# Toledo invariants of Topological Quantum Field Theories

Bertrand Deroin and Julien Marché

###### Abstract

We prove that the Fibonacci quantum representations
$\rho_{g,n}:\operatorname{Mod}_{g,n}\to\mathrm{PU}(p,q)$ for
$(g,n)\in\\{(0,4),(0,5),(1,2),(1,3),(2,1)\\}$ are holonomy representations of
complex hyperbolic structures on some compactifications of the corresponding
moduli spaces $\mathcal{M}_{g,n}$. As a corollary, the forgetful map between
the corresponding compactifications of $\mathcal{M}_{1,3}$ and
$\mathcal{M}_{1,2}$ is a surjective holomorphic map between compact complex
hyperbolic orbifolds of different dimensions higher than one, giving an answer
to a problem raised by Siu.

The proof consists in computing their Toledo invariants: we put this
computation in a broader context, replacing the Fibonacci representations with
any Hermitian modular functor and extending the Toledo invariant to a full
series of cohomological invariants beginning with the signature $p-q$.

We prove that these invariants satisfy the axioms of a Cohomological Field
Theory and compute the $R$-matrix at first order (hence the usual Toledo
invariants) in the case of the $\mathrm{SU}_{2}/\mathrm{SO}_{3}$-quantum
representations at any level.

##  1 Introduction

###  1.1 Motivation

The moduli spaces $\mathcal{M}_{g,n}$ of genus $g$ curves with $n$ marked
points, do not seem to have geometric structures in general, nor their partial
compactifications. However, some very interesting curiosities happen in
particular cases; leading examples of this kind are the compact type partial
compactification of $\mathcal{M}_{2}$ and $\mathcal{M}_{3}$ that carry
structures locally modelled on Siegel spaces (via the Jacobian of the curve),
or the examples of complex hyperbolic structures on certain partial
compactifications of $\mathcal{M}_{0,n}$ with $n\leq 8$, using hypergeometric
integrals (see [15, 41] and [32] for further developments and a nice
historical treatment to this topic). In all these examples, a key role is
played by the holonomy of the geometric structure: a linear representation of
the corresponding mapping class group $\text{Mod}_{g,n}$.

The original motivation of this work, that emerged while the second author
lectured on TQFT in Bordeaux and Paris, see [29], is to investigate whether
quantum representations provide interesting geometric structures on moduli
spaces and/or their partial compactifications: those are representations of
the mapping class groups with values in the projective linear group of vector
spaces called spaces of conformal blocks. They are associated to the data of a
simple compact Lie group $G$, a level $\ell$ (a positive integer), and some
finite set $\Lambda$ of irreducible representations of $G$ (depending on
$\ell$). This theory is extremely rich and have various aspects, one is
analytical, based on quantization of character varieties of surface groups,
the other one is combinatorial/topological, based on a modular category
(constructed from the representation theory of quantum groups or from the
Kauffman bracket), we refer to [5] for a general overview. While the two
points of view are equivalent, we will follow the combinatorial/topological
road here.

A crucial property, which has been made explicit using the topological
viewpoint as in [9] is that the space of conformal blocks is defined over a
cyclotomic field $k$ of order $\ell$ and the image of a quantum representation
takes values in the group of projective transformations that preserves a
pseudo-hermitian form on the space of conformal blocks defined over $k$ (they
also preserve an integral structure, hence taking values in an arithmetic
group, as was proved by Gilmer and Masbaum in [22]). An interesting
consequence is that whence we fix an embedding $i:k\rightarrow\mathbb{C}$, the
representation gives rise to a representation

$\rho_{g,n}^{i}(\lambda_{1},\ldots,\lambda_{n}):\text{Mod}_{g,n}\rightarrow\text{PU}(p,q)$
(1)

where $\lambda_{1},\ldots,\lambda_{n}\in\Lambda$ and $p,q$ are the integers
(depending highly on all data) so that $d=p+q$ is the dimension of the space
of conformal blocks and $\sigma=p-q$ is the signature of the hermitian form.

Denote by $\mathbb{H}^{p,q}$ the Hermitian symmetric space associated to
$\mathrm{PU}(p,q)$. The question that motivates this work is: does
$\mathcal{M}_{g,n}$, or a partial compactification of it, carries a
$\mathbb{H}^{p,q}$-structure whose holonomy is given by the quantum
representation (1)? The existence of geometric structures modelled on other
homogeneous spaces of $\text{PU}(p,q)$ is not addressed here but is certainly
very interesting.

It turns out that in general the dimension of $\mathbb{H}^{p,q}$, which is
equal to the product $pq$, is much larger than the dimension of the moduli
space $\mathcal{M}_{g,n}$, which is $3g-3+n$. To our knowledge, the only
quantum representations where the coincidence of dimension

$pq=3g-3+n>1$

holds are Fibonacci representations: the quantum representations associated to
the compact Lie group $\text{SO}(3)$, with level $\ell=5$. Notice that we do
not use the specific definition of level from conformal field theory: for us
it will simply denote the order of the root of unity necessary to carry over
the construction.

Later in the introduction, we provide an elementary construction of Fibonacci
representations for the reader which is not familiar with TQFT, but before
doing so, we present the main result of the article, namely the computation of
Toledo invariants of quantum representations, which can be performed for any
Hermitian modular functor. The computation of these invariants in the
perspective of geometrization of quantum representations is our fundamental
tool, and in some special cases permits, thanks to Siu’s rigidity theory, to
overcome the lack of naturally defined period maps associated to quantum
representations.

###  1.2 Toledo invariants of Hermitian modular functors

A fundamental property of quantum representations of level $\ell$ is that they
map any Dehn twist to an element of order $\ell$ in the group
$\text{PU}(p,q)$. So they can be thought of as representations defined on the
quotient $\operatorname{Mod}_{g,n}^{\ell}$ of the mapping class group by the
group generated by $\ell$-th powers of Dehn twists. This group is the orbifold
fundamental group of the compact orbifold
$\overline{\mathcal{M}}_{g,n}^{\ell}$ obtained from Deligne-Mumford
compactification $\overline{\mathcal{M}}_{g,n}$ of moduli space by twisting
the complex structure along the boundary divisors of
$\overline{\mathcal{M}}_{g,n}$, namely the map
$\overline{\mathcal{M}}_{g,n}^{\ell}\rightarrow\overline{\mathcal{M}}_{g,n}$
is a set-theoretic bijection but in orbifold charts, it ramifies at order
$\ell$ on the boundary divisors. In terms of stacks, this is the $\ell$-root
stack ramifying along the boundary divisors : it has already been considered
in the context of TQFT [18] and in the context of $r$-spin structures, see
[13] Section 2.1. From all this, we can see quantum representations as
representations defined on the fundamental group
$\pi_{1}(\overline{\mathcal{M}}_{g,n}^{\ell})$.

The main topological information that detects whether a representation
$\rho:\pi_{1}(X)\rightarrow\text{PU}(p,q)$ defined on the fundamental group of
a compact complex manifold is or isn’t the holonomy of a
$\mathbb{H}^{p,q}$-structure, is contained in a characteristic class belonging
to the second rational cohomology of the manifold, called the Toledo
invariant. This class is the degree two part of a higher cohomology class that
is defined in the following way: suppose $\rho$ lifts to a representation with
values in $\text{U}(p,q)$, and take a decomposition of $\mathcal{E}$, the flat
$\mathbb{C}^{p,q}$-bundle over $X$ with monodromy $\rho$ as an orthogonal sum
$\mathcal{E}=\mathcal{E}^{+}\oplus\mathcal{E}^{-}$ of a positive rank $p$
subbundle $\mathcal{E}^{+}$ and a negative rank $q$ subbundle
$\mathcal{E}^{-}$. Then the higher cohomology class is defined by

$\operatorname{sch}(\rho):=\operatorname{ch}(\mathcal{E}^{+})-\operatorname{ch}(\mathcal{E}^{-})\in
H^{*}(X,\mathbb{Q})$

where $ch$ is the Chern character and $sch$ stands for super/signed Chern
character. These invariants have been introduced in the context of Hermitian
K-theory with applications to algebraic topology, see [33]. If $X$ is an
orbifold, we can still define a higher Toledo class in the cohomology of the
underlying topological space of $X$ with rational coefficients. In the case of
a quantum representation
$\rho:\pi_{1}(\overline{\mathcal{M}}_{g,n}^{\ell})\rightarrow\text{PU}(p,q)$,
we thus have a class $\operatorname{sch}(\rho)\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})\simeq
H^{*}(\overline{\mathcal{M}}_{g,n}^{\ell},\mathbb{Q})$.

Recall that the representation $\rho$ depends on $g,n$ of course, but also on
the particular embedding $i:k\to\mathbb{C}$ and the colors
$\lambda_{1},\ldots,\lambda_{n}$ attached to the marked points. Setting
$V=\mathbb{Q}[\Lambda]$, we define a multilinear map $\omega_{g,n}:V^{n}\to
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ by putting

$\omega_{g,n}(\lambda_{1},\ldots,\lambda_{n})=\operatorname{sch}(\rho_{g,n}^{i}(\lambda_{1},\ldots,\lambda_{n})).$

######  Theorem.

For any hermitian modular functor, the family $\omega_{g,n}$ defined above
satisfies the axioms of a Cohomological Field Theory (CohFT).

A particular interesting instance of this is that the degree $0$ part of this
CohFT, namely the signature $\sigma$, defines on $V$ a structure of Frobenius
algebra which have not been studied before as far as we know. We prove that
for the $\mathrm{SU}_{2}/\mathrm{SO}_{3}$-TQFTs at any level, these
$\mathbb{Q}$-algebras are semi-simple.

To illustrate this theorem, consider the example of the Fibonacci
representation where $k=\mathbb{Q}(q)$ and $q^{5}=1$. In this case, $\Lambda$
has two elements called the trivial color and the non-trivial color and
$V=\mathbb{Q}[\Lambda]$ is a quadratic number field.

When $i(q)=e^{4i\pi/5}$, the representation $\rho^{i}_{g,n}$ becomes unitary,
hence the higher cohomological invariants vanish: we have
$\operatorname{sch}(\rho^{i}_{g,n})=d_{g,n}$, the dimension of the
representation. Then $V=\mathbb{Q}(\varphi)$ as an algebra where
$\varphi=-q-q^{-1}$ is the golden ratio and

$d_{g,n}=\operatorname{Tr}_{\mathbb{Q}(\varphi)/\mathbb{Q}}(\varphi^{n}(2+\varphi)^{g-1}).$

When $i(q)=e^{2i\pi/5}$, the representation $\rho^{i}_{g,n}$ is no longer
unitary. We have in this case $V=\mathbb{Q}(j)$ where $j^{2}+j+1=0$ and the
signature of the representation is

$\sigma_{g,n}=\operatorname{Tr}_{\mathbb{Q}(j)/\mathbb{Q}}(j^{n}(2+j)^{g-1}).$

We collect in the following table the complete signature $p|q$ of $(V,h)$:

| $n=0$ | $n=1$ | $n=2$ | $n=3$ | $n=4$ | $n=5$ | $n=6$
---|---|---|---|---|---|---|---
$g=0$ | $1|0$ | $0|0$ | $0|1$ | $1|0$ | $1|1$ | $1|2$ | $3|2$
$g=1$ | $2|0$ | $0|1$ | $1|2$ | $3|1$ | $3|4$ | $5|6$ | $10|8$
$g=2$ | $4|1$ | $1|4$ | $5|5$ | $9|6$ | $11|14$ | $20|20$ | $34|31$
$g=3$ | $9|6$ | $7|13$ | $19|16$ | $29|26$ | $42|48$ | $74|71$ | $119|116$

This table makes clearer the terminology Fibonacci. We will provide explicit
formulas for the three Frobenius algebras arising at level $7$. At prime level
$\ell$, these algebras seem to be particularly interesting number fields and
deserve a further study. They also provide a conceptual explanation to a
phenomenon observed by Funar, Pitsch and Costantino see [20].

The general theorem opens the door to a computation of the cohomological
invariants using the Givental-Teleman classification theorem, see [35]. In
particular, we give in this article an algorithm for computing the $R$-matrix
at first order that we implemented with Sage. The computation in the Fibonacci
case $(\ell=5)$ can be performed by hand and is already much more complicated
when $\ell=7$.

The computation of this $R$-matrix reduces to the computation of Toledo
invariants of representations of triangle groups in $\mathrm{PU}(p,q)$. We
provide in Appendix A a formula generalizing Meyer’s formula for the signature
of $4$-manifolds which reduces the computation to the signature of some
explicit Hermitian matrices.

These CohFTs look particularly interesting: it seems difficult to compute the
$R$-matrices at higher order or to find the spectral curve encoding it through
Topological Recursion. Notice that CohFTs already appeared in the context of
modular functors in [30, 2] where the authors computed the Chern character of
the vector bundle of conformal blocks over $\overline{\mathcal{M}}_{g,n}$. We
stress that our construction is indeed different as it highly relies on the
Hermitian structure, which plays no role in the aforementioned articles.

###  1.3 Interlude: a quick construction of Fibonacci representations

We sketch here a construction which is detailed in [29] for the case of
$\mathrm{SU}_{2}$-modular functors. The Fibonacci case which is treated here
is indeed different, but the proofs are similar. We include it so that the
unfamiliar reader get a flavour of it: we refer to [9] for a full account of
these constructions.

Let $S$ be a surface of genus $g$ and $P\subset S$ be a finite subset of
punctures. We set $k=\mathbb{Q}(q)$ to be the cyclotomic field of order $5$
where $q^{5}=1$ and define the elements $\varphi=-q-q^{-1}$ and $A=-q^{3}$.

We define $\mathcal{A}(S,P)$ as the $k$-vector space generated by isotopy
classes of finite graphs $G$ embedded in $S\setminus P$ (or equivalently
$1$-dimensional sub-cell-complexes) up to the following five local moves.

  1. 1.

Contraction-Deletion relation: $[G/e]=[G]+[G\setminus e]$.

$G/e$
---
$G$
---
$G\setminus e$
---
$e$
---
  2. 2.

Fusion relation: $[G_{\times}]=\varphi^{-1}[G_{||}]+\varphi^{-1}[G_{=}]$.

$G_{\times}$
---
$G_{||}$
---
$G_{=}$
---
$G_{c}$
---
  3. 3.

Bridge and loop relation: see Figure 1.

$=0$
---
$=\varphi$
---
Figure 1: Bridge and loop relations.

  4. 4.

Boundary relation: $\gamma=1-\varphi$ for all curves $\gamma$ surrounding a
puncture $x\in P$, that is $\gamma=\partial D^{2}$ where $D^{2}\subset S$ and
$D^{2}\cap P=\\{x\\}$.

One can easily prove that this vector space is finite dimensional and carries
an action of the mapping class group $\operatorname{Mod}(S,P)$ by the formula
$[f].[G]=[f(G)]$.

Even more, $\mathcal{A}(S,P)$ has an algebra structure given by “stacking” and
which is formally defined in the following way. Let $G_{1},G_{2}$ be two
graphs embedded in $S\setminus P$. We can ensure by an isotopy that they
intersect tranversally in a finite number of points. For any $\xi:G_{1}\cap
G_{2}\to\\{\pm 1\\}$ we define the smoothing $G_{1}\cup_{\xi}G_{2}$ by
replacing the neighborhood of each intersection point $p\in G_{1}\cap G_{2}$
by a diagram where $G_{1}$ turns lefts to $G_{2}$ at $p$ if $\xi(p)=1$, right
if $\xi(p)=-1$. We set then

$[G_{1}][G_{2}]=\sum_{\xi:G_{1}\cap G_{2}\to\\{\pm
1\\}}A^{\sum_{p}\xi(p)}[G_{1}\cup_{\xi}G_{2}].$

One can then prove that this product induces a well-defined structure of
algebra on $\mathcal{A}(S,P)$ which is preserved by $\operatorname{Mod}(S,P)$.

Traditionally, this algebra structure is described with skein modules: we put
$G_{1}$ “above” $G_{2}$ and apply at each crossing the Kauffman relation
$[G_{c}]=A[G_{||}]+A^{-1}[G_{=}]$. The construction of the Fibonacci
representation reduces to the following structure theorem:

######  Theorem.

The algebra $\mathcal{A}(S,P)$ is isomorphic to $\operatorname{End}(V)$ for
some finite dimensional $k$-vector space $V$.

Let $\Phi:\mathcal{A}(S,P)\to\operatorname{End}(V)$ be such an (non canonical)
isomorphism. As $\operatorname{Mod}(S,P)$ acts on $\mathcal{A}(S,P)$ by
algebra automorphisms, the Skolem-Noether theorem implies that this action is
given through $\Phi$ by a conjugation, hence defining the Fibonacci
representation $\rho:\operatorname{Mod}(S,P)\to\mathrm{PGL}(V)$ such that

$\Phi(f.x)=\rho(f)\Phi(x)\rho(f)^{-1}\quad\forall
f\in\operatorname{Mod}(S,P),\forall x\in\mathcal{A}(S,P).$

Finally, let $x\mapsto\overline{x}$ be the involution of $k$ satisfying
$\overline{q}=q^{-1}$. This extends uniquely to an anti-involution of
$\mathcal{A}(S,P)$ given by $\lambda[G]\mapsto\overline{\lambda}[G]$ for any
embedded graph $G\subset S\setminus P$. This anti-involution corresponds
through $\Phi$ to the adjunction with respect to a Hermitian form $h$ on $V$
preserved by $\operatorname{Mod}(S,P)$. In formulas,
$h(\Phi(x)v,w)=h(v,\Phi(\overline{x})w)$ for all $v,w\in V$.

The involution being preserved by $\operatorname{Mod}(S,P)$, the Fibonacci
representation is promoted to a representation
$\rho:\operatorname{Mod}(S,P)\to\mathrm{PU}(V)$.

To actually work with this construction, we need to find an explicit model for
$V$: it can be constructed in a way similar to $\mathcal{A}(S,P)$ from a
handlebody bounding the surface $S$, we refer to [29] for details.

###  1.4 Geometrization: cabinet de curiosités

As the reader can check in the signature table of Fibonacci representations,
the coincidence of dimension $pq=3g-3+n$ happens only in few cases that we
list here

$(g,n)\in\\{(0,4),\ (0,5),\ (1,2),\ (1,3),\ (2,1)\\}$ (2)

We prove that each of these coincidences correspond to a genuine complex
hyperbolic structure on some compactification of the corresponding moduli
space.

It turns out that in the Fibonacci case $\ell=5$, there exists an orbifold
contraction
$\overline{\mathcal{M}}_{g,n}^{5}\rightarrow\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$,
which contracts the boundary divisor consisting of stable nodal curves having
at least one elliptic tail. This contraction was already considered in the PhD
dissertation of Livne [27] in the case of $(g,n)=(1,2)$, but we provide a
generalization of this result for any $(g,n)\neq(2,0)$, see 2.3.1 (in the case
$(g,n)=(2,0)$, the contraction leads to a quadratic singularity that will be
studied in a forthcoming paper). This elliptic contraction, which might be
interesting in its own, is very well adapted to the study of Fibonacci
representations, since
$\pi_{1}(\overline{\mathcal{M}}_{g,n}^{5})\simeq\pi_{1}(\overline{\mathcal{M}}_{g,n}^{\mathcal{E}})$.
We prove (see the combination of Propositions 9, 10, 11 and 12)

######  Theorem.

In each of the cases (2), the elliptic tail contraction
$\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$ admits a complex hyperbolic
structure whose holonomy is the corresponding Fibonacci quantum
representation.

The uniformization of $\overline{\mathcal{M}}_{0,5}^{5}$ by the complex
hyperbolic plane has been made explicit by Deligne and Mostow in [15], with
the use of hypergeometric integrals. Together with Theorem Theorem this gives
a proof that the quantum representation $\rho_{0,5}^{5}$ is the monodromy of
the hypergeometric function

$F(x,y)=\int_{1}^{\infty}u^{-2/5}(u-1)^{-2/5}(u-x)^{-2/5}(u-y)^{-2/5}du.$

Hirzebruch gave an alternative more abstract argument, by computing the Chern
numbers of a convenient finite abelian smooth covering of the orbifold
$\overline{\mathcal{M}}_{0,5}^{5}$ and showed that they satisfy the equality
$c_{1}^{2}=3c_{2}$, leading to the conclusion, thanks to Yau’s theorem
(solution to the Calabi’s conjecture), that $\overline{\mathcal{M}}_{0,5}^{5}$
has a complex hyperbolic structure. This relation of Hirzebruch complex
hyperbolic orbifold with that of the orbifold
$\overline{\mathcal{M}}_{0,5}^{5}$ has been noticed by Eyssidieux and Funar,
see [18, Example 2.7].

Analogously, the construction of the elliptic tail contraction together with
its complex hyperbolic structure was discovered in the PhD’s dissertation of
Livne, [27]. What theorem Theorem says in that case is that the holonomy of
this structure is in fact the corresponding Fibonacci representation.

A nice unexpected consequence of Theorem Theorem is that it provides the
solution of a problem raised by Siu in the survey paper [40, Problem (a), p.
182]. We prove

######  Corollary.

There exists a surjective holomorphic map between connected compact complex
hyperbolic manifolds, the domain and target being respectively of dimension
$3$ and $2$.

This map is obtained by lifting the forgetful map
$\overline{\mathcal{M}}_{1,3}^{\mathcal{E}}\rightarrow\overline{\mathcal{M}}_{1,2}^{\mathcal{E}}$
to finite smooth coverings. Using forgetful maps to approach Siu’s problem was
already investigated in the context of Deligne-Mostow orbifolds in the work of
Deraux [16], although the conclusion was opposite. We notice that Koziarz and
Mok proved that such a surjective map between complex hyperbolic manifolds of
different dimensions cannot be a submersion, see [26].

Acknowledgments: We are indebted to many persons for numerous conversations
and advices around this work, including Martin Deraux, Pascal Dingoyan,
Philippe Eyssidieux, Elisha Falbel, Louis Funar, Selim Ghazouani, Alessandro
Giacchetto, Vincent Koziarz, Danilo Lewanski, Ron Livne, Gregor Masbaum, Luc
Pirio, Adrien Sauvaget, Jérémy Toulisse, Nicolas Tholozan and Dimitri
Zvonkine.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Toledo invariants of Hermitian modular functors
    3. 1.3 Interlude: a quick construction of Fibonacci representations
    4. 1.4 Geometrization: cabinet de curiosités
  2. 2 Twisted orbifold structures on $\overline{\mathcal{M}}_{g,n}$
    1. 2.1 Preliminary remarks on orbifolds
      1. 2.1.1 Orbifold versus orbispaces
      2. 2.1.2 Euler characteristic
      3. 2.1.3 The orbifold structure of $\overline{\mathcal{M}}_{1,1}^{\ell}$
      4. 2.1.4 The orbispace $\overline{\mathcal{M}}_{1,1}^{\ell}$ as a classifying space
    2. 2.2 Construction of the twisted compactification $\overline{\mathcal{M}}_{g,n}^{l}$
      1. 2.2.1 The orbifold structure $\overline{\mathcal{M}}_{g,n}^{l}$
      2. 2.2.2 The orbifold $\overline{\mathcal{M}}^{\ell}_{g,n}$ is uniformizable for $\ell\geq 5$ odd
      3. 2.2.3 Forgetful map
      4. 2.2.4 The $\ell$-twisted compactification as a classifying space
    3. 2.3 The elliptic tail contraction
      1. 2.3.1 The construction
      2. 2.3.2 Forgetful map between Fibonacci elliptic tail contractions
    4. 2.4 The canonical bundles of $\overline{\mathcal{M}}_{g,n}^{\ell}$ and $\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$
    5. 2.5 The Euler characteristic of $\overline{\mathcal{M}}^{\ell}_{g,n}$
  3. 3 Hermitian cohomological invariants
    1. 3.1 Definition of the invariants
    2. 3.2 Compatibility with operations
    3. 3.3 A differential definition of the super Chern character
    4. 3.4 Relation to the tangent bundle of the symmetric space
    5. 3.5 The Toledo class as an obstruction class
  4. 4 Hermitian modular functors
    1. 4.1 Marked surfaces
    2. 4.2 Hermitian modular functor
    3. 4.3 The associated cohomological field theory
    4. 4.4 Proof of the CohFT axioms
      1. 4.4.1 Forgetting a point
      2. 4.4.2 Non-separating gluing
    5. 4.5 Separating gluing
  5. 5 Computation of the CohFT associated to the $\mathrm{SU}_{2}/\mathrm{SO}_{3}$-modular functors
    1. 5.1 Semi-simplicity of the Frobenius algebras
      1. 5.1.1 Generalities on Frobenius algebras
      2. 5.1.2 The $\mathrm{SU}_{2}$-modular functor
      3. 5.1.3 The $\mathrm{SO}_{3}$-modular functor
    2. 5.2 Degree $2$ of a CohFT and the $R_{1}$-matrix
      1. 5.2.1 Consequences of the Givental-Teleman theorem
      2. 5.2.2 A decomposition of the $R_{1}$-matrix
      3. 5.2.3 The computation of $R_{1}^{\prime}$ for $\mathrm{SO}_{3}$-modular functors
      4. 5.2.4 The computation of $r_{1}$ for $\mathrm{SO}_{3}$-modular functors
    3. 5.3 The example of $q=\exp(2i\pi/5)$
    4. 5.4 The examples of level 7
      1. 5.4.1 The Frobenius algebras
      2. 5.4.2 The $R_{1}$-matrix
  6. 6 Complex hyperbolic structures on moduli spaces associated to Fibonacci TQFT
    1. 6.1 A criterion for uniformization
    2. 6.2 Deligne-Mostow / Hirzebruch’s example
    3. 6.3 Livne’s example
    4. 6.4 Complex hyperbolic structure on $\overline{\mathcal{M}}_{1,3}^{\mathcal{E}}$
    5. 6.5 Complex hyperbolic structure on $\overline{\mathcal{M}}_{2,1}^{\mathcal{E}}$
    6. 6.6 Solution to Siu’s problem: proof of Corollary Corollary
  7. A Meyer formula for the Toledo invariant
    1. A.1 Definition of the Meyer cocycle
    2. A.2 A relation with the Toledo class
    3. A.3 Application to triangle groups
  8. B Toledo invariants from mapping class group presentations
    1. B.1 Equivariant Hopf formula
    2. B.2 Computation of the Toledo invariant
      1. B.2.1 The easy relations: $D,B,\partial_{j},R_{i}$
      2. B.2.2 2-chain
      3. B.2.3 Lantern
      4. B.2.4 Solving the linear system

##  2 Twisted orbifold structures on $\overline{\mathcal{M}}_{g,n}$

In the context of algebraic geometry, the moduli space $\mathcal{M}_{g,n}$ and
its compactification $\overline{\mathcal{M}}_{g,n}$ are Deligne-Mumford
stacks, a notion invented specially for them. The twisted version dealed with
in this article has already been introduced in the context of TQFT in [18] and
in the context of $r$-spin structures, see [13] Section 2.1. This notion is
not very accessible, at least to the authors of this article, and is not
strictly necessary for our purposes. For these reasons we define here these
compactifications in an independent way. We present both the orbifold and
orbispace viewpoints. Even though we will concretely mostly use the first one,
the second will be useful to keep in mind for those having a more topological
background/affinities. Most of the material of this section is classical,
apart in the last subsection where a new construction of a particular
contraction of the twisted orbifold structure of
$\overline{\mathcal{M}}_{g,n}$ in the level $\ell=5$ case is described: the
elliptic tail contraction.

###  2.1 Preliminary remarks on orbifolds

####  2.1.1 Orbifold versus orbispaces

In this article, we oscillate between two points of view on orbifolds. The
first one is the usual concept of orbifold in the realm of differential
complex geometry, the other one is the notion of orbispace which belongs to
homotopy theory. Both are well-known, we refer to [23] for a nice discussion
about their interplay. For the benefit of the reader, let us recall what these
structures mean in the case of a developable orbifold, i.e a space of the form
$X/G$ where $X$ is a complex variety and $G$ is a discrete group acting
properly and holomorphically on $X$.

An orbifold chart of $X/G$ around $[x]$ is obtained by linearizing the action
of $\operatorname{Stab}(x)$ in a neighborhood $U$ of $x\in X$. This open set
is projected to a neighborhood of $[x]\in X/G$, providing the orbifold atlas
of $X/G$. A map $f:X/G\rightarrow Y/H$ between two developable orbifolds is an
orbifold map if it can be lifted to a map $\widetilde{f}:X\rightarrow Y$ which
is equivariant with respect to a morphism $G\rightarrow H$.

The emblematic example that will be considered here is the moduli space
$\mathcal{M}_{g,n}$ of algebraic curves of genus $g$ with $n$ marked points,
assuming that the stability condition $3g-3+n>0$ holds. This space is the
quotient of the Teichmüller space $\mathcal{T}_{g,n}$ by the action of the
mapping class group $\text{Mod}_{g,n}$. We recall (see e.g. [8]) that
$\mathcal{T}_{g,n}$ has a structure of smooth complex manifold of dimension
$3g-3+n$ and that the action of $\text{Mod}_{g,n}$ is properly discontinuous,
so $\mathcal{M}_{g,n}=\mathcal{T}_{g,n}/\operatorname{Mod}_{g,n}$ has a
natural structure of developable orbifold.

Another fundamental example is the Deligne-Mumford compactification
$\overline{\mathcal{M}}_{g,n}$ of $\mathcal{M}_{g,n}$. We refer to [25], [4]
or [45] for its definition as an orbifold, and provide a review of its
construction in Section 2.2. Contrary to $\mathcal{M}_{g,n}$,
$\overline{\mathcal{M}}_{g,n}$ is not developable. However, we will work with
alternative compactifications, twisted versions of
$\overline{\mathcal{M}}_{g,n}$, which are developable, see section 2.2.

This point of view is well-adapted for most geometric constructions involving
for instance the integration of differential forms. However, the algebraic
topology of $X/G$ is partially lost in the underlying topological space and
not so easy to capture from the system of orbifold charts, as for instance the
orbifold fundamental group.

For this reason, what we call the orbispace is, in this case, the homotopical
quotient, that is the space $X_{G}=EG\times X/G$ where $EG$ is a contractible
space with a free and proper action of $G$. The action of $G$ is diagonal so
that we have a natural projection $p:X_{G}\to X/G$. In the case when $G$ is a
finite group acting trivially on a point $*$, this gives $*_{G}=EG/G=BG$, the
classifying space of $G$. We observe that in general, the preimage
$p^{-1}([x])$ is a classifying space for the finite group
$\operatorname{Stab}(x)$. We refer to [23] for a general definition of
orbispace and for the construction of the orbispace associated to an orbifold.

The advantage of this second definition is that the orbifold fundamental group
of $X/G$ is the usual fundamental group of $X_{G}$ and more generally, all
invariants of $X/G$ coming from algebraic topology will be, by definition, the
usual invariants of the homotopical quotient $X_{G}$.

To sum up, an orbifold is a topological space $M$ endowed with a system of
orbifold charts that we denote by $M^{o}$. It can be converted into a
(infinite dimensional) cell-complex $M^{h}$ by gluing the homotopical
quotients of the charts. This latter space comes with a map $p:M^{h}\to M$
such that $p^{-1}(\\{x\\})\simeq B_{\operatorname{Stab}(x)}$. When no
confusion is possible, the three structures $M,M^{o},M^{h}$ will be denoted
simply by $M$.

In paragraphs 2.1.3 and 2.1.4, we describe the twisted orbifold and orbispace
structure $\overline{\mathcal{M}}_{1,1}^{\ell}$ in details, for the benefit of
the reader who is not familiar with these notions.

####  2.1.2 Euler characteristic

If a topological space $A$ is homeomorphic to the complement of a closed
subcomplex $F$ of a finite complex $X$, we set $\chi(A)=\chi(X)-\chi(F)$. This
quantity satisfies the identity $\chi(A)+\chi(B)=\chi(A\cup B)+\chi(A\cap B)$
when it makes sense and gives rise to an Eulerian integral so that
$\chi(X)=\int_{X}d\chi$, see for instance [14] for a full account.

If $\widetilde{X}\to X$ is a finite covering of degree $d$ of finite CW-
complexes, one has $\chi(\widetilde{X})=d\chi(X)$. As $EG\to BG$ is a covering
of degree $|G|$ and $EG$ is contractible, it is natural to set
$\chi(BG)=1/|G|$.

Finally, by integrating the Euler characteristic along the fibers of the map
$p:M^{h}\to M$, we are led to define

$\chi(M^{o})=\int_{M}\frac{d\chi(x)}{|\operatorname{Stab}(x)|}.$

####  2.1.3 The orbifold structure of $\overline{\mathcal{M}}_{1,1}^{\ell}$

Consider first $\mathcal{M}_{1,1}$, the moduli space of elliptic curves with
one marked point. As any pointed elliptic curve has the form
$E_{\tau}=(\mathbb{C}/\mathbb{Z}\oplus\tau\mathbb{Z},0)$ for some
$\tau\in\mathbb{H}$, two such curve $E_{\tau}$ and $E_{\tau^{\prime}}$ being
biholomorphic iff $\tau^{\prime}=\frac{a\tau+b}{c\tau+d}$ where
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z})$, we have
$\mathcal{M}_{1,1}=\mathbb{H}/\mathrm{SL}_{2}(\mathbb{Z})$ where the quotient
is understood in the orbifold sense. The underlying topological space is a
complex plane where the generic point has a stabilizer of order 2 and two
special points have order 4 and 6. This gives
$\chi(\mathcal{M}_{1,1})=-\frac{1}{12}$.

In order to compactify $\mathcal{M}_{1,1}$, we add all rational points to the
boundary of $\mathbb{H}$ to define
$\overline{\mathbb{H}}=\mathbb{H}\cup\mathbb{P}^{1}(\mathbb{Q})$ and set
$\overline{\mathcal{M}}_{1,1}=\overline{\mathbb{H}}/\mathrm{SL}_{2}(\mathbb{Z})$.
As these rational points form a single orbit, we have set theoretically
$\overline{\mathcal{M}}_{1,1}=\mathcal{M}_{1,1}\cup\\{\infty\\}$.

As $\overline{\mathbb{H}}$ is no longer a complex variety, we need to explain
how are defined the orbifold charts around the point $\infty$. The stabilizer
of $\infty\in\overline{\mathbb{H}}$ is the group $\mathbb{Z}$ of translations
by $1$: for $r>1$, the open set
$U_{r}=\\{z\in\mathbb{H},\operatorname{Im}z>r\\}\cup\\{\infty\\}$ induces a
homeomorphism $U_{r}/\mathbb{Z}\to\overline{\mathcal{M}}_{1,1}$ onto a
neighborhood $V$ of the point at infinity. We fix now $\ell$ a positive
integer and identify $U_{r}/\ell\mathbb{Z}$ with the disc $D$ of radius
$e^{-2\pi r/\ell}$ by mapping $z$ to $e^{2i\pi z/\ell}$. By construction,
there is a map $D\to V\subset\overline{\mathcal{M}}_{1,1}$ which induces a
homeomorphism $D/\mu_{\ell}\simeq V$ where $\mu_{\ell}$ is the group of
$\ell$-th roots of unity.

######  Definition 1.

The orbifold structure on $\overline{\mathcal{M}}_{1,1}^{\ell}$ is the unique
orbifold structure on $\overline{\mathcal{M}}_{1,1}$ which extends the
orbifold structure on $\mathcal{M}_{1,1}$ and such that the orbifold chart
around the point at infinity is given by the action of the group
$\mu_{\ell}\times\mathbb{Z}/2\mathbb{Z}$ on $D$ where the second factor acts
trivially.

By construction, the isotropy group at infinity is
$\mathbb{Z}/\ell\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ which gives in
particular
$\chi(\overline{\mathcal{M}}_{1,1}^{\ell})=\frac{1}{2\ell}-\frac{1}{12}$. Its
fundamental group is a double cover of the triangular group
$\Delta(2,3,\ell)=\langle\alpha,\beta,\gamma|\alpha^{2}=\beta^{3}=\gamma^{\ell}=\alpha\beta\gamma=1\rangle$.
It is well-known that this orbifold is developable, its universal covering
being the hyperbolic plane $\mathbb{H}$ if $\ell>6$, the complex plane
$\mathbb{C}$ if $\ell=6$ and the Riemann sphere $\mathbb{P}^{1}$ if $\ell<6$.
Notice that in the case $\ell=5$, the fundamental group of
$\overline{\mathcal{M}}_{1,1}^{\ell}$ is the binary icosahedral group
$\mathrm{I}2\subset\text{SU}(2)$. This orbifold plays a fundamental role in
the article, notably in section 2.3.1.

We also observe that the topological space underlying
$\overline{\mathcal{M}}_{1,1}^{\ell}$ is homeomorphic to $S^{2}$. As the map
$p:^{h}\\!\\!\overline{\mathcal{M}}_{1,1}^{\ell}\to\overline{\mathcal{M}}_{1,1}^{\ell}$
induces an isomorphism in rational (co-)homology and cohomology (as shown by
the spectral sequence of equivariant (co-)homology), we get
$H^{*}(\overline{\mathcal{M}}_{1,1}^{\ell},\mathbb{Q})\simeq
H^{*}(S^{2},\mathbb{Q}).$

####  2.1.4 The orbispace $\overline{\mathcal{M}}_{1,1}^{\ell}$ as a
classifying space

Recall that the classifying space of a category is a simplicial set whose
vertices are the objects of the category and $n$-simplices are parametrized by
chains of maps $C_{0}\to\cdots\to C_{n}$. We refer to [38] for this notion and
recall the two following basic facts. Two equivalent categories have
homotopically equivalent classifying spaces and the classifying space of the
category with one object $*$ with $\operatorname{Hom}(*,*)=G$ is the
classifying space $BG$.

Consider a category $SC_{\ell}$ whose objects consists in pairs $(S,\gamma)$
where $S$ is a closed surface of genus $1$ and $\gamma$ is an essential simple
closed curve, possibly empty.

Denote by $\Gamma_{\ell}(S,\gamma)$ the group of homeomorphisms of $S$
preserving $\gamma$ and isotopic to some power of $T_{\gamma}^{\ell}$, where
$T_{\gamma}$ denotes the Dehn twist along $\gamma$. A morphism
$f:(S,\gamma)\to(S^{\prime},\gamma^{\prime})$ is a homeomorphism such that
$f(\gamma)\subset\gamma^{\prime}$ with the relation that
$f\sim\varphi^{\prime}\circ f\circ\varphi$ for any
$\varphi\in\Gamma_{\ell}(S,\gamma)$ and
$\varphi^{\prime}\in\Gamma_{\ell}(S^{\prime},\gamma^{\prime})$.

The classsifying space of this category is precisely the space
${}^{h}\\!\\!\overline{\mathcal{M}}_{1,1}^{\ell}$. The classifying space of
the subcategory of pairs of the form $(S,\emptyset)$ is the space
$B_{\mathrm{SL}_{2}(\mathbb{Z})}$ which is homotopic to
${}^{h}\\!\\!\mathcal{M}_{1,1}$. The classifying space of the subcategory of
pairs of the form $(S,\gamma)$ with $\gamma\neq\emptyset$ is the space
$B_{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/\ell\mathbb{Z}}$.

This construction is a $\ell$-twisted version of the construction given in
[12].

###  2.2 Construction of the twisted compactification
$\overline{\mathcal{M}}_{g,n}^{l}$

####  2.2.1 The orbifold structure $\overline{\mathcal{M}}_{g,n}^{l}$

We review an analytic construction of a twisted version of Deligne-Mumford’s
orbifold, which was considered in the work of Eyssidieux and Funar [18]. Our
point of view is slightly different, and instead of using the stack road, we
use the augmented Teichmüller space.

Fix integers $g,n\geq 0$, $\ell\geq 1$, such that $2g-2+n>0$, and let $S$ be a
reference oriented closed surface of genus $g$ with a subset $P\subset S$ of
cardinality $n$. We denote by $\overline{\mathcal{T}}(S,P)$ the augmented
Teichmüller space, namely the set of equivalence classes of couples $(C,f)$
where $C$ is a stable nodal curve of genus $g$ and $f:S\to C$ is a pinching
map, which means

  1. 1.

$f:S\to C$ is a continuous map such that
$f(P)\cap\operatorname{Sing}{C}=\emptyset$ where $\operatorname{Sing}(C)$ is
the set of nodes of $C$.

  2. 2.

For all $x\in\operatorname{Sing}{C}$, $\alpha_{x}=f^{-1}(x)$ is a simple curve
and, setting $\alpha=f^{-1}\operatorname{Sing}(C)$, $f$ induces a
homeomorphism from $S\setminus\alpha$ to $C\setminus\operatorname{Sing}(C)$.

The isotopy class of $\alpha$ will be referred to as the pinched set of $f$.
The stability condition is that each component of
$C\setminus\operatorname{Sing}{C}\cup f(P)$ has negative Euler characteristic.
Finally, two pairs $(C,f)$ and $(C^{\prime},f^{\prime})$ are equivalent if
there exists a biholomorphism $\varphi:C\to C^{\prime}$ such that
$\varphi\circ f$ and $f^{\prime}$ are isotopic.

The augmented Teichmüller space is not a manifold, but it carries a natural
stratification by sets having a complex manifold structure. Given an isotopy
class of one dimensional submanifold $\alpha\subset S\setminus P$ whose
complementary regions have negative Euler characteristic, let $B_{\alpha}$ be
the stratum corresponding to curves whose pinched set is $\alpha$. Each
stratum $B_{\alpha}$ is naturally identified with a product of usual
Teichmüller spaces, and acquires a structure of complex manifold. For
instance, the strata of maximal dimension $B_{\emptyset}$ is identified with
the usual Teichmüller space $\mathcal{T}(S,P)$. For the topology on
$\overline{\mathcal{T}}(S,P)$ we refer to [1, 8] (see also the more recent
treatments [4] and [25]). The naive quotient
$\overline{\mathcal{M}}(S,P)=\overline{\mathcal{T}}(S,P)/\operatorname{Mod}(S,P)$
of the augmented Teichmüller space by the modular group is a compact space
homeomorphic to the underlying topological space of Deligne-Mumford
compactification $\overline{\mathcal{M}}_{g,n}$ of the moduli space of curves,
see [24].

We now review the complex orbifold structures on $\overline{\mathcal{M}}(S,P)$
inherited from Deligne-Mumford, and its twisted versions. Let
$\Gamma_{\alpha}\subset\operatorname{Mod}(S,P)$ be the subgroup generated by
the Dehn twists along the components of $\alpha$, and by
$\operatorname{Mod}(S,P,\alpha)\subset\operatorname{Mod}(S,P)$ the subgroup of
elements that fix $\alpha$ (the components might be permuted). Notice that
$\Gamma_{\alpha}$ is a free abelian group of rank $|\alpha|$, the number of
components of $\alpha$. We have an exact sequence

$0\rightarrow\Gamma_{\alpha}\rightarrow\operatorname{Mod}(S,P,\alpha)\rightarrow\operatorname{Mod}(S/\alpha,P)\rightarrow
0$

where $S/\alpha$ is obtained from $S$ by collapsing each connected component
of $\alpha$ to a point. If $(C,f)$ is an element of
$\overline{\mathcal{T}}(S,P)$ whose pinched set is $\alpha$, we define
$\operatorname{Aut}(C,f)$ so that it fits in the following exact sequence

$0\to\Gamma_{\alpha}\to\operatorname{Stab}(C,f)\to\operatorname{Aut}(C,f)\to
0$

Let $U_{\alpha}$ be the open subset of $\overline{\mathcal{T}}(S,P)$ formed by
stable marked curves whose pinched set is contained in $\alpha$ up to isotopy.
We denote by $V_{\alpha}$ the quotient of $U_{\alpha}$ by $\Gamma_{\alpha}$.
The following result allows to define an orbifold structure on the quotient
$\overline{\mathcal{M}}(S,P)$ which recovers Deligne-Mumford’s orbifold
$\overline{\mathcal{M}}_{g,n}$, see [25]:

######  Theorem 1.

  1. 1.

$V_{\alpha}$ has a unique structure of complex manifold so that the natural
map $\mathcal{T}(S,P)\rightarrow V_{\alpha}$ is holomorphic.

  2. 2.

The projections in $V_{\alpha}$ of the strata $B_{\alpha^{\prime}}$, with
$\alpha^{\prime}$ describing the components of $\alpha$, is a family of normal
crossing divisors.

  3. 3.

The natural projection $V_{\alpha}\rightarrow\overline{\mathcal{M}}(S,P)$ has,
locally around the class of $(C,f)$ in $V_{\alpha}$, fibers given by the
orbits of the group $\operatorname{Aut}(C,f)$.

  4. 4.

These charts provide an orbifold structure on $\overline{\mathcal{M}}(S,P)$
which is biholomorphic to Deligne-Mumford’s orbifold
$\overline{\mathcal{M}}_{g,n}$.

We now define, for any $\ell\geq 1$, a twisted orbifold structure
$\overline{\mathcal{M}}^{\ell}(S,P)$ which recovers the previous one when
$\ell=1$ and share the same underlying topological space. We define, for any
$(C,f)\in\overline{\mathcal{T}}(S,P)$ whose pinched set is $\alpha$, the group
$\operatorname{Aut}^{\ell}(C,f)=\operatorname{Stab}(C,f)/\ell\Gamma_{\alpha}$;
this group is a central extension

$0\rightarrow\Gamma_{\alpha}/\ell\Gamma_{\alpha}\rightarrow\operatorname{Aut}^{\ell}(C,f)\rightarrow\operatorname{Aut}(C,f)\rightarrow
0.$ (3)

######  Corollary 1.

  1. 1.

There is a unique complex manifold structure on
$V_{\alpha}^{\ell}=U_{\alpha}/\ell\Gamma_{\alpha}$ such that the ramified
covering $V_{\alpha}^{\ell}\rightarrow V_{\alpha}$ of group
$\Gamma_{\alpha}/\ell\Gamma_{\alpha}$ is holomorphic.

  2. 2.

The projection of the strata $B_{\alpha^{\prime}}$ in $V_{\alpha}^{\ell}$ is a
normal crossing family of divisors.

  3. 3.

For any $(C,f)\in\overline{\mathcal{T}}(S,P)$ whose pinched set is $\alpha$,
the natural quotient map
$V_{\alpha}^{\ell}\rightarrow\overline{\mathcal{M}}(S,P)$ has local fibers
around the class of $(C,f)\in V_{\alpha}^{\ell}$ given by the orbits of the
group $\operatorname{Aut}^{\ell}(C,f)$.

  4. 4.

These charts provide an orbifold structure
$\overline{\mathcal{M}}^{\ell}(S,P)$ which is biholomorphic to the
construction given by Eyssidieux and Funar in [18].

####  2.2.2 The orbifold $\overline{\mathcal{M}}^{\ell}_{g,n}$ is
uniformizable for $\ell\geq 5$ odd

We recall that an orbifold is uniformizable if it carries a finite orbifold
covering which is smooth, in the sense that the isotropy groups are trivial.
This is equivalent to saying that the orbifold is the quotient of a smooth
manifold by a finite group acting by biholomorphisms. Eyssidieux and Funar
proved that the orbifold $\overline{\mathcal{M}}_{g,n}^{\ell}$ is
uniformizable, at least if $\ell\geq 5$ is an odd integer, see [18,
Proposition 4.10] (In the case $n=0$, this is a consequence of Pikaart and de
Jong’s work [36]: the smooth covering is a moduli space of curves with
nilpotent level structures.). They notice that the $\text{SO}(3)$-quantum
representations of level $\ell$ with all colors equal to $1$ are injective in
restriction to the isotropy groups $\operatorname{Aut}^{\ell}(C,f)$ of the
orbifold $\overline{\mathcal{M}}_{g,n}^{\ell}$, defined in (3), hence this is
a consequence of Selberg’s lemma applied to the image of the relevant quantum
representation.

####  2.2.3 Forgetful map

######  Lemma 1.

The natural forgetful map
$\overline{\mathcal{M}}_{g,n+1}^{\ell}\rightarrow\overline{\mathcal{M}}_{g,n}^{\ell}$
is a holomorphic orbifold map.

###### Proof.

Given $g,n$ with $2g-2+n>0$, we fix a subset
$P=\\{x_{1},\ldots,x_{n+1}\\}\subset S_{g}$ and define a continuous forgetful
map (see [4])

$\overline{\mathcal{T}}_{g,n+1}\rightarrow\overline{\mathcal{T}}_{g,n}$ (4)

which assigns to a marked stable curve of genus $g$ with $n+1$ marked numbered
points, $(C,f,Q=f(P))$, the curve $(C^{\prime},f^{\prime},Q^{\prime})$ where
$(C^{\prime},Q^{\prime})$ is the stabilization of the curve
$(C,\\{f(x_{1}),\ldots,f(x_{n})\\})$, and $f^{\prime}$ is the composition of
$f$ with the stabilisation map $(C,Q)\rightarrow(C^{\prime},Q^{\prime})$. The
map (4) is equivariant with respect to the morphism

$\text{Mod}_{g,n+1}\rightarrow\text{Mod}_{g,n}$ (5)

which sends the subgroup $\Gamma_{g,n+1}^{\ell}\subset\text{Mod}_{g,n+1}$
generated by the $\ell$-powers of Dehn twists in $\text{Mod}_{g,n+1}$ to the
corresponding subgroup $\Gamma_{g,n}^{\ell}\subset\text{Mod}_{g,n}$. Hence,
denoting
$\overline{\mathcal{T}}_{g,n}^{\ell}:=\overline{\mathcal{T}}_{g,n}/\Gamma_{g,n}^{\ell}$,
the map (4) induces a map

$\overline{\mathcal{T}}_{g,n+1}^{\ell}\rightarrow\overline{\mathcal{T}}_{g,n}^{\ell}$

which is holomorphic with respect to the smooth complex structures on
$\overline{\mathcal{T}}_{g,n+1}^{\ell}$ and
$\overline{\mathcal{T}}_{g,n}^{\ell}$ given by Corollary 1; indeed, it is
continuous and holomorphic in restriction to
$\mathcal{T}_{g,n+1}/\operatorname{Mod}_{g,n+1}$, which is Zariski dense in
$\overline{\mathcal{T}}_{g,n+1}^{\ell}$, so this is a consequence of Riemann’s
extension theorem. We deduce that the forgetful map

$\overline{\mathcal{M}}_{g,n+1}^{\ell}\simeq\overline{\mathcal{T}}_{g,n+1}^{\ell}/\text{Mod}_{g,n+1}^{\ell}\rightarrow\overline{\mathcal{M}}_{g,n}^{\ell}\simeq\overline{\mathcal{T}}_{g,n}^{\ell}/\text{Mod}_{g,n}^{\ell}$

is a holomorphic orbifold map as we wanted to prove. ∎

####  2.2.4 The $\ell$-twisted compactification as a classifying space

We present here a construction of the homotopical version of
$\overline{\mathcal{M}}_{g,n}^{\ell}$ which is purely topological and makes
clear the formal properties of these spaces. It is sufficient for defining the
higher Toledo invariants of quantum representations and showing that they
satisfy the axioms of a CohFT.

Let $SC_{\ell}$ be the category whose objects are triples $(S,P,\alpha)$ where
$S$ is a closed oriented surface of genus $g$, $P\subset S$ a finite set of
cardinality $n$ and $\alpha\subset S\setminus P$ a collection of disjoint
simple curves such that each component of $S\setminus(P\cup\alpha)$ has
negative Euler characteristic. We denote by $\Gamma_{\alpha}^{\ell}(S,P)$ the
group of homeomorphisms of $S$ fixing $P$, preserving $\alpha$ and generated
up to isotopy by $\ell$-th powers of Dehn twists along the components of
$\alpha$. Finally we define a morphism
$(S_{1},P_{1},\alpha_{1})\to(S_{2},P_{2},\alpha_{2})$ as a homeomorphism
$f:S_{1}\to S_{2}$ mapping $P_{1}$ to $P_{2}$ and $\alpha_{1}$ into
$\alpha_{2}$, up to the relation $f\sim\varphi_{2}\circ f\circ\varphi_{1}$ for
$\varphi_{i}\in\Gamma_{\alpha_{i}}^{\ell}(S_{i},P_{i})$.

######  Proposition 1.

The classifying space of the category $SC_{\ell}$ is the orbispace associated
to the orbifold $\overline{\mathcal{M}}^{\ell}(S,P)$.

###### Proof.

This is a direct adaptation of Theorem 6.1.1 in [12]. ∎

###  2.3 The elliptic tail contraction

An alternative compactification of the moduli space will be useful to study
the Fibonacci quantum representations, namely those corresponding to the group
$\text{SO}(3)$ and the level $\ell=5$: it is obtained from
$\overline{\mathcal{M}}_{g,n}^{5}$ by contracting the elliptic tail divisor
$\delta_{1,\emptyset}^{5}$. This divisor is made of nodal curves having at
least one singular point separating the curve into two components, one of
which being an elliptic curve without marked point. The main property of this
compactification is that it still has a natural orbifold structure. In this
section we provide the construction of this compactification, prove that it
has some good functoriality properties with respect to forgetful maps, and
finally we compute the first Chern class of its canonical bundle.

####  2.3.1 The construction

######  Theorem 2.

For $(g,n)\neq(2,0)$, there exists an orbifold
$\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$ and an orbifold holomorphic
(contraction) map
$c:\overline{\mathcal{M}}_{g,n}^{5}\rightarrow\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$
whose fibers consist of equivalence classes of stable curves that are
isomorphic after taking out the elliptic tails while keeping their attaching
points. The map $c$ induces an isomorphism at the fundamental group level.

The space $\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$, as a set, might be
identified with the set of stable nodal curves of genus $g-k$ with $n+k$
marked points ($k\leq g$), $n$ of which (the marked points) being numbered,
not the $k$ remaining ones (the tail points). The map
$\overline{\mathcal{M}}_{g,n}^{5}\simeq\overline{\mathcal{M}}_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$
associates to a stable nodal curve of genus $g$ with $n$ numbered marked
points, having $k$ elliptic tails, the curve obtained by contracting each of
its elliptic tails to a tail point.

######  Remark 1.

In the case $(g,n)=(2,0)$, the elliptic tail divisor is still contractible,
but its contraction leads to a quadratic singularity. This will be
investigated in a forthcoming work.

It will be convenient to use an appropriate smooth finite orbifold Galois
covering of $\overline{\mathcal{M}}_{g,n}^{5}$, and to construct the
contraction in an equivariant way with respect to the Galois group:

######  Lemma 2.

There exists a finite Galois orbifold covering
$X\rightarrow\overline{\mathcal{M}}_{g,n}^{5}$ having the property that $X$ is
smooth, and that the preimage in $X$ of the elliptic tail divisor
$\delta_{1,\emptyset}^{5}$ is a normal crossing divisor
$\delta_{1,\emptyset}^{X}$ whose irreducible components are smooth
hypersurfaces.

###### Proof.

Let $\rho:\pi_{1}(\overline{\mathcal{M}}_{g,n}^{5})\rightarrow G$ be a
morphism to a finite group having the property that it is injective in
restriction to the isotropy groups of $\overline{\mathcal{M}}_{g,n}^{5}$, see
section 2.2.2 for its existence, and
$\rho^{\prime}:\pi_{1}(\overline{\mathcal{M}}_{g,n}^{5})\rightarrow\text{Aut}(H_{1}(S,\mathbb{Z}/5\mathbb{Z}))$
be the morphism induced by the action on the homology of $S$ modulo $5$ (it is
a priori defined on the mapping class group; the fact that it descends to a
morphism defined on $\pi_{1}(\overline{\mathcal{M}}_{g,n}^{5})$ comes from
that Dehn twists are mapped to elements of order $1$ or $5$ in
$\text{Aut}(H_{1}(S,\mathbb{Z}/5\mathbb{Z}))$).

Let $X$ be the covering of $\overline{\mathcal{M}}_{g,n}^{5}$ corresponding to
the morphism
$\rho\times\rho^{\prime}:\pi_{1}(\overline{\mathcal{M}}_{g,n}^{5})\rightarrow
G\times\text{Sp}(2g,\mathbb{Z}/5\mathbb{Z})$. Since $\rho\times\rho^{\prime}$
is injective on isotropy groups of $\overline{\mathcal{M}}_{g,n}^{5}$, the
covering $X$ is a smooth orbifold. In particular, denoting by
$\pi:X\rightarrow\overline{\mathcal{M}}_{g,n}^{5}$ the natural projection map,
the second item of Corollary 1 tells us that the preimage
$\delta_{1,\emptyset}^{X}:=\pi^{-1}(\delta_{1,\emptyset}^{5})\subset X$ is a
normal crossing divisor.

Any point $x\in X$ corresponds to an equivalence class of pinching maps
$f:S\to C_{x}$. Denote by
$f_{x}:=f_{*}:H_{1}(S,\mathbb{Z}/5\mathbb{Z})\rightarrow
H_{1}(C,\mathbb{Z}/5\mathbb{Z})$ the map induced by the pinching map. As $X$
is the quotient of the augmented Teichmüller space by the kernel of the
morphism $\rho\times\rho^{\prime}$, the map $f_{x}$ is well-defined since the
kernel of $\rho\times\rho^{\prime}$ contains the kernel of $\rho^{\prime}$.

For any symplectic subspace $E\subset H_{1}(S,\mathbb{Z}/5\mathbb{Z})$ of
dimension $2$, denote by $H_{E}\subset X$ the set of elements $x\in X$ so that
$C_{x}$ is a nodal curve having an elliptic tail whose first homology modulo
$5$ is the subspace $f_{x}(E)$; the union of all $H_{E}$’s is the divisor
$\delta_{1,\emptyset}^{X}$. So it suffices to prove that $H_{E}$ is a smooth
hypersurface to conclude the proof of the lemma.

The preimage of $H_{E}$ in the augmented Teichmüller space is the union of all
strata $B_{\alpha}$ where some component $\beta$ of $\alpha$ is a simple
closed curve that separates $S$ in two components, one of which being
homeomorphic to a torus minus a disc $T_{\beta}\subset S$ whose homology
modulo $5$ maps via inclusion onto the subspace $E$. We denote by
$\mathcal{C}_{E}$ the set of all these $\alpha$’s. The closure of $B_{\alpha}$
is the union of strata $B_{\alpha^{\prime}}$ where $\alpha^{\prime}$ contains
an element $\alpha\in\mathcal{C}_{E}$ up to isotopy. Hence it suffices to
prove that two non isotopic simple closed curves
$\beta,\beta^{\prime}\in\mathcal{C}_{E}$ cannot be components of a same
$\alpha\in\mathcal{C}_{E}$, or which is equivalent, that $\beta$ and
$\beta^{\prime}$ intersect. Suppose by contradiction that this happens. Then
$\beta^{\prime}$ being separating and not isotopic to $\beta$ it cannot be
contained in $T_{\beta}$. Reversing the role of $\beta$ and $\beta^{\prime}$
shows that $T_{\beta}$ and $T_{\beta^{\prime}}$ are disjoint; in particular
the image of their homology group in $H_{1}(S,\mathbb{Z}/5\mathbb{Z})$ are
orthogonal, which is contradictory to the fact that they are both equal to
$E$. ∎

The elliptic tail divisor $\delta_{1,\emptyset}^{5}$ is parametrized in a
natural way by the moduli space
$\overline{\mathcal{M}}_{1,1}^{5}\times\overline{\mathcal{M}}_{g-1,n+1}^{5}$
via an attaching map. Although this parametrization is not injective as soon
as $g\geq 2$, and so cannot be inverted at the level of
$\delta_{1,\emptyset}^{5}\subset\overline{\mathcal{M}}_{g,n}^{5}$, it can be
done at the level of $H_{E}\subset X$ in the following sense: one has a
natural covering map
$r_{E}:H_{E}\rightarrow\overline{\mathcal{M}}_{1,1}^{5}\times\overline{\mathcal{M}}_{g-1,n+1}^{5}$,
which assigns to an element of $H_{E}$ the unique elliptic tail whose homology
modulo $5$ is $E$ as the first coordinates in
$\overline{\mathcal{M}}_{1,1}^{5}$, and the contraction of this latter as the
second coordinates in $\overline{\mathcal{M}}_{g-1,n+1}^{5}$. Since the unique
smooth cover of $\overline{\mathcal{M}}_{1,1}^{5}$ is its universal cover
which is biholomorphic to the Riemann sphere, this equip each (smooth)
hypersurface $H_{E}$ of $\delta_{1,\emptyset}^{X}$ with a locally trivial
fibration

$\widetilde{\overline{\mathcal{M}}_{1,1}^{5}}\simeq\mathbb{P}^{1}\rightarrow
H_{E}\stackrel{{\scriptstyle f_{E}}}{{\rightarrow}}K_{E}$ (6)

over a smooth manifold $K_{E}$. The fibers of $f_{E}$ are the connected
components of $r_{E}^{-1}(\overline{\mathcal{M}}_{1,1}^{5}\times\\{*\\})$.

Those fibrations on different $H_{E}$’s are compatible in the following sense:

######  Lemma 3.

Given $E_{1},\ldots,E_{r}$ a collection of symplectic $2$-dimensional
subspaces of $H_{1}(S,\mathbb{Z}/5\mathbb{Z})$, the intersection
$H_{E_{1},\ldots,E_{r}}:=H_{E_{1}}\cap\ldots\cap H_{E_{r}}$ (it is not empty
if and only if the $E_{i}$’s are orthogonal wrt the intersection form), there
is a fibration of $H_{E_{1}}\cap\ldots\cap H_{E_{r}}$ by
$(\mathbb{P}^{1})^{r}$ (with a natural identification of the fibers with
$\widetilde{\overline{\mathcal{M}}_{1,1}^{5}}^{r}$ up to the action of
$\pi_{1}(\overline{\mathcal{M}}_{1,1}^{5})^{r}$, in particular the monodromy
does not permute the $\mathbb{P}^{1}$-factors) so that the $i$-th
$\mathbb{P}^{1}$-subfibration corresponds to the fibration $f_{E_{i}}$ of
$H_{E_{i}}$ restricted to the intersection $H_{E_{1}}\cap\ldots\cap
H_{E_{r}}$.

###### Proof.

The intersection $H_{E_{1},\ldots,E_{r}}$ has a natural map to
$(\overline{\mathcal{M}}_{1,1}^{5})^{r}\times\overline{\mathcal{M}}_{g-r,n+r}^{5}$,
whose first $r$ coordinates are given by the elliptic tails corresponding to
each $E_{i}$’s, and the last one is the contraction of those elliptic tails.
This map is an orbifold covering map, and since $H_{E_{1},\ldots,E_{r}}$ is
smooth, and the unique smooth covering of
$(\overline{\mathcal{M}}_{1,1}^{5})^{r}$ is its universal cover biholomorphic
to the $r$-power of the Riemann sphere, the lift of the fibration of
$(\overline{\mathcal{M}}_{1,1}^{5})^{r}\times\overline{\mathcal{M}}_{g-r,n+r}^{5}$
by $(\overline{\mathcal{M}}_{1,1}^{5})^{r}$ defines a fibration of
$H_{E_{1},\ldots,E_{r}}$ whose fibers hare naturally universal covers of
$(\overline{\mathcal{M}}_{1,1}^{5})^{r}$. The last statement of the lemma is
then obvious. ∎

######  Lemma 4.

For any fiber $F$ of the fibration (6), we have $H_{E}\cdot F=-1$.

###### Proof.

Although this lemma is true in general, we explain the proof only when
$(g,n)\neq(2,0)$. In the case where $(g,n)=(1,2)$, this result can be found in
Livne’s PhD dissertation [27], but for completeness we recall the proof here.
The preimage $\delta_{1,\emptyset}^{X}$ of
$\delta_{1,\emptyset}^{5}\simeq\overline{\mathcal{M}}_{1,1}^{5}$ in $X$ is a
finite union of rational curves in this case, the stabilizer of each of those
being a subgroup of $H$ isomorphic to $\pi_{1}(\mathcal{V})$, where
$\mathcal{V}$ is a small tubular neighborhood $\mathcal{V}$ of
$\delta_{1,\emptyset}^{5}$. Notice that the fundamental group of $\mathcal{V}$
is a $\mathbb{Z}/5\mathbb{Z}$-extension of the fundamental group of
$\delta_{1,\emptyset}^{5}$, which is the binary icosahedral group
$\mathrm{I}2\subset\mathrm{SU}(2)$, see Subsection 2.1.3. Denoting by $R$ a
component of $\delta_{1,\emptyset}^{X}$, we then have

$[R]^{2}=5\times|\mathrm{I}2|\times[\delta_{1,\emptyset}^{5}]^{2}.$

But $[\delta_{1,0}^{5}]=\frac{1}{5}\delta_{1,\emptyset}$, so
$[\delta_{1,\emptyset}^{5}]^{2}=\frac{1}{5^{2}}[\delta_{1,\emptyset}]^{2}=-\frac{1}{5^{2}\times
24}$, see [45, Part 2.2.2], and finally we find $[R]^{2}=-1$ as claimed since
$|\mathrm{I}2|=120$.

######  Remark 2.

Using Castelnuovo’s contraction theorem, one can deduce from this computation
that the neighborhood of the divisor $\delta_{1,\emptyset}^{5}$ in
$\overline{\mathcal{M}}_{1,2}^{5}$ is isomorphic as an orbifold to the
neighborhood of the exceptional divisor in the quotient of the blow-up of
$\mathbb{C}^{2}$ at the origin by the group generated by multiplication by
$\mu_{5}\operatorname{Id}$ and by the binary icosahedral group
$\mathrm{I}2\subset\mathrm{SU}(2)$. We leave the details for the reader.

Let us now consider the general case. First notice that if $g=0$ there is
nothing to prove since $\delta_{1,\emptyset}^{5}$ is empty. So in the sequel
we suppose that $g\geq 1$. The strategy is to construct a complex surface
$S\subset X$ which intersects $H_{E}$ transversally in a curve whose
components are fibers $F$ of $f_{E}$. Then the statement of the lemma is
equivalent to saying that $F$ is a $(-1)$-rational curve in $S$.

The construction of $S$ depends on $(g,n)$. In the sequel we assume that
$(g,n)\notin\\{(1,1),(1,2),(2,0)\\}$.

If $n\geq 1$, fix an element $C_{0}\in\overline{\mathcal{M}}_{g-1,n}^{5}$
(notice that the stability condition $2(g-1)-2+n>0$ is satisfied by our
assumptions), and define $S$ as being the pull-back in $X$ of the submanifold
$R\subset\overline{\mathcal{M}}_{g,n}^{5}$ formed by nodal curves obtained by
attaching a curve $C\in\overline{\mathcal{M}}_{1,2}^{5}$ to $C_{0}$ by
identifying fixed marked points. Then $R$ is isomorphic to
$\overline{\mathcal{M}}_{1,2}^{5}$, and under this identification, its
intersection with
$\delta_{1,\emptyset}^{5}\subset\overline{\mathcal{M}}_{g,n}^{5}$ is the
elliptic tail divisor of $\overline{\mathcal{M}}_{1,2}^{5}$. Hence, the
intersection of $S$ with $H_{E}$ are made of fibers $F$ of $f_{E}$, and the
previous considerations in the particular case $(g,n)=(1,2)$ show that in $S$
those $F$’s are $(-1)$-curves. Hence we are done in that case.

If $n=0$ and $g\geq 3$. Fix smooth curves
$C_{1}\in\overline{\mathcal{M}}_{g-1,1}^{5}$ and
$C_{2}\in\overline{\mathcal{M}}_{1,1}^{5}$, and let
$R\subset\overline{\mathcal{M}}_{g,n}^{5}$ be the surface formed by nodal
curves obtained by attaching $C_{1}$ and $C_{2}$ to a curve
$C\in\overline{\mathcal{M}}_{1,2}^{5}$. Let now $S\subset X$ be the set of
curves that project to a curve of $R$, in such a way that the homology of the
component $C$ modulo $5$ is equal to $E$. Then $S$ is a covering of
$\overline{\mathcal{M}}_{1,2}^{5}$, and its intersection with $H_{E}$ is the
pull back of the tail divisor of $\overline{\mathcal{M}}_{1,2}^{5}$. Hence the
lemma is proved in that case too. ∎

###### Proof of Theorem 2.

Suppose that we have a smooth compact complex manifold $Y$, a finite family of
normal crossing smooth hypersurfaces $H_{i}\subset Y$, and for any subset of
indices $J\subset I$, fibrations $(\mathbb{P}^{1})^{J}\rightarrow
H_{J}:=\bigcap\limits_{j\in J}H_{j}\stackrel{{\scriptstyle
f_{J}}}{{\rightarrow}}K_{J}$, in such a way that

  1. 1.

the monodromy of $f_{J}$ does not exchange the factors of the fibers
$\simeq(\mathbb{P}^{1})^{J}$, so given any $J^{\prime}\subset J$, the
$J^{\prime}$-coordinate part of the fibration $f_{J}$ is a well-defined
$(\mathbb{P}^{1})^{J^{\prime}}$-fibration ,

  2. 2.

given disjoint subsets $J_{1},J_{2}\subset I$ the intersection
$H_{J_{1},J_{2}}=H_{J_{1}}\cap H_{J_{2}}$ is invariant by the fibration
$f_{J_{1}}$ (resp. $f_{J_{2}}$), and its restriction to $H_{J_{1},J_{2}}$ is
the $J_{1}$-coordinate part (resp. the $J_{2}$-coordinate part) of the
fibration $f_{J_{1},J_{2}}$,

  3. 3.

for any $i\in I$, and any fiber $F=f_{i}^{-1}(*)\subset H_{i}$, we have
$H_{i}\cdot F=-1$.

Choose such a data and enumerate $I=\\{i_{1},\ldots,i_{r}\\}$. By [34] and its
supplement [21], we can find a contraction map $c_{1}:Y\rightarrow Y_{1}$ to a
smooth complex compact manifold, which has the property that it maps
$H_{i_{1}}$ to a codimension two submanifold of $Y_{1}$, contracting each
fibers of $f_{i_{1}}$ to a point and not more. The $c_{1}$-images of the
hypersurfaces $H_{i_{k}}$ for $k\geq 2$ form a smooth family of normal
crossing hypersurfaces in $Y_{1}$, and their intersections are naturally
endowed with fibrations that satisfy all the previous properties 1., 2. and
3.. We can then define inductively smooth compact complex spaces $Y_{k}$ and
contractions $c_{k}:Y_{k-1}\rightarrow Y_{k}$ that contract the
$\mathbb{P}^{1}$-fibration of the hypersurface $c_{k-1}\circ\ldots\circ
c_{1}(H_{i_{k}})$. At the end we obtain a space $Y_{r}$ which is topologically
the quotient of $Y$ by the equivalence class given by $x\sim y$ iff for each
$i$ we have that $f_{i}(x)=f_{i}(y)$ as soon as both $x,y\in H_{i}$. This
space does not depend on the enumeration of $I$ that we have chosen; indeed,
this is clear at the topological level, and at the analytical one this is a
consequence of Riemann’s extension theorem and of the fact that the map
$c_{r}\circ\ldots\circ c_{1}$ is injective apart from a codimension one
analytic set.

Applying this to the covering $X$ of $\overline{\mathcal{M}}_{g,n}^{5}$
constructed before, together with the family of smooth hypersurfaces $H_{E}$
and the fibrations of their intersections given by Lemma 3, we find a
contraction map $c:X\rightarrow Z$ to a smooth space $Z$, which by the
aforementioned unicity is equivariant with respect to a morphism
$\text{Gal}(X\rightarrow\overline{\mathcal{M}}_{g,n}^{5})\rightarrow\text{Aut}(Z)$.
The quotient of $Z$ by the image of the previous morphism is the desired
quotient of $\overline{\mathcal{M}}_{g,n}^{5}$. ∎

####  2.3.2 Forgetful map between Fibonacci elliptic tail contractions

In the Fibonacci case $\ell=5$, this enables to construct natural forgetful
maps between the elliptic tail contractions.

######  Lemma 5.

The forgetful map of Lemma 1 induces an orbifold holomorphic map

$\overline{\mathcal{M}}_{g,n+1}^{\mathcal{E}}\rightarrow\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$
(7)

which is onto.

###### Proof.

We consider the finite Galois orbifold coverings
$X_{g,n+1}\rightarrow\overline{\mathcal{M}}_{g,n+1}^{5}$ and
$X_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n}^{5}$ constructed in subsection
2.3.1, with smooth underlying spaces $X_{g,n+1}$ and $X_{g,n}$. As in Lemma 2,
we choose the coverings in such a way that that they cover the covering of
$\overline{\mathcal{M}}_{g,n+1}^{5}$ or $\overline{\mathcal{M}}_{g,n}^{5}$
made of pointed curves whose underlying curve is marked by
$H_{1}(S,\mathbb{Z}/5\mathbb{Z})$. In particular the pull back of the elliptic
tail divisor is the union of hypersurfaces $H_{g,n}=\cup_{E}H_{E,g,n}$, with
$E$ varying over the set of symplectic dimension two symplectic submodules of
$H_{1}(S,\mathbb{Z}/5\mathbb{Z})$, with $H_{E,g,n}$ being the subset of
$\overline{\mathcal{M}}_{g,n}^{5}$ having an elliptic tail whose homology
group maps onto $E$ by inclusion.

Up to taking a larger covering of $X_{g,n+1}$ if necessary, we can assume that
the forgetful map
$\overline{\mathcal{M}}_{g,n+1}^{5}\rightarrow\overline{\mathcal{M}}_{g,n}^{5}$
(constructed in 2.2.3) lifts to a holomorphic map $h:X_{g,n+1}\rightarrow
X_{g,n}$ equivariant wrt to the actions of the Galois groups of the coverings
$X_{g,n+1}\rightarrow\overline{\mathcal{M}}_{g,n+1}^{5}$ and
$X_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n}^{5}$. By construction, the map
$h$ maps $H_{E,g,n+1}$ to $H_{E,g,n}$, sending the $\mathbb{P}^{1}$-fibration
of $H_{E,g,n+1}$ to the one of $H_{E,g,n}$. So it induces a continuous (and
hence holomorphic by Riemann’s extension theorem) map from the contraction
$Z_{g,n+1}^{E}$ of the $H_{E,g,n+1}$’s to the contraction $Z_{g,n}^{E}$ of
$H_{E,g,n}$. Reasoning inductively as in the proof of Theorem 2 shows that the
map $h$ induces a holomorphic map from the contraction $Z_{g,n+1}$ of all
$H_{E,g,n+1}$ to the corresponding space $Z_{g,n}$. This map is equivariant
with respect to the natural actions of the Galois group of
$X_{g,n+1}\rightarrow\overline{\mathcal{M}}_{g,n+1}^{5}$ (resp. of
$X_{g,n}\rightarrow\overline{\mathcal{M}}_{g,n}^{5}$) acting on $Z_{g,n+1}$
(resp. $Z_{g,n}$). This ends the proof of the lemma.∎

######  Remark 3.

The kernel of the forgetful morphism (5) is isomorphic to the fundamental
group $F_{g,n}=\pi_{1}(S\setminus\\{x_{1},\ldots,x_{n}\\})$ of the genus $g$
surface minus $n$ points, by Birman’s exact sequence. So the kernel of the
$\ell$-twisted forgetful morphism

$\operatorname{Mod}_{g,n+1}^{\ell}\rightarrow\operatorname{Mod}_{g,n}^{\ell}$
(8)

is the quotient of $F_{g,n}$ by the group generated by $\ell$-powers of
elements freely homotopic to simple closed curves. It would be interesting to
compute this group for $(g,n)=(1,2)$ (in this case
$\overline{\mathcal{M}}_{g,n+1}^{\mathcal{E}}$ and
$\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$ are both complex hyperbolic
orbifolds of respective dimension $3$ and $2$, and the forgetful map answers
in the negative Siu’s problem [40, Problem (a)], see Corollary Corollary).

###  2.4 The canonical bundles of $\overline{\mathcal{M}}_{g,n}^{\ell}$ and
$\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$

We recall that the second cohomology group of $\overline{\mathcal{M}}_{g,n}$
with rational coefficients is generated by the $\psi$-classes $\psi_{i}$,
$i=1,\ldots,n$, the class $\kappa_{1}$, and the classes of the boundary
divisors: $\delta_{irr}$ and $\delta_{a,A}$, where $0\leq a\leq g$ and
$A\subset P$ is a subset satisfying the inequalities $2a-2+|A|\geq 0$ and
$2(g-a)-2+(n-|A|)\geq 0$ (see [3]).

In the sequel we denote by $\delta$ the sum of all boundary divisors and by
$\psi=\sum_{i}\psi_{i}$ the sum of $\psi$-classes. It will be convenient for
us to introduce the class

$\widetilde{\kappa}_{1}=\kappa_{1}-\psi$

since the Toledo invariants of quantum representations are better expressed in
the basis formed by $\psi$-classes, boundary classes, and
$\widetilde{\kappa}_{1}$.

######  Lemma 6.

$K_{\overline{\mathcal{M}}_{g,n}^{\ell}}=\frac{13}{12}\widetilde{\kappa}_{1}+\left(\frac{1}{12}-\frac{1}{\ell}\right)\delta+\psi$

###### Proof.

Harris and Mumford proved that (see [4, Theorem 7.15])

$K_{\overline{\mathcal{M}}_{g,n}}=13\lambda_{1}+\psi-2\delta$

where $\lambda_{1}$ is the first Chern class of the Hodge bundle, and this
latter is expressed in our basis by the formula (see [4, Theorem 7.6]).

$\widetilde{\kappa}_{1}=12\lambda_{1}-\delta.$

So we have:

$K_{\overline{\mathcal{M}}_{g,n}}=\frac{13}{12}\widetilde{\kappa}_{1}-\frac{11}{12}\delta+\psi.$

Now, the natural map
$\overline{\mathcal{M}}_{g,n}^{\ell}\rightarrow\overline{\mathcal{M}}_{g,n}$
is holomorphic and ramifies on the boundary divisors at the order $\ell$, so
we get

$K_{\overline{\mathcal{M}}_{g,n}^{\ell}}=K_{\overline{\mathcal{M}}_{g,n}}+(1-\frac{1}{\ell})\delta$

and the result follows. ∎

######  Lemma 7.

For $(g,n)\neq(2,0)$, denote by
$c:\overline{\mathcal{M}}_{g,n}^{5}\rightarrow\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}$
the blow-down (see Theorem 2). We have

$c^{*}K_{\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}}=K_{\overline{\mathcal{M}}_{g,n}^{5}}-\delta_{1,\emptyset}^{5}$

so its first Chern class in
$H^{2}(\overline{\mathcal{M}}_{g,n}^{5},\mathbb{Q})\simeq
H^{2}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ satisfies

$c_{1}(c^{*}K_{\overline{\mathcal{M}}_{g,n}^{\mathcal{E}}})=c_{1}(K_{\overline{\mathcal{M}}_{g,n}^{5}})-\frac{1}{5}\delta_{1,\emptyset}$

###### Proof.

If $Y$ is a smooth complex analytic space, with $\mathbb{P}^{1}$-fibered
hypersurfaces $H_{i}$ as in the proof of Theorem 2, we have
$c_{1}^{*}(K_{Y_{1}})=K_{Y}-H_{1}$. By induction, if we denote $Z=Y_{r}$,
$H=\cup_{i}H_{i}$, and $c=c_{r}\circ\cdots\circ c_{1}:Y\rightarrow Z$ the
composition of all contractions, we get $c^{*}K_{Z}=K_{Y}-H$. If $Y$, $H$ and
the $\mathbb{P}^{1}$-fibrations on the $H_{i}$’s are invariant by a finite
group $G\subset\text{Aut}(Y)$, denoting by $G^{\prime}\subset\text{Aut}(Z)$
the image of the action of $G$ on the quotient $Z$ (which is unique), we then
have

$K_{Y/G}=c^{*}K_{Z/G^{\prime}}-H/G,$

which implies the lemma by construction of the elliptic contraction. ∎

###  2.5 The Euler characteristic of $\overline{\mathcal{M}}^{\ell}_{g,n}$

Let $x=(C,f)$ be a stable curve of genus $g$ with $n$ marked points that we
think as a point in $\overline{\mathcal{M}}_{g,n}$. We define
$|\operatorname{Sing}(x)|$ to be the number of nodal points of $x$ and
$|\operatorname{Stab}^{\ell}(x)|$ to be the size of the isotropy group of $x$
in $\overline{\mathcal{M}}^{\ell}_{g,n}$ (set
$\operatorname{Stab}^{1}=\operatorname{Stab}$). From the exact sequence (3),
we get
$|\operatorname{Stab}^{\ell}(x)|=\ell^{|\operatorname{Sing}(x)|}|\operatorname{Stab}(x)|$.
This suggest to define the polynomial

$\overline{\chi}_{g,n}(\kappa)=\int_{\overline{\mathcal{M}}_{g,n}}\frac{\kappa^{|\operatorname{Sing}(x)|}}{|\operatorname{Stab}(x)|}d\chi(x)\in\mathbb{Q}[\kappa]$

so that
$\chi(\overline{\mathcal{M}}_{g,n}^{\ell})=\overline{\chi}_{g,n}(\frac{1}{\ell})$.
This polynomial satisfies the following quadratic recursion relation which
allows to compute it effectively, see [43] Theorem 3.6 and the formulas
following the theorem (notice that they use instead
$\widetilde{\chi}_{g,n}=\overline{\chi}_{g,n}/n!$):

$\overline{\chi}_{g,n}=\int_{0}^{\kappa}\Big{(}\overline{\chi}_{g-1,n+2}+\\!\\!\sum_{\underset{n+2=n_{1}+n_{2}}{g=g_{1}+g_{2}}}\\!\\!n_{1}n_{2}\binom{n}{n_{1}}\overline{\chi}_{g_{1},n_{1}}\overline{\chi}_{g_{2},n_{2}}\Big{)}d\kappa+\chi(\mathcal{M}_{g,n}).$

This formula, together with Harer-Zagier formula

$\chi(\mathcal{M}_{g,n})=(-1)^{n}\frac{(2g-1)B_{2g}}{(2g)!}(2g+n-3)!$

allows to compute the Euler characteristic of
$\overline{\mathcal{M}}_{g,n}^{\ell}$. We find for instance
$\overline{\chi}_{0,5}(\kappa)=2-10\kappa+15\kappa^{2}$ hence
$\chi(\overline{\mathcal{M}}_{0,5}^{5})=\frac{3}{5}$.

##  3 Hermitian cohomological invariants

Let $V$ be a finite dimensional complex vector space endowed with a non-
degenerate Hermitian form $h$ of signature $(p,q)$. This means that there are
coordinates $(x_{1},\ldots,x_{p},y_{1},\ldots,y_{q})$ such that

$h=\sum_{i=1}^{p}|x_{i}|^{2}-\sum_{j=1}^{q}|y_{j}|^{2}.$ (9)

For simplicity, we will often remove $h$ from the notation. Depending on the
purpose, we will denote by $\mathrm{PU}(V)$ or $\mathrm{PU}(p,q)$ the group of
projective unitary transformations of $V$. We will also write $d(V)=p+q$ and
$\sigma(V)=p-q$.

###  3.1 Definition of the invariants

Let $X$ be a connected topological space endowed with a representation
$\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$. The purpose of this section is to define
a family

$\operatorname{sch}(\rho)=\sum_{k\geq
0}\operatorname{sch}_{k}(\rho)\in\prod_{k\geq 0}H^{2k}(X,\mathbb{Q})$

which is natural in the sense that whenever there is $f:X\to Y$ and
$\rho:\pi_{1}(Y)\to\mathrm{PU}(p,q)$ then $\operatorname{sch}(\rho\circ
f_{*})=f^{*}\operatorname{sch}(\rho)$.

One may think of $\rho$ as the holonomy of a flat $\mathrm{PU}(p,q)$-principal
bundle over $X$. By forgetting the flat structure, this bundle is obtained by
pulling back a universal $\mathrm{PU}(p,q)$-principal bundle
$\mathrm{EPU}(p,q)\to\mathrm{BPU}(p,q)$ by a map $f:X\to\mathrm{BPU}(p,q)$,
well-defined up to homotopy. We are then reduced to defining a class
$\operatorname{sch}\in H^{*}(\mathrm{BPU}(p,q),\mathbb{Q})$ and set
$\operatorname{sch}(\rho)=f^{*}\operatorname{sch}$.

Recall that the natural map $\pi:\mathrm{SU}(p,q)\to\mathrm{PU}(p,q)$ fits
into the following central exact sequence

$0\to\mu_{p+q}\to\mathrm{SU}(p,q)\overset{\pi}{\to}\mathrm{PU}(p,q)\to 0$

where $\mu_{p+q}\subset\mathbb{C}^{*}$ is the group of roots of unity of order
$p+q$. This exact sequence gives rise to a fibration
$\mathrm{B}\mu_{p+q}\to\mathrm{BSU}(p,q)\to\mathrm{BPU}(p,q)$. As
$H^{k}(\mathrm{B}\mu_{p+q},\mathbb{Q})=0$ for $k>0$, a usual argument
involving the Leray-Serre spectral sequence gives that the map

$\pi^{*}:H^{*}(\mathrm{BPU}(p,q),\mathbb{Q})\to
H^{*}(\mathrm{BSU}(p,q),\mathbb{Q})$

is an isomorphism. Hence it is sufficient to define $\operatorname{sch}\in
H^{*}(\mathrm{BSU}(p,q),\mathbb{Q})$.

Consider $P\to X$ a principal $\mathrm{SU}(p,q)$-bundle and form the Hermitian
bundle $\mathcal{E}\to X$ associated to the tautological action of
$\mathrm{SU}(p,q)$ on $V$. We can find an orthogonal decomposition
$\mathcal{E}=\mathcal{E}^{+}\oplus\mathcal{E}^{-}$ such that the restriction
of the Hermitian structure to $\mathcal{E}^{+}$ (resp. $\mathcal{E}^{-}$) is
positive (resp. negative). Then we set

$\operatorname{sch}(\rho)=\operatorname{ch}(\mathcal{E}^{+})-\operatorname{ch}(\mathcal{E}^{-})$

where $\operatorname{ch}$ denotes the Chern character. In particular, we have
$\operatorname{sch}_{0}(\rho)=\operatorname{sch}_{0}(P)=p-q=\sigma(V)$. We
also observe that if $q=0$ then $\mathcal{E}=\mathcal{E}^{+}$. If this bundle
is constructed from a representation, it has a flat connection, giving
$\operatorname{sch}(\rho)=d(V)$. The same argument works if $p=0$, giving
$\operatorname{sch}(\rho)=-d(V)$. Hence in the sequel, we suppose that $pq>0$.

Denote by $\mathbb{H}(V)=\mathbb{H}^{p,q}$ the space of orthogonal
decompositions $V=V^{+}\oplus V^{-}$. It is the symmetric space associated to
$\mathrm{PU}(p,q)$, in particular it is contractible. Given $P$, one can form
the bundle $\mathcal{H}\to X$ associated to the action of $\mathrm{SU}(p,q)$
on $\mathbb{H}^{p,q}$. The decompositions
$\mathcal{E}=\mathcal{E}^{+}\oplus\mathcal{E}^{-}$ are in bijection with the
sections of $\mathcal{H}$, hence are unique up to homotopy. This shows that
the class $\operatorname{sch}(\rho)$ is well-defined.

Notice that in order to compute the higher Toledo invariants, we need to
linearize the representation $\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$, i.e. to
lift it to $\mathrm{SU}(p,q)$. In the cases we will encounter, this is not
possible unless we modify the space $X$. However, in some cases handled in the
following proposition, we can find a short-cut.

######  Proposition 2.

Let $\mathcal{E}\to X$ be a Hermitian bundle endowed with a projectively flat
connexion with holonomy $\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$. Then, given a
decomposition $\mathcal{E}=\mathcal{E}^{+}\oplus\mathcal{E}^{-}$ as before, we
have:

$\operatorname{sch}(\rho)=(\operatorname{ch}(\mathcal{E}^{+})-\operatorname{ch}(\mathcal{E}^{-}))e^{-\frac{c_{1}(\mathcal{E})}{p+q}}.$

We can check that this quantity does not change if we replace $\mathcal{E}$ by
$\mathcal{E}\otimes L$ for some Hermitian line bundle $L$. In particular, if
the representation lifts to $\mathrm{U}(p,q)$, and we take $\mathcal{E}$ to be
the associated flat bundle, then $c_{1}(\mathcal{E})=0$ and the two formulas
coincide. We insist on a crucial property: $\operatorname{sch}(\rho)$ is
independent of the lift.

###### Proof of Proposition 2.

Let $\mathcal{E}\to X$ be as in the proposition. In order to compute
$\operatorname{sch}(\rho)$, we need to linearize $\rho$, which is generally
impossible. However we can look for a map $f:\widetilde{X}\to X$ such that
$f^{*}:H^{*}(X,\mathbb{Q})\to H^{*}(\widetilde{X},\mathbb{Q})$ is an
isomorphism and such that there is a diagram

$\textstyle{\pi_{1}(\widetilde{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\rho}}$$\textstyle{\mathrm{SU}(p,q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mathrm{PU}(p,q)}$

Suppose first that we have solved this problem. The naturality of the
construction gives
$\operatorname{sch}(\widetilde{\rho})=f^{*}\operatorname{sch}(\rho)$. Hence we
are reduced to the case when $\rho$ takes its values in $\mathrm{SU}(p,q)$. In
that case, we may compare the projectively flat bundle $\mathcal{E}$ with the
flat Hermitian bundle $\mathcal{E}_{\rho}$ associated to $\rho$. The
corresponding projective bundles are isomorphic: this implies that there is a
Hermitian line bundle $L$ such that $\mathcal{E}_{\rho}=\mathcal{E}\otimes L$.
As $c_{1}(\mathcal{E}_{\rho})=0$, we get
$c_{1}(L)=-\frac{c_{1}(\mathcal{E})}{p+q}$.

We check that
$\operatorname{sch}(\mathcal{E}_{\rho})=\operatorname{ch}(\mathcal{E}_{\rho}^{+})-\operatorname{ch}(\mathcal{E}_{\rho}^{-})=\operatorname{ch}(\mathcal{E}^{+}\otimes
L)-\operatorname{ch}(\mathcal{E}^{-}\otimes
L)=(\operatorname{ch}(\mathcal{E}^{+})-\operatorname{ch}(\mathcal{E}^{-}))\operatorname{ch}(L)$.
This coincides with the formula of Proposition 2.

We now prove the existence of $f:\widetilde{X}\to X$ with a twist: we will
replace $X$ by a space homotopically equivalent to it. Recall that the
obstruction of lifting $\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$ is a class
$o(\rho)\in H^{2}(X,\mu_{p+q})$, represented by a map $f:X\to K(2,\mu_{p+q})$.
Replacing $X$ by the (homotopically equivalent) mapping path space

$E_{f}=\\{(x,\gamma),x\in X,\gamma:[0,1]\to K(2,\mu_{p+q}),\gamma(0)=f(x)\\},$

the map $g:E_{f}\to K(2,\mu_{p+q})$ given by $g(x,\gamma)=\gamma(1)$ is a
fibration homotopic to $f$. Its fiber $F$ solves the problem. Indeed, the
composition $F\to E_{f}\to K(2,\mu_{p+q})$ is constant, meaning that the
obstruction $o(\rho)$ vanishes on $F$. Moreover, as the rational cohomology of
$K(2,\mu_{p+q})$ is trivial, the inclusion $F\subset E_{f}\simeq X$ induces an
isomorphism in rational cohomology (from the Leray-Serre spectral sequence). ∎

For the sake of completeness, we study the problem of realizing a projective
representation as the holonomy of a projectively flat bundle.

######  Lemma 8.

Given any representation $\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$, there exists a
Hermitian complex bundle $\mathcal{E}\to X$ endowed with a projectively flat
connection whose monodromy is conjugate to $\rho$ if and only if some
obstruction class in $H^{3}(X,\mathbb{Z})$ vanishes.

###### Proof.

Recall that the representation $\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$ gives rise
to a flat $\mathrm{PU}(p,q)$-bundle $P\to X$. Consider a good open covering
$(U_{i})_{i\in I}$ of $X$ with trivializations of $P|_{U_{i}}$. The transition
functions are constant maps $g_{ij}:U_{i}\cap U_{j}\to\mathrm{PU}(p,q)$
satisfying a cocycle condition. A Hermitian bundle $\mathcal{E}$ may be
constructed by taking continuous maps $h_{ij}:U_{i}\cap U_{j}\to U(p,q)$
satisfing the same cocycle condition. The condition that $\mathcal{E}$ has a
projectively flat connection with monodromy $\rho$ means that $\pi\circ
h_{ij}=g_{ij}$ where $\pi:U(p,q)\to\mathrm{PU}(p,q)$ is the obvious
projection.

As $U_{i}\cap U_{j}$ is contractible, one can find such a map $h_{ij}$
independently for all $i\neq j$. The cocycle condition gives a map $U_{i}\cap
U_{j}\cap U_{k}\to\mathrm{U}(1)=\ker\pi$ which has to vanish in order to prove
the lemma. This defines a class in $H^{2}(X,C_{X}(\mathrm{U}(1)))$ where for
any topological abelian group $G$, $C_{X}(G)$ denotes the sheaf of continuous
$G$-valued functions. From the exact sequence of sheaves $0\to
C_{X}(\mathbb{Z})\to C_{X}(\mathbb{R})\to C_{X}(\mathrm{U}(1))\to 0$ and the
vanishing of $H^{*}(X,C_{X}(\mathbb{R}))$, we find an obstruction in
$H^{3}(X,C_{X}(\mathbb{Z}))=H^{3}(X,\mathbb{Z})$. ∎

###  3.2 Compatibility with operations

Let $V$ and $W$ be two finite dimensional Hermitian spaces. If we have two
representations $\rho_{V}:\pi_{1}(X)\to\mathrm{PU}(V)$,
$\rho_{W}:\pi_{1}(X)\to\mathrm{PU}(W)$, we cannot make sense of their sum but
we can make sense of their tensor product
$\rho_{V}\otimes\rho_{W}:\pi_{1}(X)\to\mathrm{PU}(V\otimes W)$.

######  Proposition 3.

Given two projective representations as above we have

$\operatorname{sch}(\rho_{V}\otimes\rho_{W})=\operatorname{sch}(\rho_{V})\smile\operatorname{sch}(\rho_{W})\in
H^{*}(X,\mathbb{Q}).$

###### Proof.

As explained in the previous section, one can suppose that the representations
are linearized in the sense that $\rho_{V}:\pi_{1}(X)\to\mathrm{SU}(V)$,
$\rho_{W}:\pi_{1}(X)\to\mathrm{SU}(W)$. One may form the associated bundle
$\mathcal{E}_{V\otimes W}$ of $\rho_{V}\otimes\rho_{W}$ by taking the tensor
product of $\mathcal{E}_{V}$ and $\mathcal{E}_{W}$, the Hermitian bundles
associated respectively to $\rho_{V}$ and $\rho_{W}$. Taking a decomposition
$\mathcal{E}_{V}=\mathcal{E}_{V}^{+}\oplus\mathcal{E}_{V}^{-}$ and
$\mathcal{E}_{W}=\mathcal{E}_{W}^{+}\oplus\mathcal{E}_{W}^{-}$, we get a
decomposition

$\mathcal{E}_{V\otimes
W}=\Big{(}\mathcal{E}_{V}^{+}\otimes\mathcal{E}_{W}^{+}\oplus\mathcal{E}_{V}^{-}\otimes\mathcal{E}_{W}^{-}\Big{)}\oplus\Big{(}\mathcal{E}_{V}^{+}\otimes\mathcal{E}_{W}^{-}\oplus\mathcal{E}_{V}^{-}\otimes\mathcal{E}_{W}^{+}\Big{)}.$

From the properties of the Chern character, we readily get
$\operatorname{ch}(\mathcal{E}_{V}^{+}\otimes\mathcal{E}_{W}^{+}\oplus\mathcal{E}_{V}^{-}\otimes\mathcal{E}_{W}^{-})-\operatorname{ch}(\mathcal{E}_{V}^{+}\otimes\mathcal{E}_{W}^{-}\oplus\mathcal{E}_{V}^{-}\otimes\mathcal{E}_{W}^{+})=(\operatorname{ch}(\mathcal{E}_{V}^{+})-\operatorname{ch}(\mathcal{E}_{V}^{-}))(\operatorname{ch}(\mathcal{E}_{W}^{+})-\operatorname{ch}(\mathcal{E}_{W}^{-}))$
from which the result follows. ∎

Let us now deal with the more subtle sum of two Hermitian spaces $V$ and $W$.
We set $\mathrm{PU}(V,W)=\mathrm{P}(\mathrm{U}(V)\times\mathrm{U}(W))$. There
are two natural projections $p_{V}:\mathrm{PU}(V,W)\to\mathrm{PU}(V)$,
$p_{W}:\mathrm{PU}(V,W)\to\mathrm{PU}(W)$ and an inclusion
$i:\mathrm{PU}(V,W)\to\mathrm{PU}(V\oplus W)$.

######  Proposition 4.

Given a representation $\rho:\pi_{1}(X)\to\mathrm{PU}(V,W)$, we have

$\operatorname{sch}(p_{V}\circ\rho)+\operatorname{sch}(p_{W}\circ\rho)=\operatorname{sch}(i\circ\rho).$

###### Proof.

Again we can suppose that $\rho$ takes its values in $\mathrm{SU}(V,W)$. Its
associated bundle is the sum $\mathcal{E}_{V}\oplus\mathcal{E}_{W}$ of the
bundle associated to the projections $p_{V}:\mathrm{SU}(V,W)\to U(V)$,
$p_{W}:\mathrm{SU}(V,W)\to U(W)$. We may decompose the bundles
$\mathcal{E}_{V}$ and $\mathcal{E}_{W}$ as usual: this gives
$\operatorname{sch}(i\circ\rho)=\operatorname{ch}(\mathcal{E}_{V}^{+})+\operatorname{ch}(\mathcal{E}_{W}^{+})-\operatorname{ch}(\mathcal{E}_{V}^{-})-\operatorname{ch}(\mathcal{E}_{W}^{-})$.
From Proposition 2 and the fact that $\mathcal{E}_{V}$ and $\mathcal{E}_{W}$
are flat, we get
$\operatorname{sch}(p_{V}\circ\rho)=\operatorname{ch}(\mathcal{E}_{V}^{+})-\operatorname{ch}(\mathcal{E}_{V}^{-})$
and the same for $W$, showing the result. ∎

###  3.3 A differential definition of the super Chern character

In this subsection, we give an alternative definition of the super Chern
character having a differential flavour, in the case of a representation
defined on the fundamental group of a smooth orbifold.

Recall that for a Hermitian vector space $V$ of signature $(p,q)$ we denoted
by $\mathbb{H}^{p,q}$ the space of either, positive $p$-dimensional subspaces
$V^{+}$, negative $q$-dimensional subspaces $V^{-}$, or orthogonal
decompositions $V=V^{+}\oplus V^{-}$. The group $\mathrm{PU}(p,q)$ acts
transitively on these decompositions and the stabilizer of $V^{+}\oplus V^{-}$
is the maximal compact subgroup
$\mathrm{PU}(V^{+},V^{-})=\mathrm{P}(\mathrm{U}(p)\times\mathrm{U}(q))$,
showing that $\mathbb{H}^{p,q}$ is the symetric space of $\mathrm{PU}(p,q)$.

Let us define a family of $\mathrm{PU}(p,q)$-invariant differential forms
$\omega_{k}$ of degree $2k$ on $\mathbb{H}^{p,q}$. The tangent space of
$\mathbb{H}^{p,q}$ at a point $V^{+}$ is naturally identified with the space
$\operatorname{Hom}(V^{+},V^{-})$ and the complex structure on this space
induces a complex structure on $\mathbb{H}^{p,q}$: together with $\omega_{1}$,
this gives the Kähler structure on $\mathbb{H}^{p,q}$. The adjonction map
$\alpha\mapsto\alpha^{*}$ gives an anti-linear isomorphism
$\operatorname{Hom}(V^{+},V^{-})\simeq\operatorname{Hom}(V^{-},V^{+})$.

######  Remark 4.

It is also possible to identify $T_{(V^{+},V^{-})}\mathbb{H}^{p,q}$ with
$\operatorname{Hom}(V^{-},V^{+})$, but it gives the opposite complex structure
on $\mathbb{H}^{p,q}$. This also corresponds to changing the Hermitian form
$h$ to its opposite, or said informally, exchanging $p$ and $q$. We have to
take great care of this subtlety which occurs everywhere in the article.

For any family $\xi_{1},\ldots,\xi_{2k}\in\operatorname{Hom}(V^{+},V^{-})$, we
set:

$\omega_{k}(\xi_{1},\ldots,\xi_{2k})=\frac{2^{1-k}}{(2i\pi)^{k}k!}\text{Tr}\sum_{\sigma\in
S_{2k}}\varepsilon(\sigma)\prod_{i=1}^{k}\Big{(}\xi_{\sigma(2i-1)}^{*}\xi_{\sigma(2i)}-\xi_{\sigma(2i)}^{*}\xi_{\sigma(2i-1)}\Big{)}.$

In this formula $S_{2k}$ is the group of permutations of $\\{1,\ldots,2k\\}$
and $\varepsilon(\sigma)$ is the signature of $\sigma\in S_{2k}$.

######  Lemma 9.

Assume that $X$ is a developable orbifold and
$\rho:\pi_{1}(X)\rightarrow\mathrm{SU}(p,q)$ is a morphism. Then, for any
smooth $\rho$-equivariant map $f:\widetilde{X}\rightarrow\mathbb{H}^{p,q}$,
the form $f^{*}\omega_{k}$, which is invariant by $\pi_{1}(X)$ and thus
descends to a differential form of degree $2k$ on $X$, is a De Rham
representative of $\operatorname{sch}_{k}(\rho)$ in $H^{2k}(X,\mathbb{R})$.

###### Proof.

Recall that by assumption, the orbifold universal cover $\widetilde{X}$ of $X$
is smooth. By [44, Theorem 2.4], there exists smooth $\rho$-equivariant maps
$f:\widetilde{X}\to\mathbb{H}^{p,q}$ and those are unique up to homotopy.

Let $\mathcal{E}^{+}\rightarrow\mathbb{H}^{p,q}$ be the rank $p$ positive
tautological vector bundle, whose fiber over the point $(V^{+},V^{-})$ is the
subspace $V^{+}$. We observe that $\mathcal{E}^{+}$ is naturally a sub-bundle
of the trivial bundle $\mathcal{V}=V\times\mathbb{H}^{p,q}$ and denote by
$\pi:\mathcal{V}\rightarrow\mathcal{E}^{+}$ the orthogonal projection with
respect to the Hermitian form. We use the trivial connection $D$ on
$\mathcal{V}$ to define a connection $\nabla$ on $\mathcal{E}^{+}$ by

$\nabla_{\xi}s=\pi D_{\xi}s,$

where $s$ is any smooth section of $\mathcal{E}^{+}$ and $\xi$ any vector
field on $\mathbb{H}^{p,q}$. An painful but elementary computation shows that
the curvature
$\Omega_{\nabla}(\xi,\eta)=\nabla_{\xi}\nabla_{\eta}-\nabla_{\eta}\nabla_{\xi}-\nabla_{[\xi,\eta]}$
of this connection is given by the simple formula

$\Omega_{\nabla}(\xi,\eta)=\eta^{*}\xi-\xi^{*}\eta,$

where as before $\xi,\eta$ are considered as elements of
$\operatorname{Hom}(\mathcal{E}^{+},\mathcal{E}^{-})$. Notice that

$\text{Tr}\exp\left(\frac{-\Omega_{\nabla}}{2i\pi}\right)=\frac{1}{2}\sum_{k\geq
0}\omega_{k}$

hence by Chern-Weil theory, the forms $\frac{1}{2}\omega_{k}$ represent the
Chern character of $\mathcal{E}^{+}$ on $\mathbb{H}^{p,q}$. Consider a smooth
$\rho$-equivariant map $f:\widetilde{X}\rightarrow\mathbb{H}^{p,q}$. Pulling
back $\mathcal{E}^{\pm}$ gives rise to orthogonal sub-bundles
$\mathcal{E}^{\pm}_{\rho}$ of the flat hermitian bundle $\mathcal{E}_{\rho}$
of fiber $V$ and monodromy $\rho$ over $X$. We have
$\operatorname{ch}(\mathcal{E}_{\rho}^{+})+\operatorname{ch}(\mathcal{E}_{\rho}^{-})=\operatorname{ch}(\mathcal{E}_{\rho})=p+q$
so
$\operatorname{sch}_{k}(\rho)=\operatorname{ch}_{k}(\mathcal{E}_{\rho}^{+})-\operatorname{ch}_{k}(\mathcal{E}_{\rho}^{-})=2f^{*}\operatorname{ch}_{k}(\mathcal{E}^{+})$
for $k>0$. We then deduce the result from the fact that the pull-back of the
connection $\nabla$ to $\mathcal{E}^{+}_{\rho}$ defines a connection whose
curvature is $f^{*}\Omega_{\nabla}$. ∎

It would be interesting to give an analogous geometric construction in the
projective case, the following section gives one possible way.

###  3.4 Relation to the tangent bundle of the symmetric space

Let $X$ be a connected topological space and
$\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$ be a representation. We choose
$f:\widetilde{X}\to\mathbb{H}^{p,q}$, a continuous $\rho$-equivariant map. We
form the complex vector bundle $\mathcal{F}$ over $X$ defined as the quotient
of $f^{*}T\mathcal{\mathbb{H}}^{p,q}$ by the action of $\pi_{1}(X)$ given by

$\gamma\cdot(x,\xi)=(\gamma(x),D_{f(x)}\rho(\gamma)\xi)$

for any $x\in\widetilde{X}$, any $\xi\in T_{f(x)}\mathbb{H}^{p,q}$, and any
$\gamma\in\pi_{1}(X)$. The map $f$ is well-defined up to $\rho$-equivariant
homotopy, so the complex vector bundle $\mathcal{F}$ is well-defined.

######  Lemma 10.

For odd $k$, we have
$\operatorname{sch}_{k}(\rho)=\frac{-2}{p+q}ch_{k}(\mathcal{F}).$

In particular $\frac{(p+q)k!}{2}\operatorname{sch}_{k}(\rho)$ is an integral
class. For even $k$ and $p\neq q$, we can also express
$\operatorname{sch}_{k}$ as a polynomial in the Chern character of
$\mathcal{F}$. For instance

$\operatorname{sch}_{2}(\rho)=\frac{-2}{p-q}\left(\operatorname{ch}_{2}(\mathcal{F})-\frac{1}{(p+q)^{2}}\operatorname{ch}_{1}(\mathcal{F})^{2}\right).$

###### Proof.

The construction being natural in $X$, we can suppose as in the proof of
Proposition 2 that the representation $\rho$ lifts to $\mathrm{SU}(p,q)$. We
can then define the two associated bundles $\mathcal{E}_{\rho}^{\pm}$ so that
$\mathcal{F}=\operatorname{Hom}(\mathcal{E}_{\rho}^{+},\mathcal{E}_{\rho}^{-})=(\mathcal{E}_{\rho}^{+})^{*}\otimes\mathcal{E}^{-}_{\rho}$.
The proof follows by inspection of the following identities;
$\operatorname{sch}(\rho)=\operatorname{ch}(\mathcal{E}_{\rho}^{+})-\operatorname{ch}(\mathcal{E}_{\rho}^{-})$,
$\operatorname{ch}(\mathcal{E}_{\rho}^{+})+\operatorname{ch}(\mathcal{E}_{\rho})^{-}=p+q$
and

$\operatorname{ch}(\mathcal{F})=\Big{(}p-\operatorname{ch}_{1}(\mathcal{E}_{\rho}^{+})+\operatorname{ch}_{2}(\mathcal{E}_{\rho}^{+})-\cdots\Big{)}\Big{(}q+\operatorname{ch}_{1}(\mathcal{E}_{\rho}^{-})+\operatorname{ch}_{2}(\mathcal{E}_{\rho}^{-})+\cdots\Big{)}.$

∎

This lemma has the following important consequence:

######  Corollary 2.

Suppose that a complex orbifold $X$ is locally modeled on the symmetric space
$\mathbb{H}^{p,q}$ and let $\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$ be its
monodromy representation. Then, denoting by $K_{X}$ the canonical bundle of
$X$, we have

$\operatorname{sch}_{1}(\rho)=\frac{2}{p+q}c_{1}(K_{X}).$

In particular, the holonomy of a $\mathbb{H}^{1,1}$-structure on a closed
oriented surface $S$ of genus $g\geq 2$ satisfies
$\int_{S}\operatorname{sch}_{1}(\rho)=2g-2$.

######  Remark 5.

Corollary 2 gives a necessary condition for the uniformization of a
representation $\rho:\pi_{1}(X)\rightarrow\text{PU}(p,q)$ by a
$\mathbb{H}^{p,q}$-structure which is almost sufficient, thanks to Siu’s
rigidity theory (see Lemma 14).

###  3.5 The Toledo class as an obstruction class

The purpose of this section is to identify the Toledo class
$\operatorname{sch}_{1}(\rho)$ with an obstruction class.

For any connected Lie group $G$, we derive from the homotopy sequence of the
fibration $G\to EG\to BG$ that $\pi_{n}(BG)=\pi_{n-1}(G)$, in particular $BG$
is simply connected and from the Hurewicz theorem, we get
$H_{2}(BG,\mathbb{Z})=\pi_{2}(BG)=\pi_{1}(G)$. The universal coefficient
theorem gives the isomorphism
$H^{2}(BG,\mathbb{Q})=\operatorname{Hom}(\pi_{1}(G),\mathbb{Q})$. Taking
$G=\mathrm{PU}(p,q)$, the class $\operatorname{sch}_{1}\in
H^{2}(\mathrm{PU}(p,q),\mathbb{Q})$ corresponds to a map
$\varphi:\pi_{1}(\mathrm{PU}(p,q))\to\mathbb{Q}$.

Recall that the maximal compact subgroup of $\mathrm{PU}(p,q)$ is
$\mathrm{P}(\mathrm{U}(p)\times\mathrm{U}(q))$ and have the same fundamental
group. The exact sequence of the fibration
$\mathrm{U}(1)\to\mathrm{U}(p)\times\mathrm{U}(q)\to\mathrm{P}(\mathrm{U}(p)\times\mathrm{U}(q))$
gives the description
$\pi_{1}(\mathrm{PU}(p,q))=\mathbb{Z}^{2}/(p,q)\mathbb{Z}$.

######  Lemma 11.

The map
$\varphi:\pi_{1}(\mathrm{PU}(p,q))=\mathbb{Z}^{2}/(p,q)\mathbb{Z}\to\mathbb{Q}$
associated to $\operatorname{sch}_{1}$ by the above procedure is

$\varphi(x,y)=\frac{2}{p+q}(xq-py).$

This lemma tells that $\operatorname{sch}_{1}(\rho)$ can be computed by the
following constructive procedure. Consider the central extension

$0\to\pi_{1}(\mathrm{PU}(p,q))\to\widetilde{\mathrm{PU}}(p,q)\to\mathrm{PU}(p,q)\to
0.$

The obstruction of lifting $\rho:\pi_{1}(X)\to\mathrm{PU}(p,q)$ to
$\widetilde{\mathrm{PU}}(p,q)$ is a class $o(\rho)\in
H^{2}(\pi_{1}(X),\pi_{1}(\mathrm{PU}(p,q))$ that we can map to
$H^{2}(X,\pi_{1}(\mathrm{PU}(p,q)))$ (using a map $f:X\to B\pi_{1}(X)$
inducing the identity on fundamental groups). The lemma claims that one has

$\operatorname{sch}_{1}(\rho)=\varphi_{*}(o(\rho)).$

###### Proof.

As $\pi_{1}(\mathrm{PU}(p,q))\otimes\mathbb{Q}$ has dimension 1, this is just
a question of normalization. Consider first the case $p=q=1$. Then
$\pi_{1}(\mathrm{PU}(1,1))=\mathbb{Z}$ and $\varphi:\mathbb{Z}\to\mathbb{Q}$
is the standard inclusion. We have to take care of the orientation here: the
loop $\gamma(\theta)=(e^{i\theta},1)\in\mathrm{PU}(1,1)$ corresponds to the
positive generator. Recall that
$\mathbb{H}^{1,1}=\\{[v]\in\mathbb{P}V,h(v)>0\\}$ is a hyperbolic disc,
naturally oriented by its complex structure: we check that $\gamma(\theta)$
acts by rotation of angle $-\theta$ on $\mathbb{H}^{1,1}$: the two
orientations disagree.

We take a surface $S$ of genus $g$ with a $\mathbb{H}^{1,1}$-structure and
holonomy representation $\rho:\pi_{1}(S)\to\mathrm{PU}(1,1)$. The obstruction
of lifting it to the universal cover is the Euler class, which in this case is
known to be equal to the Euler characteristic $2-2g$. By Corollary 2, we have
$\int_{S}\operatorname{sch}_{1}(\rho)=2g-2$. The change of sign observed in
the previous paragraph makes this formula agree: we have in this case
$\operatorname{sch}_{1}=-\rm{eu}$.

In the general case, we consider a decomposition $V\oplus W$ where $V$ has
signature $(1,1)$ and a representation $\rho:\pi_{1}(X)\to\mathrm{SU}(V)$,
trivially extended to $\mathrm{SU}(V\oplus W)$. The additivity formula of
Proposition 4 gives $\operatorname{sch}_{1}(\rho_{V\oplus
W})=\operatorname{sch}_{1}(\rho_{V})$. It suffices to check that the following
diagram commutes.

$\textstyle{\pi_{1}\mathrm{SU}(1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1,1}}$$\textstyle{\pi_{1}\mathrm{SU}(p,q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{p,q}}$$\textstyle{\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{\mathbb{Q}}$

To check it, consider an element $(x,-x)\in\pi_{1}\mathrm{SU}(1,1)$. We
compute $\varphi_{1,1}(x,-x)=2x$. When mapping $\mathrm{SU}(1,1)$ to
$\mathrm{SU}(p,q)$ as above, the element $(x,-x)$ stays equal. This time we
compute $\varphi_{p,q}(x,-x)=\frac{2}{p+q}(qx-p(-x))=2x$. This proves the
result. ∎

##  4 Hermitian modular functors

###  4.1 Marked surfaces

A marked surface is a triple $(S,\varphi,L)$ where

  1. 1.

$S$ is a compact oriented surface whose boundary is the disjoint union of the
components $\partial_{i}S$ for $i\in\pi_{0}(\partial S)$,

  2. 2.

$\varphi$ is a collection of homeomorphisms preserving the orientation
$\varphi_{i}:S^{1}\to\partial_{i}S$ for $i\in\pi_{0}(\partial S)$.

  3. 3.

$L$ is a split Lagrangian in $H_{1}(S,\mathbb{Q})$. This means that
$L=\bigoplus_{i\in\pi_{0}(S)}L_{i}$, where $L_{i}$ is a Lagrangian in
$H_{1}(\dot{S}_{i},\mathbb{Q})$ and $\dot{S}_{i}$ is the $i$-th connected
component of $S$ where each boundary curve has been collapsed to a point.

Frequently, we will denote only by $S$ the marked surface $(S,\varphi,L)$. A
morphism $(S,\varphi,L)\to(S^{\prime},\varphi^{\prime},L^{\prime})$ is a pair
$(f,s)$ where $f:S\to S^{\prime}$ is a homeomorphism preserving the
orientation and satisfying $\varphi=\varphi^{\prime}\circ f$ and
$s\in\mathbb{Z}$ is an integer. The composition of $(f,s):S_{1}\to S_{2}$ and
$(g,t):S_{2}\to S_{3}$ is

$(g\circ f,s+t-\operatorname{Maslov}((gf)_{*}L_{1},g_{*}L_{2},L_{3})).$

We can define three operations on marked surfaces: the disjoint union, the
change of orientation and the gluing operation. Only the third one deserves an
explanation. Pick $\partial_{+}S$ and $\partial_{-}S$ two components of
$\partial S$. We define $S_{\pm}$ to be the result of identifying
$\varphi_{+}(z)$ and $\varphi_{-}(\overline{z})$ for any $z\in S^{1}$. Denote
by $\dot{S}$ the surface obtained from $S$ by collapsing $\partial S_{+}$ and
$\partial S_{-}$ to a point. There are natural maps $S\to\dot{S}\leftarrow
S_{\pm}$. We define $L_{\pm}$ to be the preimage in
$H_{1}(S_{\pm},\mathbb{Q})$ of the image of $L$ in
$H_{1}(\dot{S},\mathbb{Q})$. The triple $(S_{\pm},\varphi,L_{\pm})$ is the
gluing of $S$ along $\partial_{\pm}S$.

###  4.2 Hermitian modular functor

Let $\Lambda$ be a finite set endowed with an involution
$\lambda\mapsto\lambda^{*}$ and a unit $0\in\Lambda$ satisfying $0^{*}=0$. A
$\Lambda$-coloring of a surface $S$ is a map $\lambda:\pi_{0}(\partial
S)\to\Lambda$. A Hermitian modular functor is a functor $\mathcal{V}$ from the
category of $\Lambda$-colored marked surfaces to the category of Hermitian
vector spaces satisfying the following axioms.

MF1: Monoidality (simplified). There are compatible isomorphisms

$\mathcal{V}((S,\lambda)\amalg(S^{\prime},\lambda^{\prime}))\simeq\mathcal{V}(S,\lambda)\otimes\mathcal{V}(S^{\prime},\lambda^{\prime})$

MF2: Gluing: there is a natural isomorphism

$\mathcal{V}(S_{\pm},\lambda)\simeq\bigoplus_{\mu\in\Lambda}\mathcal{V}(S,\mu,\mu^{*},\lambda)\otimes\mathcal{V}(S^{2},\mu,\mu^{*})$

MF3: Change of orientation. There is a natural perfect pairing

$\mathcal{V}(S,\lambda)\times\mathcal{V}(-S,\lambda^{*})\to\mathbb{C}$

MF4: Sphere with 1 point.

$\dim\mathcal{V}(S^{2},\lambda)=1\text{ if }\lambda=0,0\text{ otherwise}.$

MF5: Sphere with 2 points.

$\dim\mathcal{V}(S^{2},\lambda,\mu)=1\text{ if }\lambda=\mu^{*},0\text{
otherwise}.$

In these axioms, we make several shortcuts in the notation to keep it light.
When we add $\lambda$ to a marked surface, it means either that we color by
$\lambda$ part or all of the boundary components or even that we create a
boundary component that we color with $\lambda$, depending on the context. Let
us name some important constants associated to a Hermitian modular functor.

  1. 1.

For any (Hermitian) modular functor, any morphism of the form
$(\operatorname{Id},1)$ acts on $\mathcal{V}(S,\lambda)$ multiplying by
$c\in\mathbb{C}^{*}$. One can prove from the axioms that this number is
independent on $S$ and $\lambda$ and is called the _central charge_ of the
modular functor.

  2. 2.

On $\mathcal{V}(S^{2},\lambda,\lambda^{*})$ acts the Dehn twist $T_{\delta}$
along a simple curve separating $x$ and $y$. As this space is $1$-dimensional,
$(T_{\delta},0)$ acts multiplying by
$r_{\lambda}=r_{\lambda^{*}}\in\mathbb{C}^{*}$. We will call these constants
the _multipliers_ associated to the colors.

  3. 3.

On this latter space which is $1$-dimensional, the Hermitian form is definite.
We denote by $\varepsilon_{\lambda}=\varepsilon_{\lambda^{*}}$ its sign.

It is known that $c$ and $r_{\lambda}$ are always roots of unity.

######  Definition 2.

The level of a modular functor $\mathcal{V}$ is an integer $\ell$ such that
$r_{\lambda}^{\ell}=1$ for all $\lambda\in\Lambda$.

Our main example is the Fibonacci TQFT for which we have $\ell=5$. We warn the
reader that there is a shift with the level commonly used in Conformal Field
Theory.

###  4.3 The associated cohomological field theory

We set $V=\mathbb{Q}[\Lambda]$. We first define for all $g,n\geq 0$,

$\omega_{g,n}\in\operatorname{Hom}(V^{\otimes
n},H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})).$

It reduces to define for every genus $g$ and for any
$\lambda_{1},\ldots,\lambda_{n}\in\Lambda$ a class
$\omega_{g,n}(\lambda_{1},\ldots,\lambda_{n})\in
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$.

Let $S$ be a surface with genus $g$ and $n$ boundary components. We recall
that $\operatorname{Mod}(S)$ is the group of isotopy classes of homeomorphisms
of $S$ fixing the boundary pointwise. Pick a marking $\varphi$ of the boundary
of $S$ and a coloring $\lambda:\pi_{0}(\partial S)\to\Lambda$. The
automorphism group of $S$ in the category of marked surface is a central
extension of $\operatorname{Mod}(S)$ (its class is given by $\lambda_{1}\in
H^{2}(\operatorname{Mod}(S),\mathbb{Z})$ but it does not matter here). As the
modular functor is Hermitian and sends the central element to
$c\operatorname{Id}$, we get a representation

$\rho_{\lambda}:\operatorname{Mod}(S)\to\mathrm{PU}\big{(}\mathcal{V}({\bf
S},\lambda)\big{)}$

Let $\delta$ be a simple curve parallel to a boundary component of $S$ colored
by $\mu$. From the axioms, the Dehn twist $T_{\delta}$ acts by multiplication
by $r_{\mu}$, hence trivially in the projective unitary group. This means that
the representation $\rho_{\lambda}$ factors trough the group
$\operatorname{Mod}(\dot{S},P)$ where $\dot{S}$ is the surface obtained by
collapsing each boundary component of $S$ to a point, and $P$ is the set of
resulting marked points.

Finally, the axiom MF2 shows that every Dehn twist $T_{\delta}$ is
diagonalizable with eigenvalues $r_{\mu}$ for $\mu\in\Lambda$. In particular
$T_{\delta}^{\ell}$ acts trivially, hence $\rho_{\lambda}$ factors through a
representation

$\rho_{\lambda}:\operatorname{Mod}^{\ell}(\dot{S},P)\to\mathrm{PU}\big{(}\mathcal{V}(S,\lambda)\big{)}$

where $\operatorname{Mod}^{\ell}(\dot{S},P)$ is the quotient of the mapping
class group $\operatorname{Mod}(\dot{S},P)$ by the (normal) subgroup generated
by $\ell$-th powers of Dehn twists.

As we showed in Section 2 that
$\pi_{1}(\overline{\mathcal{M}}^{\ell}(\dot{S},P))=\operatorname{Mod}^{\ell}(\dot{S},P)$,
the construction of Section 3 defines a class

$\omega_{g,n}(\lambda)=\operatorname{sch}(\rho_{\lambda})\in
H^{*}(\overline{\mathcal{M}}^{\ell}(\dot{S},P),\mathbb{Q})=H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$

The last equality is due to the fact that the orbifolds
$\overline{\mathcal{M}}^{\ell}(\dot{S},P)$ and $\overline{\mathcal{M}}_{g,n}$
have the same underlying topological space, hence the same rational
cohomology.

###  4.4 Proof of the CohFT axioms

We define on $V$ the bilinear form
$\eta(\lambda,\mu)=\operatorname{sign}\mathcal{V}(S^{2},\lambda,\mu)$. By MF5,
$\eta(\lambda,\mu)=\varepsilon_{\lambda}$ if $\lambda=\mu^{*}$ and $0$
otherwise. We refer to [35] for details on the axioms of a CohFT, here we
recall them at the same time that we prove them. The first one is a
compatibility of the construction with the action of the symmetric groups
permuting the colors and the marked point. It is satisfied by construction.

We recall that the product on $V$ is defined by the formula
$\omega_{0,3}(\lambda,\mu,\nu)=\eta(\lambda\cdot\mu,\nu)$. It follows from the
axioms of a Hermitian modular functor that the unit $1\in V$ corresponds to
the color $0\in\Lambda$. We will try not to confuse the reader using both
notations.

####  4.4.1 Forgetting a point

Let $\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ the
map which forgets the last marked point. One needs to check

$\pi^{*}\omega_{g,n}(\lambda_{1},\ldots,\lambda_{n})=\omega_{g,n+1}(\lambda_{1},\ldots,\lambda_{n},0).$
(10)

Let $S$ be a marked surface and $S^{\circ}$ be the result of removing a disc
in the interior of $S$. Corresponding to $\pi$, there is a morphism
$p:\operatorname{Mod}(S^{\circ})\to\operatorname{Mod}(S)$ obtained by gluing
back the disc. This morphism $p$ induces a map
$\operatorname{Mod}^{\ell}(S_{g,n+1})\to\operatorname{Mod}^{\ell}(S_{g,n})$
which is the morphism induced on the fundamental groups by the map
$\pi:\overline{\mathcal{M}}^{\ell}_{g,n+1}\to\overline{\mathcal{M}}^{\ell}_{g,n}$.

From the axioms of the modular functor, there is a $p$-equivariant isomorphism
$\mathcal{V}(S^{\circ},\lambda,0)=\mathcal{V}(S,\lambda)$ which fits in the
following commutative diagram:

$\textstyle{\pi_{1}\overline{\mathcal{M}}^{\ell}_{g,n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Mod}^{\ell}(S_{g,n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\lambda,0}}$$\scriptstyle{\ell}$$\textstyle{\mathrm{PU}\big{(}\mathcal{V}(S,\lambda,0)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\pi_{1}\overline{\mathcal{M}}^{\ell}_{g,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Mod}^{\ell}(S_{g,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\lambda}}$$\textstyle{\mathrm{PU}\big{(}\mathcal{V}(S,\lambda)\big{)}}$

Equation (10) hence follows from the naturality of the class
$\operatorname{sch}$.

####  4.4.2 Non-separating gluing

Let $\pi:\overline{\mathcal{M}}_{g,n+2}\to\overline{\mathcal{M}}_{g+1,n}$ the
map which glue the two last points. The second axiom of a CohFT to be checked
is

$\pi^{*}\omega_{g+1,n}(\lambda_{1},\ldots,\lambda_{n})=\sum_{\mu\in\Lambda}\varepsilon_{\mu}\omega_{g,n+2}(\lambda_{1},\ldots,\lambda_{n},\mu,\mu^{*}).$
(11)

We consider this time a marked surface $S=(S,\varphi,L)$ with two special
boundary components $\partial_{+}S$ and $\partial_{-}S$. As in Section 4.2, we
denote by $S_{\pm}$ the result of gluing these components using their
parametrization. Again, there is a natural morphism
$p:\operatorname{Mod}(S)\to\operatorname{Mod}(S_{\pm})$ which induces a
morphism
$p:\operatorname{Mod}^{\ell}(S_{g,n+2})\to\operatorname{Mod}^{\ell}(S_{g+1,n})$.
This morphism is the one induced by
$\pi:\overline{\mathcal{M}}^{\ell}_{g,n+2}\to\overline{\mathcal{M}}^{\ell}_{g+1,n}$
on fundamental groups. We get hence a picture very similar to the preceding
section.

The main difference is that axiom MF2 gives a decomposition of
$\mathcal{V}(S_{\pm},\lambda)$ which is preserved by the action of
$\operatorname{Mod}(S_{\pm},\delta)=p(\operatorname{Mod}(S_{g,n+2}))$ where
$\delta$ is the common image of the glued boundaries. This decomposition
corresponds to the eigenspace decomposition of the Dehn twist $T_{\delta}$.
The situation is better visualized in the following diagram:

$\textstyle{\pi_{1}\overline{\mathcal{M}}^{p}_{g,n+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Mod}^{p}(S_{g,n+2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum\rho_{\lambda,\mu,\mu^{*}}}$$\scriptstyle{p}$$\textstyle{\mathrm{PU}\big{(}\mathcal{V}({\bf
S},\lambda,\mu,\mu^{*})_{\mu\in\Lambda}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\pi_{1}\overline{\mathcal{M}}^{p}_{g+1,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Mod}^{p}(S_{g+1,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\lambda}}$$\textstyle{\mathrm{PU}\big{(}\mathcal{V}({\bf
S}_{\pm},\lambda)\big{)}}$

The formula (11) follows then from the additivity formula of Proposition 4,
taking into account that each factor $\mathcal{V}(S,\lambda,\mu,\mu^{*})$
appears tensored by the one-dimensional space $\mathcal{V}(S^{2},\mu,\mu^{*})$
which has sign $\varepsilon_{\mu}$. This factor acts on the class
$\operatorname{sch}$ by the global sign $\varepsilon_{\mu}$ (as a consequence
of the multiplicativity property).

###  4.5 Separating gluing

Let
$\pi:\overline{\mathcal{M}}_{g_{1},n_{1}+1}\times\overline{\mathcal{M}}_{g_{2},n_{2}+1}\to\overline{\mathcal{M}}_{g_{1}+g_{2},n_{1}+n_{2}}$
the map obtained by gluing the last points. This time we must check that

$\pi^{*}\omega_{g,n}(\lambda_{1},\lambda_{2})=\sum_{\mu\in\Lambda}\varepsilon_{\mu}\omega_{g_{1},n_{1}+1}(\lambda_{1},\mu)\otimes\omega_{g_{2},n_{2}+1}(\lambda_{2},\mu^{*}).$
(12)

Here the tensor product makes sense using the Künneth formula.

Again, consider two marked surfaces $S_{1},S_{2}$ with respective genus
$g_{1},g_{2}$ and respectively $n_{1}+1$ and $n_{2}+1$ boundary components.
The operation of gluing the last components produces a surface $S$ of genus
$g=g_{1}+g_{2}$ and $n=n_{1}+n_{2}$ boundary components together with a map
$p:\operatorname{Mod}(S_{1})\times\operatorname{Mod}(S_{2})\to\operatorname{Mod}(S)$
inducing a map
$\operatorname{Mod}^{\ell}(S_{g_{1},n_{1}+1})\times\operatorname{Mod}^{\ell}(S_{g_{2},n_{2}+1})\to\operatorname{Mod}^{\ell}(S_{g,n})$.
The situation is very similar to the one of the previous section: this time
the group $\operatorname{Mod}^{\ell}(S_{g,n},\delta)$ which is the image of
$p$ preserves the decomposition

$\mathcal{V}(S,\lambda_{1},\lambda_{2})=\bigoplus_{\mu\in\Lambda}\mathcal{V}(S_{1},\lambda_{1},\mu)\otimes\mathcal{V}(S_{2},\lambda_{2},\mu^{*})\otimes\mathcal{V}(S^{2},\mu,\mu^{*}).$

As in the previous section, and using this time the multiplicative property of
$\operatorname{sch}$ given in Proposition 3, we obtain a proof of Equation
(12), which ends the proof that $\omega_{g,n}$ satisfies the axioms of a
CohFT.

##  5 Computation of the CohFT associated to the
$\mathrm{SU}_{2}/\mathrm{SO}_{3}$-modular functors

The purpose of this section is to give some detail on two interesting families
for which the construction of the preceding section applies. The degree 0 part
of those CohFTs (usually called Topological Field Theories or Frobenius
algebras) are already interesting and new as they provide formulas for the
signatures of TQFT as investigated in [20].

###  5.1 Semi-simplicity of the Frobenius algebras

####  5.1.1 Generalities on Frobenius algebras

Let us start with generalities about Frobenius $\mathbb{Q}$-algebras. They are
by definition finite dimensional $\mathbb{Q}$-algebras $V$ endowed with a
linear form $\varepsilon:V\to\mathbb{Q}$ such that the bilinear form
$\eta(x,y)=\varepsilon(xy)$ is non-degenerate.

Consider its inverse $\eta^{-1}\in V\otimes V$ : composing with the
multiplication $m:V\otimes V\to V$, we get an element $\Omega=m(\eta^{-1})\in
V$. It is well-known and easy to check that any TFT $\omega_{g,n}$ with
underlying Frobenius algebra $V$ satisfies

$\omega_{g,n}(v_{1},\ldots,v_{n})=\varepsilon(v_{1}\cdots v_{n}\Omega^{g})$

In particular, the signature of the Hermitian vector space associated to a
genus $g$ surface by a modular functor of Frobenius algebra $V$ is
$\omega_{g,0}=\varepsilon(\Omega^{g})$. This is a generalization of the
Verlinde formula.

A crucial property of a Frobenius algebra is its semi-simplicity, holding if
and only if it is isomorphic to a product of number fields. Denote by
$M_{x}\in\operatorname{End}(V)$ the operator of multiplication by $x\in V$ and
by $\operatorname{Tr}_{V}:V\to\mathbb{Q}$ the trace form given by
$\operatorname{Tr}_{V}(x)=\operatorname{Tr}(M_{x})$. A property equivalent to
semi-simplicity is that the bilinear pairing
$(x,y)\mapsto\operatorname{Tr}_{V}(xy)$ is non-degenerate.

Hence a Frobenius structure on a semi-simple algebra is given by an invertible
element $\alpha\in V^{\times}$ satisfying
$\varepsilon(x)=\operatorname{Tr}_{V}(\alpha x)$. It looks like in the most
interesting cases of $\mathrm{SO}_{3}$-modular functors of prime level, the
algebra $V$ is a number field.

######  Lemma 12.

If $V$ is a semi-simple Frobenius algebra associated to $\alpha\in V^{\times}$
then $\Omega=\alpha^{-1}$. In particular,

$\omega_{g,n}(v_{1},\ldots,v_{n})=\operatorname{Tr}_{V}(v_{1}\cdots
v_{n}\alpha^{1-g}).$

###### Proof.

By Artin-Wedderburn theorem, we can reduce to the case when $V$ is a number
field. The computation of $\Omega$ can be done in $V\otimes\mathbb{C}$ which
is isomorphic to $\mathbb{C}^{n}$ via the map
$x\mapsto(\varphi_{1}(x),\ldots,\varphi_{n}(x))$ where
$\varphi_{1},\ldots,\varphi_{n}$ denote the embeddings
$V\hookrightarrow\mathbb{C}$. The linear form $\varepsilon$ on the $i$-th
factor is the multiplication by $\varphi_{i}(\alpha)$, hence the element
$\Omega$ on the $i$-th factor is $\varphi_{i}(\alpha^{-1})$, proving the
lemma. ∎

A nice example is given by the celebrated Verlinde formula which compute the
dimension of the modular functors. In the next sections, considering the
$\mathrm{SO}(3)$-modular functor associated to a specific root of unity
$\zeta$ of prime order, we will find a unitary modular functor whose CohFT
reduces to its degree $0$ part. In that case, $V$ is the subfield of
$\mathbb{Q}(\zeta)^{+}$ fixed by the involution $\zeta\mapsto\zeta^{-1}$ and
$\alpha=-\frac{(\zeta-\zeta^{-1})^{2}}{\ell}$.

Using the formula
$\operatorname{Tr}_{V}(x)=\sum\limits_{i=1}^{n}\varphi_{i}(x)$, we get the
Verlinde formula:

$\dim\mathcal{V}_{\ell}(S_{g})=\Big{(}\frac{\ell}{4}\Big{)}^{g-1}\sum_{m=1}^{\frac{\ell-1}{2}}\sin\Big{(}\frac{2m\pi}{\ell}\Big{)}^{2-2g}.$

We will get similar formulas for the signatures with the twist that the
conjugates $\varphi_{i}(\alpha)$ will no longer have an explicit expression.

####  5.1.2 The $\mathrm{SU}_{2}$-modular functor

We set $r\geq 2$ and choose $A$ to be a primitive $4r$-th root of unity. The
construction in [9] produces a Hermitian modular functor $\mathcal{V}_{A}$
from this data.

The set of colors is $\Lambda=\\{0,1,\ldots,r-2\\}$ with the trivial
involution. The multiplicators are $r_{i}=(-1)^{i}A^{i(i+2)}$ and the signs
are $\varepsilon_{i}=\operatorname{sign}((-1)^{i}[i+1])$ where
$[n]=\frac{A^{2n}-A^{-2n}}{A^{2}-A^{-2}}$. The level of this theory in our
sense is $\ell=4r$.

In the sequel, we denote by
$V_{A}=\mathbb{Q}e_{0}\oplus\cdots\oplus\mathbb{Q}e_{r-2}$ the Frobenius
algebra underlying the CohFT associated to $\mathcal{V}_{A}$.

Recall that the bilinear form $\eta$ is diagonal in this basis and satisfies
$\eta(e_{i},e_{i})=\varepsilon_{i}$. As in any Frobenius algebra the product
is given by

$\eta(e_{i}e_{j},e_{k})=\omega_{0,3}(e_{i},e_{j},e_{k})=\operatorname{sign}\mathcal{V}_{A}(S^{2},i,j,k).$

The space $\mathcal{V}(S^{2},i,j,k)$ is one dimensional if one can write
$i=b+c,j=a+c,k=a+b$ for some integers $a,b,c\in\mathbb{N}$ and $0$ if we
cannot. In the first case, we find in [9, Lemma 4.2] the formula
$\omega_{0,3}(i,j,k)=\operatorname{sign}\langle i,j,k\rangle$ where

$\langle
i,j,k\rangle=(-1)^{a+b+c}\frac{[a+b+c+1]![a]![b]![c]!}{[i]![j]![k]!}.$

Here we used the quantum factorial $[n]!=[1][2]\cdots[n]$.

######  Proposition 5 ($\mathrm{SU}_{2}$ case).

For any $4r$-th root of unity $A$, $V_{A}$ is semi-simple.

###### Proof.

We check from the above formulas that
$e_{1}e_{i}=\frac{\varepsilon_{i}}{\varepsilon_{i-1}}e_{i-1}+e_{i+1}$ if we
set $e_{-1}=e_{r-1}=0$. This proves that $e_{1}$ generates $V_{A}$ as an
algebra and hence the natural surjection $\mathbb{Q}[t]/P(t)\to V_{A}$ is an
isomorphism where $P(t)=\det(M_{e_{1}}-t\operatorname{Id})$ and $M_{e_{1}}$ is
the matrix of the multiplication by $e_{1}$ on $V_{A}$. This matrix has the
simple form

$M_{e_{1}}=\begin{pmatrix}0&\varepsilon_{1}/\varepsilon_{0}&0&0&0\\\
1&0&\varepsilon_{2}/\varepsilon_{1}&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\\\
0&0&1&0&\varepsilon_{r-2}/\varepsilon_{r-3}\\\ 0&0&0&1&0\end{pmatrix}.$

It is an exercise, left to the reader, that these kind of Jacobi matrices have
a simple spectrum, which implies that the algebra $V_{A}$ is semi-simple. ∎

When $A=\pm e^{\frac{i\pi}{2r}}$ we get $\varepsilon_{i}=(-1)^{i}$ and
$\omega_{0,3}(e_{i},e_{j},e_{k})=(-1)^{(i+j+k)/2}$ when it is non zero. In
this case the modular functor is Hermitian in the standard sense (the
Hermitian form is definite) and the CohFT constructed above reduces to its
degree 0 part. The Frobenius algebra we thus obtained is the Verlinde fusion
algebra, described in many places, see [6, 9].

It would be interesting to investigate the properties of these Frobenius
algebras. Here, we directly skip to the $\mathrm{SO}_{3}$-case which gives
lower dimensional and often simple Frobenius algebras. Moreover the
corresponding representations of the mapping class group are irreducible, have
good arithmetic properties if the level is prime, and contain the main example
of this article, Fibonacci modular functor.

####  5.1.3 The $\mathrm{SO}_{3}$-modular functor

We choose $A$ to be a primitive $2\ell$-th root of unity where, this time,
$\ell$ is odd. This corresponds in [9] to a modular functor with group
$\mathrm{SO}_{3}$ and our main example concerns the case when $\ell=5$. In
this case, the set of colors is $\Lambda=\\{0,2,\ldots,\ell-3\\}$, the
involution is trivial and the multiplicators and the signs are given by the
same formulas as above. Precisely, we have $\varepsilon_{2i}=[2i+1]$ and
$\mu_{2i}=A^{4i(i+1)}$ and we check that $\mu_{2i}^{\ell}=1$ for all $i$ so
that $\ell$ is the level of this theory.

Set $q=A^{2}$ and
$V_{q}=\mathbb{Q}e_{0}\oplus\mathbb{Q}e_{1}\oplus\cdots\oplus\mathbb{Q}e_{\frac{\ell-3}{2}}$
where $e_{i}$ represents the color $2i$. This time, the Frobenius algebra
depends only on $q$, hence the notation. We have
$\omega_{0,3}(e_{i},e_{j},e_{k})=\operatorname{sign}\langle 2i,2j,2k\rangle$
if $i,j,k$ satisfy

$i\leq j+k,\,j\leq i+k,\,k\leq i+j\text{ and }i+j+k<l-1$ (T)

and $0$ otherwise. The root giving a Hermitian theory is $q=e^{\pm
2i\pi\frac{\ell-1}{\ell}}$. We claim that Proposition 5 also holds when $\ell$
is odd.

######  Proposition 6 ($\mathrm{SO}_{3}$ case).

For any $\ell$-th root of unity $q$, $V_{q}$ is semi-simple.

###### Proof.

We compute this time that

$e_{1}e_{i}=\operatorname{sign}\Big{(}\frac{[2i+1]}{[2i-1]}\Big{)}e_{i-1}-\operatorname{sign}\Big{(}\frac{[2i+2]}{[2i][2]}\Big{)}e_{i}+e_{i+1}.$

This shows that the matrix of multiplication by $e_{1}$ is tridiagonal with
non-zero entries. Hence the argument of the preceding proof repeats, showing
that $V_{q}$ is semi-simple for any root $q$ of odd order. ∎

We have no proof for the following properties that we checked numerically for
$\ell<100$ prime and $q^{\ell}=1$.

  1. 1.

$V_{q}$ is a number field.

  2. 2.

The fields associated to $q=\exp(2i\pi k/\ell)$ and $q^{\prime}=\exp(2i\pi
k^{\prime}/\ell)$ are isomorphic if and only if $kk^{\prime}=\pm p^{\prime}$
where $p^{\prime}=\frac{p+1}{4}$ if $p=-1[4]$ and $p^{\prime}=\frac{p-1}{4}$
if $p=1[4]$. We say that $q$ and $q^{\prime}$ are conjugate.

  3. 3.

The ring linearly generated by $e_{0},\ldots,e_{(\ell-3)/2}$ is equal to the
ring of integers of $V_{q}$ except possibly for one pair of conjugate
$\ell$-th roots.

We will describe all Frobenius algebras of level $5$ and $7$ in Sections 5.3
and 5.4.

###  5.2 Degree $2$ of a CohFT and the $R_{1}$-matrix

####  5.2.1 Consequences of the Givental-Teleman theorem

Suppose we have a semi-simple CohFT $\omega_{g,n}:V^{\otimes n}\to
H^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$. We denote by $\eta$ its non-
degenerate bilinear form and by $1$ its unit. We recall the formula
$\omega_{0,3}(v_{1},v_{2},v_{3})=\eta(v_{1}v_{2},v_{3})$.

Denote by $\sigma$ and $\tau$ the degree $0$ and $2$ terms of $\omega$. This
notation is suggested by our examples where they correspond respectively to
the signature and the Toledo invariant of the Hermitian modular functor.

The celebrated Givental-Teleman classification theorem says that the CohFT
$\omega_{g,n}$ can be reconstructed from the degree $0$ part $\sigma$ and a
$R$-matrix $R\in\operatorname{End}(V)[[z]]$ that we write
$R(z)=\operatorname{Id}+zR_{1}+o(z)$. In this article, we will use it only to
express $\tau$ in terms of $\sigma$ and $R_{1}$ so that we recall only the
parts of the theorem necessary for our purposes. We refer to [35] for the full
statement.

The $R$-matrix satisfies the so-called symplectic condition
$R(z)R^{*}(-z)=\operatorname{Id}$ where $A^{*}$ is the adjoint of $A$ with
respect to the bilinear form $\eta$. This condition implies in degree 1 that
$R_{1}$ satisfies $R_{1}^{*}=R_{1}$ or matricially,
$\eta^{-1}R_{1}^{T}=R_{1}\eta^{-1}$. We also set
$T(z)=z(1-R(z)1)=-z^{2}R_{1}(1)+o(z^{3})\in V[[z]]$. Givental-Teleman’s
theorem state that $R$ and $T$ act on CohFTs in such a way that one has

$\omega=RT\sigma.$

Compute first $T\sigma$ at first order, denoting by
$p_{1}=\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ the
forgetful map and setting $\kappa_{1}=(p_{1})_{*}\psi_{n+1}^{2}$, we get from
Definition 6 of [35]:

$(T\sigma)_{g,n}(v_{1},\ldots,v_{n})|_{\deg=2}=-\omega_{g,n+1}(v_{1},\ldots,v_{n},R_{1}(1))\kappa_{1}.$

Then from the definition of $R\sigma$ (Equation (2) in [35]) we get, writing
the symmetric form $R_{1}\eta^{-1}=\sum r_{\mu\nu}\mu\otimes\nu\in V\otimes
V$:

$\displaystyle(R\sigma)_{g,n}(v_{1},\ldots,v_{n})|_{\deg=2}=\sum_{i=1}^{n}\sigma_{g,n}(v_{1},\ldots,R_{1}(v_{i}),\ldots,v_{n})\psi_{i}$
$\displaystyle-\sum_{\mu,\nu}r_{\mu,\nu}\sigma_{g-1,n+2}(v_{1},\ldots,v_{n},\mu,\nu)\delta_{\mathrm{irr}}$
$\displaystyle-\sum_{*}\sum_{\mu,\nu}r_{\mu,\nu}\sigma_{g_{1},n_{1}+1}(v_{i_{1}},\ldots,v_{i_{n_{1}}},\mu)\sigma_{g_{2},n_{2}+1}(v_{j_{1}},\ldots,v_{j_{n_{2}}},\nu)\delta_{g_{1},I}$

In this last formula, the sum $\Sigma_{*}$ is over decompositions
$g=g_{1}+g_{2}$ and partitions $I\amalg J=\\{1,\ldots,n\\}$ where
$I=\\{i_{1},\ldots,i_{n_{1}}\\}$ and $J=\\{j_{1},\ldots,j_{n_{2}}\\}$.

Using the Frobenius algebra structure, we recast this formula in the case when
$(g,n)=(0,4)$ or $(1,1)$ in the following proposition.

######  Proposition 7.

Let $\omega=\sigma+\tau+(\deg>2)$ be a semi-simple CohFT and $R_{1}$ be its
$R$-matrix at first order. We have

$\displaystyle\tau_{0,4}$
$\displaystyle(v_{1},\ldots,v_{4})=\sum_{i=1}^{4}\eta(R_{1}v_{i},\prod_{j\neq
i}v_{j})\psi_{i}-\eta(v_{1}v_{2}v_{3}v_{4},R_{1}(1))\kappa_{1}$
$\displaystyle-\eta(R_{1}(v_{1}v_{2}),v_{3}v_{4})\delta_{12}-\eta(R_{1}(v_{1}v_{3}),v_{2}v_{4})\delta_{13}-\eta(R_{1}(v_{1}v_{4}),v_{2}v_{3})\delta_{14}.$
$\displaystyle\tau_{1,1}$
$\displaystyle(v)=\eta(\Omega,R_{1}(v))\psi_{1}-\eta(\Omega,vR_{1}(1))\kappa_{1}-\operatorname{Tr}(R_{1}M_{v})\delta_{\mathrm{irr}}.$

In this formula, $\Omega\in V$ is the value of the punctured torus: it equals
$\Omega=\sum_{i=1}^{n}v_{i}^{2}$ for any orthonormal basis
$v_{1},\ldots,v_{n}$ of $V\otimes\mathbb{C}$.

We simplify further these formulas by observing that
$H^{2}(\overline{\mathcal{M}}_{0,4},\mathbb{Q})$ and
$H^{2}(\overline{\mathcal{M}}_{1,1},\mathbb{Q})$ are 1-dimensional. Hence, we
can replace the classes with their integrals, using

$\int_{\overline{\mathcal{M}}_{0,4}}\psi_{i}=\int_{\overline{\mathcal{M}}_{0,4}}\kappa_{1}=\int_{\overline{\mathcal{M}}_{0,4}}\delta_{ij}=1$

and

$\int_{\overline{\mathcal{M}}_{1,1}}\psi_{1}=\int_{\overline{\mathcal{M}}_{1,1}}\kappa_{1}=\frac{1}{24}\text{
and }\int_{\overline{\mathcal{M}}_{1,1}}\delta_{\mathrm{irr}}=\frac{1}{2}.$

####  5.2.2 A decomposition of the $R_{1}$-matrix

Let $\mathcal{S}(V)=\\{A\in\operatorname{End}(V),A^{*}=A\\}$ be the space of
rational endomorphisms of $V$, symmetric with respect to $\eta$. From the
axioms of Frobenius algebras, the map $v\mapsto M_{v}$ embeds $V$ into
$\mathcal{S}(V)$.

We endow $\mathcal{S}(V)$ with the bilinear form $\langle
A,B\rangle=\operatorname{Tr}(AB)$: by semi-simplicity, its restriction to $V$
is non-degenerate, hence we have a decomposition $\mathcal{S}(V)=V\oplus
V^{\perp}$ which allows to decompose any $R_{1}$-matrix in the form

$R_{1}=M_{r_{1}}+R_{1}^{\prime},\quad r_{1}\in V,\quad R_{1}^{\prime}\in
V^{\perp}.$

Plugging this decomposition into the formula of Proposition 7, we observe that
the contribution of $r_{1}$ in $\tau_{0,4}$ cancels: knowing $\tau_{0,4}$ is
equivalent to knowing $R_{1}^{\prime}$. A standard way to do so is to
decompose the matrix into the idempotent basis but it seems to be more
efficient to use a fixed element $w$, that we will call the _pivot_ , and try
to extract $R_{1}^{\prime}$ from the endomorphism $A_{w}$ defined for all
$u,v\in V$ by

$\tau_{0,4}(w,w,u,v)=\eta(A_{w}(u),v).$

A computation using Proposition 7 gives

$\displaystyle A_{w}(v)$ $\displaystyle=$ $\displaystyle
2R^{\prime}_{1}(w)wv+R^{\prime}_{1}(w^{2})v+w^{2}R^{\prime}_{1}(v)$ (13)
$\displaystyle-R^{\prime}_{1}(1)w^{2}v-R^{\prime}_{1}(w^{2})v-2wR^{\prime}_{1}(wv)$
$\displaystyle=$
$\displaystyle[[R^{\prime}_{1},M_{w}],M_{w}](v)-[[R^{\prime}_{1},M_{w}],M_{w}](1)v$

This shows that if $M_{w}$ is semi-simple, we can indeed extract
$R_{1}^{\prime}$ from $A_{w}$.

If we decompose $R_{1}$ in the formula expressing $\tau_{1,1}(v)$ we get from
the equality
$\operatorname{Tr}(R_{1}M_{v})=\operatorname{Tr}(M_{r_{1}}M_{v})=\operatorname{Tr}_{V}(r_{1}v)$
the expression:

$\displaystyle\tau_{1,1}(v)$ $\displaystyle=$
$\displaystyle\frac{1}{24}\operatorname{Tr}_{V}(R_{1}(v)-R_{1}(1)v)-\frac{1}{2}\operatorname{Tr}_{V}(r_{1}v)$
$\displaystyle=$
$\displaystyle\frac{1}{24}\operatorname{Tr}_{V}(R_{1}^{\prime}(v)-R_{1}^{\prime}(1)v)-\frac{1}{2}\operatorname{Tr}_{V}(r_{1}v).$

This last equation shows how to compute $r_{1}$ from $R_{1}^{\prime}$ and
$\tau_{1,1}$.

####  5.2.3 The computation of $R_{1}^{\prime}$ for $\mathrm{SO}_{3}$-modular
functors

Let $q$ be a primitive root of unity of order $\ell=2r+1$ and
$\mathcal{V}_{q}$ be the associated modular functor. We recall that its
Frobenius algebra has basis $1=e_{0},e_{1},\ldots,e_{r-1}=w$. From the
formulas of Section 5.1.3, the pivot $w$ acts by

$we_{i}=\operatorname{sign}\Big{(}\frac{[2i+2]}{[2]}\Big{)}e_{r-1-i}-\operatorname{sign}\Big{(}\frac{[2i+1]}{[2]}\Big{)}e_{r-i}$

from which it follows that $e_{1}=w^{2}+1$. As $e_{1}$ has a simple spectrum,
the same is true for $w$ and the strategy of the preceding section works for
$w$.

######  Remark 6.

Specialists in TQFT may notice that $w$ corresponds to the color $1$ in the
basis of “small colors”, see [9]. It is then quite expected that it plays a
prominent role.

We can compute the dimension of the vector space
$\mathcal{V}_{q}(S^{2},2i,2j,2r-2,2r-2)$ by applying the axiom MF2 along a
curve $\gamma$ which separates the colors $2i,2j$ from the colors $2r-2,2r-2$.