR17].

Open/closed TFTs on unoriented—and possibly non-orientable—manifolds play a
central role in orientifold string theory [HW08] and related mathematics
[You20, FH21, GI21, NY22]. In condensed matter physics, unoriented TFTs in
general, and Dijkgraaf–Witten theory in particular, model topological phases
of matter with time reversal symmetry [FM13, KT17, BBC+20]. The main result of
this paper is an algebraic construction of a class of unoriented lifts of the
oriented open/closed Dijkgraaf–Witten theories $\mathcal{Z}_{(G,\theta)}$.

######  Theorem A (Theorem 3.7).

A triple $(\hat{G},\hat{\theta},\lambda)$ consisting of a short exact sequence
of finite groups

$1\rightarrow
G\rightarrow\hat{G}\xrightarrow[]{\pi}C_{2}=\\{1,-1\\}\rightarrow 1,$

a $\pi$-twisted $2$-cocycle $\hat{\theta}$ on $B\hat{G}$ which restricts to
$\theta$ on $BG$ and a character $\lambda:\hat{G}\rightarrow U(1)$ defines a
two dimensional unoriented open/closed topological field theory
$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ whose oriented sector is a
subtheory of $\mathcal{Z}_{(G,\theta)}$.

A number of authors have studied
$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ under the assumption that
$\hat{G}=G\times C_{2}$ is the trivial extension, $\hat{\theta}$ is in the
image of the map $H^{2}(BG;C_{2})\rightarrow H^{2}(B\hat{G};U(1)_{\pi})$ and
$\lambda$ is trivial [KM97, AN06, Tur07, LN11, Sny17]. For general
$(\hat{G},\hat{\theta})$ and trivial $\lambda$, a topological construction of
the closed sector of $\mathcal{Z}_{(\hat{G},\hat{\theta},1)}$, and its higher
dimensional analogues, was given in [You20] while a $G$-equivariant extension
of the closed sector of $\mathcal{Z}_{(\hat{G},\hat{\theta},1)}$ was given in
[KT17]. We emphasize that for the applications of unoriented Dijkgraaf–Witten
theory mentioned before Theorem A, general input data
$(\hat{G},\hat{\theta},\lambda)$ is required; see Remark 3.8. As explained
below, general input data is also natural from the representation theoretic
and $K$-theoretic perspectives.

Theorem A is proved using an algebraic characterization of unoriented TFTs,
Theorem 3.4, which builds off characterizations of oriented closed and
open/closed TFTs [Dij89, Abr96, Laz01, MS06, AN06, LP08], unoriented closed
TFTs [TT06] and unoriented open/closed TFTs with a single boundary condition
[AN06]. The algebraic data required to define an unoriented open/closed TFT
includes:

  * •

A commutative Frobenius algebra $A$; this defines the oriented closed sector.

  * •

A Calabi–Yau category $\mathcal{B}$; this defines the oriented open sector.

  * •

An isometric involution $p:A\rightarrow A$ and a _crosscap state_ $Q\in A$,
the latter corresponding to the value of the TFT on the compact Möbius strip;
this defines the unoriented closed sector.

  * •

A strict contravariant involution of $\mathcal{B}$, that is, a functor
$P:\mathcal{B}^{\textup{\text{op}}}\rightarrow\mathcal{B}$ which squares to
the identity, which is moreover required to be the identity on objects; this
defines the unoriented open sector.

The data (and that which we have omitted here) is required to satisfy a number
of coherence conditions. The oriented theory $\mathcal{Z}_{(G,\theta)}$ is
defined by the commutative Frobenius algebra
$HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G))\simeq
Z(\mathbb{C}^{\theta^{-1}}[G])$ with the Haar bilinear form
$\langle-,-\rangle_{G}$ and Calabi–Yau category
$\textup{\text{Rep}}^{\theta}(G)$. Motivated by the search for the data
required to define the unoriented lift
$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$, in Section 2 we construct and
study a contravariant involution
$(P^{(\hat{G},\hat{\theta},\lambda)},\Theta^{(\hat{G},\hat{\theta},\lambda)})$
of $\textup{\text{Rep}}^{\theta}(G)$. The functor
$P^{(\hat{G},\hat{\theta},\lambda)}$ acts non-trivially on objects and so is
not an admissible choice for the defining data of
$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$. A key representation theoretic
observation is that the homotopy fixed points of
$(P^{(\hat{G},\hat{\theta},\lambda)},\Theta^{(\hat{G},\hat{\theta},\lambda)})$
is the category of $(\hat{G},\hat{\theta},\lambda)$-twisted Real
representations of $G$. The Real representation theory of $G$ was originally
studied by Wigner [Wig59] and Dyson [Dys62] as a generalization of real and
quaternionic representation theory in the context of anti-unitary symmetries
in quantum mechanics. More recently, Real representation theory has been
developed from the related perspective of twisted equivariant $KR$-theory
[AS69, Kar70, FM13, NY22] and categorical representation theory [You21, RY21,
RT22]. In the $K$-theoretic setting, general pairs $(\hat{\theta},\lambda)$
are required to realize all $KR$-theory twists. Motivated by the above
perspectives, we consider the element

$\nu_{(\hat{G},\hat{\theta},\lambda)}=\sum_{\varsigma\in\hat{G}\setminus
G}\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma|\varsigma])}l_{\varsigma^{2}}\in
HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G)).$

The role of $\nu_{(\hat{G},\hat{\theta},\lambda)}$ in Real representation
theory is summarized by the next result.

######  Theorem B (Theorem 2.7 and Corollary 2.8).

Let $V$ be a $\theta$-twisted representation of $G$ with character $\chi_{V}$.
Then $\langle\chi_{V},\nu_{(\hat{G},\hat{\theta},\lambda)}\rangle_{G}$ is
equal to the trace of the involution

$\textup{\text{Hom}}_{G}(V,P^{(\hat{G},\hat{\theta},\lambda)}(V))\rightarrow\textup{\text{Hom}}_{G}(V,P^{(\hat{G},\hat{\theta},\lambda)}(V)).\qquad
f\mapsto
P^{(\hat{G},\hat{\theta},\lambda)}(f)\circ\Theta^{(\hat{G},\hat{\theta},\lambda)}_{V}.$

In particular, if $V$ is irreducible, then

$\langle\chi_{V},\nu_{(\hat{G},\hat{\theta},\lambda)}\rangle_{G}=\begin{cases}1&\mbox{if
and only if $V$ lifts to a $(\hat{G},\hat{\theta},\lambda)$-twisted Real
representation},\\\ -1&\mbox{if and only if $V$ lifts to a
$(\hat{G},\delta\hat{\theta},\lambda)$-twisted Real representation},\\\
0&\mbox{otherwise},\end{cases}$

where $\delta$ is a representative of the generator of
$H^{2}(BC_{2};U(1)_{\pi})\simeq C_{2}$.

The element $\nu_{(\hat{G},\hat{\theta},\lambda)}$ recovers under various
specializations of the data $(\hat{G},\hat{\theta},\lambda)$ other generalized
Frobenius–Schur elements [FS06, Gow79, Tur07, IT23]. In particular, the second
statement in Theorem B shows that $\nu_{(\hat{G},\hat{\theta},\lambda)}$ is a
generalization to twisted Real representation theory of the classical
Frobenius–Schur element. Theorem B and a complete understanding of the
$\theta$-twisted representation theory of $G$ suffices to understand the
$(\hat{G},\hat{\theta},\lambda)$-twisted Real representation theory of $G$.

Returning to the proof of Theorem A, we take for $\mathcal{B}$ the Calabi–Yau
category of $(\hat{G},\hat{\theta},\lambda)$-twisted Real representations of
$G$ and their $G$-equivariant linear maps. We view this as an orientifold-type
construction, with $\textup{\text{Rep}}^{\theta}(G)$ seen as the category of
$D$-branes in an oriented string theory and $\mathcal{B}$ the category of
$D$-branes which survive the orientifold projection defined by
$(P^{(\hat{G},\hat{\theta},\lambda)},\Theta^{(\hat{G},\hat{\theta},\lambda)})$.
The category twisted Real representations is a non-full subcategory of
$\mathcal{B}$ and the forgetful functor
$\mathcal{B}\rightarrow\textup{\text{Rep}}^{\theta}(G)$ respects Calabi–Yau
structures. Moreover, $\mathcal{B}$ inherits a contravariant involution which
is the identity on objects and
$A=HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G))$ inherits an isometric
involution $p$. We take for the crosscap state $Q$ the generalized
Frobenius–Schur element $\nu_{(\hat{G},\hat{\theta},\lambda)}$. It remains to
verify the coherence conditions. A mild generalization of the first equality
in Theorem B (proved in Theorem 2.7) is the unoriented counterpart of the
famous Cardy condition, asserting the equality of two ways of evaluating a
Möbius strip diagram with boundary condition $V$. The remaining coherence
conditions required of the crosscap state, involution $p$ and boundary-bulk
and bulk-boundary maps are verified using the calculus of twisted cocycles. In
Section 3.3, we compute partition functions of
$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$.

### Acknowledgements

The work of M. B. Y. was supported by National Science Foundation grant
DMS-2302363 and a Simons Foundation Collaboration Grant for Mathematicians
(Award ID 853541).

##  1\. Background material

Throughout the paper the ground field is $\mathbb{C}$ and vector spaces are
finite dimensional. Linear duality is
$(-)^{\vee}=\textup{\text{Hom}}_{\mathbb{C}}(-,\mathbb{C})$. Denote by $U(1)$
the group of unit norm complex numbers and $C_{n}$ the cyclic group of order
$n$, seen as a multiplicative group.

###  1.1. Group cohomology

Let $K$ be a finite group and $M$ a left $K$-module. We regard the underlying
abelian group of $M$ as multiplicative. Let $C^{\bullet}(BK;M)$ be the complex
of normalized simplicial cochains on $BK$ with coefficients in $M$. An element
$\theta\in C^{n}(BK;M)$ is a function

$\theta:K^{n}\rightarrow
M,\qquad(k_{n},\dots,k_{1})\mapsto\theta([k_{n}|\cdots|k_{1}])$

whose value is the identity if any $k_{i}$ is the identity. The differential
$d\theta$ of an $(n-1)$-cochain $\theta$ is defined so that
$d\theta([k_{n}|\cdots|k_{1}])$ is equal to

$k_{n}\cdot\theta([k_{n-1}|\cdots|k_{1}])\prod_{j=1}^{n-1}\theta([k_{n}|\cdots|k_{j+1}k_{j}|\cdots|k_{1}])^{(-1)^{n-j}}\times\theta([k_{n}|\cdots|k_{2}])^{(-1)^{n}}.$

Write $Z^{\bullet}(BK;M)$ and $H^{\bullet}(BK;M)$ for the cocycles and
cohomologies of $C^{\bullet}(BK;M)$.

When $M=U(1)$ with trivial $K$-action, write $C^{\bullet}(BK)$ for
$C^{\bullet}(BK;M)$. When $\pi:\hat{G}\rightarrow C_{2}$ is a group
homomorphism and $M=U(1)$ with $\hat{G}$-action $\omega\cdot
z=z^{\pi(\omega)}$, write $C^{\bullet+\pi}(B\hat{G})$ for
$C^{\bullet}(B\hat{G};M)$. If $\pi:C_{2}\rightarrow C_{2}$ is the identity
map, then $H^{2+\pi}(BC_{2})\simeq C_{2}$; a cocycle representative $\delta$
for the generator is given by

$\delta([\varsigma_{2}|\varsigma_{1}])=\begin{cases}-1&\mbox{if
}\varsigma_{1}=\varsigma_{2}=-1,\\\ 0&\mbox{otherwise}.\end{cases}$

We use the same notation for $\delta$ and its image under
$\pi^{*}:Z^{2+\pi}(BC_{2})\rightarrow Z^{2+\pi}(B\hat{G})$.

######  Lemma 1.1.

Let $\pi:\hat{G}\rightarrow C_{2}$ be a $C_{2}$-graded finite group and
$\hat{\theta}\in Z^{2+\pi}(B\hat{G})$. For all $g_{i}\in G$,
$\omega\in\hat{G}$ and $\varsigma\in\hat{G}\setminus G$, the following
equalities hold:

$\frac{\hat{\theta}([\omega g_{2}\omega^{-1}|\omega
g_{1}\omega^{-1}])}{\hat{\theta}([g_{2}|g_{1}])^{\pi(\omega)}}=\frac{\hat{\theta}([\omega
g_{2}\omega^{-1}|\omega])}{\hat{\theta}([\omega|g_{2}])}\frac{\hat{\theta}([\omega
g_{1}\omega^{-1}|\omega])}{\hat{\theta}([\omega|g_{1}])}\left(\frac{\hat{\theta}([\omega
g_{2}g_{1}\omega^{-1}|\omega])}{\hat{\theta}([\omega|g_{2}g_{1}])}\right)^{-1}$
(1)
$\frac{\hat{\theta}([\omega\varsigma\omega^{-1}|\omega\varsigma\omega^{-1}])}{\hat{\theta}([\varsigma|\varsigma])^{-\pi(\omega)}}=\frac{\hat{\theta}([\omega|\varsigma^{2}])}{\hat{\theta}([\omega\varsigma^{2}\omega^{-1}|\omega])}.$
(2)

###### Proof.

Both equalities follow from repeated use of the $2$-cocycle condition on
$\hat{\theta}$. ∎

###  1.2. Twisted representation theory

We recall background on twisted representation theory following [Kar85]. Let
$G$ be a finite group and $\theta\in Z^{2}(BG)$.

######  Definition 1.2.

A _$\theta$ -twisted_ (or _$\theta$ -projective_) _representation of $G$_ is
pair $(V,\rho)$ consisting of a vector space $V$ and a map $\rho:G\rightarrow
GL(V)$ which satisfies $\rho(e)=\textup{\text{id}}_{V}$ and

$\rho(g_{2})\circ\rho(g_{1})=\theta([g_{2}|g_{1}])\rho(g_{2}g_{1}),\qquad
g_{1},g_{2}\in G.$

We often write $V$ or $\rho_{V}$ for $(V,\rho)$. The category
$\textup{\text{Rep}}^{\theta}(G)$ of $\theta$-twisted representations and
their $G$-equivariant linear maps is $\mathbb{C}$-linear finite semisimple.

The $\theta$-twisted group algebra $\mathbb{C}^{\theta}[G]$ is the
$\mathbb{C}$-algebra with basis $\\{l_{g}\mid g\in G\\}$ and multiplication
$l_{g_{2}}\cdot l_{g_{1}}=\theta([g_{2}|g_{1}])l_{g_{2}g_{1}}$. The category
of finite dimensional $\mathbb{C}^{\theta}[G]$-modules is equivalent to
$\textup{\text{Rep}}^{\theta}(G)$. We sometimes interpret
$\mathbb{C}^{\theta}[G]$ as functions on $G$, in which case $l_{g}$ the
$\delta$-function at $g$. The centre $Z(\mathbb{C}^{\theta}[G])$ consists of
elements $\sum_{g\in G}a_{g}l_{g}$ whose coefficients satisfy

$a_{hgh^{-1}}=\uptau(\theta)([h]g)^{-1}a_{g},\qquad g,h\in G.$

Here $\uptau(\theta)([h]g)=\frac{\theta([hgh^{-1}|h])}{\theta([h|g])}$ are the
components of a $1$-cocycle $\uptau(\theta)$ on the loop groupoid of $BG$
called the _loop transgression_ of $\theta$ [Wil08, Theorem 3]. Define a non-
degenerate symmetric bilinear form on $\mathbb{C}^{\theta}[G]$ by

$\langle\sum_{g\in G}a_{g}l_{g},\sum_{h\in
G}b_{h}l_{h}\rangle_{G,\theta}=\frac{1}{|G|}\sum_{g\in
G}\theta([g^{-1}|g])a_{g^{-1}}b_{g}.$

The character of $(V,\rho)\in\textup{\text{Rep}}^{\theta}(G)$ is the function
$\chi_{V}:G\rightarrow\mathbb{C}$, $g\mapsto\textup{\text{tr}}_{V}\,\rho(g)$.
A short calculation shows that
$\chi_{V}(hgh^{-1})=\uptau(\theta)([h]g)\chi_{V}(g)$. Functions
$G\rightarrow\mathbb{C}$ with this conjugation equivariance are elements of
$Z(\mathbb{C}^{\theta^{-1}}[G])$ and are called $\theta$-twisted class
functions. Characters of irreducible $\theta$-twisted representations form an
orthonormal basis of $Z(\mathbb{C}^{\theta^{-1}}[G])$ with respect to
$\langle-,-\rangle_{G}:=\langle-,-\rangle_{G,\theta^{-1}}$.

Given $(V,\rho_{V})\in\textup{\text{Rep}}^{\theta}(G)$, define
$(V^{\vee},\rho_{V^{\vee}})\in\textup{\text{Rep}}^{\theta^{-1}}(G)$ by
$\rho_{V^{\vee}}(g)=(\rho_{V}(g)^{-1})^{\vee}$. For ease of notation, we write
$\rho_{V}(g)^{-\vee}$ for $(\rho_{V}(g)^{-1})^{\vee}$.

###  1.3. Categories with duality

######  Definition 1.3.

  1. (1)

A _category with duality_ is a triple $(\mathcal{C},P,\Theta)$ consisting of a
category $\mathcal{C}$, a functor
$P:\mathcal{C}^{\textup{\text{op}}}\rightarrow\mathcal{C}$ and a natural
isomorphism $\Theta:\textup{\text{id}}_{\mathcal{C}}\Rightarrow P\circ
P^{\textup{\text{op}}}$ whose components satisfy

$P(\Theta_{V})\circ\Theta_{P(V)}=\textup{\text{id}}_{P(V)},\qquad
V\in\mathcal{C}.$ (3)

The duality structure $(P,\Theta)$ is _strict_ if $\Theta$ is the identity
natural transformation.

  2. (2)

A _homotopy fixed point_ of $(\mathcal{C},P,\Theta)$ is a pair $(V,\psi_{V})$
consisting of an object $V\in\mathcal{C}$ and an isomorphism
$\psi_{V}:V\rightarrow P(V)$ which satisfies
$P(\psi_{V})\circ\Theta_{V}=\psi_{V}$.

We interpret $(P,\Theta)$ as defining a categorical $C_{2}$-action on
$\mathcal{C}$ in which the generator acts contravariantly. Motivated by this,
let $\mathcal{C}^{hC_{2}}$, $\mathcal{C}^{\tilde{h}C_{2}}$ be the categories
with objects homotopy fixed points and morphisms

$\textup{\text{Hom}}_{\mathcal{C}^{hC_{2}}}((V,\psi_{V}),(W,\psi_{W}))=\\{\phi\in\textup{\text{Hom}}_{\mathcal{C}}(V,W)\mid\psi_{V}=P(\phi)\circ\psi_{W}\circ\phi\\},$
$\textup{\text{Hom}}_{\mathcal{C}^{\tilde{h}C_{2}}}((V,\psi_{V}),(W,\psi_{W}))=\textup{\text{Hom}}_{\mathcal{C}}(V,W).$

Let
$P^{\tilde{h}C_{2}}:(\mathcal{C}^{\tilde{h}C_{2}})^{\textup{\text{op}}}\rightarrow\mathcal{C}^{\tilde{h}C_{2}}$
be the identity on objects and send a morphism
$\phi:(V,\psi_{V})\rightarrow(W,\psi_{W})$ to
$P^{\tilde{h}C_{2}}(\phi)=\psi_{V}^{-1}\circ P(\phi)\circ\psi_{W}$. Let
$\Theta^{\tilde{h}C_{2}}:\textup{\text{id}}_{\mathcal{C}^{\tilde{h}C_{2}}}\Rightarrow
P^{\tilde{h}C_{2}}\circ(P^{\tilde{h}C_{2}})^{\textup{\text{op}}}$ be the
identity natural transformation.

######  Lemma 1.4.

The triple
$(\mathcal{C}^{\tilde{h}C_{2}},P^{\tilde{h}C_{2}},\Theta^{\tilde{h}C_{2}})$ is
a category with strict duality. Moreover, $P^{\tilde{h}C_{2}}$ is the identity
on objects.

##  2\. A Frobenius–Schur indicator for twisted Real representation theory

###  2.1. Twisted Real representation theory

The Real representation theory of a finite group has been studied by many
authors as a generalization of representation theory over $\mathbb{R}$ or
$\mathbb{H}$ [Wig59, Dys62, AS69, Kar70, FM13, You21]. We establish relevant
aspects of the twisted form of this theory following [You21, §3.2].

Let $\pi:\hat{G}\rightarrow C_{2}$ be a $C_{2}$-graded finite group with $\pi$
surjective. Fix $\hat{\theta}\in Z^{2+\pi}(B\hat{G})$ and a character
$\lambda:\hat{G}\rightarrow U(1)$. Note that $\lambda$ can be interpreted as
an element of $Z^{1}(B\hat{G})$. Denote by $G=\ker\pi$ and $\theta\in
Z^{2}(BG)$ the restriction of $\hat{\theta}$ along $BG\rightarrow B\hat{G}$.

An element $\varsigma\in\hat{G}\backslash G$ determines a $\mathbb{C}$-linear
exact duality structure
$(P^{(\hat{\theta},\lambda,\varsigma)},\Theta^{(\hat{\theta},\lambda,\varsigma)})$
on $\textup{\text{Rep}}^{\theta}(G)$. On objects, we have
$P^{(\hat{\theta},\lambda,\varsigma)}(V,\rho)=(V^{\vee},\rho^{(\hat{\theta},\lambda,\varsigma)})$,
where

$\rho^{(\hat{\theta},\lambda,\varsigma)}(g)=\frac{\lambda(g)}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)}\rho(\varsigma
g^{-1}\varsigma^{-1})^{\vee},\qquad g\in G.$

The coefficients

$\uptau^{\textup{\text{refl}}}_{\pi}(\hat{\theta})([\omega]g)=\hat{\theta}([g^{-1}|g])^{\frac{\pi(\omega)-1}{2}}\frac{\hat{\theta}([\omega
g^{\pi(\omega)}\omega^{-1}|\omega])}{\hat{\theta}([\omega|g^{\pi(\omega)}])},\qquad
g\in G,\;\omega\in\hat{G}$

are best understood in terms of orientation-twisted loop transgression [NY22,
Theorem 2.8], which is a cochain map

$\uptau^{\textup{\text{refl}}}_{\pi}:C^{\bullet+\pi}(B\hat{G})\rightarrow
C^{\bullet-1}(B(G/\\!\\!/_{R}\hat{G})).$

The codomain is simplicial cochains on the classifying space of the quotient
groupoid $G/\\!\\!/_{R}\hat{G}$ resulting from the Real conjugation action of
$\hat{G}$ on $G$: $\omega\cdot g=\omega g^{\pi(\omega)}\omega^{-1}$,
$\omega\in\hat{G},\;g\in G$. In geometric terms, $G/\\!\\!/_{R}\hat{G}$ is the
unoriented loop groupoid of $B\hat{G}$, that is, the quotient of the loop
groupoid of $BG$ by the $C_{2}$-action which reverses orientation of loops and
acts on $BG$ by deck transformations. Continuing, on morphisms
$P^{(\hat{\theta},\lambda,\varsigma)}$ is $\mathbb{C}$-linear duality. The
natural isomorphism $\Theta^{(\hat{\theta},\lambda,\varsigma)}$ is defined by
its components

$\Theta^{(\hat{\theta},\lambda,\varsigma)}_{V}=\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma|\varsigma])}\textup{\text{ev}}_{V}\circ\rho(\varsigma^{2})^{-1},\qquad(V,\rho)\in\textup{\text{Rep}}^{\theta}(G)$

where $\textup{\text{ev}}_{V}:V\rightarrow V^{\vee\vee}$ is the evaluation
isomorphism of underlying vector spaces. The normalization of
$\Theta^{(\hat{\theta},\lambda,\varsigma)}_{V}$ ensures that the coherence
condition (3) holds.

######  Definition 2.1.

The _category $\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ of
$(\hat{\theta},\lambda)$-twisted Real representations of $G$_ is the homotopy
fixed point category $\textup{\text{Rep}}^{\theta}(G)^{hC_{2}}$ of
$(P^{(\hat{\theta},\lambda,\varsigma)},\Theta^{(\hat{\theta},\lambda,\varsigma)})$.

Up to equivalence,
$(P^{(\hat{\theta},\lambda,\varsigma)},\Theta^{(\hat{\theta},\lambda,\varsigma)})$
depends only on $(\hat{G},[\hat{\theta}],\lambda)$. The same is therefore true
of $\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ and we drop $\varsigma$
from the notation if it is fixed or the particular realization of the duality
structure is not important.

Concretely, an object
$(V,\psi_{V})\in\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ is an
isomorphism $\psi_{V}:V\rightarrow P^{(\hat{\theta},\lambda)}(V)$ in
$\textup{\text{Rep}}^{\theta}(G)$ which satisfies
$P^{(\hat{\theta},\lambda)}(\psi_{V})\circ\Theta^{(\hat{\theta},\lambda)}_{V}=\psi_{V}$.
A morphism $\phi:(V,\psi_{V})\rightarrow(W,\psi_{W})$ in
$\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ is a morphism in
$\textup{\text{Rep}}^{\theta}(G)$ which satisfies
$P^{(\hat{\theta},\lambda)}(\phi)\circ\psi_{W}\circ\phi=\psi_{V}$. Note that
$\phi$ is necessarily injective and
$\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(\hat{G})$ is neither linear nor
abelian.

A more standard representation theoretic interpretation of
$\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ is as follows. Given a
vector space $V$ and sign $\epsilon\in C_{2}$, introduce the notation

${{}^{\epsilon}}V=\begin{cases}V&\mbox{if }\epsilon=1,\\\ V^{\vee}&\mbox{if
}\epsilon=-1\end{cases}$

with similar notation for linear maps. A $(\hat{\theta},\lambda)$-twisted Real
representation of $G$ is then a vector space $V$ together with linear maps
$\rho(\omega):\prescript{\pi(\omega)}{}{V}\rightarrow V$, $\omega\in\hat{G}$,
which satisfy $\rho(e)=\textup{\text{id}}_{V}$ and

$\rho(\omega_{2})\circ\prescript{\pi(\omega_{2})}{}{\rho(\omega_{1})}^{\pi(\omega_{2})}\circ\textup{\text{ev}}_{V}^{\delta_{\pi(\omega_{1}),\pi(\omega_{2}),-1}}=\lambda(\omega_{1})^{\frac{\pi(\omega_{2})-1}{2}}\hat{\theta}([\omega_{2}|\omega_{1}])\rho(\omega_{2}\omega_{1}).$
(4)

The notation
$\textup{\text{ev}}_{V}^{\delta_{\pi(\omega_{1}),\pi(\omega_{2}),-1}}$
indicates that $\textup{\text{ev}}_{V}$ is included exactly when
$\pi(\omega_{1})=\pi(\omega_{2})=-1$. The equivalence of this interpretation
with that of homotopy fixed points follows from noting that a homotopy fixed
point $((V,\rho_{V}),\psi_{V})$ determines an extension of $\rho_{V}$ to
$\hat{G}\setminus G$ by

$\rho_{V}(\omega)=\hat{\theta}([\omega\varsigma^{-1}|\varsigma])^{-1}\rho_{V}(\omega\varsigma^{-1})\circ\psi_{V}^{-1},\qquad\omega\in\hat{G}\setminus
G.$

A third interpretation of twisted Real representations will also be useful.

######  Proposition 2.2.

Fix $\varsigma\in\hat{G}\setminus G$. A $(\hat{\theta},\lambda)$-twisted Real
representation of $G$ is equivalent to the data of a $\theta$-twisted
representation of $G$ on $V$ together with a non-degenerate bilinear form
$\langle-,-\rangle:V\times V\rightarrow\mathbb{C}$ which satisfies the twisted
$G$-invariance condition

$\langle\rho(g)v_{1},\rho(\varsigma
g\varsigma^{-1})v_{2}\rangle=\lambda(g)\frac{\hat{\theta}([\varsigma|g])}{\hat{\theta}([\varsigma
g\varsigma^{-1}|\varsigma])}\langle v_{1},v_{2}\rangle,\qquad g\in G$

and the twisted symmetry condition

$\langle
v_{1},v_{2}\rangle=\lambda(\varsigma)\theta([\varsigma^{-1}|\varsigma^{-1}])\langle\rho(\varsigma^{-2})v_{2},v_{1}\rangle$

for all $v_{1},v_{2}\in V$.

###### Proof.

Let $(V,\rho)$ be a $(\hat{\theta},\lambda)$-twisted Real representation of
$G$. Fix $\varsigma\in\hat{G}\setminus G$ and define a non-degenerate bilinear
form on $V$ by

$\langle v_{1},v_{2}\rangle=\rho(\varsigma^{-1})^{-1}(v_{1})v_{2}.$ (5)

With this definition, $\langle\rho(g)v_{1},\rho(\varsigma
g\varsigma^{-1})v_{2}\rangle$ is equal to

$\displaystyle\rho(\varsigma^{-1})^{-1}(\rho(g)(v_{1}))(\rho(\varsigma
g\varsigma^{-1})v_{2})$ $\displaystyle=$ $\displaystyle\lambda(\varsigma
g)\hat{\theta}([\varsigma^{-1}|\varsigma g])^{-1}\rho(\varsigma
g)^{-\vee}(\textup{\text{ev}}_{V}(v_{1}))(\rho(\varsigma
g\varsigma^{-1})v_{2})$ $\displaystyle=$ $\displaystyle\lambda(\varsigma
g)\lambda(\varsigma^{-1})\hat{\theta}([\varsigma^{-1}|\varsigma
g])^{-1}\hat{\theta}([\varsigma g|\varsigma^{-1}])^{-1}\rho(\varsigma
g\varsigma^{-1})^{-\vee}\rho(\varsigma^{-1})^{-1}(v_{1})(\rho(\varsigma
g\varsigma^{-1})v_{2})$ $\displaystyle=$
$\displaystyle\lambda(g)\frac{\hat{\theta}([\varsigma|g])}{\hat{\theta}([\varsigma
g\varsigma^{-1}|\varsigma])}\langle v_{1},v_{2}\rangle.$

The first two equalities follow from equation (4) and the third from the
$2$-cocycle condition on $\hat{\theta}$. Similarly, we compute

$\displaystyle\langle v_{1},v_{2}\rangle$ $\displaystyle=$
$\displaystyle\lambda(\varsigma)\hat{\theta}([\varsigma^{-1}|\varsigma])^{-1}\rho(\varsigma)^{-\vee}(\textup{\text{ev}}_{V}(v_{1}))v_{2}$
$\displaystyle=$
$\displaystyle\lambda(\varsigma)\hat{\theta}([\varsigma^{-1}|\varsigma])^{-1}\rho(\varsigma^{-2})^{-\vee}\circ\rho(\varsigma)^{-\vee}(\textup{\text{ev}}_{V}(v_{1}))(\rho(\varsigma^{-2})v_{2})$
$\displaystyle=$
$\displaystyle\lambda(\varsigma)\hat{\theta}([\varsigma^{-1}|\varsigma])^{-1}\hat{\theta}([\varsigma^{-2}|\varsigma])^{-1}\textup{\text{ev}}_{V}(v_{1})(\rho(\varsigma^{-1})^{-1}\circ\rho(\varsigma^{-2})v_{2})$
$\displaystyle=$
$\displaystyle\lambda(\varsigma)\hat{\theta}([\varsigma^{-1}|\varsigma^{-1}])\langle\rho(\varsigma^{-2})v_{2},v_{1}\rangle.$

Conversely, given $(V,\rho)\in\textup{\text{Rep}}^{\theta}(G)$ with non-
degenerate bilinear form $\langle-,-\rangle$ satisfying the conditions of the
lemma, define $\rho(\varsigma^{-1})$ by equation (5) and set

$\rho(\omega)=\hat{\theta}([\omega\varsigma|\varsigma^{-1}])^{-1}\rho(\omega\varsigma)\circ\rho(\varsigma^{-1}),\qquad\omega\in\hat{G}\setminus
G.$

The verification that $\rho$ is a $(\hat{\theta},\lambda)$-twisted Real
representation of $G$ mirrors the calculations from the previous paragraph. ∎

A $(\hat{\theta},\lambda)$-twisted Real representation is called _irreducible_
if it has no non-trivial Real subrepresentations. The direct sum
$(V,\psi_{V})\oplus(W,\psi_{W})=(V\oplus W,\psi_{V}\oplus\psi_{W})$ allows for
the following formulation of a Real analogue of Maschke’s lemma.

######  Proposition 2.3.

Let $V\in\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ be irreducible.
Then the restriction of $V$ to $G$ is irreducible or of the form $U\oplus
P^{(\hat{\theta},\lambda)}(U)$ for an irreducible
$U\in\textup{\text{Rep}}^{\theta}(G)$.

###### Proof.

Interpret $V$ as a $\theta$-twisted representation of $G$ with compatible
bilinear form $\langle-,-\rangle$, as in Proposition 2.2, and suppose that the
restriction $V_{|G}$ has a non-trivial irreducible $\theta$-twisted
subrepresentation $U$. The twisted $G$-invariance of $\langle-,-\rangle$
implies that the orthogonal complement $U^{\perp}$ is a $\theta$-twisted
subrepresentation of $V_{|G}$ and $V_{|G}=U\oplus U^{\perp}$ as
$\theta$-twisted representations. Since $V$ is irreducible, the map
$\rho(\varsigma):V^{\vee}\rightarrow V$ restricts to a map
$\rho(\varsigma):U^{\vee}\rightarrow U^{\perp}$ which defines an isomorphism
$P^{(\hat{\theta},\lambda)}(U)\xrightarrow[]{\sim}U^{\perp}$ of
$\theta$-twisted representations. ∎

###  2.2. A Frobenius–Schur indicator

Keep the notation of Section 2.1.

######  Definition 2.4.

The _$(\hat{\theta},\lambda)$ -twisted Frobenius–Schur element_ is

$\nu_{(\hat{\theta},\lambda)}=\sum_{\varsigma\in\hat{G}\setminus
G}\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma|\varsigma])}l_{\varsigma^{2}}\in\mathbb{C}^{\theta^{-1}}[G].$

When $(\hat{\theta},\lambda)$ is clear from the context, we write $\nu$ for
$\nu_{(\hat{\theta},\lambda)}$. Note that

$\nu_{(\hat{\theta},\lambda)}=-\nu_{(\delta\hat{\theta},\lambda)}=-\nu_{(\hat{\theta},\pi\lambda)}.$

######  Lemma 2.5.

The element $\nu_{(\hat{\theta},\lambda)}$ is a $\theta$-twisted class
function on $G$.

###### Proof.

The statement amounts to the identity

$\hat{\theta}([h\varsigma h^{-1}|h\varsigma
h^{-1}])^{-1}=\uptau(\theta)([h]\varsigma^{2})\hat{\theta}([\varsigma|\varsigma])^{-1},\qquad
h\in G,\;\varsigma\in\hat{G}\backslash G,$

which is seen to hold using equation (2). ∎

We require the following elementary result from linear algebra.

######  Lemma 2.6.

Let $V$ be a finite dimensional vector space and
$\phi\in\textup{\text{Hom}}_{\mathbb{C}}(V,V)$. Then
$\textup{\text{tr}}_{V}\,\phi$ is equal to the trace of the map

$\iota_{\phi}:\textup{\text{Hom}}_{\mathbb{C}}(V,V^{\vee})\rightarrow\textup{\text{Hom}}_{\mathbb{C}}(V,V^{\vee}),\qquad
f\mapsto f^{\vee}\circ\textup{\text{ev}}_{V}\circ\phi.$

###### Proof.

Let $\dim_{\mathbb{C}}V=v$. Fix a basis of $V$ with induced basis
$\\{E_{ij}\\}_{i,j=1}^{v}$ of $\textup{\text{Hom}}_{\mathbb{C}}(V,V)$. Writing
$\phi=\sum_{i,j=1}^{v}\phi_{ij}E_{ij}$ in this basis, we compute
$\iota_{\phi}(E_{ij})=\sum_{k=1}^{v}\phi_{ik}E_{jk}$ so that

$\textup{\text{tr}}_{\textup{\text{Hom}}_{\mathbb{C}}(V,V^{\vee})}\,\iota_{\phi}=\sum_{i,j=1}^{v}\iota_{\phi}(E_{ij})_{ij}=\sum_{i,j,k=1}^{v}\phi_{ik}(E_{jk})_{ij}=\sum_{i,j,k=1}^{v}\phi_{ik}\delta_{ji}\delta_{kj}=\textup{\text{tr}}_{V}\,\phi.\qed$

Let $V\in\textup{\text{Rep}}^{\theta}(G)$ and
$\phi\in\textup{\text{Hom}}_{G}(V,V)$. Consider the map

$\iota_{\phi}:\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda,\varsigma)}(V))\rightarrow\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda,\varsigma)}(V)),\qquad
f\mapsto
P^{(\hat{\theta},\lambda,\varsigma)}(f)\circ\Theta^{(\hat{\theta},\lambda,\varsigma)}_{V}\circ\phi.$

Independence of the duality structure up to equivalence on
$\varsigma\in\hat{G}\setminus G$ implies that $\iota_{\phi}$ is independent of
$\varsigma$. The coherence condition (3) implies that
$\iota:=\iota_{\textup{\text{id}}_{V}}$ is an involution.

For each $V\in\textup{\text{Rep}}^{\theta}(G)$, define

$\tau^{V}:\textup{\text{Hom}}_{G}(V,V)\rightarrow
Z(\mathbb{C}^{\theta^{-1}}[G]),\qquad\phi\mapsto\sum_{g\in
G}\textup{\text{tr}}_{V}(\phi\circ\rho_{V}(g))l_{g}.$

Note that $\tau^{V}(\textup{\text{id}}_{V})=\chi_{V}$.

######  Theorem 2.7.

For each $V\in\textup{\text{Rep}}^{\theta}(G)$ and
$\phi\in\textup{\text{Hom}}_{G}(V,V)$, there is an equality

$\textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda)}(V))}\,\iota_{\phi}=\langle\tau^{V}(\phi),\nu_{(\hat{\theta},\lambda)}\rangle_{G}.$

###### Proof.

Write $(P,\Theta)$ for
$(P^{(\hat{\theta},\lambda,\varsigma)},\Theta^{(\hat{\theta},\lambda,\varsigma)})$.
We compute

$\displaystyle\textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P(V))}\,\iota_{\phi}$
$\displaystyle=$
$\displaystyle\textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P(V))}\,(f\mapsto\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma|\varsigma])}f^{\vee}\circ\textup{\text{ev}}_{V}\circ\rho(\varsigma^{2})^{-1}\circ\phi)$
$\displaystyle=$
$\displaystyle\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma|\varsigma])}\textup{\text{tr}}_{V}\,(\rho(\varsigma^{2})^{-1}\circ\phi),$

the second equality following from Lemma 2.6. Since
$\textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P(V))}\,\iota_{\phi}$ is
independent of the choice $\varsigma\in\hat{G}\setminus G$ used in the
definition of $\iota_{\phi}$, we average over all such choices to obtain

$\textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P(V))}\,\iota_{\phi}=\frac{1}{|G|}\sum_{\varsigma\in\hat{G}\backslash
G}\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma|\varsigma])}\textup{\text{tr}}_{V}\,(\rho(\varsigma^{2})^{-1}\circ\phi).$

On the other hand, we have

$\displaystyle\langle\tau^{V}(\phi),\nu\rangle_{G}$ $\displaystyle=$
$\displaystyle\frac{1}{|G|}\sum_{\varsigma\in\hat{G}\backslash
G}\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma|\varsigma])}\theta([\varsigma^{2}|\varsigma^{-2}])^{-1}\textup{\text{tr}}_{V}\,\left(\phi\circ\rho_{V}(\varsigma^{-2})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{|G|}\sum_{\varsigma\in\hat{G}\backslash
G}\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma|\varsigma])}\textup{\text{tr}}_{V}\,\left(\rho_{V}(\varsigma^{2})^{-1}\circ\phi\right),$

thereby proving the desired equality. ∎

Recall that $\delta$ is a cocycle representative of the generator of
$H^{2+\pi}(BC_{2})\simeq C_{2}$.

######  Corollary 2.8.

Let $V$ be an irreducible $\theta$-twisted representation of $G$. Then

$\langle\chi_{V},\nu_{(\hat{\theta},\lambda)}\rangle_{G}=\begin{cases}1&\mbox{if
and only if $V$ lifts to a $(\hat{\theta},\lambda)$-twisted Real
representation},\\\ -1&\mbox{if and only if $V$ lifts to a
$(\delta\hat{\theta},\lambda)$-twisted Real representation},\\\
0&\mbox{otherwise}.\end{cases}$

When $\langle\chi_{V},\nu_{(\hat{\theta},\lambda)}\rangle_{G}=\pm 1$, the
twisted Real structure on $V$ is unique up to isomorphism.

###### Proof.

Schur’s Lemma for $\textup{\text{Rep}}^{\theta}(G)$ implies that
$\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda)}(V))\simeq\mathbb{C}$ if
$P^{(\hat{\theta},\lambda)}(V)\simeq V$ and
$\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda)}(V))=0$ otherwise. Hence,
if $\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda)}(V))=0$, then $V$ does
not lift to a Real representation and the statement follows by applying
Theorem 2.7 with $\phi=\textup{\text{id}}_{V}$. If
$\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda)}(V))\simeq\mathbb{C}$,
then a non-zero element
$\psi_{V}\in\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda)}(V))$ is an
isomorphism which, by Theorem 2.7, satisfies

$P^{(\hat{\theta},\lambda)}(\psi_{V})\circ\Theta_{V}=\langle\chi_{V},\nu_{(\hat{\theta},\lambda)}\rangle_{G}\cdot\psi_{V}.$

The first statement of the corollary now follows from the homotopy fixed point
interpretation of twisted Real representations. Uniqueness of the Real
structure up to isomorphism follows from one dimensionality of
$\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda)}(V))$. ∎

In the setting of Corollary 2.8, if
$\langle\chi_{V},\nu_{(\hat{\theta},\lambda)}\rangle_{G}=0$, then $V$
determines an irreducible $(\hat{\theta},\lambda)$-twisted Real representation
by $H^{(\hat{\theta},\lambda)}(V)=V\oplus P^{(\hat{\theta},\lambda)}(V)$ with
its hyperbolic homotopy fixed point structure [You21, §7.3]. Note that
$H^{(\hat{\theta},\lambda)}(V)\simeq
H^{(\hat{\theta},\lambda)}(P^{(\hat{\theta},\lambda)}(V))$.

######  Corollary 2.9.

There are finitely many isomorphism classes of irreducible
$(\hat{\theta},\lambda)$-twisted Real representations.

###### Proof.

This follows from Proposition 2.3, finiteness of isomorphism classes of
irreducible $\theta$-twisted representations and the final statement of
Corollary 2.8. ∎

######  Corollary 2.10.

There is an equality

$\nu_{(\hat{\theta},\lambda)}=\sum_{\begin{subarray}{c}V\in\textup{\text{Irr}}^{\theta}(G)\\\
P^{(\hat{\theta},\lambda)}(V)\simeq
V\end{subarray}}\langle\chi_{V},\nu_{(\hat{\theta},\lambda)}\rangle_{G}\chi_{V}.$

###### Proof.

This follows from the second part of Corollary 2.8 and the fact that the set
$\\{\chi_{V}\\}_{V\in\textup{\text{Irr}}^{\theta}(G)}$ of irreducible
$\theta$-twisted characters is an orthonormal basis of the space of
$\theta$-twisted class function on $G$. ∎

Various instances of the element $\nu_{(\hat{\theta},\lambda)}$ and Corollary
2.8 are known:

  1. (1)

The classical setting of Frobenius and Schur [FS06] corresponds to taking
$\hat{G}=G\times C_{2}$ with $\pi$ the projection to the second factor and the
cohomological data $\hat{\theta}$ and $\lambda$ trivial. The conditions on the
bilinear form $\langle-,-\rangle$ from Proposition 2.2 reduce to
$G$-invariance and symmetry; if $\hat{\theta}=\delta$, then
$\langle-,-\rangle$ is skew-symmetric. Corollary 2.8 then gives the standard
necessary and sufficient condition for $V$ to admit a $G$-invariant bilinear
form and so be defined over $\mathbb{R}$ (in the symmetric case) or
$\mathbb{H}$ (in the skew-symmetric case).

  2. (2)

Taking $\hat{\theta}$ and $\lambda$ to be trivial recovers Gow’s generalized
Frobenius–Schur element used in the character theoretic study of
$2$-regularity of finite groups [Gow79, §2]. For representation theoretic
applications, see [RT21].

  3. (3)

Take $\hat{G}=G\times C_{2}$. In this case, there is a homomorphism
$G\rightarrow\hat{G}$ which splits $\pi$. When $\hat{\theta}$ is in the image
of the resulting map $H^{2}(BG;C_{2})\rightarrow H^{2+\pi}(B\hat{G})$ and
$\lambda$ is trivial, $\nu_{(\hat{\theta},1)}$ recovers Turaev’s generalized
Frobenius–Schur element studied in the context of closed unoriented TFT
[Tur07]. See [IT23] for a generalization in the setting of closed
$\textup{\text{Pin}}_{2}^{-}$ TFT.

  4. (4)

When $\phi=\textup{\text{id}}_{V}$ the trace of Theorem 2.7 is an instance of
a Shimizu’s Frobenius–Schur indicator in a category with duality [Shi12].

######  Example 2.11.

Let $G=C_{n}$ with generator $r$ and $\zeta=e^{\frac{2\pi\sqrt{-1}}{n}}$. The
one dimensional representations $\\{\rho_{k}\mid 0\leq k\leq n-1\\}$, defined
by $\rho_{k}(r)=\zeta^{k}$, constitute a complete set of irreducible
representations of $G$. Take $\hat{\theta}$ and $\lambda$ to be trivial in
this example.

  1. (1)

Let $\hat{G}=C_{n}\times C_{2}$ with $\pi$ projection to the second factor. We
have

$\langle\chi_{k},\nu\rangle_{G}=\begin{cases}1&\mbox{if $k=0$ or
$k=\frac{n}{2}$},\\\ 0&\text{otherwise},\end{cases}$

whence the trivial and sign representation (which exists when $n$ is even)
admit Real structures. These are precisely the irreducible representations
which are defined over $\mathbb{R}$.

  2. (2)

Let $\hat{G}=C_{2n}$ with generator $\varsigma$ satisfying $\varsigma^{2}=r$
and $C_{2}$-grading $\pi:\hat{G}\rightarrow C_{2}$ determined by
$\pi(\varsigma)=-1$. Assume that $n$ is even, as otherwise $\hat{G}\simeq
C_{n}\times C_{2}$ as $C_{2}$-graded groups. We have
$\nu=2\sum_{j=0}^{\frac{n}{2}-1}l_{r^{2j}}$ from which we compute

$\langle\chi_{k},\nu\rangle_{G}=\frac{2}{n}\sum_{j=0}^{\frac{n}{2}-1}\zeta^{2kj}\\\
=\begin{cases}1&\mbox{if }k=0,\\\ -1&\mbox{if }k=\frac{n}{2},\\\
0&\mbox{otherwise}.\end{cases}$

The Real structure on $\rho_{0}$ is given by
$\rho_{0}(\varsigma)(1^{\vee})=1$. The same formula gives the $\delta$-twisted
Real structure on $\rho_{\frac{n}{2}}$.

  3. (3)

Let $\hat{G}$ be the dihedral group $D_{2n}=\langle r,s\mid
r^{n}=s^{2}=e,\,srs=r^{-1}\rangle$ with $\pi:\hat{G}\rightarrow C_{2}$
determined by $\pi(r)=1$ and $\pi(s)=-1$. We have $\nu=nl_{e}$ from which we
compute $\langle\chi_{k},\nu\rangle_{G}=1$. Each irreducible representation
$\rho_{k}$ can therefore be extended to a Real representation by the formula
$\rho_{k}(s)(1^{\vee})=1$.∎

######  Example 2.12.

Let $\hat{G}=Q_{8}$ be the quaternion group with $C_{2}$-grading given on the
standard generators by $\pi(i)=1$ and $\pi(j)=-1$. Then $G\simeq C_{4}$ is
generated by $i$. We have $\nu=4l_{-1}$ so that
$\langle\chi_{k},\nu\rangle=(-1)^{k}$. The Real structure on $\rho_{k}$, which
is $\delta$-twisted precisely when $k$ is even, is determined by
$\rho_{k}(j)(1^{\vee})=1$. ∎

######  Example 2.13.

Let $G=A_{4}$ be the alternating group on $4$ letters. The irreducible
representations of $G$ are the trivial representation $U$, two non-trivial one
dimensional representations $U^{\prime}$ and $U^{\prime\prime}$ and a three
dimensional representation $V$. Writing $\zeta=e^{\frac{2\pi\sqrt{-1}}{3}}$,
we take the convention that their characters are

$\displaystyle\chi_{U^{\prime}}(123)$ $\displaystyle=\zeta,$
$\displaystyle\chi_{U^{\prime}}(132)$ $\displaystyle=\zeta^{2},$
$\displaystyle\chi_{U^{\prime}}((12)(34))$ $\displaystyle=1,$
$\displaystyle\chi_{U^{\prime\prime}}(123)$ $\displaystyle=\zeta^{2},$
$\displaystyle\chi_{U^{\prime\prime}}(132)$ $\displaystyle=\zeta,$
$\displaystyle\chi_{U^{\prime\prime}}((12)(34))$ $\displaystyle=1,$
$\displaystyle\chi_{V}(123)$ $\displaystyle=0,$ $\displaystyle\chi_{V}(132)$
$\displaystyle=0,$ $\displaystyle\chi_{V}((12)(34))$ $\displaystyle=-1.$

  1. (1)

Taking $\hat{G}=A_{4}\times C_{2}$ with $\pi$ the projection to the second
factor gives $\nu=4l_{(1)}+\sum_{3\mbox{\tiny-cycles}\;\sigma}l_{\sigma}$.
Using this, we compute

$\langle\chi_{U},\nu\rangle=1,\qquad\langle\chi_{U^{\prime}},\nu\rangle=0,\qquad\langle\chi_{U^{\prime\prime}},\nu\rangle=0,\qquad\langle\chi_{V},\nu\rangle=1.$

Hence, only $U$ and $V$ admit real structures.

  2. (2)

Taking $\hat{G}=S_{4}$ the symmetric group with $\pi$ the sign representation
gives $\nu=6l_{(1)}+2(l_{(12)(34)}+l_{(13)(24)}+l_{(14)(23)})$. Using this, we
compute

$\langle\chi_{U},\nu\rangle=1,\qquad\langle\chi_{U^{\prime}},\nu\rangle=1,\qquad\langle\chi_{U^{\prime\prime}},\nu\rangle=1,\qquad\langle\chi_{V},\nu\rangle=0.$

Hence, all one dimensional representations admit Real structures. Taking
$\lambda$ to be non-trivial, that is, $\lambda=\pi$, replaces $\nu$ with its
negative and leads to $\delta$-twisted Real structures on the one dimensional
representations.∎

##  3\. Two dimensional unoriented open/closed topological field theory

###  3.1. Algebraic characterization

Following Lazaroiu [Laz01] and Moore and Segal [MS06], we begin by recalling
an algebraic characterization of two dimensional oriented open/closed
topological field theories (TFTs). See also [AN06, LP08]. In topological
terms, such a TFT is a symmetric monoidal functor
$\mathcal{Z}:\textup{\text{Bord}}_{2}^{\textup{\text{or}},D}\rightarrow\textup{\text{Vect}}_{\mathbb{C}}$.
Here $\textup{\text{Bord}}_{2}^{\textup{\text{or}},D}$ two dimensional
open/closed bordism category [LP08, §3]. Objects are compact oriented
$1$-manifolds with boundary components labelled by elements of a given set
$D$. Morphisms are isomorphism classes of oriented bordisms with corners whose
free boundaries are $D$-labelled compatibly with the incoming and outgoing
boundaries. The monoidal structure of
$\textup{\text{Bord}}_{2}^{\textup{\text{or}},D}$ is disjoint union.

######  Theorem 3.1 ([MS06, Theorem 1]).

Two dimensional oriented open/closed TFTs are classified by the following
data:

  1. (1)

A commutative Frobenius algebra $A$ with identity $1_{A}$ and trace
$\langle-\rangle_{0}:A\rightarrow\mathbb{C}$.

  2. (2)

A Calabi–Yau category $\mathcal{B}$, that is, $\mathbb{C}$-linear additive
category with cyclic traces
$\langle-\rangle_{V}:\textup{\text{Hom}}_{\mathcal{B}}(V,V)\rightarrow\mathbb{C}$,
$V\in\mathcal{B}$, whose associated pairings

$\langle-,-\rangle_{V,W}:\textup{\text{Hom}}_{\mathcal{B}}(W,V)\otimes\textup{\text{Hom}}_{\mathcal{B}}(V,W)\xrightarrow[]{\circ}\textup{\text{Hom}}_{\mathcal{B}}(V,V)\xrightarrow[]{\langle-\rangle_{V}}\mathbb{C}$

are non-degenerate.

  3. (3)

For each $V\in\mathcal{B}$, a linear _boundary-bulk_ map
$\tau^{V}:\textup{\text{Hom}}_{\mathcal{B}}(V,V)\rightarrow A$ and linear
_bulk-boundary_ map
$\tau_{V}:A\rightarrow\textup{\text{Hom}}_{\mathcal{B}}(V,V)$.

This data is required to satisfy the following conditions:

  1. (i)

$\tau_{V}$ is a unital algebra homomorphism.

  2. (ii)

$\tau_{W}(a)\circ\phi=\phi\circ\tau_{V}(a)$ for all $a\in A$ and
$\phi\in\textup{\text{Hom}}_{\mathcal{B}}(V,W)$.

  3. (iii)

$\langle\phi,\tau_{V}(a)\rangle_{V,V}=\langle\tau^{V}(\phi),a\rangle_{0}$ for
all $a\in A$ and $\phi\in\textup{\text{Hom}}_{\mathcal{B}}(V,V)$.

  4. (iv)

(The _oriented Cardy condition_) Let $\\{\psi_{i}\\}_{i}$ be a basis of
$\textup{\text{Hom}}_{\mathcal{B}}(V,W)$ and $\\{\psi^{i}\\}_{i}$ the basis of
$\textup{\text{Hom}}_{\mathcal{B}}(W,V)$ which is dual with respect to
$\langle-,-\rangle_{V,W}$. Then $\tau_{V}\circ\tau^{W}$ is equal to the map

$\textup{\text{Hom}}_{\mathcal{B}}(W,W)\rightarrow\textup{\text{Hom}}_{\mathcal{B}}(V,V),\qquad\phi\mapsto\sum_{i}\psi^{i}\circ\phi\circ\psi_{i}.$

######  Remarks 3.2.

  1. (1)

When $\mathcal{B}$ has a single object, the algebraic data of Theorem 3.1 is
called a _Cardy–Frobenius_ or _knowledgeable_ Frobenius algebra [AN06, LP08].

  2. (2)

Let $\mathcal{Z}$ be an oriented open/closed TFT with object set $D$. The
category111Since $\mathcal{B}$ is assumed to be additive, it may be required
to formally add some elements to $D$ to ensure the existence of direct sums.
See [MS06, §2.5]. $\mathcal{B}$ has objects $D$, morphisms
$\textup{\text{Hom}}_{\mathcal{B}}(V,W)$ given by the value of $\mathcal{Z}$
on the closed interval labelled by $V$ and $W$ and oriented from $V$ to $W$
and composition defined by the value of $\mathcal{Z}$ on the flattened pair of
pants. The value of $\mathcal{Z}$ on the flattened cap defines the Calabi–Yau
traces.

  3. (3)

By non-degeneracy of the Calabi–Yau pairings, the oriented Cardy condition
holds if and only if

$\textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,W)}\,(f\mapsto\psi\circ
f\circ\phi)=\langle\tau^{W}(\psi),\tau^{V}(\phi)\rangle_{0}$ (6)

for all $\phi\in\textup{\text{Hom}}_{\mathcal{B}}(V,V)$ and
$\psi\in\textup{\text{Hom}}_{\mathcal{B}}(W,W)$. Following [CW10, §7.4], we
refer to equation (6) as the _baggy oriented Cardy condition_. Topologically,
the oriented Cardy condition asserts the equality of two ways of evaluating
the TFT on the annulus with boundary components labelled by $V$ and $W$.

We are interested in the extension of Theorem 3.1 to the unoriented bordism
category $\textup{\text{Bord}}_{2}^{D}$, defined analogously to
$\textup{\text{Bord}}_{2}^{\textup{\text{or}},D}$ except that objects and
morphisms are unoriented. Upon restriction to the closed sector, the extension
is known.

######  Theorem 3.3 ([TT06, Proposition 2.9]).

Two dimensional unoriented TFTs are classified by the data of an _unoriented
Frobenius algebra_ , that is, a commutative Frobenius algebra
$(A,1_{A},\langle-\rangle_{0})$ with an isometric algebra involution
$p:A\rightarrow A$ and an element $Q\in A$, the _crosscap state_ , which
satisfy the following conditions:

  1. (i)

$p(Qa)=Qa$ for all $a\in A$.

  2. (ii)

(_The Klein condition_) Given a basis $\\{a_{i}\\}_{i}$ of $A$ with basis
$\\{a^{i}\\}_{i}$ of $A$ dual with respect to $\langle-\rangle_{0}$, the
equality $Q^{2}=\sum_{i}p(a^{i})a_{i}$ holds.

In terms of bordisms, $Q$ is the image under $\mathcal{Z}$ of the compact
Möbius strip $\mathbb{RP}^{2}\setminus\mathring{D}^{2}$,

$\leavevmode\hbox to57.5pt{\vbox
to41.79pt{\pgfpicture\makeatletter\hbox{\hskip 57.50414pt\lower-3.84514pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{tikz@color}{rgb}{0,0,0}\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{}{}}}{{}}{}{{{}{}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.45184pt}{12.44807pt}\pgfsys@curveto{-56.90414pt}{12.44807pt}{-56.90414pt}{37.34317pt}{-28.45184pt}{37.34317pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}{}}}{{}}{}{{{}{}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}\pgfsys@color@rgb@stroke{0}{0}{0.8}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0.8}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0,0,0.8}{}\pgfsys@moveto{-28.45184pt}{12.44807pt}\pgfsys@curveto{-17.78252pt}{12.44807pt}{-17.78252pt}{37.34317pt}{-28.45184pt}{37.34317pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}} {{{ {\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{-3.04pt}{3.84514pt}\pgfsys@curveto{-2.4846pt}{1.53802pt}{-1.24696pt}{0.4486pt}{0.0pt}{0.0pt}\pgfsys@curveto{-1.24696pt}{-0.4486pt}{-2.4846pt}{-1.53802pt}{-3.04pt}{-3.84514pt}\pgfsys@stroke\pgfsys@endscope}}
}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.00003}{1.0}{-1.0}{-0.00003}{-20.44978pt}{24.29616pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}
{{{}}} }{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}} {{}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}{}}}{{}}{}{{{}{}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}\pgfsys@color@rgb@stroke{0}{0}{0.8}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0.8}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0,0,0.8}\pgfsys@stroke@opacity{0.2}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.2}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.45184pt}{12.44807pt}\pgfsys@curveto{-39.12117pt}{12.44807pt}{-39.12117pt}{37.34317pt}{-28.45184pt}{37.34317pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-44.8127pt}{26.67404pt}\pgfsys@lineto{-41.25595pt}{23.11725pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-44.8127pt}{23.11725pt}\pgfsys@lineto{-41.25595pt}{26.67404pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-43.03447pt}{24.89658pt}\pgfsys@moveto{-39.47787pt}{24.89658pt}\pgfsys@curveto{-39.47787pt}{26.86084pt}{-41.0702pt}{28.45317pt}{-43.03447pt}{28.45317pt}\pgfsys@curveto{-44.99873pt}{28.45317pt}{-46.59106pt}{26.86084pt}{-46.59106pt}{24.89658pt}\pgfsys@curveto{-46.59106pt}{22.93231pt}{-44.99873pt}{21.33998pt}{-43.03447pt}{21.33998pt}\pgfsys@curveto{-41.0702pt}{21.33998pt}{-39.47787pt}{22.93231pt}{-39.47787pt}{24.89658pt}\pgfsys@closepath\pgfsys@moveto{-43.03447pt}{24.89658pt}\pgfsys@stroke\pgfsys@invoke{
} \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}}:\varnothing\rightarrow
S^{1},$

and $p$ is the image of the mapping cylinder of circle reflection,

$\leavevmode\hbox to73.51pt{\vbox
to22.54pt{\pgfpicture\makeatletter\hbox{\hskip 4.15585pt\lower-11.26912pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{tikz@color}{rgb}{0,0,0}\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{7.11345pt}{10.66913pt}\pgfsys@lineto{56.90483pt}{10.66913pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{7.11345pt}{-10.66913pt}\pgfsys@lineto{56.90483pt}{-10.66913pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}{}{{{}{}}}{{}}{}{{{}{}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}\pgfsys@color@rgb@stroke{0}{0}{0.8}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0.8}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0,0,0.8}{}\pgfsys@moveto{7.11345pt}{10.66913pt}\pgfsys@curveto{17.78278pt}{10.66913pt}{17.78278pt}{-10.66913pt}{7.11345pt}{-10.66913pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}} {{{ {\pgfsys@beginscope{{}}
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{3.04pt}{3.84514pt}\pgfsys@curveto{2.4846pt}{1.53802pt}{1.24696pt}{0.4486pt}{0.0pt}{0.0pt}\pgfsys@curveto{1.24696pt}{-0.4486pt}{2.4846pt}{-1.53802pt}{3.04pt}{-3.84514pt}\pgfsys@stroke\pgfsys@endscope}}
}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.00003}{-1.0}{1.0}{0.00003}{15.1153pt}{3.64047pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}
{{{}}} }{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}} {{}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}{}}}{{}}{}{{{}{}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}\pgfsys@color@rgb@stroke{0}{0}{0.8}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0.8}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0,0,0.8}{}\pgfsys@moveto{7.11345pt}{10.66913pt}\pgfsys@curveto{-3.55586pt}{10.66913pt}{-3.55586pt}{-10.66913pt}{7.11345pt}{-10.66913pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}{}}}{{}}{}{{{}{}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}\pgfsys@color@rgb@stroke{0}{0}{0.8}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0.8}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0,0,0.8}{}\pgfsys@moveto{56.90483pt}{10.66913pt}\pgfsys@curveto{67.57416pt}{10.66913pt}{67.57416pt}{-10.66913pt}{56.90483pt}{-10.66913pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.00003}{-1.0}{1.0}{0.00003}{64.90677pt}{0.60048pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}
{{{}}} }{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}} {{}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}{}}}{{}}{}{{{}{}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}\pgfsys@color@rgb@stroke{0}{0}{0.8}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0.8}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0,0,0.8}\pgfsys@stroke@opacity{0.2}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.2}\pgfsys@invoke{
}{}\pgfsys@moveto{56.90483pt}{10.66913pt}\pgfsys@curveto{46.2355pt}{10.66913pt}{46.2355pt}{-10.66913pt}{56.90483pt}{-10.66913pt}\pgfsys@stroke\pgfsys@invoke{
} \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}}:S^{1}\rightarrow
S^{1}.$

The Klein condition is illustrated in Figure 1.

=

Figure 1. The equality of bordisms responsible for the Klein condition.

We now come the main classification result.

######  Theorem 3.4.

Two dimensional unoriented open/closed TFTs are classified by the data of an
underlying closed theory, as in Theorem 3.3, together with the data of a
$\mathbb{C}$-linear strict duality $P$ on $\mathcal{B}$. This data is required
to satisfy the following conditions:

  1. (i)

The functor $P$ is the identity on objects.

  2. (ii)

$\langle P(\phi)\rangle_{V}=\langle\phi\rangle_{V}$ for all
$\phi\in\textup{\text{Hom}}_{\mathcal{B}}(V,V)$.

  3. (iii)

$P\circ\tau_{V}=\tau_{V}\circ p$ for all $V\in\mathcal{B}$.

  4. (iv)

$p\circ\tau^{V}=\tau^{V}\circ P$ for all $V\in\mathcal{B}$.

  5. (v)

(The _unoriented Cardy condition_) Let $\\{\psi_{i}\\}_{i}$ be a basis of
$\textup{\text{Hom}}_{\mathcal{B}}(V,V)$ with dual basis $\\{\psi^{i}\\}_{i}$
with respect to $\langle-,-\rangle_{V,V}$. Then there is an equality

$\tau_{V}(Q)=\sum_{i}\psi^{i}\circ P(\psi_{i}).$ (7)

###### Proof.

The theorem is proved in [AN06, §4] under the assumption that $\mathcal{B}$
has a single object, where the above algebraic data is known as a _structure
algebra_. This proof generalizes immediately to allow for $\mathcal{B}$ to
have many objects, in the same way as the analogous generalization in the
oriented case [LP08, §5]. ∎

Topologically, $P$ is the image under $\mathcal{Z}$ of the mapping cylinder of
reflection of the closed interval so that
$P_{V,W}:\textup{\text{Hom}}_{\mathcal{B}}(V,W)\rightarrow\textup{\text{Hom}}_{\mathcal{B}}(W,V)$
comes from the bordism

$V$$W$$V$$W$=$V$$W$$W$$V$.

As indicated on the right, we will picture this bordism as embedded in
$\mathbb{R}^{3}$ with a half-twist. That $P$ is a strict involution follows
from the fact that reflection of the closed interval is an involution. We
record two basic consequences of Theorem 3.4.

######  Proposition 3.5.

  1. (1)

The equality $\langle Q^{2}\rangle_{0}=\textup{\text{tr}}_{A}\,p$ holds.

  2. (2)

For any $V\in\mathcal{B}$ and $\phi\in\textup{\text{Hom}}_{\mathcal{B}}(V,V)$,
the equality

$\langle\tau^{V}(\phi),Q\rangle_{0}=\textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,V)}\,\iota_{\phi}$

holds, where $\iota_{\phi}$ is defined analogously to Section 2.2.

###### Proof.

Since $p$ is an involution, there exists a basis $\\{a_{i}\\}_{i}$ of $A$ such
that $p(a_{i})=s_{i}a_{i}$ with $s_{i}\in\\{1,-1\\}$. Let $\\{a^{i}\\}_{i}$ be
a dual basis, so that $\langle a^{j},a_{i}\rangle_{0}=\delta^{j}_{i}$. Since
$p$ is an isometry of $\langle-\rangle_{0}$, we have $p(a^{i})=s_{i}a^{i}$.
With these preliminaries, we compute

$\langle
Q^{2}\rangle_{0}=\langle\sum_{i}p(a^{i})a_{i}\rangle_{0}=\sum_{i}s_{i}\langle
a^{i}a_{i}\rangle_{0}=\sum_{i}s_{i}=\textup{\text{tr}}_{A}\,p.$

For the second statement, we compute

$\langle\tau^{V}(\phi),Q\rangle_{0}=\langle\phi,\tau_{V}(Q)\rangle_{V,V}=\sum_{i}\langle\phi\circ\psi^{i}\circ
P(\psi_{i})\rangle_{V}=\\\
\sum_{i}\langle\psi^{i}\circ\iota_{\phi}(\psi_{i})\rangle_{V}=\textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,V)}\,\iota_{\phi}.$

The first equality is the adjointness of $\tau^{V}$ and $\tau_{V}$, the second
is the unoriented Cardy condition and the third is cyclicity of traces. ∎

By non-degeneracy of the Calabi–Yau pairings, the unoriented Cardy condition
is equivalent to the second equality from Proposition 3.5, which we term the
_baggy unoriented Cardy condition_. The unoriented Cardy condition reflects
the equality of two ways of evaluating the TFT on the Möbius strip with
boundary component labelled by $V$. See Figure 2.

The next result constructs the algebraic input of Theorem 3.4 from a
Calabi–Yau category with a contravariant involution which need not act
trivially on objects.

=

Figure 2. The equality of bordisms responsible for the unoriented Cardy
condition (7). All boundaries are labelled by the object $V\in\mathcal{B}$.

######  Proposition 3.6.

Let $(\mathcal{B},\tau^{\bullet},\tau_{\bullet},A)$ define a two dimensional
oriented open/closed TFT and $(p,Q)$ an unoriented lift of $A$. Let
$(P,\Theta)$ be a duality structure on $\mathcal{B}$ such that $\langle
P(\phi)\rangle_{P(V)}=\langle\phi\rangle_{V}$ for all
$\phi\in\textup{\text{Hom}}_{\mathcal{B}}(V,V)$ and
$P\circ\tau_{V}=\tau_{P(V)}\circ p$ and $p\circ\tau^{V}=\tau^{P(V)}\circ P$
for all $V\in\mathcal{B}$. If the equality

$\textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,P(V))}\,\iota_{\phi}=\langle\tau^{V}(\phi),Q\rangle_{0}$
(8)

holds for all $\phi\in\textup{\text{Hom}}_{\mathcal{B}}(V,V)$, then
$(\mathcal{B}^{\tilde{h}C_{2}},\tau^{\bullet},\tau_{\bullet},A,p,Q)$ defines a
two dimensional unoriented open/closed TFT.

###### Proof.

By Lemma 1.4, the triple
$(\mathcal{B}^{\tilde{h}C_{2}},P^{\tilde{h}C_{2}},\Theta^{\tilde{h}C_{2}})$ is
a category with strict duality and $P^{\tilde{h}C_{2}}$ acts trivially on
objects. The category $\mathcal{B}^{\tilde{h}C_{2}}$ inherits a Calabi–Yau
structure from $\mathcal{B}$ with traces
$\langle-\rangle_{(V,\psi_{V})}:=\langle-\rangle_{V}$. Define boundary-bulk
and bulk-boundary maps for $\mathcal{B}^{\tilde{h}C_{2}}$ by
$\tau^{(V,\psi_{V})}=\tau^{V}$ and $\tau_{(V,\psi_{V})}=\tau_{V}$. The
assumption that $P$ preserves the Calabi–Yau structure and that $P$ and $p$
are compatible with $\tau_{\bullet}$ and $\tau^{\bullet}$ verifies conditions
(ii)-(iv) of Theorem 3.4 for $P^{\tilde{h}C_{2}}$. It remains to verify the
unoriented Cardy condition. Let $(V,\psi_{V})\in\mathcal{B}^{\tilde{h}C_{2}}$.
Let $\\{\psi_{i}\\}_{i}$ be a basis of
$\textup{\text{Hom}}_{\mathcal{B}}(V,P(V))$ with dual basis
$\\{\psi^{i}\\}_{i}$ of $\textup{\text{Hom}}_{\mathcal{B}}(P(V),V)$. Then
$\\{\psi_{V}^{-1}\circ\psi_{i}\\}_{i}$ is a basis of
$\textup{\text{Hom}}_{\mathcal{B}}(V,V)$ with dual basis
$\\{\psi^{i}\circ\psi_{V}\\}_{i}$. We compute

$\tau_{(V,\psi_{V})}(Q)=\sum_{i}\psi^{i}\circ P(\psi_{i})\circ\Theta_{V}=\\\
\sum_{i}\psi^{i}\circ\psi_{V}\circ\psi^{-1}_{V}\circ P(\psi_{i})\circ
P(\psi_{V}^{-1})\circ\psi_{V}=\sum_{i}\psi^{i}\circ\psi_{V}\circ
P^{\tilde{h}C_{2}}(\psi_{V}^{-1}\circ\psi_{i}).$

For the first equality, note that the discussion proceeding Proposition 3.5
shows that equation (8) implies that $\tau_{V}(Q)=\sum_{i}\psi^{i}\circ
P(\psi_{i})\circ\Theta_{V}$. The second equality follows from the coherence
condition on homotopy fixed points and the final equality from the definition
of $(P^{\tilde{h}C_{2}},\Theta^{\tilde{h}C_{2}})$. ∎

We comment on the physical interpretation of Proposition 3.6. As mentioned in
the introduction, the Calabi–Yau category $\mathcal{B}$ should be seen as a
model for the category of D-branes in an oriented string theory. With this
interpretation, a duality structure $(P,\Theta)$ which preserves the
Calabi–Yau pairings is the categorical data of the orientifold construction;
see [DGRKS07, HW08] in the setting of orientifolds of IIB string theory and
Landau–Ginzburg theory. In this context, the quantity
$\textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,P(V))}\,\iota_{\phi}$
is a _parity-twisted Witten index_ [BH04, §2] and it is through its
computation via closed sector quantities, namely equation (8), that the
crosscap state $Q$ naturally appears. The D-branes which survive the
orientifold projection are the homotopy fixed points of $(P,\Theta)$, that is,
objects of the category $\mathcal{B}^{\tilde{h}C_{2}}$ above. With these
remarks in mind, Proposition 3.6 is an orientifold-type construction of an
unoriented open/closed TFT from an oriented open/closed TFT.

###  3.2. The Frobenius–Schur element as a crosscap state

We give an algebraic construction of a two dimensional unoriented open/closed
TFT from twisted Real representation theory. When $\hat{G}=G\times C_{2}$ and
the cohomological data $(\hat{\theta},\lambda)$ is trivial, this generalizes
results of [AN06, LN11]. When $\lambda$ is trivial, a topological construction
of the closed sector of this theory was given in [You20, §4.4].

Fix group theoretic data $(\hat{G},\hat{\theta},\lambda)$ as in Section 2.1.
Let $A=Z(\mathbb{C}^{\theta^{-1}}[G])$ with Frobenius pairing
$\langle-,-\rangle_{G}$ and $\mathcal{B}=\textup{\text{Rep}}^{\theta}(G)$ the
Calabi–Yau category with traces
$\langle\phi\rangle_{V}=\frac{1}{|G|}\textup{\text{tr}}_{V}\,\phi$. The
boundary-bulk map $\tau^{V}$ is as in Section 2.2 and the bulk-boundary map is
defined by

$\tau_{V}\Big{(}\sum_{g\in G}a_{g}l_{g}\Big{)}=\sum_{g\in
G}a_{g}\theta([g|g^{-1}])^{-1}\rho_{V}(g^{-1}).$

This data defines a two dimensional oriented open/closed TFT
$\mathcal{Z}_{(G,\theta)}$ via Theorem 3.1. See [MS06, Tur07, Kho11]. The main
axiom to be verified is the oriented Cardy condition which, in the present
setting, is a mild generalization of the orthogonality of characters of
irreducible $\theta$-twisted representations.

######  Theorem 3.7.

The data $(\hat{G},\hat{\theta},\lambda)$ defines a two dimensional unoriented
open/closed TFT $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ whose oriented
sector is a sub TFT of $\mathcal{Z}_{(G,\theta)}$.

We will prove Theorem 3.7 using the orientifold construction of Proposition
3.6. We take $(P^{(\hat{\theta},\lambda)},\Theta^{(\hat{\theta},\lambda)})$
for the duality structure on $\mathcal{B}=\textup{\text{Rep}}^{\theta}(G)$ and
$Q=\nu_{(\hat{\theta},\lambda)}$ for the candidate crosscap state. We compute

$\langle
P(\phi)\rangle_{P(V)}=\frac{1}{|G|}\textup{\text{tr}}_{P(V)}\,P(\phi)=\frac{1}{|G|}\textup{\text{tr}}_{V^{\vee}}\,\phi^{\vee}=\langle\phi\rangle_{V},$

which verifies the open sector assumption of Proposition 3.6. The remainder of
the proof of Theorem 3.7 is divided into closed sector computations and
verification of the open/closed coherence conditions required to apply
Proposition 3.6.

The oriented open sector of $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ is
the full theory $\mathcal{Z}_{(G,\theta)}$ precisely when the forgetful
functor
$\textup{\text{Rep}}^{\theta}(G)^{\tilde{h}C_{2}}\rightarrow\textup{\text{Rep}}^{\theta}(G)$
is essentially surjective. By Corollary 2.8, this is the case when
$\langle\chi_{V},\nu_{(\hat{\theta},\lambda)}\rangle_{G}=1$ for each
irreducible $V\in\textup{\text{Rep}}^{\theta}(G)$. Otherwise, the oriented
open sector is a strict subtheory of $\mathcal{Z}_{(G,\theta)}$. In the
context of Example 2.11, the forgetful functor is essentially surjective only
in subexample (3).

######  Remark 3.8.

We comment on the relation of $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$
to categories of $D$-branes in orientifold string theory on global quotients.
Recall that the spacetime of an orientifold string theory is an orbifold
double cover $\pi:\mathcal{X}\rightarrow\hat{\mathcal{X}}$. Additional data
required to define the theory includes the (gauge equivalence class of a)
$B$-field $\check{B}\in\check{H}^{3+\pi}(\hat{\mathcal{X}})$ [DFM11], which is
a class in the $\pi$-twisted differential cohomology of $\hat{\mathcal{X}}$,
and a complex line bundle with connection
$\check{L}\in\check{H}^{2}(\hat{\mathcal{X}})$ [GH10, §8.4.1]. The underlying
(oriented) orbifold string theory depends only on
$(\mathcal{X},\pi^{*}\check{B})$. Consider now the particular case in which
the spacetime is a global quotient $\pi:X/\\!\\!/G\rightarrow
X/\\!\\!/\hat{G}$ associated to a finite $C_{2}$-graded group $\hat{G}$ acting
on a smooth manifold $X$. A special class of $B$-fields arises through the
composition

$H^{2+\pi}(B\hat{G})\rightarrow
H^{2+\pi}(X/\\!\\!/\hat{G})\hookrightarrow\check{H}^{3+\pi}(X/\\!\\!/G),\qquad\hat{\theta}\mapsto\check{B}_{\hat{\theta}},$

where the first map is restriction along the the canonical morphism
$X/\\!\\!/\hat{G}\rightarrow B\hat{G}$ and the second is the inclusion of flat
$B$-fields. Similarly, a class $\lambda\in H^{1}(B\hat{G})$ defines a flat
line bundle $\check{L}_{\lambda}\in\check{H}^{2}(X/\\!\\!/\hat{G})$. The pair
$(\check{B}_{\hat{\theta}},\check{L}_{\lambda})$ can be seen as defining
universal twists for global $\hat{G}$-orientifolds. The unoriented TFT
$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ is a precise mathematical
description of the affects of the twists
$(\check{B}_{\hat{\theta}},\check{L}_{\lambda})$ on partition functions. See
[BS02], [Sha11, §5], [NY22, §4.5] for detailed discussions of these affects in
the closed sector.

We return to the proof of Theorem 3.7.

####  3.2.1. Closed sector

Denote by
$\textup{\text{Aut}}^{\textnormal{gen}}(\mathbb{C}^{\theta^{-1}}[G])$ the
group of algebra automorphisms and algebra anti-automorphisms of
$\mathbb{C}^{\theta^{-1}}[G]$. The group
$\textup{\text{Aut}}^{\textnormal{gen}}(\mathbb{C}^{\theta^{-1}}[G])$ is
$C_{2}$-graded by sending anti-automorphisms to $-1$.

######  Lemma 3.9.

The function
$p:\hat{G}\rightarrow\textup{\text{Aut}}^{\textnormal{gen}}(\mathbb{C}^{\theta^{-1}}[G])$,
$\omega\mapsto p^{\omega}$, where

$p^{\omega}(l_{g})=\lambda(g)^{\frac{\pi(\omega)-1}{2}}\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]g)l_{\omega
g^{\pi(\omega)}\omega^{-1}},\qquad g\in G$

is a $C_{2}$-graded group homomorphism. Moreover, each $p^{\omega}$ is an
isometry of $\langle-\rangle_{G}$.

###### Proof.

We prove that $p^{\omega}$, $\omega\in\hat{G}\backslash G$, is an anti-
automorphism and omit the easier calculation that $p^{g}$, $g\in G$, is an
automorphism. For $g,h\in G$, direct calculations give

$p^{\omega}(l_{g}\cdot
l_{h})=\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]gh)}{\lambda(gh)\theta([g|h])}l_{\omega(gh)^{-1}\omega^{-1}}$

and

$p^{\omega}(l_{h})\cdot
p^{\omega}(l_{g})=\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]h)\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]g)}{\lambda(g)\lambda(h)\theta([\omega
h^{-1}\omega^{-1}|\omega g^{-1}\omega^{-1}])}l_{\omega
h^{-1}g^{-1}\omega^{-1}}.$

It therefore suffices to prove that

$\frac{\theta([g|h])}{\theta([\omega h^{-1}\omega^{-1}|\omega
g^{-1}\omega^{-1}])}=\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]gh)}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]h)\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]g)}.$

A short calculation using equation (1) shows that this identity indeed holds.

That $p^{\omega}$ is an isometry follows from the equalities

$\langle
p^{\omega}(l_{g})\rangle_{G}=\frac{1}{|G|}\delta_{e,g}\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})}{\lambda(e)}([\omega]e)=\frac{\delta_{e,g}}{|G|}=\langle
l_{g}\rangle_{G}.$

It remains to prove the homomorphism property, $p^{\omega_{2}}\circ
p^{\omega_{1}}=p^{\omega_{2}\omega_{1}}$. Recall from Section 2.1 that
$\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})$ is a $1$-cocycle on the
groupoid $G/\\!\\!/_{R}\hat{G}$ whose objects are elements of $G$ and whose
morphisms are $\omega:g\rightarrow\omega g^{\pi(\omega)}\omega^{-1}$,
$\omega\in\hat{G}$. With this description, closedness of
$\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})$ becomes the equalities

$\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega_{2}]\omega_{1}g^{\pi(\omega_{1})}\omega_{1}^{-1})\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega_{1}]g)=\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega_{2}\omega_{1}]g),\qquad
g\in G,\;\omega_{i}\in\hat{G}$

which are immediately seen to imply the homomorphism property. ∎

######  Proposition 3.10.

For each $\varsigma\in\hat{G}\backslash G$, the map $p^{\varsigma}$ restricts
to an algebra involution of $Z(\mathbb{C}^{\theta^{-1}}[G])$. Moreover, this
involution is independent of $\varsigma$.

###### Proof.

Using the explicit descriptions of the centre $Z(\mathbb{C}^{\theta^{-1}}[G])$
from Section 1.2 and the $G$-action on $\mathbb{C}^{\theta^{-1}}[G]$ from
Lemma 3.9 we see that
$\mathbb{C}^{\theta^{-1}}[G]^{G}=Z(\mathbb{C}^{\theta^{-1}}[G])$. It follows
that the generalized $\hat{G}$-action on $\mathbb{C}^{\theta^{-1}}[G]$ from
Lemma 3.9 induces an action of $C_{2}\simeq\hat{G}/G$ by algebra automorphisms
on $Z(\mathbb{C}^{\theta^{-1}}[G])$. ∎

Denote by $p$ the algebra involution of $Z(\mathbb{C}^{\theta^{-1}}[G])$
induced by any $\varsigma\in\hat{G}\backslash G$.

######  Remark 3.11.

Using functoriality of Hochschild homology and invariance under taking
opposites, we form the composition

$HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G))\xrightarrow[]{\sim}HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G)^{\textup{\text{op}}})\xrightarrow[]{HH_{\bullet}(P^{(\hat{\theta},\lambda)})}HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G)).$
(9)

Since $\textup{\text{Rep}}^{\theta}(G)$ is finite semisimple,
$HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G))$ is concentrated in degree
zero, where is it isomorphic to $Z(\mathbb{C}^{\theta^{-1}}[G])$. Under this
isomorphism, the map (9) is $p$. The $+1$ (resp. $-1$) eigenspace of $p$ is
then the involutive (resp. skew-involutive) Hochschild homology of
$(\textup{\text{Rep}}^{\theta}(G),P^{(\hat{\theta},\lambda)},\Theta^{(\hat{\theta},\lambda)})$.
See [Bra14, Theorem 2.14] for an analogous result in the setting of strictly
involutive $A_{\infty}$-algebras. The first part of Proposition 3.5 therefore
shows that the Klein condition computes the difference in dimensions of
involutive and skew-involutive Hochschild homologies.

######  Proposition 3.12.

The element $\nu_{(\hat{\theta},\lambda)}\in Z(\mathbb{C}^{\theta^{-1}}[G])$
is $p$-invariant.

###### Proof.

We have seen in Lemma 2.5 that $\nu_{(\hat{\theta},\lambda)}\in
Z(\mathbb{C}^{\theta^{-1}}[G])$. For $p$-invariance, we have

$p^{\varsigma}(\nu_{(\hat{\theta},\lambda)})=\sum_{\mu\in\hat{G}\backslash
G}\frac{\lambda(\mu)\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]\mu^{2})}{\lambda(\mu^{2})\hat{\theta}([\mu|\mu])}l_{\varsigma\mu^{-2}\varsigma^{-1}}.$

Equation (2) gives
$\hat{\theta}([\varsigma\mu^{-1}\varsigma^{-1}|\varsigma\mu^{-1}\varsigma^{-1}])=\frac{\hat{\theta}([\varsigma|\mu^{-2}])}{\hat{\theta}([\mu^{-1}|\mu^{-1}])\hat{\theta}([\varsigma\mu^{-2}\varsigma^{-1}|\varsigma])}$
so that $p^{\varsigma}(\nu_{(\hat{\theta},\lambda)})$ is equal to

$\sum_{\mu\in\hat{G}\backslash
G}\lambda(\mu)^{-1}\frac{\hat{\theta}([\varsigma\mu^{-2}\varsigma^{-1}|\varsigma])}{\hat{\theta}([\mu|\mu])\hat{\theta}([\mu^{2}|\mu^{-2}])\hat{\theta}([\varsigma|\mu^{-2}])}l_{\varsigma\mu^{-2}\varsigma^{-1}}\\\
=\sum_{\mu\in\hat{G}\backslash
G}\lambda(\mu)^{-1}\hat{\theta}([\mu|\mu])^{-1}\hat{\theta}([\mu^{2}|\mu^{-2}])^{-1}\hat{\theta}([\mu^{-1}|\mu^{-1}])^{-1}\hat{\theta}([\varsigma\mu^{-1}\varsigma^{-1}|\varsigma\mu^{-1}\varsigma^{-1}])^{-1}l_{\varsigma\mu^{-2}\varsigma^{-1}}.$

A short calculation shows that
$\hat{\theta}([\mu^{-1}|\mu^{-1}])\hat{\theta}([\mu|\mu])\hat{\theta}([\mu^{2}|\mu^{-2}])=1$,
whence

$p^{\varsigma}(\nu_{(\hat{\theta},\lambda)})=\sum_{\mu\in\hat{G}\backslash
G}\lambda(\mu^{-1})\hat{\theta}([\varsigma\mu^{-1}\varsigma^{-1}|\varsigma\mu^{-1}\varsigma^{-1}])^{-1}l_{\varsigma\mu^{-2}\varsigma^{-1}}=\nu_{(\hat{\theta},\lambda)}.\qed$

######  Lemma 3.13.

The following equality holds for all $g\in G$ and $\mu\in\hat{G}\backslash G$:

$\hat{\theta}([\mu|\mu])^{-1}l_{\mu^{2}}\cdot
a_{g}l_{g}=\lambda(g)p^{\mu}(a_{g}l_{g})\cdot\hat{\theta}([\mu g|\mu
g])^{-1}l_{(\mu g)^{2}}.$

###### Proof.

This can be verified directly from the twisted $2$-cocycle condition on
$\hat{\theta}$. ∎

######  Proposition 3.14.

The equality $p(\nu_{(\hat{\theta},\lambda)}f)=\nu_{(\hat{\theta},\lambda)}f$
holds for all $f\in Z(\mathbb{C}^{\theta^{-1}}[G])$.

###### Proof.

Write $\sum_{g\in G}a_{g}l_{g}$ for $f\in Z(\mathbb{C}^{\theta^{-1}}[G])$.
Lemma 3.13 gives

$\nu_{(\hat{\theta},\lambda)}\sum_{g\in
G}a_{g}l_{g}=\sum_{\begin{subarray}{c}g\in G\\\ \mu\in\hat{G}\backslash
G\end{subarray}}\lambda(\mu g)p^{\mu}(a_{g}l_{g})\hat{\theta}([\mu g|\mu
g])^{-1}l_{(\mu g)^{2}}$

from which we find that
$p^{\varsigma}(\nu_{(\hat{\theta},\lambda)}\sum_{g}a_{g}l_{g})$ is equal to

$\displaystyle\sum_{\begin{subarray}{c}g\in G\\\ \mu\in\hat{G}\backslash
G\end{subarray}}\lambda(\mu
g)p^{\varsigma}(p^{\mu}(a_{g}l_{g})\cdot\hat{\theta}([\mu g|\mu
g])^{-1}l_{(\mu g)^{2}})$ $\displaystyle=$
$\displaystyle\sum_{g,\mu}\lambda(\mu g)p^{\varsigma}(\hat{\theta}([\mu g|\mu
g])^{-1}l_{(\mu g)^{2}})p^{\varsigma}p^{\mu}(a_{g}l_{g})$ $\displaystyle=$
$\displaystyle\sum_{g,\mu}\lambda(\mu g)^{-1}\hat{\theta}([\varsigma
g^{-1}\mu^{-1}\varsigma^{-1}|\varsigma
g^{-1}\mu^{-1}\varsigma^{-1}])^{-1}l_{\varsigma(g^{-1}\mu^{-1})^{2}\varsigma^{-1}}p^{\varsigma}p^{\mu}(a_{g}l_{g})$
$\displaystyle=$ $\displaystyle\sum_{g,\mu}\lambda(\mu
g)\hat{\theta}([\varsigma g^{-1}\mu^{-1}\varsigma^{-1}|\varsigma
g^{-1}\mu^{-1}\varsigma^{-1}])^{-1}l_{\varsigma(g^{-1}\mu^{-1})^{2}\varsigma^{-1}}\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma\mu]g)^{-1}a_{g}l_{\varsigma\mu
g(\varsigma\mu)^{-1}}$ $\displaystyle=$ $\displaystyle\sum_{g,\mu}\lambda(\mu
g)\hat{\theta}([\varsigma g^{-1}\mu^{-1}\varsigma^{-1}|\varsigma
g^{-1}\mu^{-1}\varsigma^{-1}])^{-1}l_{\varsigma(g^{-1}\mu^{-1})^{2}\varsigma^{-1}}a_{\varsigma\mu
g(\varsigma\mu)^{-1}}l_{\varsigma\mu g(\varsigma\mu)^{-1}}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}h\in G\\\ \eta\in\hat{G}\backslash
G\end{subarray}}\frac{\lambda(\eta)}{\hat{\theta}([\eta|\eta])}l_{\eta^{2}}a_{h}l_{h},$

which is $\nu_{(\hat{\theta},\lambda)}\sum_{h\in G}a_{h}l_{h}$. The first
equality follows from the fact that $p^{\varsigma}$ is an anti-homomorphism
(Lemma 3.9), the second from Proposition 3.12, the third from Lemma 3.9 and
the definition of $p$, the fourth from the assumed centrality of
$\sum_{g}a_{g}l_{g}$ and the fifth from the change of variables
$\eta=\varsigma g^{-1}\mu^{-1}\varsigma^{-1}$ and $h=\varsigma\mu
g\mu^{-1}\varsigma^{-1}$. ∎

Recall that a conjugacy class $\mathcal{O}\subset G$ is called _$\theta$
-regular_ if $\frac{\theta([g|h])}{\theta([h|g])}=1$ for all $g\in\mathcal{O}$
and $h\in C_{G}(g)$.

######  Proposition 3.15.

The Klein condition holds.

###### Proof.

The vector space $Z(\mathbb{C}^{\theta^{-1}}[G])$ has a basis
$\\{l_{\mathcal{O}}\\}_{\mathcal{O}}$ labelled by $\theta$-regular conjugacy
classes of $G$. For convenience, set $l_{\mathcal{O}}=0$ if $\mathcal{O}$ is
not $\theta$-regular. Writing
$l_{\mathcal{O}}=\sum_{g\in\mathcal{O}}a_{g}l_{g}$ and
$l_{\mathcal{O}^{-1}}=\sum_{h\in\mathcal{O}}b_{h^{-1}}l_{h^{-1}}$, we have
$\langle
l_{\mathcal{O}},l_{\mathcal{O}^{-1}}\rangle_{G}=\frac{1}{|G|}\sum_{g\in\mathcal{O}}\theta([g|g^{-1}])^{-1}a_{g}b_{g^{-1}}.$
Centrality of $l_{\mathcal{O}^{\pm 1}}$ implies that the function
$\mathcal{O}\rightarrow\mathbb{C}$,
$g\mapsto\theta([g|g^{-1}])^{-1}a_{g}b_{g^{-1}}$, is constant; denote its
(necessarily non-zero) value by $c_{\mathcal{O}}$. We also have $\langle
l_{\mathcal{O}},l_{\mathcal{O}^{\prime}}\rangle_{G}=0$ if
$\mathcal{O}^{\prime}\neq\mathcal{O}^{-1}$. It follows that
$l_{\mathcal{O}}^{\vee}=\frac{|G|}{c_{\mathcal{O}}|\mathcal{O}|}l_{\mathcal{O}^{-1}}$.
With this notation, the right hand side of the Klein condition is
$R:=\sum_{\mathcal{O}\in\pi_{0}(G/\\!\\!/G)}l_{\mathcal{O}}p^{\varsigma}(l_{\mathcal{O}}^{\vee})$.
We compute

$\displaystyle R$ $\displaystyle=$
$\displaystyle|G|\sum_{\mathcal{O}\in\pi_{0}(G/\\!\\!/G)}\sum_{g,h\in\mathcal{O}}\frac{a_{g}l_{g}p^{\varsigma}(b_{h^{-1}}l_{h^{-1}})}{c_{\mathcal{O}}|\mathcal{O}|}$
$\displaystyle=$
$\displaystyle|G|\sum_{\mathcal{O}\in\pi_{0}(G/\\!\\!/G)}\sum_{g,h\in\mathcal{O}}\frac{a_{g}b_{h^{-1}}\lambda(h)\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]h^{-1})l_{g\varsigma
h\varsigma^{-1}}}{c_{\mathcal{O}}|\mathcal{O}|\hat{\theta}([g|\varsigma
h\varsigma^{-1}])}$ $\displaystyle=$
$\displaystyle\sum_{\mathcal{O}\in\pi_{0}(G/\\!\\!/G)}\sum_{g\in\mathcal{O}}\sum_{t\in
G}\frac{a_{g}b_{tg^{-1}t^{-1}}\lambda(g)\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]tg^{-1}t^{-1})l_{g\varsigma
tgt^{-1}\varsigma^{-1}}}{c_{\mathcal{O}}\hat{\theta}([g|\varsigma
tgt^{-1}\varsigma^{-1}])}$ $\displaystyle=$ $\displaystyle\sum_{g,t\in
G}\frac{a_{g}b_{tg^{-1}t^{-1}}}{c_{\mathcal{O}}}\lambda(g)\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]tg^{-1}t^{-1})}{\hat{\theta}([g|\varsigma
tgt^{-1}\varsigma^{-1}])}l_{g\varsigma tgt^{-1}\varsigma^{-1}}.$

Above we have set $h=tgt^{-1}$. As
$b_{tg^{-1}t^{-1}}=\uptau(\theta)([t]g^{-1})b_{g^{-1}}$, we can write

$\displaystyle R$ $\displaystyle=$ $\displaystyle\sum_{g,t\in
G}\frac{a_{g}b_{g^{-1}}}{c_{\mathcal{O}}}\lambda(g)\frac{\uptau(\theta)([t]g^{-1})\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]tg^{-1}t^{-1})}{\hat{\theta}([g|\varsigma
tgt^{-1}\varsigma^{-1}])}l_{g\varsigma tg^{-1}t^{-1}\varsigma^{-1}}$
$\displaystyle=$ $\displaystyle\sum_{g,t\in
G}\lambda(g)\frac{\theta([g|g^{-1}])}{\hat{\theta}([g|\varsigma
tgt^{-1}\varsigma^{-1}])}\uptau(\theta)([t]g^{-1})\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]tg^{-1}t^{-1})l_{g\varsigma
tgt^{-1}\varsigma^{-1}}$ $\displaystyle=$ $\displaystyle\sum_{g,t\in
G}\lambda(g)\frac{\theta([g|g^{-1}])}{\hat{\theta}([g|\varsigma
tgt^{-1}\varsigma^{-1}])}\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma
t]g^{-1})l_{g\varsigma tgt^{-1}\varsigma^{-1}}.$

The second equality follows from the definition of $c_{\mathcal{O}}$ and the
final from closedness of $\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})$,
as in the proof of Lemma 3.9. Define $\mu,\xi\in\hat{G}\backslash G$ by
$\mu=g\varsigma t$ and $\xi=t^{-1}\varsigma^{-1}$, so that
$t=\varsigma^{-1}\xi^{-1}$ and $g=\mu\xi$. Making these substitutions, the
coefficient of $l_{g\varsigma tgt^{-1}\varsigma^{-1}}=l_{\mu^{2}\xi^{2}}$ in
$R$ is

$\lambda(\mu\xi)\hat{\theta}([\mu\xi|\xi^{-1}\mu\xi^{2}])^{-1}\frac{\hat{\theta}([\xi^{-1}\mu\xi^{2}|\xi^{-1}])}{\hat{\theta}([\xi^{-1}|\mu\xi])}=\lambda(\mu\xi)\hat{\theta}([\mu|\mu])^{-1}\hat{\theta}([\xi|\xi])^{-1}\hat{\theta}([\mu^{2}|\xi^{2}])^{-1}.$

It follows that

$R=\sum_{\mu,\xi\in\hat{G}\backslash
G}\lambda(\mu\xi)\hat{\theta}([\mu|\mu])^{-1}\hat{\theta}([\xi|\xi])^{-1}\hat{\theta}([\mu^{2}|\xi^{2}])^{-1}l_{\mu^{2}\xi^{2}},$

which is exactly $\nu_{(\hat{\theta},\lambda)}^{2}$. ∎

####  3.2.2. Open/closed coherence

Note that Theorem 2.7 verifies equation (8).

######  Proposition 3.16.

The maps $\tau^{\bullet}$, $\tau_{\bullet}$, $P$ and $p$ satisfy the
assumptions of Proposition 3.6.

###### Proof.

We compute

$P\circ\tau_{V}(p(\sum_{g\in G}a_{g}l_{g}))=\sum_{g\in
G}a_{g}\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)}{\lambda(g)}\rho_{V}(\varsigma
g^{-1}\varsigma^{-1})^{\vee}=\tau_{P(V)}(\sum_{g\in G}a_{g}l_{g}),$

that is, $P\circ\tau_{V}\circ p=\tau_{P(V)}$. Since $p$ is an involution, this
implies $P\circ\tau_{V}=\tau_{P(V)}\circ p$.

We also have

$\displaystyle p\circ\tau^{V}(\phi)$ $\displaystyle=$ $\displaystyle\sum_{g\in
G}\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)}{\lambda(g)}\textup{\text{tr}}_{V}(\phi\circ\rho_{V}(g^{-1}))l_{\varsigma
g^{-1}\varsigma^{-1}}$

and

$\displaystyle\tau^{P(V)}(P(\phi))$ $\displaystyle=$ $\displaystyle\sum_{g\in
G}\frac{\lambda(g)}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)}\textup{\text{tr}}_{V^{\vee}}(\phi^{\vee}\circ\rho_{V}(\varsigma
g^{-1}\varsigma^{-1})^{\vee})l_{g}$ $\displaystyle=$ $\displaystyle\sum_{g\in
G}\frac{\lambda(\varsigma
g^{-1}\varsigma^{-1})}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})(\varsigma
g^{-1}\varsigma^{-1})}\textup{\text{tr}}_{V^{\vee}}(\phi^{\vee}\circ\rho_{V}(\varsigma^{2}g\varsigma^{-2})^{\vee})l_{\varsigma
g^{-1}\varsigma^{-1}}.$

Closedness of $\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})$ implies

$\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]\varsigma
g^{-1}\varsigma^{-1})=\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma^{2}]g).$

Since $\phi$ is $G$-equivariant and

$\rho_{V}(\varsigma^{2})\circ\rho_{V}(g)\circ\rho_{V}(\varsigma^{-2})=\theta([\varsigma^{2}|g])\theta([\varsigma^{2}g|\varsigma^{-2}])\rho_{V}(\varsigma^{2}g\varsigma^{-2}),$

we have

$\textup{\text{tr}}_{V^{\vee}}(\phi^{\vee}\circ\rho_{V}(\varsigma^{2}g\varsigma^{-2})^{\vee})=\frac{\theta([\varsigma^{-2}|\varsigma^{2}])}{\theta([\varsigma^{2}|g])\theta([\varsigma^{2}g|\varsigma^{-2}])}\textup{\text{tr}}_{V}(\phi\circ\rho_{V}(g)).$

The coefficient of $\textup{\text{tr}}_{V}(\phi\circ\rho_{V}(g))l_{\varsigma
g^{-1}\varsigma^{-1}}$ in $\tau^{P(V)}(P(\phi))$ is therefore

$\lambda(g)^{-1}\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma^{2}]g)}\frac{\theta([\varsigma^{-2}|\varsigma^{2}])}{\theta([\varsigma^{2}|g])\theta([\varsigma^{2}g|\varsigma^{-2}])}=\frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)}{\lambda(g)}.$

We conclude that $p\circ\tau^{V}=\tau^{P(V)}\circ P$. ∎

This completes the proof of Theorem 3.7.

###  3.3. Partition functions

The algebraic construction of $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$
allows for the explicit computation of the partition function of an arbitrary
surface.

####  3.3.1. Closed surfaces

For the real projective plane, we have

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{RP}^{2})=\langle\nu_{(\hat{\theta},\lambda)}\rangle_{G}=\frac{1}{|G|}\sum_{\begin{subarray}{c}\mu\in\hat{G}\backslash
G\\\ \mu^{2}=e\end{subarray}}\frac{\lambda(\mu)}{\hat{\theta}([\mu|\mu])},$

the first equality reflecting that $\mathbb{RP}^{2}$ is a Möbius strip glued
to a disk. In particular,
$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{RP}^{2})$ vanishes
unless $\pi:\hat{G}\rightarrow C_{2}$ splits. Realizing the Klein bottle as
two cylinders glued together, with one gluing by circle reflection, and using
that $Z(\mathbb{C}^{\theta^{-1}}[G])=\mathbb{C}^{\theta^{-1}}[G]^{G}$ (see the
proof of Proposition 3.10), we compute

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{K})=\frac{1}{|G|}\sum_{h\in
G}\textup{\text{tr}}_{\mathbb{C}^{\theta^{-1}}[G]}p^{h\varsigma}=\frac{1}{|G|}\sum_{\begin{subarray}{c}g\in
G\\\ \omega\in\hat{G}\backslash G\\\ \omega
g^{-1}\omega^{-1}=g\end{subarray}}\frac{1}{\lambda(g)\hat{\theta}([g^{-1}|g])}\frac{\hat{\theta}([g|\omega])}{\hat{\theta}([\omega|g^{-1}])}.$

In general, a formula for the partition function of a closed connected non-
orientable surface $\Sigma$ can be written in terms of $\hat{\theta}$-weighted
counts of $C_{2}$-graded homomorphisms from the fundamental group of the
orientation double cover of $\Sigma^{\textup{\text{or}}}\rightarrow\Sigma$ to
$\hat{G}$. See [You20, §4.4]. Alternatively, the primitive orthogonal
idempotents of the semisimple algebra $Z(\mathbb{C}^{\theta^{-1}}[G])$ can be
used to evaluate the partition functions. Proceeding in this way and writing
the crosscap state as in Corollary 2.10, we find

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma)=\sum_{\begin{subarray}{c}V\in\textup{\text{Irr}}^{\theta}(G)\\\
P^{(\hat{\theta},\lambda)}(V)\simeq
V\end{subarray}}\left(\frac{\langle\chi_{V},\nu_{(\hat{\theta},\lambda)}\rangle_{G}\dim_{\mathbb{C}}V}{|G|}\right)^{\chi(\Sigma)}.$

For example,

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{RP}^{2})=\frac{1}{|G|}\sum_{\begin{subarray}{c}V\in\textup{\text{Irr}}^{\theta}(G)\\\
P^{(\hat{\theta},\lambda)}(V)\simeq
V\end{subarray}}\langle\chi_{V},\nu_{(\hat{\theta},\lambda)}\rangle_{G}\dim_{\mathbb{C}}V$

and

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{K})=|\\{V\in\textup{\text{Irr}}^{\theta}(G)\mid
P^{(\hat{\theta},\lambda)}(V)\simeq V\\}|.$

Equating these expressions for the partition function of $\Sigma$ relates
weighted counts of $C_{2}$-graded homomorphisms
$\pi_{1}(\Sigma^{\textup{\text{or}}})\rightarrow\hat{G}$ to Real character
theoretic sums. Various specializations of these identities are known [FS06,
KM97, MY05, Sny17, BBC+20, You20] and provide non-orientable counterparts of
Mednykh’s formulae [Med78].

####  3.3.2. Surfaces with boundary

Let $\Sigma$ be a compact connected non-orientable surface with $b\geq 1$
boundary components. To begin, label each boundary component by the same
irreducible twisted Real representation
$V\in\textup{\text{Rep}}^{\theta}(G)^{\tilde{h}C_{2}}$. The partition function

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma;V):\textup{\text{Hom}}_{\textup{\text{Rep}}^{\theta}(G)^{\tilde{h}C_{2}}}(V,V)^{\otimes
b}\rightarrow\mathbb{C}$

can be computed as follows. By Proposition 2.3, there are two cases to
consider. If $V$ is irreducible as a twisted representation, then the
primitive orthogonal idempotent of $\mathbb{C}^{\theta^{-1}}[G]$ corresponding
to $V$ is $e_{V^{\vee}}=\frac{\dim_{\mathbb{C}}V}{|G|}\chi_{V}$, whence
$\chi_{V}^{b}=\left(\frac{\dim_{\mathbb{C}}V}{|G|}\right)^{-b}e_{V^{\vee}}$.
Using this and the fact that
$\langle\chi_{V},\nu_{(\hat{\theta},\lambda))}\rangle_{G}=1$ (see Corollary
2.8), we compute

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma;V)(\textup{\text{id}}_{V}^{\otimes
b})=\left(\frac{\dim_{\mathbb{C}}V}{|G|}\right)^{\chi(\Sigma)}.$

If instead the underlying twisted representation of $V$ is reducible, then
$V\simeq H^{(\hat{\theta},\lambda)}(U)$ with
$U\in\textup{\text{Rep}}^{\theta}(G)$ irreducible. It follows that
$\chi_{H^{(\hat{\theta},\lambda)}(U)}=\frac{|G|}{\dim_{\mathbb{C}}U}(e_{U^{\vee}}+e_{P^{(\hat{\theta},\lambda)}(U)^{\vee}})$
and

$\chi_{H^{(\hat{\theta},\lambda)}(U)}^{b}=\left(\frac{\dim_{\mathbb{C}}U}{|G|}\right)^{-b}(e_{U^{\vee}}+e_{P^{(\hat{\theta},\lambda)}(U)^{\vee}}).$

There are two further sub-cases:

  * •

$P^{(\hat{\theta},\lambda)}(U)\not\simeq U$, in which case
$\langle\chi_{U},\nu_{(\hat{\theta},\lambda)}\rangle_{G}=\langle\chi_{P^{(\hat{\theta},\lambda)}(U)}\nu_{(\hat{\theta},\lambda)}\rangle_{G}=0$.
Since $\nu_{(\hat{\theta},\lambda)}$ has no $\chi_{U}$ or
$\chi_{P^{(\hat{\theta},\lambda)}(U)}$ components, we find

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma;H^{(\hat{\theta},\lambda)}(U))(\textup{\text{id}}_{H^{(\hat{\theta},\lambda)}(U)}^{\otimes
b})=0.$

  * •

$P^{(\hat{\theta},\lambda)}(U)\simeq U$, in which case
$\langle\chi_{U},\nu_{(\hat{\theta},\lambda)}\rangle_{G}=-1$ and

$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma;H^{(\hat{\theta},\lambda)}(U))(\textup{\text{id}}_{H^{(\hat{\theta},\lambda)}(U)}^{\otimes
b})=2\left(-\frac{\dim_{\mathbb{C}}U}{|G|}\right)^{\chi(\Sigma)}.$

A formula in the general case, with boundary components labelled by arbitrary
twisted Real representations $V_{i}$, $i=1,\dots,b$, can be deduced from the
previous formulae by writing $\chi_{V_{i}}$ as a linear combination of
primitive idempotents.

## References

  * [Abo10] M. Abouzaid, _A geometric criterion for generating the Fukaya category_ , Publ. Math. Inst. Hautes Études Sci. (2010), no. 112, 191–240.
  * [Abr96] L. Abrams, _Two-dimensional topological quantum field theories and Frobenius algebras_ , J. Knot Theory Ramifications 5 (1996), no. 5, 569–587.
  * [AN06] A. Alexeevski and S. Natanzon, _Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves_ , Selecta Math. (N.S.) 12 (2006), no. 3-4, 307–377.
  * [AS69] M. Atiyah and G. Segal, _Equivariant $K$-theory and completion_, J. Differential Geometry 3 (1969), 1–18.
  * [BBC+20] M. Barkeshli, P. Bonderson, M. Cheng, C.-M. Jian, and K. Walker, _Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies_ , Comm. Math. Phys. 374 (2020), no. 2, 1021–1124.
  * [BCT09] A. Blumberg, R. Cohen, and C. Teleman, _Open-closed field theories, string topology, and Hochschild homology_ , Alpine perspectives on algebraic topology, Contemp. Math., vol. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 53–76.
  * [BH04] I. Brunner and K. Hori, _Orientifolds and mirror symmetry_ , J. High Energy Phys. (2004), no. 11, 005, 119 pp.
  * [Bra14] C. Braun, _Involutive $A_{\infty}$-algebras and dihedral cohomology_, J. Homotopy Relat. Struct. 9 (2014), no. 2, 317–337.
  * [BS02] V. Braun and B. Stefanski, Jr., _Orientifolds and $K$-theory_, Cargese 2002, Progress in String, Field and Particle Theory, NATO Science Series II: Mathematics, Physics and Chemistry, Springer Netherlands, 2002, pp. 369–372.
  * [Cos07] K. Costello, _Topological conformal field theories and Calabi-Yau categories_ , Adv. Math. 210 (2007), no. 1, 165–214.
  * [CW10] A. Căldăraru and S. Willerton, _The Mukai pairing. I. A categorical approach_ , New York J. Math. 16 (2010), 61–98.
  * [DFM11] J. Distler, D. Freed, and G. Moore, _Spin structures and superstrings_ , Surveys in differential geometry. Volume XV. Perspectives in mathematics and physics, Surv. Differ. Geom., vol. 15, Int. Press, Somerville, MA, 2011, pp. 99–130.
  * [DGRKS07] D.-E. Diaconescu, A. Garcia-Raboso, R. Karp, and K. Sinha, _D-brane superpotentials in Calabi-Yau orientifolds_ , Adv. Theor. Math. Phys. 11 (2007), no. 3, 471–516.
  * [Dij89] R. Dijkgraaf, _A geometrical approach to two-dimensional conformal field theory_ , 1989, Thesis (Ph.D.)–Utrecht University.
  * [DW90] R. Dijkgraaf and E. Witten, _Topological gauge theories and group cohomology_ , Comm. Math. Phys. 129 (1990), no. 2, 393–429.
  * [Dys62] F. Dyson, _The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics_ , Journal of Mathematical Physics 3 (1962), no. 6, 1199–1215.
  * [FH21] D. Freed and M. Hopkins, _Consistency of M-theory on non-orientable manifolds_ , Q. J. Math. 72 (2021), no. 1-2, 603–671.
  * [FM13] D. Freed and G. Moore, _Twisted equivariant matter_ , Ann. Henri Poincaré 14 (2013), no. 8, 1927–2023.
  * [Fre94] D. Freed, _Higher algebraic structures and quantization_ , Comm. Math. Phys. 159 (1994), no. 2, 343–398.
  * [FS06] G. Frobenius and I. Schur, _Uber die reellen Darstellungen der endlichen Gruppen_ , Sitzungsberichte der königlich preussichen Akademi der Wissenschaften zu Berlin (1906), 198–208.
  * [GH10] D. Gao and K. Hori, _On the structure of the Chan–Paton factors for D-branes in type II orientifolds_ , arXiv:1004.3972, 2010.
  * [GI21] P. Georgieva and E.-N. Ionel, _A Klein TQFT: the local real Gromov-Witten theory of curves_ , Adv. Math. 391 (2021), Paper No. 107972, 70.
  * [Gow79] R. Gow, _Real-valued and $2$-rational group characters_, J. Algebra 61 (1979), no. 2, 388–413.
  * [HLS21] T.-C. Huang, Y.-H. Lin, and S. Seifnashri, _Construction of two-dimensional topological field theories with non-invertible symmetries_ , J. High Energy Phys. (2021), no. 12, 43 pp.
  * [HW08] K. Hori and J. Walcher, _D-brane categories for orientifolds—the Landau-Ginzburg case_ , J. High Energy Phys. 4 (2008), 030, 36.
  * [IT23] T. Ichikawa and Y. Tachikawa, _The super Frobenius–Schur indicator and finite group gauge theories on $\textnormal{Pin}^{-}$ surfaces_, Comm. Math. Phys. 400 (2023), no. 1, 417–428.
  * [Kar70] M. Karoubi, _Sur la $K$-théorie équivariante_, Séminaire Heidelberg-Saarbrücken-Strasbourg sur la K-théorie (1967/68), Lecture Notes in Mathematics, Vol. 136, Springer, Berlin, 1970, pp. 187–253.
  * [Kar85] G. Karpilovsky, _Projective representations of finite groups_ , Monographs and Textbooks in Pure and Applied Mathematics, vol. 94, Marcel Dekker, Inc., New York, 1985.
  * [Kho11] V. Khoi, _On Turaev’s theorem about Dijkgraaf-Witten invariants of surfaces_ , J. Knot Theory Ramifications 20 (2011), no. 6, 837–846.
  * [KM97] V. Karimipour and A. Mostafazadeh, _Lattice topological field theory on nonorientable surfaces_ , J. Math. Phys. 38 (1997), no. 1, 49–66.
  * [KR04] A. Kapustin and L. Rozansky, _On the relation between open and closed topological strings_ , Comm. Math. Phys. 252 (2004), no. 1-3, 393–414.
  * [KT17] A. Kapustin and A. Turzillo, _Equivariant topological quantum field theory and symmetry protected topological phases_ , J. High Energy Phys. (2017), no. 3, 006, front matter+19.
  * [Laz01] C. Lazaroiu, _On the structure of open-closed topological field theory in two dimensions_ , Nuclear Phys. B 603 (2001), no. 3, 497–530.
  * [LN11] S. Loktev and S. Natanzon, _Klein topological field theories from group representations_ , SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 070, 15.
  * [LP08] A. Lauda and H. Pfeiffer, _Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras_ , Topology Appl. 155 (2008), no. 7, 623–666.
  * [Med78] A. Mednykh, _Determination of the number of nonequivalent coverings over a compact Riemann surface_ , Dokl. Akad. Nauk SSSR 239 (1978), no. 2, 269–271.
  * [MS06] G. Moore and G. Segal, _D-branes and $K$-theory in 2D topological field theory_, arXiv:hep-th/0609042, 2006.
  * [MY05] M. Mulase and J. Yu, _Non-commutative matrix integrals and representation varieties of surface groups in a finite group_ , Ann. Inst. Fourier (Grenoble) 55 (2005), no. 6, 2161–2196.
  * [NY22] B. Noohi and M. Young, _Twisted loop transgression and higher Jandl gerbes over finite groupoids_ , Algebr. Geom. Topol. 22 (2022), no. 4, 1663–1712.
  * [RT21] D. Rumynin and J. Taylor, _Real representations of $C_{2}$-graded groups: the antilinear theory_, Linear Algebra Appl. 610 (2021), 135–168.
  * [RT22] by same author, _Real representations of $C_{2}$-graded groups: the linear and hermitian theories_, High. Struct. 6 (2022), no. 1, 359–374.
  * [RY21] D. Rumynin and M. Young, _Burnside rings for Real 2-representation theory: The linear theory_ , Commun. Contemp. Math. 23 (2021), no. 5, 2050012, 54.
  * [Sha11] E. Sharpe, _Notes on discrete torsion in orientifolds_ , J. Geom. Phys. 61 (2011), no. 6, 1017–1032.
  * [Sha15] by same author, _Notes on generalized global symmetries in QFT_ , Fortschr. Phys. 63 (2015), no. 11-12, 659–682.
  * [Shi12] K. Shimizu, _Frobenius–Schur indicator for categories with duality_ , Axioms 1 (2012), no. 3, 324–364.
  * [Sny17] N. Snyder, _Mednykh’s formula via lattice topological quantum field theories_ , Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 46, Austral. Nat. Univ., Canberra, 2017, pp. 389–398.
  * [SR17] K. Shiozaki and S. Ryu, _Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in $(1+1)$ dimensions_, J. High Energy Phys. (2017), no. 4, 100, front matter+46.
  * [TT06] V. Turaev and P. Turner, _Unoriented topological quantum field theory and link homology_ , Algebr. Geom. Topol. 6 (2006), 1069–1093.
  * [Tur07] V. Turaev, _Dijkgraaf-Witten invariants of surfaces and projective representations of groups_ , J. Geom. Phys. 57 (2007), no. 11, 2419–2430.
  * [Wig59] E. Wigner, _Group theory and its application to the quantum mechanics of atomic spectra_ , Pure and Applied Physics. Vol. 5, Academic Press, New York-London, 1959.
  * [Wil08] S. Willerton, _The twisted Drinfeld double of a finite group via gerbes and finite groupoids_ , Algebr. Geom. Topol. 8 (2008), no. 3, 1419–1457.
  * [You20] M. Young, _Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf–Witten theory_ , Comm. Math. Phys. 374 (2020), no. 3, 1645–1691.
  * [You21] by same author, _Real representation theory of finite categorical groups_ , High. Struct. 5 (2021), no. 1, 18–70.