# Abelian TQFTS and Schrödinger local systems

Anna Beliakova  Institut für Mathematik, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich<EMAIL_ADDRESS>and  Christian
Blanchet  Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG,
F-75006 Paris, France<EMAIL_ADDRESS>
###### Abstract.

In this paper we construct an action of 3-cobordisms on the finite dimensional
Schrödinger representations of the Heisenberg group by Lagrangian
correspondences. In addition, we review the construction of the abelian
Topological Quantum Field Theory (TQFT) associated with a $q$-deformation of
$U(1)$ for any root of unity $q$. We prove that for 3-cobordisms compatible
with Lagrangian correspondences, there is a normalization of the associated
Schrödinger action that reproduces the abelian TQFT. Restricting to mapping
cylinders, our construction yields two projective representations of the
mapping class group. We show that their linearizations do not coincide by
analysing the corresponding $2$-cocycles.

To the memory of Vaughan Jones,

the founder of quantum topology.

##  1\. Introduction

The discovery of the Jones polynomial revolutionized low–dimensional topology.
The new link invariants constructed by Jones, Kauffman, HOMFLY-PT,
Reshetikhin–Turaev etc. were extended to mapping class group representations,
later shown to be asymptotically faithful, and to 3-manifold invariants. These
developments have reached their peak in constructions of Topological Quantum
Field Theories (TQFTs) [30, 10]. The scope of ideas initiated by Vaughan Jones
built the foundations for the new domain of mathematics – the quantum
topology. One of the main open problems in quantum topology is to understand
the topological nature of quantum invariants.

In the 90s, Lawrence [25] initiated a program aiming at homological
interpretation of quantum invariants. In 2001 Bigelow was able to read the
Jones polynomial from the intersection pairing on the twisted homology of the
configuration space $\mathrm{Conf}_{n}(\mathbb{D}^{2}_{m})$ of $n$ points in
$m$-punctured disc $\mathbb{D}^{2}_{m}$. This construction led to a family of
representations (indexed by $n$) of the braid group $B_{m}$, that recovers for
$n=1$ the Burau representation. A spectacular achievement was the proof by
Bigelow [8] and Krammer [24] that this braid group representation for $n=2$ is
faithful, showing the linearity of the braid group. Bigelow’s construction was
extended later to other quantum link invariants [9, 1, 2].

Recently homological mapping class group representations were constructed by
the second author together with Palmer and Shaukat [12]. The idea here was to
use a Heisenberg cover of the space ${\mathrm{Conf}}_{n}(\Sigma)$ of unordered
$n$ configurations in a 1-punctured surface $\Sigma$, whose group of deck
transformations is the Heisenberg group $\mathcal{H}(\Sigma)$. Recall that
$\mathcal{H}(\Sigma)=\mathbb{Z}\times H_{1}(\Sigma,\mathbb{Z})$ has the group
law

$(k,x)(l,y)=(k+l+\,x.y,x+y)$

where $x.y$ is the intersection pairing. Since the surface braid group
$B_{n}(\Sigma):=\pi_{1}(\mathrm{Conf}_{n}(\Sigma))$ surjects onto
$\mathcal{H}(\Sigma)$, the Heisenberg cover
$\widetilde{\mathrm{Conf}}_{n}(\Sigma)$ is determined by the kernel of this
map.

Since the group of deck transformations $\mathcal{H}(\Sigma)$ acts on the
chain groups of the Heisenberg cover, any module $M$ over
$\mathbb{C}[\mathcal{H}(\Sigma)]$ can be used as a local system to define a
twisted homology $H_{\bullet}(\mathrm{Conf}_{n}(\Sigma),M)$ with coefficients
in $M$ as homology of the complex

$C_{\bullet}(\widetilde{\mathrm{Conf}}_{n}(\Sigma))\otimes_{\mathbb{C}[B_{n}(\Sigma)]}M$

where $\mathbb{C}[B_{n}(\Sigma)]$ is the group algebra of the surface braid
group. An interesting choice of $M$ provides a finite dimensional Schrödinger
representation $W_{q}(L)$ of a finite quotient of $\mathcal{H}(\Sigma)$, which
depends on a choice of a Lagrangian $L\subset H_{1}(\Sigma,\mathbb{Z})$ and a
root of unity $q$. If the order of $q$ is odd, the resulting mapping class
group representations were recently shown to contain the quantum
representations arising from the non-semisimple TQFT for the small quantum
$\mathfrak{sl}_{2}$ by De Renzi and Martel [15]. In particular, they defined
the action of the quantum $\mathfrak{sl}_{2}$ on the Schrödinger homology
explicitly and showed that it commutes with the action of the mapping class
group.

To complete Lawrence–Bigelow program we are lacking homological interpretation
of quantum 3-manifold invariants and of the action of 3-cobordism on the
Schrödinger homologies. This paper is a first step in this direction. Here we
construct the symplectic action of 3-cobordisms on the Schrödinger local
systems by Lagrangian correspondences. In addition, we show that on a certain
subcategory of extended 3-cobordisms and after a suitable normalization this
action recovers the abelian TQFT.

Abelian TQFTs are functorial extensions of $3$-manifold invariants constructed
by Murakami–Ohtsuki–Okada from linking matrices [29]. Their connections with
theta functions and Schrödinger representations, in the case when the quantum
parameter (called $t$ in these papers) is a root of unity of order divisible
by $4$, were extensively studied by Gelca and collaborators [20, 19, 18, 17].
Here we work with an arbitrary root of unity. We show that interesting cases
are if the order is either odd or divisible by $4$. In the latter case we
complete the work of Gelca and al. by constructing TQFTs via modularization
functor. In addition, we discuss refined TQFTs corresponding to the choice of
the spin structure or a first cohomology class on 3-manifolds.

Our preferred cobordism category is the Crane–Yetter category $3\mathrm{Cob}$
of connected oriented 3-cobordism between connected 1-punctured surfaces with
the boundary connected sum as a monoidal structure. This category has a
beautiful algebraic presentation: it is monoidally generated by the Habiro
Hopf algebra object — the punctured torus [7, 13]. By the result of [6], for
any finite unimodular ribbon category $\mathcal{C}$, there exits a monoidal
TQFT functor $F:3\mathrm{Cob}^{\sigma}\to\mathcal{C}$ defined by sending the
punctured torus to the end of $\mathcal{C}$. Here $3\mathrm{Cob}^{\sigma}$ is
the category of extended 3-cobordisms, whose objects are connected 1-punctured
surfaces equipped with a choice of Lagrangian, and morphisms are 3-cobordisms
equipped with natural numbers called weights. The composition includes a
correction term given by a Maslov index. Note that if $\mathcal{C}$ is the
category of modules over a unimodular ribbon Hopf algebra $H$, then
$\mathrm{end}(\mathcal{C})=(H,\rhd)$ where $\rhd$ denotes the adjoint action.

Let us define a subcategory $3\mathrm{Cob}^{\rm{LC}}$ of
$3\mathrm{Cob}^{\sigma}$ having the same objects, but a smaller set of
morphisms. A cobordism $C=(C,0)$ belongs to
$3\mathrm{Cob}^{\rm{LC}}\\!\\!\left((\Sigma_{-},L_{-}),(\Sigma_{+},L_{+})\right)$
if and only if

$L_{C}.L_{-}=L_{+}\quad\text{where}\quad
L_{C}=\mathrm{Ker}\left(i_{*}:H_{1}(\partial
C,\mathbb{Z})=H_{1}(-\Sigma_{-},\mathbb{Z})\oplus
H_{1}(\Sigma_{+},\mathbb{Z})\to H_{1}(C,\mathbb{Z})\right)$

is the Lagrangian correspondence determined by $C$ which acts on $L_{-}\subset
H_{1}(\Sigma_{-},\mathbb{Z})$ by

$L_{C}.L_{-}=\\{y\in H_{1}(\Sigma_{+})\,|\,\exists\,x\in L_{-},\ (x,y)\in
L_{C}\\}.$

In this subcategory all anomalies vanish. We get a linear representation of
the subgroup of the mapping class group fixing a Lagrangian. The full mapping
class group is replaced by a groupoid whose objects are Lagrangian and
morphisms are compatible mapping classes. This action groupoid is a
subcategory in $3\mathrm{Cob}^{\rm{LC}}$.

Assume $q\in\mathbb{C}$ is a primitive $p$-th root of unity of order $p\geq
3$. Let $p^{\prime}=p$ if $p$ is odd, and $p^{\prime}=p/2$ otherwise. If
$p\not\equiv 2\pmod{4}$, we can define a finite quotient of the Heisenberg
group $\mathcal{H}(\Sigma)$ as

$\mathcal{H}_{p}(\Sigma)=\mathbb{Z}_{p}\times
H_{1}(\Sigma,\mathbb{Z}_{p^{\prime}})$

where $\mathbb{Z}_{p}=\mathbb{Z}/p\mathbb{Z}$ through this paper. Given a
Lagrangian submodule $L\subset H_{1}(\Sigma,\mathbb{Z})$, let
$L_{p}=L\otimes\mathbb{Z}_{p^{\prime}}\subset
H_{1}(\Sigma,\mathbb{Z}_{p^{\prime}})$, and
$\widetilde{L}_{p}=\mathbb{Z}_{p}\times L_{p}\subset\mathcal{H}_{p}(\Sigma)$
be a maximal abelian subgroup. Denote by $\mathbb{C}_{q}$ a $1$-dimensional
representation of $\widetilde{L}_{p}$, where $(k,x)$ acts by $q^{k}$. Then
inducing from $\mathbb{C}_{q}$ we obtain

$W_{q}(L)=\mathbb{C}[\mathcal{H}_{p}(\Sigma)]\otimes_{\mathbb{C}[\widetilde{L}_{p}]}\mathbb{C}_{q}$

a ${p^{\prime}}^{g}$-dimensional Schrödinger representation of
$\mathcal{H}_{p}(\Sigma)$. Note that as
$\mathbb{C}[\mathcal{H}_{p}(\Sigma)]$-module $W_{q}(L)$ is generated by
$\mathbf{1}\in\mathbb{C}_{q}$. Given a cobordism in the category
$3\mathrm{Cob}^{\rm{LC}}$,
$C:(\Sigma_{-},L_{-})\rightarrow(\Sigma_{+},L_{+})$, $L_{+}=L_{C}.L_{-}$, we
have a Schrödinger representation $W(L_{C})$ of the Heisenberg group
$\mathcal{H}(\partial C)$ which can be considered as a
$(\mathbb{C}[\mathcal{H}(\Sigma_{+})],\mathbb{C}[\mathcal{H}(\Sigma_{-})])$-bimodule,
after identifying the subgroup
$\mathcal{H}(-\Sigma_{-})\subset\mathcal{H}(\partial C)$ with
$\mathcal{H}(\Sigma_{-})^{op}$, and defining a right action of
$\mathcal{H}(\Sigma_{-})$ on $W_{q}(L_{C})$ as the left action of the same
element of $\mathcal{H}(-\Sigma_{-})$.

The main results of this paper can be formulated as follows.

######  Theorem 1.

Assume $p\not\equiv 2\pmod{4}$. For any cobordism $C$ from
$(\Sigma_{-},L_{-})$ to $(\Sigma_{+},L_{+})$ in $3\mathrm{Cob}^{\rm{LC}}$
there exists an isomorphism of $\mathbb{Z}[\mathcal{H}(\Sigma_{+})]$-modules

$\psi_{C}:W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})\xrightarrow{\sim}W_{q}(L_{+})$

sending $\mathbf{1}\otimes\mathbf{1}$ to $\mathbf{1}$.

By composing the map $W_{q}(L_{-})\rightarrow
W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})$
with the isomorphism $\psi_{C}$ we associate to a cobordism $C$ a map between
the Shrödinger representations of the input and output surfaces. We are able
to normalise this map so that it is functorial, producing a functor isomorphic
to the abelian TQFT. The normalising coefficient, denoted by $Z(\check{C})$,
is actually the Murakami–Ohtsuki–Okada invariant of a closed 3-manifold
$\check{C}$ obtained from $C$ by gluing of two standard handlebodies
$(H_{\pm},L_{\pm})$ with $\partial H_{\pm}=\Sigma_{\pm}$ and $L_{\pm}$
generated by meridians, along diffeomorphisms identifying the Lagrangians.

######  Theorem 2.

The map between Schrödinger local systems induced by
$C:(\Sigma_{-},L_{-})\to(\Sigma_{+},L_{+})$

$\displaystyle F_{C}:W_{q}(L_{-})$ $\displaystyle\to W_{q}(L_{+})$
$\displaystyle w$ $\displaystyle\mapsto
Z(\check{C})\,\psi_{C}(\mathbf{1}\otimes w)\in W_{q}(L_{+})$

extends to a functor $F:3\mathrm{Cob}^{\rm{LC}}\to\mathrm{Vect}_{\mathbb{C}}$
which is equivalent to the abelian TQFT at $q$ on $3\mathrm{Cob}^{\rm{LC}}$.

Observe that the normalization coefficient $Z(\check{C})=0$ if and only if
there exists $\alpha\in H^{1}(\check{C},\mathbb{Z}_{p^{\prime}})$ with non
vanishing triple product $\alpha\cup\alpha\cup\alpha$ [29, Thm. 3.2], however
the Schrödinger action is always non trivial.

We plan to use these results to construct an action of cobordisms on
Schrödinger homology and provide a homological interpretation of the
Kerler–Lyubashenko TQFTs. Our long term goal will be to use infinite
dimensional Schrödinger representations to construct TQFTs with generic
quantum parameter $q$, rather than at a root of unity. An existence of such
TQFTs was predicted by physicists. They are expected to play a crucial role in
the categorification of quantum 3-manifold invariants [22]. Lagrangian Floer
homology may serve as an inspiration for this purpose.

The paper is organized as follows. In Section 2 we review representation
theoretical and skein constructions of abelian TQFTs, we discuss
modularization functors, refinements as well as the action of the mapping
class group and its extensions. In Section 3 we define Schrödinger local
systems and compare two different mapping class group actions on them. In
Section 4 we prove the two main theorems.

##  2\. Abelian TQFTs

###  2.1. Algebraic approach

Let $q\in\mathbb{S}^{1}\subset\mathbb{C}$ be a primitive $p$-th root of $1$
and $p\geq 3$ is an integer. Let $p^{\prime}=p$ if $p$ is odd and
$p^{\prime}=p/2$ if $p$ even. Consider the group algebra
$H=\mathbb{C}[K]/(K^{p}-1)$ of the cyclic group. This algebra can be
identified with the Cartan part of the quantum $\mathfrak{sl}_{2}$ at $q$ by
extending the group monomorphism

$\displaystyle U(1)$ $\displaystyle\to SL(2,\mathbb{C})$ $\displaystyle z$
$\displaystyle\to\left(\begin{matrix}z&0\\\ 0&\bar{z}\end{matrix}\right)$

For this reason, abelian TQFTs are also called $U(1)$ TQFTs.

The algebra $H$ has a natural Hopf algebra structure with a grouplike
generator, i.e. $\Delta(K)=K\otimes K$, $S(K)=K^{-1}$. Moreover, $H$ is a
ribbon Hopf algebra with $R$-matrix and its inverse given by

$R=\frac{1}{p}\sum_{0\leq i,j\leq p-1}q^{-ij}K^{i}\otimes K^{j},\quad\quad
R^{-1}=\frac{1}{p}\sum_{0\leq i,j\leq p-1}q^{ij}K^{-i}\otimes K^{-j},$

the ribbon elements

$v=\frac{1}{p}\sum_{0\leq i,j\leq p-1}q^{i(j-i)}K^{j},\quad\quad
v^{-1}=\frac{1}{p}\sum_{0\leq i,j\leq p-1}q^{i(i-j)}K^{-j}$

and the trivial pivotal structure.

Similarly to the $U_{q}(\mathfrak{sl}_{2})$ case, the representation category
$H\mathrm{-mod}$ has $p$ simple modules $V_{k}$ for $0\leq k\leq p-1$.
However, here $V_{k}$ is the $1$-dimensional representation determined by its
character $K\mapsto q^{k}$. Also in our case, the fusion rules are very
simple: $V_{i}\otimes V_{j}=V_{i+j}$ where the index $i+j$ is taken module
$p$. Hence, all objects $V_{j}$ are invertible, meaning that for each $j$
there exists $k=p-j$ such that $V_{j}\otimes V_{k}=V_{0}$, where $V_{0}$ is
the tensor unit of $H\mathrm{-mod}$. The $R$-matrix is acting by $q^{kl}$ on
$V_{k}\otimes V_{l}$.

###  2.2. Skein approach

For explicit computations, it is more convenient to work with a skein
theoretic construction.

Consider the skein relations depicted in Figure 1.

Figure 1. $U(1)$ skein relations.

Given a 3-manifold $M$, a skein module $S(M)$ is a $\mathbb{C}$-vector space
generated by links in $M$ modulo the skein relations. For a surface $F$ it is
custom to denote by $S(F)$ the skein $S(F\times[0,1])$. We will usually
identify a coloring of a component $K$ of a framed link with an element of the
skein $S(A)$, where the annulus $A$ is embedded along $K$ by using the
framing. For example, $V_{j}$-coloring is represented by an element $y^{j}$
where $y$ is the core of $A$. Here we use the usual algebra structure on
$S(A)$ to identify $y^{j}$ with $j$ parallel copies of $y$. The Kirby color is

$\Omega=\sum_{j=0}^{p-1}y^{j}\in S(A).$

The $\Omega$-colored $(+1)$-framed unknot gets value

$G=\sum^{p-1}_{k=0}q^{k^{2}}=\left\\{\begin{array}[]{ll}\varepsilon\sqrt{p}\,(1+\sqrt{-1})&\text{
if $p\equiv 0$ (mod $4$)}\\\ \pm\sqrt{p}&\text{ if $p\equiv 1$ (mod $4$)},\\\
0&\text{ if $p\equiv 2$ (mod $4$)}\\\ \pm\sqrt{-p}&\text{ if $p\equiv 3$ (mod
$4$)}\end{array}\right.$ (1)

by the well-known Gauss formula, where $\varepsilon$ is a $4$-th root of 1.
For $p$ odd, we write $|G|=\eta^{-1}$, $\kappa=\eta G$. For $p=0\pmod{4}$, we
explain in the next section, why the sums for $G$ and $\Omega$ should be taken
till $p^{\prime}-1$, and we denote them by

$g=\sum^{p^{\prime}-1}_{k=0}q^{k^{2}}\quad\text{and}\quad\omega=\sum_{j=0}^{p^{\prime}-1}y^{j}.$
(2)

Using that in our case $q^{k^{2}}=q^{(k+p^{\prime})^{2}}$, we obtain
$|g|=\eta^{-1}=\frac{|G|}{2}=\sqrt{p^{\prime}}$. Hence, for all $p$ except
$p^{\prime}=2\pmod{4}$, we can define the invariant of a closed 3-manifold $M$
obtained by surgery on $S^{3}$ along a framed $n$ component link $L$ as
follows

$Z(M)=\kappa^{-\mathrm{sign}(L)}\eta\langle\eta\omega,\dots,\eta\omega\rangle_{L}$

where $\mathrm{sign}(L)$ is the signature of the linking matrix and $\langle
x_{1},\dots,x_{n}\rangle_{L}$ denotes the evaluation of of $L$ whose $i$th
component is colored by $x_{i}$ in the skein $S(\mathbb{R}^{3})$.

The normalization is chosen in such a way that

$Z(S^{2}\times S^{1})=\eta^{2}\sum^{p^{\prime}-1}_{i=0}\langle
y^{i}\rangle=1\quad\text{and}\quad Z(S^{3})=\eta=1/\sqrt{p^{\prime}}.$

The right Dehn twist along a curve $\gamma$ is represented by coloring the
curve $\gamma$ with

$\eta\omega_{-}=\eta\,\sum_{j=0}^{p^{\prime}-1}q^{-j^{2}}y^{j}.$

If $p\equiv 0\pmod{4}$, then we split $\omega=\omega_{0}+\omega_{1}$ into odd
and even colors. Then we define an additional topological structure on $M$
that determines a $\mathbb{Z}/2\mathbb{Z}$-grading on the components of $L$,
and thus a $\mathbb{Z}/2\mathbb{Z}$-grading on their colorings. In particular,
for $p^{\prime}\equiv 4\pmod{8}$, we construct an invariant of the pair
$(M,s)$

$Z(M,s)=\kappa^{-\mathrm{sign}(L)}\eta\langle\eta\omega_{s_{1}},\dots,\eta\omega_{s_{n}}\rangle_{L}$

where $(s_{1},\dots,s_{n})\in(\mathbb{Z}/2\mathbb{Z})^{n}$ satisfying

$\sum^{n}_{j=1}L_{ij}s_{j}=L_{ii}\pmod{2}\quad$

determines a characteristic sublink of $L$ corresponding to the spin structure
$s$ on $M$. Analogously, for $p^{\prime}\equiv 0\pmod{8}$ we construct
invariants of a pair $(M,h)$

$Z(M,h)=\kappa^{-\mathrm{sign}(L)}\eta\langle\eta\omega_{h_{1}},\dots,\eta\omega_{h_{n}}\rangle_{L}$

where $(h_{1},\dots,h_{n})\in(\mathbb{Z}/2\mathbb{Z})^{n}$ satisfying

$\sum^{n}_{j=1}L_{ij}h_{j}=0\pmod{2}\quad$

determines the first cohomology class $h\in H^{1}(M,\mathbb{Z}/2\mathbb{Z})$.
In both cases, $Z(M)$ is the sum of the refined invariants over all choices of
the additional structure.

###  2.3. Modularization and refinements

A $\mathbb{C}$-linear ribbon category with a finite number of dominating
simple objects is called premodular. If in addition, the monodromy $S$-matrix
is invertible, then the category is modular. In our case the $S$-matrix, whose
$(i,j)$ component is the invariant $q^{2ij}$ of the $(i,j)$-colored Hopf link,
is invertible only for odd $p$, and in this case $H\mathrm{-mod}$ is modular,
providing an abelian TQFT by standard constructions [30] or [10].

We call a premodular category $\mathcal{C}$ modularizable, if there exists a
braided monoidal essentially surjective functor from $\mathcal{C}$ to a
modular category, sending the subcategory of transparent objects to the tensor
unit. In [14, Prop. 4.2] Bruguières gave a simple criterion for a premodular
category to be modularizable, see also [28]. In particular, such category
cannot contain transparent objects with twist coefficient $-1$. Recall that an
object is called transparent, if it has trivial braiding with any other
object. Observe that the row in the $S$-matrix corresponding to the
transparent object is colinear with the one for the tensor unit.

If $p$ is even, $H\mathrm{-mod}$ is a premodular category. The object
$V_{p^{\prime}}$ is transparent and has twist coefficient $q^{p^{\prime 2}}$,
which is $1$ if $p^{\prime}$ is even and $-1$ is $p^{\prime}$ is odd. Using
results of [14], we deduce that in the case when $p\equiv 0\pmod{4}$,
$H\mathrm{-mod}$ is modularizable. The resulting modular category has
$p^{\prime}$ simple objects, that are all invertible. The new Kirby color is
given in (2). Hence, we have $\eta=|g|^{-1}=(\sqrt{p^{\prime}})^{-1}$ in all
cases when invariant is defined.

Furthermore, if $p^{\prime}\equiv 4\pmod{8}$, the object $V_{p^{\prime}/2}$
has twist coefficient $-1$. From [4] we deduce that our category in this case
is actually spin modular, hence providing an abelian spin TQFT for
3-cobordisms equipped with a spin structure. Analogously, if $p^{\prime}\equiv
0\pmod{8}$, we can construct a refined TQFT that gives rise to invariants of
3-cobordisms equipped with first cohomology classes over
$\mathbb{Z}/2\mathbb{Z}$. We refer to [4] for details about the construction
of the refined invariants and their properties.

In the case $p\equiv 2\pmod{4}$, $H\mathrm{-mod}$ is not modularizable. The
best we can do in this case to obtain 3-manifold invariants is to consider the
degree $0$ subcategory with respect to the $\mathbb{Z}/2\mathbb{Z}$-grading
given by the action of $K^{p^{\prime}}$. The corresponding invariants will
coincide with those obtained with the quantum parameter of odd order equal to
$p^{\prime}$.

To construct a map associated by an abelian TQFT with a 3-cobordism
$C:\Sigma_{-}\to\Sigma_{+}$, we first need to choose parametrizations of
surfaces $\Sigma_{\pm}$, i.e. diffeomorphisms
$\phi_{\pm}:\Sigma_{g_{\pm}}\to\Sigma_{\pm}$ where $\Sigma_{g}$ is the
standard genus $g$ surface. If $p\not\equiv 2\pmod{4}$, the TQFT vector space
associated with $\Sigma_{g}$ has dimension ${p^{\prime}}^{g}$. A basis
$\\{y^{\bf i},{\bf i}=(i_{1},\dots,i_{g}),0\leq i_{j}\leq p^{\prime}-1\\}$ is
given by $p^{\prime}$ colorings of $g$ cores of the 1-handles of a bounding
handlebody $H_{g}$. The $({\bf i},{\bf j})$-matrix element of the TQFT map is
constructed as follows: We glue the standard handlebodies $H_{g_{\pm}}$ to $C$
along the parametrizations. Inside $H_{-}$ we put the link $y^{\bf j}$ and
inside $H_{+}$ the link $y^{\bf i}$. The result is a closed $3$-manifold
$\check{M}=S^{3}(L)$ with a collection of cirles $c^{+}\cup c^{-}$ inside,
then

$Z(C)^{\mathbf{i}}_{\mathbf{j}}:=\kappa^{-\mathrm{sign}(L)}\eta^{g_{+}}\langle\eta\omega,\dots,\eta\omega,y^{\bf
i},y^{\bf j}\rangle_{L\cup c_{+}\cup c_{-}}.$

By using the universal construction [10], this map can also be computed by
gluing just one handlebody $(H_{g_{-}},y^{\bf j})$ to $C$ and by evaluating
the result in the skein of $C\cup H_{g_{-}}$. The parametrization reduces in
this approach to the choice of Lagrangian $L\in H_{1}(\Sigma,\mathbb{Z})$,
which is equal to $\ker:H_{1}(\Sigma,\mathbb{Z})\to H_{1}(H_{g},\mathbb{Z})$,
and its complement $L^{\vee}$. Since all curves representing elements of $L$
are trivial in the skein of $H_{g}$, the basis curves $y^{\mathbf{j}}$ of the
TQFT vector space are parametrized by a basis of $L^{\vee}$.

In this paper we will be particularly interested in the Crane–Yetter category
$3\mathrm{Cob}$ of connected 3-cobordisms between connected 1-punctured
surfaces. In this category the monoidal product is given by the boundary
connected sum rather than by the disjoint union, thus leading to a rich
algebraic structure [13]. By the result of [6], for any finite unimodular
ribbon category $\mathcal{C}$, there exits a TQFT functor
$F:3\mathrm{Cob}^{\sigma}\to\mathcal{C}$ defined by sending the 1-punctured
torus to the end of $\mathcal{C}$. In our case, for odd $p$

$\mathrm{end}(H\mathrm{-mod})=\oplus^{p-1}_{j=0}\,V_{j}.$

Modularization creates an isomorphism $V_{k}\cong V_{k}\otimes
V_{p^{\prime}}$, hence for $p\equiv 0\pmod{4}$ we have

$\mathrm{end}(H\mathrm{-mod})=\oplus^{p^{\prime}-1}_{j=0}\,V_{j}.$

In both cases, the vector space associated by $F$ to a genus $g$ surface with
one boundary component has dimension $p^{\prime g}$.

For even $p^{\prime}$, refined TQFTs on $3\mathrm{Cob}^{\sigma}$ can be
constructed along the lines of [5]. On the standard cobordism category this
was done in [3, 11].

###  2.4. Extended cobordisms and Lagrangian correspondence

Let us recall that the skein or Reshetikhin–Turaev TQFT constructions give
rise to projective representations of the mapping class group and the gluing
formula has a so-called framing anomaly which can be resolved by using
extended cobordisms. The later are given by a pair: a 3-cobordism between
surfaces equipped with Lagrangian subspaces in the first homology group and a
natural number. This approach leads to a representation of a certain central
extension of the mapping class group.

If $p$ is odd, then the framing anomaly $\kappa$, defined as the argument of
the Gauss sum $g$ in (1), is a $4$-th root of $1$. From [21, Remark 6.9] we
can deduce that the corresponding TQFT contains a native representation of the
mapping class group. This is because, the central generator of the extension
acts by $\kappa^{4}=1$, hence the index $4$ subgroup described in [21] is the
trivial extension. Recall that the metaplectic group $\mathrm{Mp}_{2g}$ is the
non trivial double cover of the symplectic group $\mathrm{Sp}_{2g}$. The
metaplectic mapping class group is the pull back of this double cover using
the symplectic action. In the case $p\equiv 0$ (mod $4$) the framing anomaly
$\kappa$ is a primitive $8$-th root of unity and the above argument shows that
the TQFT contains a native representation of the metaplectic mapping class
group.

To avoid anomaly issues in general, we will work with a subcategory
$3\mathrm{Cob}^{\rm{LC}}$ of the category of connected extended 3-cobordisms
between connected 1-punctured surfaces with Lagrangians. Objects of
$3\mathrm{Cob}^{\rm{LC}}$ are pairs: a connected $1$-punctured surface
$\Sigma$ and a Lagrangian subspace $L\subset H_{1}(\Sigma,\mathbb{Z})$. Recall
that a Lagrangian is a maximal submodule with vanishing intersection pairing.
A 3-cobordism $C:\Sigma_{-}\rightarrow\Sigma_{+}$ defines a Lagrangian
correspondence

$L_{C}=\mathrm{Ker}\left(i_{*}:H_{1}(\partial
C,\mathbb{Z})=H_{1}(-\Sigma_{-},\mathbb{Z})\oplus
H_{1}(\Sigma_{+},\mathbb{Z})\to H_{1}(C,\mathbb{Z})\right).$

This Lagrangian correspondence gives an action of $C$ on the Lagrangian
subspace $L_{-}\subset H_{1}(\Sigma_{-},\mathbb{Z})$ by

$C.L_{-}=\\{y\in H_{1}(\Sigma_{+})\,|\,\exists\,x\in L_{-},\ (x,y)\in
L_{C}\\}.$

The cobordism $C$ belongs to
$3\mathrm{Cob}^{\rm{LC}}\\!\\!\left((\Sigma_{-},L_{-}),(\Sigma_{+},L_{+})\right)$
if and only if $L_{C}.L_{-}=L_{+}$. If we restrict to mapping cylinders we
obtain the so called action groupoid of the mapping class group action on
Lagrangian subspaces.

Restriction of the TQFT functor to $3\mathrm{Cob}^{\rm{LC}}$ kills all Maslov
indices needed to compute framing anomalies in gluing formulas (compare [21,
Sec.2].

##  3\. Schrödinger local systems on surface configurations

###  3.1. Heisenberg group as a quotient of the surface braid group

Let $\Sigma$ be an oriented surface of genus $g$ with one boundary component.
For $n\geq 2$, the unordered configuration space of $n$ points in $\Sigma$ is

$\mathrm{Conf}_{n}(\Sigma)=\\{\\{c_{1},\dots,c_{n}\\}\subset\Sigma\mid
c_{i}\neq c_{j}\text{ for $i\neq j$}\\}.$

The surface braid group is then defined as
${B}_{n}(\Sigma)=\pi_{1}(\mathrm{Conf}_{n}(\Sigma),*)$. To construct a
presentation, we fix based loops,
$\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g}$ on $\Sigma$, as
depicted in Figure 2. The base point $*_{1}$ on $\Sigma$ belongs to the base
configuration $*$ in $\mathrm{Conf}_{n}(\Sigma)$. By abuse of notation, we use
$\alpha_{r}$, $\beta_{s}$ also for the loops in $\mathrm{Conf}_{n}(\Sigma)$
where only the first point is moving along the corresponding curve. We write
composition of loops from right to left.

Figure 2. Model for $\Sigma$.

The braid group ${B}_{n}(\Sigma)$ has generators
$\alpha_{1},\ldots,\alpha_{g}$, $\beta_{1},\ldots,\beta_{g}$ together with the
classical braid generators $\sigma_{1},\ldots,\sigma_{n-1}$, and relations:

$\begin{cases}[\sigma_{i},\sigma_{j}]=1&\text{for }\lvert i-j\rvert\geq 2,\\\
\sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{i}\sigma_{j}&\text{for
}\lvert i-j\rvert=1,\\\ [\zeta,\sigma_{i}]=1&\text{for }i>1\text{ and all
}\zeta\text{ among the }\alpha_{r},\beta_{s},\\\
[\zeta,\sigma_{1}\zeta\sigma_{1}]=1&\text{for all }\zeta\text{ among the
}\alpha_{r},\beta_{s},\\\ [\zeta,\sigma^{-1}_{1}\eta\sigma_{1}]=1&\text{for
all }\zeta\neq\eta\text{ among the }\alpha_{r},\beta_{s},\text{ with}\\\
&\\{\zeta,\eta\\}\neq\\{\alpha_{r},\beta_{r}\\},\\\
\sigma_{1}\beta_{r}\sigma_{1}\alpha_{r}\sigma_{1}=\alpha_{r}\sigma_{1}\beta_{r}&\text{for
all }r.\end{cases}$ (3)

We denote by $x.y$ the standard intersection form on
$H_{1}(\Sigma,\mathbb{Z})$. The Heisenberg group $\mathcal{H}(\Sigma)$ is the
central extension of the homology group $H_{1}(\Sigma,\mathbb{Z})$ induced by
the $2$-cocycle $(x,y)\mapsto x.y$. As a set $\mathcal{H}(\Sigma)$ is equal to
$\mathbb{Z}\times H_{1}(\Sigma,\mathbb{Z})$, with the group structure

$(k,x)(l,y)=(k+l+\,x.y,x+y).$ (4)

We use the notation $a_{r}$, $b_{s}$ for the homology classes of $\alpha_{r}$,
$\beta_{s}$, respectively. Let us denote by $[\sigma_{1},{B}_{n}(\Sigma)]$ the
normal subgroup of the surface braid group $B_{n}(\Sigma)$ generated by the
commutators $\\{[\sigma_{1},x]$, $x\in{B}_{n}(\Sigma)\\}$. From the
presentation above we obtain the following (see [12] for more details).

######  Proposition 3.

For each $g\geq 0$ and $n\geq 2$, the quotient

${B}_{n}(\Sigma)/[\sigma_{1},{B}_{n}(\Sigma)]\xrightarrow{\sim}\mathcal{H}(\Sigma)$

is isomorphic to the Heisenberg group. An isomorphism is induced by the
surjective homomorphism

$\phi\colon{B}_{n}(\Sigma)\relbar\joinrel\rightarrow\mathcal{H}(\Sigma)$

sending each $\sigma_{i}$ to $u=(1,0)$, $\alpha_{r}$ to
$\tilde{a}_{r}=(0,a_{r})$, $\beta_{s}$ to $\tilde{b}_{s}=(0,b_{s})$.

It follows that any representation of the Heisenberg group
$\mathcal{H}(\Sigma)$ is also a representation of the surface braid group
${B}_{n}(\Sigma)=\pi_{1}(\mathrm{Conf}_{n}(\Sigma),*)$ and hence provides a
local system on the configuration space $\mathrm{Conf}_{n}(\Sigma)$.

Let us denote by $\mathrm{Aut}^{+}(\mathcal{\mathcal{H}}(\Sigma))$ the group
of automorphisms of $\mathcal{H}(\Sigma))$ acting by identity on the center.
By [12, Lemma 15] we have the following split short exact sequence

$1\rightarrow
H^{1}(\Sigma,\mathbb{Z})\xrightarrow{j}\mathrm{Aut}^{+}(\mathcal{H}(\Sigma))\xrightarrow{l}\mathrm{Sp}(H_{1}(\Sigma))\to
1$

where $j(c)=[(k,x)\to(k+c(x),x)]$ and $\mathrm{Sp}(H_{1}(\Sigma))$ is the
symplectic group preserving the intersection pairing. The homomorphism $l$ has
a section

$s:g\mapsto[(k,x)\mapsto(k,g(x))]$ (5)

providing a semi-direct decomposition
$\mathrm{Aut}^{+}(\mathcal{H}(\Sigma))\cong\mathrm{Sp}(H_{1}(\Sigma))\ltimes
H^{1}(\Sigma;\mathbb{Z})$.

Let us denote by $\mathrm{Mod}(\Sigma)$ the mapping class group. Its action on
$H_{1}(\Sigma,\mathbb{Z})$ preserves the symplectic form, and hence using the
section $s$ from (5) we get a symplectic action of the mapping class group on
the Heisenberg group, where $f\in\mathrm{Mod}(\Sigma)$ acts by

$(k,x)\mapsto(k,f_{*}(x)).$ (6)

On the other hand, the quotient map $\phi:B_{n}(\Sigma)\to\mathcal{H}(\Sigma)$
induces a different action of $\mathrm{Mod}(\Sigma)$ on $\mathcal{H}(\Sigma)$.
The following proposition is proved in [12, Section 3].

######  Proposition 4.

For $f\in\mathrm{Mod}(\Sigma)$, there exists a unique homomorphism
$f_{\mathcal{H}}\colon\mathcal{H}(\Sigma)\rightarrow\mathcal{H}(\Sigma)$ such
that the following square commutes:

${B_{n}(\Sigma)}$${B_{n}(\Sigma)}$${\mathcal{H}(\Sigma)}$${\mathcal{H}(\Sigma)}$$\scriptstyle{\phi}$$\scriptstyle{f_{B_{n}(\Sigma)}}$$\scriptstyle{\phi}$$\scriptstyle{f_{\mathcal{H}}}$
(7)

We obtain an action of $\mathrm{Mod}(\Sigma)$ on the Heisenberg group
$\mathcal{H}(\Sigma)$ given by

$\displaystyle\mathrm{Mod}(\Sigma)$
$\displaystyle\longrightarrow\mathrm{Aut}^{+}(\mathcal{H}(\Sigma))$ (8)
$\displaystyle f$ $\displaystyle\mapsto
f_{\mathcal{H}}:(k,x)\mapsto(k+\theta_{f}(x),f_{*}(x))$

where the map $\theta:\mathrm{Mod}(\Sigma)\to H^{1}(\Sigma,\mathbb{Z})$
sending $f$ to $\theta_{f}\in\mathrm{Hom}(H_{1}(\Sigma),\mathbb{Z})$ is called
crossed homomorphism, satisfying $\theta(fg)=\theta(f)+f_{*}(\theta(g))$.
Clearly, both actions coincide on $\mathrm{Sp}(H_{1}(\Sigma))$, i.e.
$l(f_{\mathcal{H}})=f_{*}$.

###  3.2. Finite dimensional Schrödinger representations

Let us fix an integer $p\geq 3$, $p\not\equiv 2\text{ mod }4$, with
$p^{\prime}=p$ if $p$ is odd and $p^{\prime}=p/2$ if $p$ even. Then we can
define a finite quotient of the Heisenberg group $\mathcal{H}(\Sigma)$ as
follows

$\mathcal{H}_{p}(\Sigma)=\mathbb{Z}_{p}\times
H_{1}(\Sigma,\mathbb{Z}_{p^{\prime}}).$

Given a Lagrangian submodule $L\subset H_{1}(\Sigma,\mathbb{Z})$, let
$L_{p}=L\otimes\mathbb{Z}_{p^{\prime}}$ and
$\widetilde{L}_{p}=\mathbb{Z}_{p}\times L_{p}\subset\mathcal{H}_{p}(\Sigma)$
is a maximal abelian subgroup. Let $q$ be a primitive $p$-th root of unity.
Denote by $\mathbb{C}_{q}$ a $1$-dimensional representation of
$\mathcal{H}_{p}(\Sigma)$, where $(k,x)$ acts by $q^{k}$. Then inducing from
$\mathbb{C}_{q}$ we obtain

$W_{q}(L)=\mathbb{C}[\mathcal{H}_{p}(\Sigma)]\otimes_{\mathbb{C}[\widetilde{L}_{p}]}\mathbb{C}_{q}$

a ${p^{\prime}}^{g}$-dimensional Schrödinger representation of the finite
Heisenberg group $\mathcal{H}_{p}(\Sigma)$.

The following finite dimensional version of the famous Stone–von Neumann
theorem holds.

######  Theorem 5 (Stone–von Neumann).

For $q$ a root of unity of order $p$, $p\geq 3$, $p\not\equiv 2\text{ mod }4$,
$W_{q}(L)$ is the unique irreducible unitary representation of
$\mathcal{H}_{p}(\Sigma)$, up to unitary isomorphism, where the central
generator $u=(1,0)$ acts by $q$.

A proof for even $p$ can be found in [20, Theorem 2.4]. The odd case works
similarly. The Schrödinger representation $W_{q}(L)$ can be twisted by an
automorphim $\tau\in\mathrm{Aut}^{+}(\mathcal{H}(\Sigma))$. We denote by
${}_{\tau}\\!W_{q}(L)$ this twisted representation where
$h\in\mathcal{H}(\Sigma)$ acts by $\tau(h)$. The above Stone–von Neumann
theorem provides an isomorphism $W_{q}(L)\cong{}_{\tau}\\!W_{q}(L)$ defined up
to a complex number of absolute value $1$.

Using the Stone–von Neumann theorem, for a mapping class $f$ we obtain a
unitary isomorphism
$\mathcal{S}_{\mathcal{H}}(f):W_{q}(L)\xrightarrow{\sim}{}_{f_{\mathcal{H}}}\\!W_{q}(L)$
defined, up to a scalar in $\mathbb{S}^{1}\subset\mathbb{C}$, by the following
commutative diagram

${W_{q}(L)}$${{}_{f_{\mathcal{H}}}\\!W_{q}(L)}$${W_{q}(L)}$${{}_{f_{\mathcal{H}}}\\!W_{q}(L)}$$\scriptstyle{\mathcal{S}_{\mathcal{H}}(f)}$$\scriptstyle{\rho_{W}(k,x)}$$\scriptstyle{\rho_{W}(f_{\mathcal{H}}(k,x))}$$\scriptstyle{\mathcal{S}_{\mathcal{H}}(f)}$

where $\rho_{W}:\mathcal{H}(\Sigma)\rightarrow U(W_{q}(L)$ is the Shrödinger
representation. This provides an homomorphism

$\mathcal{S}_{\mathcal{H}}:\mathrm{Mod}(\Sigma)\rightarrow\mathrm{PU}(W_{q}(L)),\quad\text{where}\quad\mathrm{PU}(W_{q}(L))=\mathrm{U}(W_{q}(L))/\mathbb{S}^{1}$

is the projective unitary group.

Denote by ${}_{f_{*}}\\!W_{q}(L)$ the Shrödinger representation twisted with
the symplectic action, we also have an isomorphism
$\mathcal{S}(f):W_{q}(L)\xrightarrow{\sim}{}_{f_{*}}\\!W_{q}(L)$ defined, up
to a scalar in $\mathbb{S}^{1}\subset\mathbb{C}$, by the condition

$\rho_{W}(k,f_{*}(x))\circ\mathcal{S}(f)=\mathcal{S}(f)\circ\rho_{W}(k,x),\
\text{for any}\ \ (k,x)\in\mathcal{H}_{p}(\Sigma).$ (9)

This provides another homomorphism

$\mathcal{S}:\mathrm{Mod}(\Sigma)\rightarrow\mathrm{PU}(W_{q}(L))$

The following theorem was essentially proven by Gelca with collaborators [20,
Theorem 8.1], [19], [17, Chapter 7].

######  Theorem 6.

The homomorphism
$\mathcal{S}:\mathrm{Mod}(\Sigma)\rightarrow\mathrm{PU}(W_{q}(L))$ given by
the symplectic action is isomorphic to the one resulting from the abelian TQFT
described in Section 2.

In general, any projective representation of a group $G$ can be linearised on
an appropriate central extension. Given an homomorphism $R:G\rightarrow
PGL(V)$, where $V$ is a complex vector space, a choice of lift (as a set map)
$\tilde{R}:G\rightarrow GL(V)$ defines a defect map $c:G\times
G\rightarrow\mathbb{C}^{*}$, by
$\tilde{R}(gg^{\prime})=c(g,g^{\prime})\tilde{R}(g)\tilde{R}(g^{\prime})$. In
the case of a projective unitary representation the map $c$ takes values in
$\mathbb{S}^{1}$. It is well known from basic group cohomology theory that $c$
is a $2$-cocycle defining a central extension of $G$ on which $R$ can be
linearised. This central extension is classified by the class $[c]\in
H^{2}(G,\mathbb{C}^{*})$. If this class can be reduced to a subgroup, then the
linearisation already arises on a smaller extension. If $[c]=0$, the minimal
extension is $G$ itself.

Projective actions of $\mathrm{Mod}(\Sigma)$ on Schrödinger representations
are naturally equipped with such cohomology classes, determined by the
Stone–von Neumann isomorphisms. We will show that the extension which
linearises the projective representation $\mathcal{S}_{\mathcal{H}}$ is non
trivial by computing its classifying class in odd case.

From now on in this section we suppose that $p$ is odd. As explained in
Section 2.4, the homomorphism $\mathcal{S}$ can be linearised and we use the
same notation for a linearisation $\mathcal{S}:\mathrm{Mod}(\Sigma)\rightarrow
U(W_{q}(L))$. A key observation is that, for a mapping class $f$, the
automorphism
$f_{\mathcal{H}}:\mathcal{H}(\Sigma)\rightarrow\mathcal{H}(\Sigma)$ is equal
to the symplectic one composed with an inner automorphism

$f_{\mathcal{H}}(k,x)=(k+\theta_{f}(x),f_{*}(x))=(0,f_{*}(t_{f}))(k,f_{*}(x)(0,-f_{*}(t_{f})),$

where $2t_{f}\in H_{1}(\Sigma,\mathbb{Z}_{p})$ is the Poincaré dual of
$\theta_{f}$. Here we use that $2$ is invertible modulo $p$ and that
$f_{*}(t_{f}).f_{*}(x)=t_{f}.x$. Acting on $W_{q}(L)$ we get the following
commutative diagram

${W_{q}(L)}$${{}_{f_{*}}\\!W_{q}(L)}$${{}_{f_{\mathcal{H}}}\\!W_{q}(L)}$${W_{q}(L)}$${{}_{f_{*}}\\!W_{q}(L)}$${{}_{f_{\mathcal{H}}}\\!W_{q}(L)}$$\scriptstyle{\mathcal{S}(f)}$$\scriptstyle{\rho_{W}(k,x)}$$\scriptstyle{\rho_{W}(0,f_{*}(t_{f}))}$$\scriptstyle{\rho_{W}(k,f_{*}(x))}$$\scriptstyle{\rho_{W}(f_{\mathcal{H}}(k,x))}$$\scriptstyle{\mathcal{S}(f)}$$\scriptstyle{\rho_{W}(0,f_{*}(t_{f}))}$

Hence the two projective actions are related as follows

$\mathcal{S}_{\mathcal{H}}(f)=\rho_{W}(0,f_{*}(t_{f}))\circ\mathcal{S}(f)=\mathcal{S}(f)\circ\rho_{W}(0,t_{f})\
.$

We can now compute the cocycle from the intertwinning isomorphism

$W_{q}(L)\cong{}_{(fg)_{\mathcal{H}}}\\!W_{q}(L)={}_{g_{\mathcal{H}}}\\!\\!\left({}_{f_{\mathcal{H}}}\\!W_{q}(L)\right)\
.$
$\displaystyle\mathcal{S}_{\mathcal{H}}(f)\circ\mathcal{S}_{\mathcal{H}}(g)$
$\displaystyle=\mathcal{S}(f)\circ\rho_{W}(0,t_{f})\circ\mathcal{S}(g)\circ\rho_{W}(0,t_{g})$
$\displaystyle=\mathcal{S}(f)\circ\mathcal{S}(g)\circ\rho_{W}(0,g_{*}^{-1}(t_{f}))\circ\rho_{W}(0,t_{g})$
$\displaystyle=\mathcal{S}(f)\circ\mathcal{S}(g)\circ\rho_{W}(g_{*}^{-1}(t_{f}).t_{g},g_{*}^{-1}(t_{f})+t_{g})$
$\displaystyle=q^{g_{*}^{-1}(t_{f}).t_{g}}\mathcal{S}_{\mathcal{H}}(fg)$

Here we used that the crossed homomorphism property
$\theta_{fg}=\theta_{g}+g^{*}(\theta_{f})$ implies for the Poincaré dual
$t_{fg}=t_{g}+g_{*}^{-1}(t_{f})$. Using
$t_{gg^{-1}}=t_{g^{-}1}+g_{*}(t_{g})=0$, we get that the cocycle is equal to
$q^{c(f,g)}$ where
$c(f,g)=g_{*}^{-1}(t_{f}).t_{g}=t_{f}.g_{*}(t_{g})=t_{f}.t_{g^{-1}}$.

Morita studied in [27] the intersection cocycle $(f,g)\mapsto
c_{\mathrm{Mor}}(f,g)=t_{f^{-1}}.t_{g}=c(g,f)$ which represents $12c_{1}$
where $c_{1}$ is the Chern class generating
$H^{2}(\mathrm{Mod}(\Sigma),\mathbb{Z})=\mathbb{Z}$ for surfaces of genus at
least $3$. The Meyer cocycle $\tau(f,g)$ is the signature of the oriented
$4$-dimensional manifold defined as the surface bundle over the pair of pants
with monodromy $f$ and $g$ on $2$ boundary components. This definition is
symmetric in $f$ and $g$ so that we have $\tau(f,g)=\tau(f,g)$. From Morita
work we have that $[c_{\mathrm{Mor}}]=3[\tau]=12c_{1}$. By switching the
variable we get $[c]=3[\tau]=12c_{1}$. It follows that for odd $p$ the
projective action $\mathcal{S}_{\mathcal{H}}:\mathrm{Mod}(\Sigma)\rightarrow
PU(W_{q}(L)$ cannot be linearised on the mapping class group while the
symplectic action does.

##  4\. Proofs

In the previous section we have shown that finite dimensional Schrödinger
representations provide local systems on surface configuration spaces. Here we
will show that morphisms of $3\mathrm{Cob}^{\rm{LC}}$ act naturally on these
local systems by extending the symplectic action of mapping cylinders.

Let $C\in 3\mathrm{Cob}^{\rm{LC}}((\Sigma_{-},L_{-}),(\Sigma_{+},L_{+}))$ be a
cobordism compatible with Lagrangian correspondence. This means that the
kernel of the inclusion map $H_{1}(\partial C,\mathbb{Z})\rightarrow
H_{1}(C,\mathbb{Z})$ is a Lagrangian submodule $L_{C}\subset H_{1}(\partial
C,\mathbb{Z})\cong H_{1}(-\Sigma_{-},\mathbb{Z})\oplus
H_{1}(\Sigma_{+},\mathbb{Z})$ and $L_{C}.L_{-}=L_{+}$. Then we have three
Heisenberg groups $\mathcal{H}(\Sigma_{-})$, $\mathcal{H}(\Sigma_{+})$ and
$\mathcal{H}(\partial C)$, and respective Shrödinger representations
$W_{q}(L_{-})$, $W_{q}(L_{+})$ and $W_{q}(L_{C})$ for a $p$-th root of unity
$q$, with $p\geq 3$, $p\not\equiv 2\pmod{4}$.

Using that $\partial C=-\Sigma_{-}\cup_{{S}^{1}}\Sigma_{+}$ and the inclusions
$H_{1}(-\Sigma_{-},\mathbb{Z})\rightarrow H_{1}(\partial C,\mathbb{Z})$,
$H_{1}(\Sigma_{+},\mathbb{Z})\rightarrow H_{1}(\partial C,\mathbb{Z})$ we have
commuting actions of $\mathcal{H}(-\Sigma_{-})$ and $\mathcal{H}(\Sigma_{+})$
on $W_{q}(L_{C})$. Actually, $W_{q}(L_{C})$ can be viewed as a
$(\mathbb{C}[\mathcal{H}(\Sigma_{+})],\mathbb{C}[\mathcal{H}(\Sigma_{-})])$-bimodule,
after identifying the group $\mathcal{H}(-\Sigma_{-})$ with
$\mathcal{H}(\Sigma_{-})^{op}$, and defining a right action of
$\mathcal{H}(\Sigma_{-})$ on $W_{q}(L_{C})$ as the left action of the same
element of $\mathcal{H}(-\Sigma_{-})$. Then we can form the tensor product
$W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}(\Sigma_{-})]}W_{q}(L_{-})$ and
compare it with $W_{q}(L_{+})$.

Recall that the Shrödinger representation $W_{q}(L)$ is induced from
$\mathbb{C}=\mathbb{C}_{q}$ considered as a $1$-dimensional representation of
$\widetilde{L}_{p}=\mathbb{Z}_{p}\oplus(L\otimes\mathbb{Z}_{p^{\prime}})\subset\mathcal{H}_{p}(\Sigma)$,
and we denote by $\mathbf{1}$ the canonical generator of $W_{q}(L)$ as
$\mathbb{C}[\mathcal{H}(\Sigma)]$-module. Moreover, throughout this section to
simplify notation we denote $L\otimes\mathbb{Z}_{p^{\prime}}$ by ${\sf L}$.

Any Lagrangian ${L}^{\vee}\subset H_{1}(\Sigma,\mathbb{Z})$ complementary to
${L}$ provides a basis for $W_{q}(L)$ indexed by ${\sf L}^{\vee}$. Given
$b\in{\sf L}^{\vee}$ we denote by $v_{b}$ the corresponding basis vector. In
this basis the left action of the finite Heisenberg group is as follows.

  * •

The central generator $u=(1,0)$ acts by $v_{b}\mapsto qv_{b}$.

  * •

For $y\in{\sf L}^{\vee}$, $(0,y)$ acts by translation: $v_{b}\mapsto v_{b+y}$.

  * •

For $x\in{\sf L}$, $(0,x)$ acts by $v_{b}\mapsto q^{2x.b}\,v_{b}$.

In the last step we used the rule $(0,x)(0,b)=(x.b,x+b)=(0,b)(2x.b,x)$.

Our main results provide a new model for the abelian TQFT based on Schrödinger
local systems. Let us prove our main theorems.

###### Proof of Theorem 1.

Any morphism in $3\mathrm{Cob}^{\rm{LC}}$ can be decomposed into mapping
cylinders and elementary index $1$ or $2$ surgeries. This is a consequence of
the existence of a Morse datum [16, 23]. We will first prove the isomorphism
$W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})\cong
W_{q}(L_{+})$ for the elementary cobordisms and then argue that the bimodule
associated with a composition of cobordisms is the expected tensor product of
bimodules. In addition, for every elementary cobordism we compute the induced
map $\check{F}_{C}:W_{q}(L_{-})\to W_{q}(L_{+})$, that will be used to prove
Theorem 2. The reader interested in Theorem 1 only may skip this part.

For a mapping cylinder
$C_{f}:(\Sigma_{-},L_{-})\rightarrow(\Sigma_{+},L_{+})$, where the
diffeomorphism $f:\Sigma_{-}\rightarrow\Sigma_{+}$ sends $L_{-}$ to
$f_{*}(L_{-})=L_{+}$, we have

${L}_{C_{f}}=\\{(-x,f_{*}(x)),\ x\in H_{1}(\Sigma_{-},\mathbb{Z})\\}$

We choose a Lagrangian ${L}_{-}^{\vee}$ complementary to ${L}_{-}$. Then
${L}_{+}^{\vee}=f_{*}({L}_{-}^{\vee})$ is complementary to ${L}_{+}$. The
submodule

${L}_{C_{f}}^{\vee}={L}_{-}\oplus{L}_{+}^{\vee}\subset H_{1}(\partial
C_{f},\mathbb{Z})$

is Lagrangian and complementary to ${L}_{C_{f}}$. Indeed, if $(-x,f_{*}(x))$
belongs to ${L}_{C_{f}}^{\vee}$, then $x\in{L}_{-}$ and
$f_{*}(x)\in{L}^{\vee}_{+}\cap{L}_{+}=\\{0\\}$, showing that
${L}_{C_{f}}^{\vee}\cap{L}_{C_{f}}=\\{0\\}$. Recall that for all kinds of
Lagrangians $L$, the notation ${\sf L}$ means
$L\otimes\mathbb{Z}_{p^{\prime}}$. A $\mathbb{C}$-basis $b^{+}_{y}$ for
$W_{q}(L_{+})$ is labelled by elements $y\in{\sf L}_{+}^{\vee}$.

We have bases $\\{B_{z},z\in{\sf L}_{C_{f}}^{\vee}\\}$ for $W_{q}(L_{C_{f}})$,
and $\\{b^{-}_{x},x\in{\sf L}_{-}^{\vee}\\}$ for $W_{q}(L_{-})$. As a vector
space the tensor product is generated by

$\\{B_{z}\otimes b_{x},z\in{\sf L}_{C_{f}}^{\vee},x\in{\sf L}_{-}^{\vee}\\}$

with relations coming from the action by elements in
$\mathcal{H}_{p}(\Sigma_{-})$. We write $z\in{\sf L}_{C_{f}}^{\vee}$ as
$z=(z_{-},z_{+})$, $z_{-}\in{\sf L}_{-}$, $z^{+}\in{\sf
L}_{+}^{\vee}=f_{*}({\sf L}_{-}^{\vee})$.

For an element $y\in{\sf L}_{-}$ we get the relation

$q^{2y.x}B_{(z_{-},z_{+})}\otimes b^{-}_{x}=B_{(z_{-},z_{+})}(0,y)\otimes
b^{-}_{x}=(0,(y,0))B_{(z_{-},z_{+})}\otimes
b^{-}_{x}=B_{(z_{-}+y,z_{+})}\otimes b^{-}_{x}$

This reduces the set of generators to $\\{B_{(0,z_{+})}\otimes
b^{-}_{x},x\in{\sf L}^{\vee}_{-},z_{+}\in{\sf L}_{+}^{\vee}\\}$.

For an element $x\in{\sf L}_{-}^{\vee}$ we get another relation

$\displaystyle B_{(0,z_{+})}\otimes b^{-}_{x}=$ $\displaystyle
B_{(0,z_{+})}(0,x)\otimes\mathbf{1}=(0,(x,0))B_{(0,z_{+})}\otimes\mathbf{1}$
(10) $\displaystyle=$
$\displaystyle(0,(0,f_{*}(x))(0,(x,-f_{*}(x))B_{(0,z_{+})}\otimes\mathbf{1}=q^{-2f_{*}(x).z_{+}}B_{(0,z_{+}+f_{*}(x))}\otimes\mathbf{1}$

where the intersection is written on the positively oriented $\Sigma_{+}$.
This further reduces the generators to
$\\{B_{(0,z_{+})}\otimes\mathbf{1},z_{+}\in{\sf L}_{+}^{\vee}\\}$. Since any
relation coming from any element in $\mathcal{H}(\Sigma_{-})$ can be deduced
from the previously written ones, we get that
$\\{B_{(0,y)}\otimes\mathbf{1},y\in{\sf L}_{+}^{\vee}\\}$ represents a
$\mathbb{C}$-basis for the tensor product
$W_{q}({L}_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})$.
It follows that the $\mathbb{C}[\mathcal{H}(\Sigma_{+})]$-module map

$\psi_{C}:W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})\rightarrow
W_{q}(L_{+})$

which sends $\mathbf{1}\otimes\mathbf{1}$ to $\mathbf{1}$ is an isomorphism.
Moreover, the map

$\displaystyle\check{F}_{C}:W_{q}(L_{-})$ $\displaystyle\rightarrow
W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})\cong
W_{q}(L_{+})$ $\displaystyle b^{-}_{x}$
$\displaystyle\mapsto\psi_{C}(\mathbf{1}\otimes b^{-}_{x})$

sends a basis vector $b^{-}_{x}$ to $b^{+}_{f_{*}(x)}$, for any $x\in{\sf
L}_{-}^{\vee}$, by using (10) with $z_{+}=0$.

In the case of an elementary cobordism
$C:(\Sigma_{-},L_{-})\rightarrow(\Sigma_{+},L_{+})$ corresponding to an index
$1$ surgery, the genus increases by $1$. The Lagrangian correspondence is

$L_{C}=\\{(-x_{-},x_{+}),\ x_{-}\in
H_{1}(\Sigma_{-},\mathbb{Z})\\}\oplus\mathbb{Z}(0,\mu)$

where $\mu$ is a meridian of the new handle and $x_{+}$ is the class $x_{-}$
pushed in $\Sigma_{+}$. Let $\lambda$ be a longitude for the new handle. We
choose a Lagrangian $L_{-}^{\vee}$ complementary to $L_{-}$. By pushing
through the cobordism, we may also consider $L_{-}^{\vee}$ as a subspace in
$H_{1}(\Sigma_{+},\mathbb{Z})$. The span of $L_{-}^{\vee}$ and $\lambda$ gives
a Lagrangian $L^{\vee}_{+}$ complementary to $L_{+}$. Then
$L_{C}^{\vee}=L_{-}^{\vee}\oplus L_{+}^{\vee}$ is complementary to $L_{C}$ and
the previous argument constructs the isomorphism. Here the map

$\check{F}_{C}:W_{q}(L_{-})\rightarrow
W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})\cong
W_{q}(L_{+})$

sends a basis vector $b^{-}_{x}$ to $b^{+}_{x}$, where $x\in{\sf
L}_{-}^{\vee}=L_{-}^{\vee}\otimes\mathbb{Z}_{p^{\prime}}$.

Let us consider an elementary cobordism
$C:(\Sigma_{-},L_{-})\rightarrow(\Sigma_{+},L_{+})$ corresponding to an index
$2$ surgery on a curve $\gamma$. Let $\delta$ be a curve in $\Sigma_{-}$ such
that $\gamma.\delta=1$. The curves $\gamma$ and $\delta$ determine a genus one
subsurface $\Sigma_{1}$. Outside $\Sigma_{1}$ the cobordism is trivial. Denote
by $\Sigma\subset\Sigma_{-}$ the complement of $\Sigma_{1}$ which we consider
also as a subsurface of $\Sigma_{+}$. We arrange the splitting so that
$\Sigma_{-}=\Sigma\ \natural\ \Sigma_{1}$ is a boundary connected sum. Then
all Lagrangian subspaces and Schrödinger modules split. Over $\Sigma$ the
cobordism is trivial and the expected result is clear, so that it is enough to
compute in the genus $1$ case, $\Sigma_{-}=\Sigma_{1}$ and $\Sigma_{+}=D^{2}$.
The Lagrangian $L_{-}$ is generated by a simple curve $m$. A complementary
Lagrangian $L_{-}^{\vee}$ is generated by $l$ with $m.l=1$. We have
$\gamma=\alpha m+\beta l$, $\gcd(\alpha,\beta)=1$. The Lagrangian
correspondence is

$L_{C}=\mathbb{Z}(\gamma,0)\quad\text{with complement}\quad
L_{C}^{\vee}=\mathbb{Z}(\delta,0)$

where $\delta=um+vl$, $\alpha v-\beta u=1$. Then $m=v\gamma-\beta\delta$,
$l=-u\gamma+\alpha\delta$. We have bases $B_{k\delta}$ and $b_{\nu l}$, $0\leq
k,\nu<p^{\prime}$ for $W(L_{C})$ and $W(L_{-})$, respectively. Using $l$ we
get the relation

$B_{k\delta}\otimes b_{(\nu+1)l}=B_{k\delta}(0,-u\gamma+\alpha\delta)\otimes
b_{\nu l}=(0,\alpha\delta)(u\alpha,-u\gamma)B_{k\delta}\otimes b_{\nu
l}=q^{u\alpha+2ku}B_{(\alpha+k)\delta}\otimes b_{\nu l}\ $

where we used intersection on $-\Sigma_{-}$. This reduces the set of
generators to $B_{k\delta}\otimes\mathbf{1}$, $0\leq k<p^{\prime}$. The
relation coming from $m$ then gives

$B_{k\delta}\otimes\mathbf{1}=B_{k\delta}(0,v\gamma-\beta\delta)\otimes\mathbf{1}=(0,-\beta\delta)(\beta
v,v\gamma)B_{k\delta}\otimes\mathbf{1}=q^{\beta
v-2kv}B_{(k-\beta)\delta}\otimes\mathbf{1}.$ (11)

If the surgery curve $\gamma$ is in $L_{-}$, we can choose $m=\gamma$,
$l=\delta$. The last relation gives
$B_{k\delta}\otimes\mathbf{1}=q^{-2k}B_{k\delta}\otimes\mathbf{1}$. Hence we
have $B_{k\delta}\otimes\mathbf{1}=0$ for $0<k<p^{\prime}$ and the tensor
product is $\mathbb{C}$-generated by $\mathbf{1}\otimes\mathbf{1}$. The
equalities

$B_{k\delta}\otimes\mathbf{1}=\begin{cases}1&\text{if}\quad k\equiv
0\pmod{p^{\prime}}\\\ 0&\text{else}\end{cases}$

define an isomorphism
$W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})\cong\mathbb{C}_{q}$.
In particular,

$\check{F}_{C}(b_{kl})=\left\\{\begin{array}[]{ll}1&\text{ if}\quad k\equiv
0\pmod{p^{\prime}}\\\ 0&\text{ else}\end{array}\right.$

If the surgery curve $\gamma$ is not in $L_{-}$ then $\beta\neq 0$. Let
$d=\gcd(\beta,p^{\prime})$, then the order of $\beta$ modulo $p^{\prime}$ is
$a=\frac{p^{\prime}}{d}$. Hence, relation (11) reduces the generators to
$\\{B_{k\delta}\otimes\mathbf{1},0\leq k<d\\}$. Finally, the action of
$(0,am)_{-\Sigma_{-}}$ gives the following relation

$B_{k\delta}\otimes\mathbf{1}=(0,av\gamma-a\beta\delta)B_{k\delta}\otimes\mathbf{1}=(0,-a\beta\delta)(a^{2}\beta
v,av\gamma)B_{k\delta}\otimes\mathbf{1}=q^{-2kav}B_{k\delta}\otimes\mathbf{1}$

since $q^{a^{2}\beta}=1$ and the intersection pairing is taken on
$-\Sigma_{-}$. It follows $B_{k\delta}\otimes\mathbf{1}=0$ unless $k$ is
divisible by $d$, hence $\mathbf{1}\otimes\mathbf{1}$ generates
$W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma_{-})]}W_{q}(L_{-})\cong\mathbb{C}_{q}$,
as expected.

We are left with computing $\check{F}_{C}$. The action of
$(0,a\gamma)=(0,a\alpha m+a\beta l)$ gives

$\mathbf{1}\otimes b_{kl}=\mathbf{1}\otimes(0,a\beta
l)(a^{2}\alpha\beta,a\alpha m)b_{kl}=q^{-2ka\alpha}\mathbf{1}\otimes b_{kl}$

implying that $\mathbf{1}\otimes b_{kl}=0$ if $d\nmid k$. If $d\,|\,k$ we set
$k\alpha=k^{\prime}\beta$ and compute

$\mathbf{1}\otimes
b_{kl}=\mathbf{1}(0,-ku\gamma+k\alpha\delta)\otimes\mathbf{1}=(0,k\alpha\delta)(k^{2}u\alpha,-ku\gamma)\mathbf{1}\otimes\mathbf{1}=q^{k^{2}u\alpha}B_{k\alpha\delta}\otimes\mathbf{1}=q^{kk^{\prime}\beta
u}B_{k^{\prime}\beta\delta}\otimes\mathbf{1}.$

From the action of $(0,k^{\prime}m)$ we get

$\displaystyle\mathbf{1}\otimes b_{kl}=$ $\displaystyle q^{kk^{\prime}\beta
u}(0,k^{\prime}v\gamma-k^{\prime}\beta\delta)B_{k^{\prime}\beta\delta}\otimes\mathbf{1}=q^{kk^{\prime}\beta
u}(-(k^{\prime})^{2}v\beta,k^{\prime}v\gamma)(0,-k^{\prime}\beta\delta)B_{k^{\prime}\beta\delta}\otimes\mathbf{1}$
$\displaystyle=$ $\displaystyle q^{kk^{\prime}\beta u-kk^{\prime}\alpha
v}\mathbf{1}\otimes\mathbf{1}=q^{-kk^{\prime}}\mathbf{1}\otimes\mathbf{1}=q^{-\alpha
k^{2}/\beta}\mathbf{1}\otimes\mathbf{1}.$

We deduce that

$\check{F}_{C}(b_{kl})=\left\\{\begin{array}[]{ll}0&\text{ if $d\nmid k$}\\\
q^{-\alpha k^{2}/\beta}&\text{ else.}\end{array}\right.$

In order to complete the proof, we need to consider a composition of
elementary cobordisms. A modification of the previous proof shows that for a
cobordism $C^{\prime}:(\Sigma_{-},L_{-})\rightarrow(\Sigma,L)$ and an
elementary cobordism $C:(\Sigma,L)\rightarrow(\Sigma_{+},L_{+})$, both in
$3\mathrm{Cob}^{\rm{LC}}$, we have an isomorphism of
$(\mathbb{Z}[\mathcal{H}(\Sigma_{+}),\mathbb{Z}[\mathcal{H}(\Sigma_{-}])$-bimodules

$\psi_{C}^{C^{\prime}}:W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma)]}W_{q}(L_{C^{\prime}})\xrightarrow{\sim}W_{q}(L_{C^{\prime}\cup_{\Sigma}C})$

sending $\mathbf{1}\otimes\mathbf{1}$ to $\mathbf{1}$. The key point is that
$W_{q}(L_{C^{\prime}})$ can be written as a direct sum of copies of the
Schrödinger representation of $\mathcal{H}(\Sigma)$. The general case of the
theorem is then obtained by induction on the length of the decomposition,
where the inductive step follows from the three cases we computed above. ∎

It remains to show that the above construction recovers the abelian TQFT.

###### Proof of Theorem 2.

We will use the skein model from Section 2. Following [20, Theorem 4.5] the
Heisenberg group algebra $\mathbb{C}[\mathcal{H}(\Sigma)]$ can be identified
with the $U(1)$-skein algebra $S(\Sigma)$. This makes the TQFT vector space
$V(\Sigma,L)$ to a module over $\mathbb{C}[\mathcal{H}(\Sigma)]$. Actually, it
is isomorphic to the Schrödinger representation, see [20, Theorem 4.7] for
even $p$.

A priori, the Stone-von Neumann theorem provides the isomorphism up to a
complex number in $\mathbb{S}^{1}\subset\mathbb{C}$. Here we prefer to
construct the isomorphism explicitly. Let us denote by $S_{p}(\Sigma)$ the
reduced $U(1)$ skein module, where $q$ is specified to the $p$th root of
unity. Then $S_{p}(\Sigma)$ is identified with
$\mathbb{C}[\mathcal{H}_{p}(\Sigma)]$ by sending a simple closed curve
$\gamma$ with blackboard framing to

$(0,[\gamma])\in\mathbb{Z}_{p}\times
H_{1}(\Sigma,\mathbb{Z}_{p^{\prime}})=\mathcal{H}_{p}(\Sigma)$

Let $H$ be a handlebody with boundary $\Sigma$ such that ${\sf L}$ is the
kernel of the inclusion $H_{1}(\Sigma,\mathbb{Z})\hookrightarrow
H_{1}(H,\mathbb{Z})$. Then the TQFT vector space $V(\Sigma,L)$ is the quotient
of $S_{p}(\Sigma)$ by the subspace generated by $\gamma-1$ where $\gamma$ is a
simple curve that bounds in $H$ or equivalently such that $[\gamma]\in{\sf
L}$. Using the isomorphism $S(\Sigma)\cong\mathbb{C}[\mathcal{H}(\Sigma)]$, we
deduce that the quotients $V(\Sigma,L)$ and $W_{q}(L)$ are isomorphic.

A basis $\\{b_{x},x\in{\sf L}^{\vee}\\}$ for $W_{q}(L)$ can be represented by
skein elements $\\{y_{x},x\in{\sf L}^{\vee}\\}$ in $H$ providing a basis for
$V(\Sigma,L)$. Here for an embedded curve $x$ in $\Sigma$, the element $y_{x}$
is obtained by pushing $x$ in $H$ with blackboard framing and then by taking
its skein class. For example, the element $y_{3x}$ correspond to the three
parallel copies of $y_{x}$ obtained by using the blackboard framing. We are
now able to compare $F_{C}=Z(\check{C})\check{F}_{C}$ with the TQFT map on
elementary cobordisms.

Let us consider a mapping cylinder
$C_{f}:(\Sigma_{-},L_{-})\rightarrow(\Sigma_{+},L_{+})$ with $g_{-}=g_{+}=g$.
A basis for the TQFT vector space identified with the Shrödinger
representation $W_{q}(L_{-})$ is represented by a handlebody $H_{-}$, with
$\partial H_{-}=\Sigma_{-}$ and with the cores $l_{i}$, $1\leq i\leq g$, of
its handles colored by $y^{k}$, $0\leq k\leq p^{\prime}$. The TQFT map is
represented by gluing the mapping cylinder $C_{f}$ to the handlebody $H_{-}$.
This results in a handlebody $H_{+}$ with boundary $\Sigma_{+}$. Moreover when
pushing the colored curve $l$ across the cylinder we get a curve parallel to
$f(l)$. Hence, the TQFT map sends $y^{k}$ in $H_{-}$ to $f(y^{k})$ in $H_{+}$,
matching $F_{C_{f}}$. Note that $\check{C}_{f}$ is a connected sum of $g$
copies of $S^{2}\times S^{1}$, since $f$ preserves the Lagrangians. Hence, in
our normalization $Z(\check{C}_{f})=1$.

In the case of an index $1$ surgery
$C:(\Sigma_{-},L_{-})\rightarrow(\Sigma_{+},L_{+})$, the TQFT map is
represented by the inclusion of a handlebody $H_{-}\hookrightarrow
H_{+}=H_{-}\cup_{\Sigma_{-}}C$, where $\partial H_{-}=\Sigma_{-}$ and

$\ker\left(H_{1}(\Sigma,\mathbb{Z})\hookrightarrow
H_{1}(H_{-},\mathbb{Z})\right)={L}_{-}.$

This inclusion map matches again $F_{C}$ with $Z(\check{C})=1$.

In the case of an index $2$ surgery on a curve $\gamma$, we only need to
consider the case where $\Sigma_{-}$ is a genus $1$ surface. Then the TQFT map
$Z(C):V(\Sigma_{-},L_{-})\rightarrow\mathbb{C}_{q}$ is given by the evaluation
of the skein element $(H_{-},x)$ inside
$M_{\gamma}=(H_{-}\cup_{\Sigma_{-}}C)\cup_{S^{2}}D^{3}$. If $\gamma=m$,
$M_{\gamma}=S^{1}\times S^{2}$ and the evaluation reduces to a Hopf link with
one Kirby-colored component, which is zero unless $x=0$, when it is $1$.

If $\gamma=l$, $M_{\gamma}=S^{3}$ and the evaluation is $1$ for all $x=y^{k}$.
Hence, in both cases we recover $F_{C}$.

More generally, for $\gamma=\alpha m+\beta l$ with $\beta\neq 0$, the manifold
$M_{\gamma}$ is the lens space $L(\beta,\alpha)$. Let us choose a continued
fraction decomposition $\beta/\alpha=[m_{1},\dots,m_{n}]$ as in [26]. Then a
surgery link $L$ for $M_{\gamma}$ is the length $n$ Hopf chain with framings
$m_{i}$. Hence, the TQFT map sends $y^{k}$ to the following number

$Z(C)_{k}=\kappa^{-\text{\rm
sign}(L)}\;\eta^{n}\sum^{p^{\prime}}_{j_{1},\dots,j_{n}=1}\;q^{\sum^{n}_{i=1}m_{i}j^{2}_{i}}\;q^{2kj_{1}}\;q^{2\sum^{n-1}_{i=1}j_{i}j_{i+1}}$

$y^{k}$$\eta\omega$$\eta\omega$$\eta\omega$$\eta\omega$$\eta\omega$$0$$m_{1}$$m_{2}$$m_{3}$$m_{n-1}$$m_{n}$
Figure 3. Surgery link for the lens space where the upper indices correspond
to the framings and the lower ones to the colors.

Since a recursive computation of this sum was done in [26], we present here
just the result.

$Z(C)_{k}=\begin{cases}0&d\nmid k\\\ q^{-\frac{\alpha
k^{2}}{\beta}}Z(L(\beta,\alpha))&\text{else}\end{cases}$

where $d=\text{\rm gcd}(\beta,p^{\prime})$. This coincides with $F_{C}$ on
this cobordism with $Z(\check{C})=Z(L(\beta,\alpha))$. Since the TQFT map for
any cobordism is a composition of maps for elementary ones and the same works
for $F_{C}$, for any cobordism $C$ we have that the TQFT map

$Z(C):V(\Sigma_{-})\cong W_{q}(L_{-})\rightarrow V(\Sigma_{+})\cong
W_{q}(L_{+})$

is equal, up to a coefficient, to the inclusion map $W_{q}(L_{-})\rightarrow
W_{q}(L_{C})\otimes_{\mathbb{C}[\mathcal{H}_{p}(\Sigma)]}W_{q}(L_{-})$
composed with the isomorphism from Theorem 1. Closing with handlebodies
compatible with the Lagrangians we get that the coefficient is $Z(\check{C})$
which completes the proof. ∎

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