aainstitutetext: Albert Einstein Center for Fundamental Physics
Institute for Theoretical Physics, University of Bern,
Switzerlandbbinstitutetext: East African Institute for Fundamental Research
(EAIFR)
University of Rwanda, Kigali, Rwandaccinstitutetext: Abdus Salam International
Centre for Theoretical Physics
Strada Costiera 11 Trieste, Italy

# 3 Definitions of BF Theory on Homology 3-Spheres

Matthias Blau b,c Mbambu Kakona c George Thompson<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

3-dimensional BF theory with gauge group $G$ (= Chern-Simons theory with non-
compact gauge group $TG$) is a deceptively simple yet subtle topological gauge
theory. Formally, its partition function is a sum/integral over the moduli
space $\mathcal{M}$ of flat connections, weighted by the Ray-Singer torsion.
In practice, however, this formal expression is almost invariably singular and
ill-defined.

In order to improve upon this, we perform a direct evaluation of the path
integral for certain classes of 3-manifolds (namely integral and rational
Seifert homology spheres). By a suitable choice of gauge, we sidestep the
issue of having to integrate over $\mathcal{M}$ and reduce the partition
function to a finite-dimensional Abelian matrix integral which, however,
itself requires a definition. We offer 3 definitions of this integral, firstly
via residues, and then via a large $k$ limit of the corresponding $G\times G$
or $G_{\mathbb{C}}$ Chern-Simons matrix integrals (obtained previously). We
then check and discuss to which extent the results capture the expected
sum/integral over all flat connections.

##  1 Introduction

Gauge theories with non-compact gauge groups are notoriously difficult to make
sense of, issues including questions of convergence and propagators with the
wrong signs which make unitarity and hence the physical meaning of the theory
far from clear. One set of theories for which one may make some headway are
topological field theories where a particle interpretation is not required.
Our first investigation in that direction where the partition function can be
evaluated exactly BKT-MRST was to consider the Schwarz type theories Schwarz-
Partition ; Schwarz ; Schwarz-Tyupkin formally representing the Ray-Singer
Torsion Ray-Singer ; Ray-Singer-Analytic which have non-compact symmetries.
These symmetries though are essentially Abelian and we would like to consider
now a class of theories with non-Abelian non-compact gauge groups. In
particular we have in mind the topological $BF$ theories BT-RST ; Horowitz ;
BT-BF ; BT-Metric based on a compact gauge group $G$. In two dimensions such a
theory is the zero coupling limit of Yang-Mills theory with gauge group $G$
and as such does not have an associated non-compact symmetry. In dimension
greater than or equal to 3, however, $BF$ theories possess additional non-
compact shift symmetries. We will focus on 3-dimensional $BF$ theory here,
where the combined gauge group turns out to be the non-compact group $TG$, the
tangent bundle group of $G$.

Witten WBF first introduced 3-dimensional $BF$ theories as a variant of Chern-
Simons theory Witten-CS for non-compact gauge groups (in fact, it can be
regarded as a Chern-Simons theory for the non-compact gauge group $TG$). The
action is

$I_{BF}=\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B\wedge F_{A}\right)}$ (1)

where $A$ is a connection on a $G$-bundle over $M$ and $B$ is a Lie algebra
valued 1-form. The action enjoys the usual compact $G$ gauge symmetry, as well
as a non-compact shift gauge symmetry $B\rightarrow B+d_{A}\Lambda$. This
theory is deceptively simple. The path integral over $B$ yields a delta
function constraint on the curvature 2-form $F_{A}$, so that formally the path
integral for the partition fuction of $BF$ theory is

$Z_{BF}[M,G]=\int
DADB\,\exp{\left(i\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B\wedge
F_{A}\right)}\right)}\simeq\int DA\delta\left(F_{A}\right)$ (2)

This seems to suggest that the path integral simply reduces to an integral
over the moduli space of flat connections, with some measure. In general,
however, this is not correct. Rather, the complete classical equations of
motion are

$F_{A}=0,\;\;\;d_{A}B=0$ (3)

which taken together (and modulo gauge transformations) can formally be viewed
as describing the tangent bundle of the moduli space $\mathfrak{M}[M,G]$ of
flat $G$ connections on the 3-manifold $M$, $T\mathfrak{M}[M,G]$. The
partition function is then an integral over this space (and ghost zero modes
etc.), again with some measure to be determined. However, in general this
space can be very singular and the finite dimensional integral is ill-defined.
The various types of singularities that lurk inside the path integral
associated with the action (1) include issues with reducible connections
(manifested as ghost zero modes) as well as the non-compactness of the moduli
space itself (due to $B$ zero modes).

One situation in which it is pretty clear what sort of contribution one should
find from the path integral is when one expands the path integral around an
isolated irreducible flat connection. In that case, there are no non-trivial
solutions to the equations of motion for $B$ and no ghost zero modes, and
following Witten WBF , the contribution of such an acyclic connection $\omega$
on a 3-manifold $M$ can be shown to be precisely the Ray-Singer analytic
torsion $\tau_{M}(\omega)$ Ray-Singer ; Ray-Singer-Analytic of that
connection. Thus on manifolds on which the moduli space of flat connections
includes such acyclic flat connections, one should perhaps expect the
partition function to take the form

$Z_{BF}[M,G]=\sum_{\mathrm{acyclic}}\tau_{M}(\omega_{(\alpha)})+\ldots$ (4)

where the ellipses indicate the contributions from the non-acyclic flat
connections. If $M$ is an Integral Homology Sphere ($\mathbb{Z}$ Homology
Sphere, $\mathrm{H}_{1}(M,\mathbb{Z})=0$), then for $G=SU(2)$ the trivial
connection is the only reducible flat connection, and there are indeed
$\mathbb{Z}$ Homology Spheres for which all the flat connections are isolated.
In such a situation one should expect the $BF$ partition function to be
largely captured by the above expression. One of the aims of this paper will
be to check to which extent this expectation is borne out by actual explicit
evaluations of the path integral of $BF$ theory.

Moreover, by using the Massive Ray-Singer Torsion $\tau_{M}(\omega,\mu)$
introduced in BKT-MRST (among other reasons precisely with its application to
$BF$ theory in mind), one can extend the considerations of Witten to the case
that the flat connections are not necessarily irreducible and isolated, and
thus obtain a formal expression for the regularised partition function of $BF$
theory (section 2), which takes the form (26)

$Z_{BF}[M,G;\mu]=\int\mathllap{\sum}\,\tau_{M}(\omega_{(\alpha)},\mu)$ (5)

of a sum / integral over the moduli space of flat connections. This expression
can in turn be expanded in the mass-parameter $\mu$ to reproduce different
kinds of contributions to the partition function BKT-MRST . However, this
approach outlined above to defining the partition function of $BF$ theory is
rather formal: one would need to determine by hand the spaces of flat
connections and zero modes associated with them, calculate the (massive) Ray-
Singer torsion by some means, etc. So this does not all by itself lead to an
evaluation of the $BF$ partition function via path integral methods.

In order to be able to verify to which extent an actual evaluation of the $BF$
path integral reproduces the above formal expectations, one needs a class of
manifolds for which explicit path integral calculations can be performed. To
that end we will concentrate on Seifert fibred 3-manifolds. These spaces allow
for a significant simplification in the calculations via suitable gauge
choices. The first of these is that the ‘time’ (fibre) components of both $A$
and $B$ (the components of $A$ and $B$ along the fibres of the defining
Seifert fibrations) can be taken to be constant along the fibre. This is a
generalisation of the usual temporal gauge $\partial_{0}A_{0}=0$ of a gauge
field on a circle, and allows one to partially push the calculation down a
dimension, to the base of the Seifert fibration. And now further
simplification comes from the approach BT-CS ; BT-Trieste-1993 ; BT-S1 ; BT-
Seifert ; BKNT ; BT-CCS to evaluating the path integral through
Abelianisation. For Chern-Simons theory with gauge group $G$ the end result is
a finite dimensional integral over the Cartan subalgebra $\mathfrak{t}$ of the
Lie algebra $\mathfrak{g}$ of $G$. One of the characteristic features of this
approach is that the reduction to a finite-dimensional integral (over a linear
space) completely bypasses the need to define (and integrate over) the moduli
space of flat connections. Nevertheless, this approach can be shown to
reproduce the results of perturbative or localisation calculations (when
available), which in principle require an exact evaluation of the latter.

Returning to $BF$ theory, the main technical result of this paper is to show
that this calculational method can be extended to $BF$ theory with its non-
compact gauge group $TG$. The result is a finite-dimensional integral over the
space $\mathfrak{t}\oplus\mathfrak{t}$, which plays the role of the Cartan
subalgebra of the Lie algebra of $TG$. This once again bypasses the need to
integrate over the moduli space of flat connections. The finite-dimensional
integrals that we find are similar to those for Chern-Simons theory, but they
are more singular than their Chern-Simons counterparts, having poles on the
integration contour an integral over $\lambda$ which is a distribution, and
hence require a definition. For example, the partition function for $SU(2)$
and $M$ a $\mathbb{Z}$ Homology Sphere devolves to

$Z_{BF}[M,SU(2)]\simeq\sum_{n_{0}}\int_{\mathbb{R}\times\mathbb{R}}\tau_{M}\left(\phi\right)\,.\,\exp{\left(i\lambda\left(\phi/P+2\pi
in_{0}\right)\right)}$ (6)

(see the body of the paper for all the definitions). The important point to
note here is that the Ray-Singer Torsion $\tau_{M}\left(\phi\right)$ has poles
at $\phi=m\pi$ $m\neq sP$ and zero when $m$ is proportional to $P$. The
$\lambda$ integral sets $\phi=Pn_{0}$ which formally avoids the poles. Still
we can only conclude what the integral should be after it is properly
regularised (defined).

In the case of Chern-Simons theory the integrals that we find there also have
poles, however, there we could argue that the contributions at the poles
should not be included (lying on the walls of the Weyl chamber) and we
introduced a mass regulator that essentially allowed us to avoid them.
Unfortunately, in the present setting of $BF$ theory, we do not have a guiding
principle and so we offer 3 different possible definitions of $BF$ theory and
discuss their advantages as well as their shortcomings:

  1. 1.

Direct Definition via Residues

The first, and the one we spend most time on (section 4), is through the
direct definition of the finite-dimensional integral (6). In order to get a
handle on the poles of the Ray-Singer Torsion on
$\mathfrak{t}\times\mathfrak{t}$, one defines the theory in such a way that
that it is given by the residue of all the possible poles including delta
function contributions. We find that, rather remarkably, this does reproduce
the expected contributions of reducible flat connections which we can follow
by making use of the Massive Ray-Singer Torsion. In particular we show in
detail how this gives the expected results, as a sum over contributions from
the isolated non-trivial flat connections, for Lens spaces. Indeed, this
definition appears to capture the essence of the contribution of reducible
connections in general. However, this approach does not reproduce the expected
result (4) for connections that are isolated and irreducible. We note that,
suprisingly, some of the poles on the contour correspond to irreducible flat
connections, while others correspond to ‘complex’ flat connections. We offer a
brief attempt at an explanation (related to gauge fixing) for why the
irreducible connections arise in this way.

In view of this, we approach the problem from a different point of view in
section 5, based on the fact that the gauge group $TG$ can be regarded as a
contraction of either the compact group $G\times G$ or the complex group
$G_{\mathbb{C}}$. Correspondingly, the other 2 definitions that we consider
are to regard $BF$ theory as arising either in the large $k$ limit of $G\times
G$ Chern-Simons theory at levels $k$ and $-k$, or in certain limits of the
coupling $t=k+is$ of $G_{\mathbb{C}}$ Chern-Simons theory. This regularises
the theory to some extent and also side-steps the gauge-fixing issue alluded
to above.

  1. 2.

Definition via a large $k$ Limit of $G\times G$ Compact Chern-Simons Theory

The advantage of using the compact $G\times G$ theory is that the Hilbert
space of states is finite dimensional and only goes over to the infinite
dimensional Hilbert space of states of $BF$ theory in the limit. This
therefore acts as a natural regulator. A disadvantage is that certain
conditions must be met by the connections so that the correspondence that we
need exists. We offer examples of Brieskorn $\mathbb{Z}$ Homology Spheres
(Seifert manifolds with 3 exceptional fibres) Freed-Gompf where the conditions
are met and one obtains the partition function of $BF$ theory in the expected
form (4). These conditions are not met by all manifolds, however. For example,
the equality between $BF$ theory and the large $k$ limit of the $G\times G$
theory fails for certain Lens spaces. As the Lens spaces are $\mathbb{Q}$
Homology Spheres one would like to conjecture that the correspondence holds
for isolated flat connections of $\mathbb{Z}$ Homology Spheres in general.

  2. 3.

Definition via a large $k$ Limit of $G_{\mathbb{C}}$ Complex Chern-Simons
Theory

As for $G_{\mathbb{C}}$ Chern-Simons theory Witten-CCS ; Gukov ; DGLZ with
complex level $k+is$, there are many formal correspondences with $BF$ theory.
The most obvious is to set $k=0$ and let $s\rightarrow\infty$, which leads
directly to the $BF$ action (1) and its non-compact symmetries (8). Given the
difficulties we face with this action, this is not the limit that we consider.
Rather, we take $s=0$ and $k\rightarrow\infty$. An advantage of this approach
is that the finite dimensional integrals that arise are slight variants of
those discussed by Lawrence and Rozansky LR for $SU(2)$ and for general $G$ by
Mariño Marino in the context of $G$ Chern-Simons theory. In principle then the
same strategies that apply there can be used here, though we leave that for
the future.

To the extent that the perturbative large $k$ evaluation of Chern-Simons
theory reduces to a sum over contributions from flat connections, this is then
also true for the partition function of $BF$ theory. Overall, the definition
of $BF$ theory via $G_{\mathbb{C}}$ Chern-Simons theory appears to be the most
complete (however, one needs to ensure that one does not count complex flat
connections that cannot be conjugated into flat $G$ connections).

One approach not followed here but which is certainly of interest is
resurgence Witten-Bonn . Gukov, Mariño and Putrov GMP-resurge show that for
$SU(2)$ one can start with the Abelian contribution to the Chern-Simons
partition function in the large $k$ limit and Borel resum in order to obtain
contributions from non-Abelian flat connections. Once in the complex space of
connections there is the possibility that $SL(2,\mathbb{C})$ connections which
are not (conjugate to) $SU(2)$ flat connections will contribute. They show
that miraculously this does not happen. The importance of this for us is that,
as we will see, we appear to obtain contributions from connections in our
first approach that would, from this point of view be considered to be
‘complex’.

This paper is organised as follows. Section 2 follows Witten’s WBF approach to
obtaining the partition function for isolated irreducible flat connections.
There we also consider what formally happens when the connections are either
not isolated or are reducible. In these cases we can profitably make use of
the massive Ray-Singer Torsion introduced in BKT-MRST . This allows us to
express our expectations for the form of the partition function in concrete
terms.

In Section 3 we formulate $BF$ theory on Seifert 3-manifolds which are either
$\mathbb{Z}$ or $\mathbb{Q}$ homology spheres. Particular attention is paid to
issues with gauge fixing of $\phi$ and $\lambda$, the components of $A$ and
$B$ along the fibre respectively. We will shows that $\phi$ and (with an
assumption about $\phi$) $\lambda$ can be chosen to be constant along the
fibre (temporal gauge), and to simultaneously take values in a fixed Cartan
subalgebra $\mathfrak{t}\oplus\mathfrak{t}$ of $TG$ (Abelianisation in $TG$).
We then give a brief outline of how to pass from the functional integral to a
finite dimensional integral over $\mathfrak{t}\oplus\mathfrak{t}$.

As already mentioned, the integral in question has poles on the integration
contour and so requires a definition. In section 4 we thus give our first
definition of $BF$ theory, as a residue of a particular function related to
the integrand of the finite dimensional integral, and analyse its
consequences. Given the shortcomings of that definition, the alternative
Definitions 2 and 3 described above are explored in section 5.

Certain technical details are relegated to an Appendix (A for information
about the group $TG$ and its Lie algebra, and B for details about a certain
useful discrete symmetry of complex Chern-Simons theory on Seifert manifolds).

##  2 $\boldsymbol{BF}$-Theory on a 3-Manifold and Path Integrals

In this section we give a brief review of 3-dimensional $BF$ theory and
Witten’s formula for the path integral which holds when there are no zero
modes. This is followed by a discussion of a generalisation which takes zero
modes into account.

###  2.1 $BF$ Theory as a $TG$ Chern-Simons Theory

As recalled in the Introduction, 3-dimensional $BF$ theory on a 3-manifold $M$
is defined by the action (1)

$I_{BF}=\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B\wedge F_{A}\right)}$ (7)

with $A$ a connection on a $G$-bundle over $M$ and
$B\in\Omega^{1}(M,\mathfrak{g})$. The gauge symmetries of this action are the
usual $G$ gauge symmetry acting on $A$ and $B$, as well as a shift gauge
symmetry for the field $B$,

$(A,B)\rightarrow(g^{-1}Ag+g^{-1}dg,\,g^{-1}\left(B+d_{A}\Lambda\right)g)$ (8)

This shift gauge symmetry is non-compact, parametrised by
$\Lambda\in\Omega^{0}(M,\mathfrak{g})$, and thus, even for $G$ a compact Lie
group, 3-dimensional $BF$ theory is an example of a gauge theory with a non-
compact gauge group (and this is the main reason we are interested in this
theory here). In fact, $BF$-theory is not just some non-compact analogue of
Chern-Simons theory: in a precise sense it is a Chern-Simons theory for the
non-compact gauge group $TG$, the tangent bundle group of $G$. As it will
frequently be useful to have this perspective in the back of one’s mind in the
following, here (and in more detail in Appendix A) we quickly review the
relevant facts regarding the group $TG$, its Lie algebra $\mathfrak{tg}$, and
Chern-Simons theory based on it.

Thus, consider a compact semi-simple Lie group $G$ (throughout we will also
assume that $G$ is connected and simply-connected) with Lie algebra
$\mathfrak{g}$, a basis of generators $j_{a}$ and commutation relations
$[j_{a},j_{b}]=f_{ab}^{c}j_{c}$. Then the Lie algebra $\mathfrak{tg}$ of the
Lie group $TG$ has generators $(j_{a},p_{a})$ and commutation relations (151)

$[j_{a},j_{b}]=f_{ab}^{c}j_{c}\quad,\quad[j_{a},p_{b}]=[p_{a},j_{b}]=f_{ab}^{c}p_{c}\quad,\quad[p_{a},p_{b}]=0\;\;.$
(9)

For the considerations of section 5 it will be useful to keep in mind, that
this algebra can be obtained as a contraction of the Lie algebra of $G\times
G$ or $G_{\mathbb{C}}$ (see (155)).

Given an invariant non-degenerate scalar product
$g_{ab}=\mathop{\mathrm{Tr}}\nolimits j_{a}j_{b}$ on $\mathfrak{g}$, an
invariant and non-degenerate scalar product on $\mathfrak{tg}$ is (161)

$\ll j_{a},j_{b}\gg=\ll p_{a},p_{b}\gg=0\quad,\quad\ll
j_{a},p_{b}\gg=g_{ab}\;\;.$ (10)

A $TG$-gauge field can be expanded as

$C=A^{a}j_{a}+B^{a}p_{a}\;\;,$ (11)

with field strength

$F_{C}=dC+{\textstyle\frac{1}{2}}[C,C]=F_{A}^{a}j_{a}+d_{A}B^{a}p_{a}\;\;.$
(12)

The Chern-Simons action for $C$ with respect to the above scalar product is
(with a for present purposes convenient choice of normalisation, and with an
integration by parts)

$I_{CS}\equiv{\textstyle\frac{1}{2}}\int_{M}\ll
C,dC+\tfrac{1}{3}[C,C]\gg\;=\int_{M}\mathop{\mathrm{Tr}}\nolimits B\wedge
F_{A}=I_{BF}$ (13)

The equations of motion (3) of $BF$-theory are then evidently simply the
flatness conditions for the connection $C$,

$F_{C}=0\quad\Leftrightarrow\quad F_{A}=0,\;\;\;d_{A}B=0$ (14)

Moreover, the gauge symmetries (8) of $BF$-theory are precisely the $TG$ gauge
symmetries of the $TG$-connection $C$.

###  2.2 Path Integral for $BF$ Theory: Formal Considerations

Evidently (and as recalled in the Introduction), formally, the path integral
for the partition fuction of $BF$ theory is

$\displaystyle Z_{BF}[M,G]$ $\displaystyle=\int
DADB\,\exp{\left(i\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B\wedge
F_{A}\right)}\right)}$ $\displaystyle\simeq\int DA\delta\left(F_{A}\right)$
(15)

so that we may expand about flat connections to give a more complete
evaluation. Following Witten WBF , we use the standard Faddeev-Popov covariant
gauge-fixing procedure around these flat connections. More details about gauge
fixing Chern-Simons theory with non-compact gauge groups can be found in Bar-
Natan-Witten . The action and symmetries about a flat background connection
$\omega_{(\alpha)}$ are

$I_{BF}=\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B\wedge
F_{A+\omega_{(\alpha)}}\right)}$ (16)
$(A+\omega_{(\alpha)},B)\rightarrow(g^{-1}(A+\omega_{(\alpha)})g+g^{-1}dg,\,g^{-1}\left(B+d_{A+\omega_{(\alpha)}}\Lambda\right)g)$
(17)

while the covariant background field gauge fixing conditions

$d_{{\omega_{(\alpha)}}}*A=0.\quad d_{{\omega_{(\alpha)}}}*B=0$ (18)

require us to make a choice of metric on $M$. Then the ghost and gauge fixing
action is

$I_{GF}=\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(ud_{{\omega_{(\alpha)}}}*B+vd_{{\omega_{(\alpha)}}}*A+\overline{f}d_{{\omega_{(\alpha)}}}*d_{({\omega_{(\alpha)}}+A)}f+\overline{g}d_{{\omega_{(\alpha)}}}*d_{({\omega_{(\alpha)}}+A)}g\right)}$
(19)

If the space of flat connections is not made up of isolated points then one
would need to integrate over them which one could do through the introduction
of collective coordinates. One should also take into account the possible zero
modes of both $B$ and of the ghosts. Those zero modes form non-compact spaces
$\mathrm{H}^{1}_{\omega_{\alpha}}(M,\mathfrak{g})$, and
$\mathrm{H}^{0}_{\omega_{\alpha}}(M,\mathfrak{g})$ respectively. Taking the
above caveats into account the path integral formally becomes

$\displaystyle Z_{BF}[M,G]$ $\displaystyle=\sum_{\alpha}\int
DADB\,\exp{\left(iI_{BF}+iI_{GF}\right)}$
$\displaystyle=\int\mathllap{\sum_{T\mathfrak{M}_{\alpha}}}\,\tau_{M}(\omega_{(\alpha)})$
(20)

Here the sum and integral symbol over $T\mathfrak{M}_{\alpha}$ is meant to
indicate that one sums over distinct components of the moduli spaces of
classical solutions and integrates over the moduli of continuous families of
solutions. These spaces include the zero modes of $A$ and $B$ as well as of
those of the ghosts $f$ and $g$ (the multiplier field and anti-ghost zero
modes canonically cancel each other). The space $T\mathfrak{M}_{\alpha}$ space
can be very singular and so (20) as it stands is rather symbolic in general.

On the 3-manifolds of interest in this paper, the moduli space includes
acyclic flat connections, i.e. flat connections which are isolated and
irreducible, so that there are no zero modes at all at these solutions. The
path integral around such a connection simplifies by using the following
scaling argument. About such an isolated and irreducible connection send

$A\rightarrow tA,\quad B\rightarrow t^{-1}B,\quad t\in\mathbb{R}_{+}$ (21)

(with compensating scaling for the multiplier fields). This transformation has
trivial Jacobian, and the action remains well defined in the limit
$t\rightarrow 0$. Indeed, in this limit,

$\displaystyle I_{BF}$
$\displaystyle\rightarrow\int_{M}Bd_{{\omega_{(\alpha)}}}A$ $\displaystyle
I_{GF}$
$\displaystyle\rightarrow\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(ud_{{\omega_{(\alpha)}}}*B+vd_{{\omega_{(\alpha)}}}*A+\overline{f}*\Delta_{{\omega_{(\alpha)}}}f+\overline{g}*\Delta_{{\omega_{(\alpha)}}}g\right)}$
(22)

and the partition function around this connection is simply a standard path
integral representation Schwarz ; BT-BF of the Ray-Singer torsion Ray-Singer
$\tau_{M}(\omega_{(\alpha)})$ of the flat connection $\omega_{(\alpha)}$,

$\tau_{M}(\omega_{(\alpha)})=\left(\mathop{\mathrm{Det}}\nolimits{\Delta^{0}_{{\omega_{(\alpha)}}}}\right)^{3/2}.\left(\mathop{\mathrm{Det}}\nolimits{\Delta^{1}_{{\omega_{(\alpha)}}}}\right)^{-1/2}$
(23)

(the superscripts on the twisted Laplacians indicating the degrees of the
forms on which they act). Thus on such 3-manifolds, the partition function now
becomes

$Z_{BF}[M,G]=\sum_{\mathrm{acyclic}}\tau_{M}(\omega_{(\alpha)})+\ldots$ (24)

where the ellipses indicate the rest of the contributions to the path
integral. As we have already explained, there may be zero modes to deal with
in the extra terms (24) which manifest themselves as zeros of the determinants
in (23) and which invalidate the simplifications that we were allowed to make
for the isolated irreducible connections.

One way to proceed is to adopt the prescription of Ray and Singer Ray-Singer-
Analytic by first excising the zero modes and then adding a correction term to
ensure metric independence. This ‘extra’ gauge fixing of the zero modes may be
achieved by a BRST procedure BT-BF the details of which have been eplained in
some detail in BKT-MRST . The advantage of this method is that then the
torsion is a natural measure for the finite dimensional integral that appears
in (20). While this defines the integrand, one is still confronted with the
problem of determining and defining the space over which this is to be
integrated. Thus, while formally this appears to be a good definition, at a
practical level it seems somewhat intractable at the moment.

###  2.3 Path Integral for $BF$ Theory and Massive Ray-Singer Torsion

An alternative method for regularising such zero modes, and for keeping track
of the ellipses in the formula (24) was advocated in BKT-MRST . The idea is to
add a mass to the Laplacians that appear in (23), thus lifting all the zero
modes and side-stepping (or at least initially bypassing) the problem of
having to deal with them directly. As a first step, what this means is that
instead of using the twisted Laplacian $\Delta_{\omega}$ in the definition of
the Ray-Singer Torsion we instead use the massive Laplacian
$\Delta_{\omega}+\mu^{2}$ which now manifestly has a positive definite
spectrum. The Massive Ray-Singer Torsion, on a 3-manifold, for a flat
connection $\omega$, is then defined to be

$\tau_{M}(\omega,\mu)=\left(\mathop{\mathrm{Det}}\nolimits{(\Delta_{\omega}^{0}+\mu^{2})}\right)^{3/2}\,.\,\left(\mathop{\mathrm{Det}}\nolimits{(\Delta_{\omega}^{1}+\mu^{2})}\right)^{-1/2}$
(25)

One can then define a regularised partition function by

$Z_{BF}[M,G;\mu]=\int\mathllap{\sum}\,\tau_{M}(\omega_{(\alpha)},\mu)$ (26)

where we do not necessarily attempt to integrate over the tangent space as
those modes have been lifted. Here a caveat is required: in BKT-MRST we showed
that for manifolds which are mapping tori it is possible to maintain complete
gauge invariance even with the introduction of a mass. However, a mass term
cannot be introduced in a gauge invariant way on a general 3-manifold. We
accept that the gauge symmetry is broken and may be reinstated if required by
some renormalisation.

The way in which one obtains the actual Ray-Singer Torsion from the massive
Ray-Singer torsion is to take a limit

$\tau_{M}(\omega)\equiv\lim_{\mu\rightarrow
0}\mu^{-3\dim{\mathrm{H}^{0}_{\omega}}+\dim{\mathrm{H}^{1}_{\omega}}}\,\tau_{M}(\omega,\mu)$
(27)

In this way the $BF$ partition function will then take the form

$Z_{BF}[M,G,\mu]=\int\mathllap{\sum}\,\mu^{3\dim{\mathrm{H}^{0}_{\omega}}-\dim{\mathrm{H}^{1}_{\omega}}}\left(\tau_{M}(\omega)+\dots\right)$
(28)

and we will need to specify which terms we are interested in. The ellipses in
this formula refer to terms higher order in $\mu$ than the zero-th order
Torsion (essentially constants and derivatives of the Ray-Singer Torsion).

At a formal level this is, perhaps, as far as one may go. It must be said that
this situation is not very satisfying and has calculational drawbacks. The
formula (26) is very implicit and requires knowledge outside of the path
integral in order to be used. The flat connections need to be found, the
cohomology groups about the flat connections must be determined and a formula
for the Massive Ray-Singer Torsion must be given.

It is therefore of considerable interest to consider 3-manifolds for which the
BF partition function can also be calculated directly and explicitly, and
where the result can be compared with the formal expectations for the
partition function outlined above. One such class of manifolds is Seifert
manifolds.

###  2.4 Expectations for the Partition Function on Seifert Manifolds

The Ray-Singer torsion for Seifert 3-manifolds is known (through its
equivalence to Reidemeister Torsion Fried ). About an acyclic flat connection,
i.e. with trivial twisted cohomology, it is

$\tau_{M}(\omega)=\tau_{S^{1}}(\phi)^{2-N}\prod_{i=1}^{N}\tau_{S^{1}}(\phi/a_{i})$
(29)

where $\phi$ is the component of the connection in the direction of the fibre
of the Seifert 3-manifold and $N$ (the number of orbifold points) and $a_{i}$
(the order of the isotropy group at the $i$’th orbifold point) are integers
that are part of the defining data of a Seifert 3-manifold (see section 3.1).
We should also specify the representation of the group $G$ for which we are
evaluating the Torsion; however, as throughout we fix on the adjoint
representation, we do not need to indicate it in the notation.

Including the contributions from the non-acyclic connections, one expects the
partition function to take the form (24)

$Z_{BF}[M,G]\simeq\sum_{\text{acyclic}}\,\tau_{S^{1}}(\phi)^{2-N}\prod_{i=1}^{N}\tau_{S^{1}}(\phi/a_{i})+\dots$
(30)

and, as in the general situation above, one way to keep track of the ellipses
in the above formula is to use the massive Ray-Singer torsion. In the case of
Seifert manifolds all we need is the massive Ray-Singer torsion on the circle,
which is well understood BKT-MRST . Thus a suitably regularised version of
(30) is

$Z_{BF}[M,G;\mu]=\sum\,\tau_{S^{1}}(\phi,\mu)^{2-N}\prod_{i=1}^{N}\tau_{S^{1}}(\phi/a_{i},\mu)+\dots$
(31)

One can now use a mass expansion to identify the individual contributions to
the regularised partition function.

The analysis that we have presented thus far, in the covariant gauges, is an
extension of that of Witten to those flat connections which are not
necessarily flat and isolated. One shortcoming with this approach, as we have
already explained, is that with a background field approach one must, by
independent means, find the flat connections and determine the cohomology
groups about them. One would prefer an approach that evaluates the path
integral directly and does not require the split between the classical and
quantum fields. The rest of the paper is devoted to producing just such an
approach.

In particular, we will show in section 3 that the procedure developed in BT-CS
; BT-Trieste-1993 ; BT-S1 ; BT-Seifert ; BT-CCS ; BKNT to reduce the partition
function of Chern-Simons theory for a compact gauge group $G$ to a finite-
dimensional integral (over the Cartan subalgebra $\mathfrak{t}$ of
$\mathfrak{g}$) can be extended to the case at hand, namely $BF$ theory, or
Chern-Simons theory with gauge group $TG$.

The finite dimensional integrals have poles on the contour of integration and
so it then remains to define and give a meaning to them. As we will see in
section 4, a direct calculation of this finite-dimensional integral reproduces
the expected form of the partition function only partially. In particular it
captures the contributions of reducible flat connections as expected but
surprisingly does not give the expected results for the isolated and
irreducible flat connections.

This prompts us to investigate two other possible definitions in Section 5
which do lead to the expected form of the partition function discussed in this
section.

##  3 $\boldsymbol{BF}$-Theory on a Seifert Fibred 3-Manifold

In this section we specialise to Seifert fibred 3-manifolds. The extra
structure afforded by the $S^{1}$ principal bundle structure (over an orbifold
base) allows for convenient choices of gauge as well as for regularising the
Ray-Singer Torsion with the introduction of a mass term BKT-MRST . We are very
brief with the background material on Seifert manifolds as it has appeared
before. We spend some time on the gauge fixing as this is quite new for the
$TG$ theories, while the evaluation of the determinants is so close to that of
the determinants evaluated in BT-CCS that we borrow liberally from there.

###  3.1 Seifert 3-Manifolds Briefly

From now on $M$ denotes a Seifert 3-manifold. Then $M$ is a 3-manifold which
is a circle $V$-bundle over the 2 dimensional orbifold $\Sigma_{V}$ of genus
$g$ with $N$ orbifold points. A general Seifert manifold is written as
$M[\deg{\mathcal{L}_{M}},\,g,\,(a_{1},\,b_{1}),\dots,(a_{N},\,b_{N})]$ where
the $a_{i}$ are the isotropies of the orbifold points, the $b_{i}$ are the
weights of the line $V$-bundle at the orbifold points and
$\deg{\mathcal{L}_{M}}$ is the degree of that line bundle. As the Seifert
3-manifold is the circle V-bundle of the line V-bundle $\mathcal{L}_{M}$ it is
also designated by $S(\mathcal{L}_{M})$.

Let $\mathcal{L}_{0}$ be a topological line bundle at some smooth point on
$\Sigma_{V}$ with degree 1 and $\mathcal{L}_{i}$ be the line V-bundles at the
$i$-th orbifold points with degree $1/a_{i}$ then

$\mathcal{L}_{M}=\mathcal{L}_{0}^{b_{0}}\otimes\mathcal{L}_{1}^{b_{1}}\otimes\dots\otimes\mathcal{L}_{N}^{b_{N}}$
(32)

and $M$ is the circle bundle of $\mathcal{L}_{M}$ Furuta-Steer (we use
normalised Seifert data so that $1\leq b_{i}<a_{i}$).

The Seifert 3-manifold is smooth iff $\mathrm{gcd}(a_{i},\,b_{i})=1$ for each
$i$. It is a $\mathbb{Z}$ Homology Sphere ($\mathrm{H}_{1}(M,\,\mathbb{Z})=0$)
iff the line bundle $\mathcal{L}_{M}$ that defines it satisfies

$g=0,\;\;\;c_{1}(\mathcal{L}_{M})=b_{0}+\sum_{i=1}^{N}\frac{b_{i}}{a_{i}}=\pm\frac{1}{a_{1}\dots
a_{N}}\equiv\pm\frac{1}{P}$ (33)

(here we have introduced the notation $P=a_{1}\dots a_{N}$). One consequence
of these conditions is that $\mathrm{gcd}(a_{i},\,a_{j})=1$ for $i\neq j$. If
one takes a tensor power of this line V-bundle,
$\mathcal{L}_{M_{d}}=\mathcal{L}_{M}^{\otimes d}$, then the Seifert manifold
$M_{d}=M/\mathbb{Z}_{d}$ is a $\mathbb{Q}$ Homology Sphere
($\mathrm{H}_{1}(M_{d},\,\mathbb{Q})=0$) with

$g=0,\;\;\;c_{1}(\mathcal{L}_{M_{d}})=c_{1}(\mathcal{L}_{M}^{\otimes
d})=\pm\frac{d}{a_{1}\dots a_{N}}=\pm\frac{d}{P}$ (34)

and

$|d|=|\mathrm{H}_{1}(M_{d},\,\mathbb{Z})|$ (35)

These are the 3-manifolds that we will exclusively concentrate on. The reason
for this choice of $M$ is two fold.

Firstly, if $M$ is neither a $\mathbb{Z}$ nor a $\mathbb{Q}$ homology sphere,
then it will necessarily have a non-zero dimensional moduli space of Abelian
flat connections which in turn means that there will be Abelian $B$ zero modes
and hence non-compact directions to deal with.111Even if $M$ is a $\mathbb{Z}$
or $\mathbb{Q}$ homology sphere, there may be non-zero dimensional components
of the moduli space of flat connections, but there are many $M$ for which the
moduli space of flat connections is made up of a finite number of points. We
do not want to have to worry about such a situation here, as it is somewhat
tangential to the other issues that we wish to address.

Far from all $\mathbb{Z}$ or $\mathbb{Q}$ homology spheres are Seifert. The
reason for choosing $M$ to be more specifically a Seifert $\mathbb{Z}$ or
$\mathbb{Q}$ homology sphere is a more pragmatic one. Having $M$ a Seifert
manifold means that we have Fourier mode decomposition of all the fields, so
their components can be ultimately viewed as living on $\Sigma_{V}$, and, as
shown in BT-CS ; BT-Trieste-1993 ; BT-S1 ; BT-Seifert ; BKNT ; BT-CCS , there
are specific gauge choices available that allow one to significantly simplify
the evaluation of the partition function.

###  3.2 $BF$ Action and Gauge Transformations on a Seifert Manifold

As the Seifert fibred 3-manifold $M$ is a principal $U(1)$ bundle, it also
comes equipped with the fundamental vector field $\xi$ which generates the
$U(1)$ action. We also equip $M$ with a (nowhere vanishing) connection 1-form
$\kappa$, i.e.

$\iota_{\xi}\kappa=1\quad,\quad L_{\xi}\kappa=\iota_{\xi}d\kappa=0\;\;,$ (36)

and

$\int_{M}\kappa\wedge d\kappa=c_{1}\left(\mathcal{L}_{M}\right)\;\;.$ (37)

Note that, acting on a form of any degree, one has

$\iota_{\xi}\kappa+\kappa\iota_{\xi}=1$ (38)

so that $\iota_{\xi}\kappa$ and $\kappa\iota_{\xi}$ are projection operators
onto the space of horizontal and vertical forms of fixed degree respectively.

Given a trivial $G$-bundle on a Seifert 3-manifold $M$, we correspondingly
decompose the connection $A$ and 1-form $B$ of $BF$ theory as

$A=A_{\Sigma}+\kappa\phi,\;\;\;B=B_{\Sigma}+\kappa\lambda$ (39)

where

$A_{\Sigma}=\iota_{\xi}\kappa
A,\;\;\;\phi=\iota_{\xi}A,\;\;\;B_{\Sigma}=\iota_{\xi}\kappa
B,\;\;\;\lambda=\iota_{\xi}B$ (40)

We may also decompose the exterior derivative as

$d=\iota_{\xi}\kappa d+\kappa\iota_{\xi}d=d^{\Sigma}+\kappa\iota_{\xi}d$ (41)

with twisted versions

$d_{A}=\iota_{\xi}\kappa d_{A}+\kappa\iota_{\xi}d_{A}$ (42)

which acts on 0-forms as

$d_{A}\beta=d^{\Sigma}_{A}\beta+\kappa D_{\phi}\beta$ (43)

In terms of this decomposition, the gauge transformations (8) take the form,
with $t$ a local fibre coordinate

$\displaystyle A_{\Sigma}\rightarrow
g^{-1}d_{A}^{\Sigma}g,\;\;\;\phi\rightarrow g^{-1}(\partial_{t}+\phi)g$
$\displaystyle B_{\Sigma}\rightarrow
g^{-1}(B_{\Sigma}+d^{\Sigma}_{A}\Lambda)g,\;\;\;\lambda\rightarrow
g^{-1}\left(\lambda+D_{\phi}\Lambda\right)g$ (44)

and the action (7) becomes

$I_{BF}=\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B_{\Sigma}\wedge\kappa
D_{\phi}A_{\Sigma}+\kappa_{\Sigma}B\wedge d^{\Sigma}\phi+\lambda\kappa\wedge
F_{A_{\Sigma}}+\kappa\wedge d\kappa\phi\lambda\right)}$ (45)

with horizontal curvature

$F_{A_{\Sigma}}=d^{\Sigma}A_{\Sigma}+A_{\Sigma}^{2}$ (46)

###  3.3 Gauge Fixing 1: Temporal Gauge for $A$ and $B$

One of the great benefits of considering a Seifert manifold is that there are
very useful non-covariant gauges available. In particular, on a Seifert
manifold we can always choose the “temporal gauge” that the fibre-component
$\phi$ of $A$ is constant along the fibre,

$\iota_{\xi}d\phi=0\;\;.$ (47)

It turns out that (for generic $\phi$) one can also impose the analogouus
condition on $\lambda$,

$\iota_{\xi}d\lambda=0\;\;.$ (48)

In local coordinates with $t$ a fibre coordinate, these conditions simply read

$\frac{\partial}{\partial
t}\phi=0,\quad\mathrm{and}\quad\frac{\partial}{\partial t}\lambda=0$ (49)

The first of these can be achieved by having $\phi$ gauge equivalent to some
‘time’ independent field $\phi_{0}$; i.e. we have to solve

$g(t)^{-1}(\partial_{t}+\phi)g(t)=\phi_{0}\quad\Leftrightarrow\quad\partial_{t}g(t)=g(t)\phi_{0}-\phi(t)g(t)$
(50)

This equation has the solution

$g(t)=P\exp{\left(\int_{t}^{0}\phi(s)ds\right)}\;\mbox{e}^{\,\textstyle
t\phi_{0}}$ (51)

(where we have without loss of generality chosen $g(0)=1$). Periodicity
$g(1)=g(0)$ of the gauge parameter now determines $\phi_{0}$ to be the average
value of $\phi$ in the sense that

$\mbox{e}^{\,\textstyle\phi_{0}}=P\exp{\left(\int_{0}^{1}\phi(s)ds\right)}$
(52)

The second gauge condition, $\partial_{t}\lambda=0$, is perhaps not familiar
and as there is a hidden subtlety we will deal with it in some detail. Suppose
we have already implemented the first gauge choice (thus in the following
$\phi=\phi_{0}$ is taken to satisfy $\partial_{t}\phi=0$). To arrive at
$\lambda$ constant along the fibre we need to solve

$\lambda+D_{\phi}\Lambda=\lambda_{0}$ (53)

for some $\lambda_{0}$ constant in $t$ to be determined. Writing

$\partial_{t}\left(\mbox{e}^{\,\textstyle
t\phi}\Lambda\mbox{e}^{\,\textstyle-t\phi}\right)=\mbox{e}^{\,\textstyle
t\phi}(D_{\phi}\Lambda)\mbox{e}^{\,\textstyle-t\phi}$ (54)

one sees that this equation has the solution.

$h(t)\Lambda(t)h^{-1}(t)=\int_{0}^{t}\mbox{e}^{\,\textstyle
s\phi}\,(\lambda_{0}-\lambda(s))\,\mbox{e}^{\,\textstyle-s\phi}\,ds+\Lambda(0)$
(55)

where

$h(t)=\mbox{e}^{\,\textstyle t\phi},\quad h=\mbox{e}^{\,\textstyle\phi}$ (56)

Periodicity in the gauge parameter $\Lambda(1)=\Lambda(0)$ gives us the
equation

$(\mathsf{Ad}(h)-1)\Lambda(0)=\int_{0}^{1}\mbox{e}^{\,\textstyle
s\phi}\,(\lambda_{0}-\lambda(s))\,\mbox{e}^{\,\textstyle-s\phi}\,ds$ (57)

Before going on we note that a shift of $\Lambda$ by a constant (in $t$)
$\Lambda_{0}$ simply amounts to a shift in $\lambda_{0}$,

$\Lambda\rightarrow\Lambda+\Lambda_{0}\quad\Rightarrow\quad\lambda_{0}\rightarrow\lambda_{0}+[\phi,\Lambda_{0}]$
(58)

Without loss of generality we can therefore assume that $\Lambda(0)=0$, so
that the periodicity condition (57) implies

$\int_{0}^{1}\mbox{e}^{\,\textstyle
s\phi}\,(\lambda_{0}-\lambda(s))\,\mbox{e}^{\,\textstyle-s\phi}\,ds=0\;\;.$
(59)

The integral involving $\lambda_{0}$ can be done explicitly, with the result

$\frac{\mbox{e}^{\,\textstyle\mathsf{ad}(\phi)}-1}{\mathsf{ad}(\phi)}\lambda_{0}=\int^{1}_{0}\mbox{e}^{\,\textstyle
s\,\mathsf{ad}(\phi)}\,\lambda(s)ds\;\;.$ (60)

For generic values of $\phi$, the operator on the left-hand side can be
inverted to give

$\lambda_{0}=\mathsf{ad}(\phi)(\mbox{e}^{\,\textstyle\mathsf{ad}(\phi)}-1)^{-1}\int^{1}_{0}\mbox{e}^{\,\textstyle
s\,\mathsf{ad}(\phi)}\,\lambda(s)ds$ (61)

which tells us in which sense $\lambda_{0}$ can be regarded as a
$\phi$-weighted average of $\lambda$ over the fibre.

However, we see that this solution fails when $\phi$ is not suitably generic,
so that the operator $\exp\mathsf{ad}(\phi)-1$ has zero-modes. This should not
come as a suprise, as this is precisely the condition that the operator
$D_{\phi}=\partial_{t}+\mathsf{ad}(\phi)$ has zero-modes, so that the gauge
fixing condition (53) cannot be solved for all $\lambda(t)$. Indeed, for a
zero mode $\psi$ of $D_{\phi}$ one has (see also the discussion in section
2.3.4 of BKT-MRST )

$D_{\phi}\psi=0\quad\Rightarrow\quad\psi(t)=\mbox{e}^{\,\textstyle
t\mathsf{ad}(\phi)}\psi(0)\;\;,$ (62)

and periodicity $\psi(1)=\psi(0)$ implies

$\psi(1)=\psi(0)\quad\Rightarrow\quad\left(\mbox{e}^{\,\textstyle\mathsf{ad}(\phi)}-1\right)\psi(0)=0\;\;.$
(63)

For $\phi$ taking values in the Cartan subalgebra $\mathfrak{t}$ of
$\mathfrak{g}$, this is simply the condition that $\phi$ has a component that
is an element of the integral lattice $I(G)$ of $G$.

Normally this kind of constraint arising only for highly non-generic field
configurations would not be an issue as one is integrating over $\phi$ and the
points where $\phi$ is integral have measure zero. Unfortunately, in $BF$
theory we get delta function support onto particular values of $\phi$. In
particular we will see when we are considering $\mathbb{Z}$ Homology Spheres
that these integral points are the ones that are selected (cf. (91)).

###  3.4 Gauge Fixing 2: Abelianisation and Emergence of Non-Trivial Line
Bundles

After having chosen the gauges (49), we still have gauge transformations
available that are constant along the fibre. For a compact gauge group, one
can locally use this residual gauge feedom to conjugate the field $\phi$ into
the Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{g}$, i.e. one can impose
the gauge condition

$\phi^{\mathfrak{k}}=0\quad\Leftrightarrow\quad\phi=\phi^{\mathfrak{t}}\;\;,$
(64)

where $\mathfrak{k}$ is the orthogonal complement of the Cartan subalgebra
$\mathfrak{t}$ of $\mathfrak{g}$,

$\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{k}\;\;.$ (65)

We will come back to the global issues involved in this choice of gauge below.

However, first of all we need to address the question if we can do something
analogous in the case at hand, with gauge group $TG$. Since $TG$ is neither
compact nor semi-simple, this is not a priori obvious. However, we show in
Appendix A.2 that one can use the adjoint action of $TG$ on its Lie algebra to
conjugate any element into an element of $\mathfrak{t}\oplus\mathfrak{t}$,
which plays the role of the Cartan subalgebra of the Lie algebra of $TG$. From
one perspective, the reason we are able to do this is that $TG$ arises as the
Inönü-Wigner contraction of a group where this is certainly possible, namely
$G\times G$.

Explicitly, this means that we can use the gauge transformation

$(\phi,\lambda)\mapsto(g,\Lambda)^{-1}.(\phi,\lambda).(g,\Lambda)=(g^{-1}\phi
g,g^{-1}(\lambda+[\phi,\Lambda])g)$ (66)

to (locally) impose simultaneously the Abelianising gauge conditions

$\phi^{\mathfrak{k}}=0\quad,\quad\lambda^{\mathfrak{k}}=0\quad\Leftrightarrow\quad\phi=\phi^{\mathfrak{t}}\quad,\quad\lambda=\lambda^{\mathfrak{t}}$
(67)

Now let us turn to the global issues regarding this gauge choice, which
implies that we will be dealing with an Abelian gauge theory on $\Sigma_{V}$.
As has been described in detail in BT-Trieste-1993 ; BT-DIA , there are
topological obstructions to imposing the gauge choice on $\phi$ globally; in
particular, conjugating $\phi$ to the Cartan subalgebra $\mathfrak{t}$ on
$\Sigma_{V}$ is not possible with smooth gauge transformations. However, if we
insist on doing so anyway, the price to be paid is that we must sum over all
available line bundles on $\Sigma_{V}$. The reason for the emergence of this
sum has been explained a number of times BT-DIA ; BT-CS ; BT-Trieste-1993 ;
BT-S1 ; BT-Seifert ; BKNT .

Since $TG$ is contractible to $G$ (and there are no topological issues
involved in the shift transformation
$\lambda\rightarrow\lambda+[\phi,\Lambda]$ beyond those involving $\phi$
itself), there are no additional topological obstructions arising from
diagonalisation in $TG$.

In practice there are two ways to introduce the non-trivial line bundles and
the sum over them. They both have merits and in any case are equivalent. The
first makes contact with flat connections rather straightforwardly while the
other is computationally easier.

We begin by describing the method using background fields described in more
detail in BKNT . The available line V-bundles on $\Sigma_{V}$ are

$\mathcal{L}=\mathcal{L}_{0}^{\otimes n_{0}}\otimes\mathcal{L}_{1}^{\otimes
n_{1}}\otimes\dots\otimes\mathcal{L}_{N}^{\otimes n_{N}}$ (68)

and the basic idea is to allow for a non-trivial background connection on
patches about each divisor, with the divisors being the singular points and
with one divisor being a regular point of $\Sigma_{V}$. It was shown in BKNT
that in the tubular neighbourhood $V_{(a_{i},b_{i})}$ of the $i$-th singular
point one could introduce a connection 1-form $\kappa_{i}$ which would glue to
a global one form $\kappa$ on $M$. In terms of these local forms and the local
surgery data

$a_{i}s_{i}-b_{i}r_{i}=-1,\quad\mathrm{with}\quad(a_{i},b_{i})=1$ (69)

the local orbifold data, the background connection takes the simple form222A
word on notation. Throughout we write connections in a basis of simple roots
$\alpha_{i}$. For semi-simple simply laced Lie algebras we have that the
natural inner product $\langle\alpha_{i},\alpha_{j}\rangle=2\delta_{ij}$. The
bold facing of integers indicates that we mean $\mathbf{m}=m^{i}\alpha_{i}$
and for each $i$ we have $m^{i}\in\mathbb{Z}$. We will write that
$\mathbf{m}\in I(G)$ where $I(G)$ is the integral lattice of $G$. As $G$ is
simply connected we have identified the root lattice with $I(G)$.

$\mathcal{A}_{B}=2\pi
i\left(\mathbf{n}_{0}\kappa_{0}+\sum_{i=1}^{N}\mathbf{n}_{i}r_{i}\kappa_{i}\right)$
(70)

where the $\mathbf{n}$ are Hermitian and lie in $\mathfrak{t}$. With an abuse
of notation, we may write the background as

$\kappa\,\phi_{B}=2\pi
i\left(\mathbf{n}_{0}\kappa_{0}+\sum_{i=1}^{N}\mathbf{n}_{i}r_{i}\kappa_{i}\right)$
(71)

Each of the local forms $\kappa_{i}$ extends outside of the tubular
neighbourhood to $d\theta$ on the fibre. One also has that

$\int_{V_{(a_{i},b_{i})}}\kappa_{i}d\kappa_{i}=\frac{b_{i}}{a_{i}}$ (72)

with the regular point having weight $a_{0}=1$. Glued together the
$\kappa_{i}$ form a global, smooth 1-form
$\kappa=\kappa_{0}\cup\kappa_{1}\cup\dots\cup\kappa_{N}$ that defines a
principal bundle structure on $M$ with

$c_{1}(\mathcal{L}_{M})=\int_{M}\kappa\wedge
d\kappa=b_{0}+\sum_{i=1}^{N}\frac{b_{i}}{a_{i}}$ (73)

With the background field in place, one makes the substitution

$\kappa\phi\rightarrow\kappa\phi+\kappa\phi_{B}$ (74)

everywhere in the path integral, with the second term on the right given by
(71).

Alternatively BT-CS ; BT-Trieste-1993 ; BT-S1 ; BT-Seifert one sets all the
$\mathbf{n}_{i}$ $i\neq 0$ to zero and $\kappa_{0}\rightarrow\kappa$ (and this
is the natural procedure from the point of view of obstruction theory outlined
in BT-DIA and extended here to the manifolds that we are considering).
Otherwise one keeps the substitution (74) in the path integral. In Appendix B
we show that there is enough symmetry in the theory to pass between these
different formulations.

###  3.5 Reduction of the Partition Function to a Finite-Dimensional Integral

With all the preliminaries in place we can now outline how to perform the path
integral for the partition function.

Firstly the action (45), taking into account the non-trivial bundles that
arise on Abelianisation, goes over to

$I_{BF}=\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B_{\Sigma}\wedge\kappa
D_{\phi+\phi_{B}}A_{\Sigma}+\kappa B_{\Sigma}\wedge
d^{\Sigma}\phi+\lambda\kappa\wedge F_{A_{\Sigma}}+\kappa\wedge
d\kappa(\phi+\phi_{B})\lambda\right)}$ (75)

with $\kappa\phi_{B}$ and $\kappa d\kappa\phi_{B}$ suitably understood. Indeed
we note that the part of $B_{\Sigma}$ that is constant along the fibre and
takes values in the Cartan subalgebra $\mathfrak{t}$ only appears through the
term

$\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(\kappa B_{\Sigma}\wedge
d^{\Sigma}\phi\right)}\;\;.$ (76)

Thus integrating out these modes of $B_{\Sigma}$ imposes the condition that
$d^{\Sigma}\phi=0$. Given the gauge conditions $\partial_{t}\phi=0$, this
implies that $\phi$ is constant on $M$. Likewise the component of $A_{\Sigma}$
constant along the fibre and in the Cartan subalgebra only appears in

$\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(\kappa A_{\Sigma}\wedge
d^{\Sigma}\lambda\right)}$ (77)

so integrating over that mode of $A_{\Sigma}$ fixes $d^{\Sigma}\lambda=0$
which, together with $\partial_{t}\lambda=0$, tells us that $\lambda$ is also
constant on $M$. The other components of $A^{\mathfrak{t}}_{\Sigma}$ and
$B^{\mathfrak{t}}_{\Sigma}$ are paired with each other and lead to an overall
constant when integrated over.

We should also take care of any possible harmonic modes of the components of
$A$ and $B$ that we have integrated over. If there any such harmonic modes of
$B$, they would be non-compact directions and would lead to divergences. In
order to avoid this we fix on the genus of $\Sigma_{V}$ to be zero, as in (33)
and (34) thus ensuring that there are no $B^{\mathfrak{t}}_{\Sigma}$ zero
modes to contend with. The stronger choice that we have made of having $M$ be
a $\mathbb{Z}$ or $\mathbb{Q}$ homology sphere is to ensure that there are no
moduli of Abelian connections on $M$ as then we would necessarily have Abelian
$B$ zero modes.

The form that we now have for the action is

$I_{BF}=\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B^{\mathfrak{k}}_{\Sigma}\wedge\kappa
D_{\phi+\phi_{B}}A^{\mathfrak{k}}_{\Sigma}+\kappa\wedge\lambda
A^{\mathfrak{k}}_{\Sigma}A^{\mathfrak{k}}_{\Sigma}\right)}+\mathop{\mathrm{Tr}}\nolimits{\lambda\left(c_{1}(\mathcal{L}_{M})\phi+2\pi
i\widehat{\mathbf{q}}\right)}$ (78)

where

$\widehat{\mathbf{q}}=\sum_{i=0}^{N}\frac{1}{a_{i}}\mathbf{n}_{i}$ (79)

We are now ready to integrate out the fields $A^{\mathfrak{k}}_{\Sigma}$ and
$B^{\mathfrak{k}}_{\Sigma}$ that appear in (78). There are also ghost terms of
the same type to take into account. The functional determinant that we obtain
and need to evaluate is

$\left.\mathop{\mathrm{Det}}\nolimits{\left(\begin{array}[]{cc}\mathsf{ad}_{\lambda}&D_{\phi+\phi_{B}}\\\
-D_{\phi+\phi_{B}}&0\end{array}\right)}\right|_{\Omega^{1}_{H}(M,\mathfrak{k})\oplus\Omega^{1}_{H}(M,\mathfrak{k})}^{-1/2}$
(80)

There is a similar determinant from the ghost terms. As usual we will expand
in Fourier modes along the $S^{1}$ fibre and we will regulate with a
$\zeta$-function regularisation. What one finds is, even upon regularisation,
that the $\mathsf{ad}_{\lambda}$ part of the determinant does not contribute.
In this way what finally needs to be evaluated is

$\left|\mathop{\mathrm{Det}}\nolimits{\left(D_{\phi+\phi_{B}}\right)}\right|_{2\Omega^{0}_{H}(M,\mathfrak{k})\ominus\Omega^{1}_{H}(M,\mathfrak{k})}$
(81)

For every Fourier mode
$2\Omega^{0}_{H}(M,\mathfrak{k})\ominus\Omega^{1}_{H}(M,\mathfrak{k})$ is
roughly the Euler characteristic of $\Sigma_{V}$. This, almost, cancellation
of all the modes in the functional determinant can be used to evaluate it.
However, the field $\phi_{B}$ is not constant on $\Sigma_{V}$ unlike $\phi$.
This issue was circumvented in BKNT by using a density form of the index
theorem originally found in BT-CS . The determinant in BKNT is the square root
of (81), so we may straightforwardly borrow the result from there. In this way
we see that the absolute value (81) is

$\tau_{M}\left(\phi;\,\mathbf{n}_{i}\right)=\tau_{S^{1}}(\phi+2\pi
i\mathbf{n}_{0})^{2-2g-N}.\prod_{i=1}^{N}\tau_{S^{1}}((\phi+2\pi
ir_{i}\mathbf{n}_{i})/a_{i})$ (82)

where $\tau_{M}$ is the Ray-Singer torsion of $M$. The form of $\tau_{M}$
shows that it is given by circles that is in terms of the Ray-Singer torsion
of $S^{1}$’s Fried . For completeness we note that

$\tau_{S^{1}}(\varphi)=\mathrm{det}_{\mathfrak{k}}\left(I-\mathsf{Ad}{(\exp{\varphi})}\right)$
(83)

so that $\mathbf{n}_{0}$ drops out of (82) since it has integral (lattice)
entries.

We regularise the Ray-Singer Torsion by introducing a mass BKT-MRST . In the
present context the mass is introduced into the determinants (80). This is not
consistent with the $TG$ symmetry but does preserve the Abelian part of it.
One exchanges (83) with

$\tau_{S^{1}}(\varphi,\mu)=\mathrm{det}_{\mathfrak{g}}\left(I-\mbox{e}^{\,\textstyle-2\pi\mu}\mathsf{Ad}{(\exp{\varphi})}\right)$
(84)

and it is understood that we use the massive form for the Ray-Singer Torsion
of the torsion on the circles on the right hand side of (82). For the details
of how this comes about see Example 2.5 in BKT-MRST . We should note we have
introduced extra factors of $\mu$ (namely by the factor
$(1-\mbox{e}^{\,\textstyle-2\pi\mu})^{\dim T}$) by passing to (84) and hence
we have also changed the normalisation of the path integral

Putting everthing together, we see that we have now managed to reduce the path
integral for the $BF$ partition function to a finite dimensional integral over
the (by now constant) fields $\phi$ and $\lambda$,

$Z_{BF}[M,G]=\sum_{\mathbf{n}_{0}}\dots\sum_{\mathbf{n}_{N}}\int_{\mathfrak{t}\oplus\mathfrak{t}}\tau_{M}\left(\phi;\,\mathbf{n}_{i}\right)\,.\,\exp{\left(i\mathop{\mathrm{Tr}}\nolimits{\lambda\left(c_{1}(\mathcal{L}_{M})\phi+2\pi
i\widehat{\mathbf{q}}\right)}\right)}$ (85)

This is the integral that we will look at in detail in the the following
section.

##  4 Definition 1: Direct Evaluation of the $\boldsymbol{BF}$ (Path) Integral

As we saw in the last section we are able to perform the functional integral
representing the partition function $Z_{BF}$ to the point that we are left
with finite dimensional integrals. In this section we analyse the remaining
integrals and notice that there are issues with them. Indeed this prompts us
to search for a definition for $Z_{BF}$. This definition does not agree with
the perturbative result of Witten WBF in the case of isolated and irreducible
connections and we end the section by explaining where there could be possible
issues with one of the gauge conditions that we have chosen. The formulae
obtained capture those Abelian connections in $\mathbb{Q}$ Homology Spheres
and in these cases reproduce the expected result (31).

###  4.1 Symmetries and other Properties of the Finite Dimensional Integrals

At this point we have collected all the pieces and the $BF$ partition
function, for the 3-manifolds under consideration, is now a finite dimensional
integral

$Z_{BF}[M,G]=\sum_{\mathbf{n}_{0}}\dots\sum_{\mathbf{n}_{N}}\int_{\mathfrak{t}\oplus\mathfrak{t}}\tau_{M}\left(\phi;\,\mathbf{n}_{i}\right)\,.\,\exp{\left(i\mathop{\mathrm{Tr}}\nolimits{\lambda\left(d\phi/P+2\pi
i\widehat{\mathbf{q}}\right)}\right)}$ (86)

where the integral over $\mathfrak{t}\oplus\mathfrak{t}$ is that over the
constant Abelian “fields” $\phi$ and $\lambda$ and we have used the fact that
$c_{1}(\mathcal{L}_{M})=d/P$ for the 3-manifolds we are considering as given
in (33) and (34) which in particular means that $g=0$. There are no sums over
the $\mathbf{n}_{i}$ $i\neq 0$ if we have set them to zero, in which case we
also have that $\widehat{\mathbf{q}}=\mathbf{n}_{0}$. Note that as the gauge
group $G$ is compact connected and simply connected the $G$ bundles over $M$
are trivial and, on Abelianisation, the line V bundles at the smooth point are
‘honest’ line bundles so that $\mathbf{n}_{0}\in\mathbb{Z}^{\mathbf{rk}}$.
Consequently, in this case, $\mathbf{n}_{0}$ only appears in the action in
(86) (for either formulation of background).

In order to understand and ultimately evaluate the integral 86, we note the
following properties of the integrand:

  1. 1.

both the action and the Ray-Singer torsion are invariant under the discrete
symmetry

$\phi\rightarrow\phi+2\pi
iP\mathbf{r},\;\;\;\mathbf{n}_{0}\rightarrow\mathbf{n}_{0}-d\mathbf{r}\;\;\;\mathrm{with}\;\;\mathbf{r}\in\mathbb{Z}^{\mathbf{rk}(G)},\;\;P=\prod_{i=1}^{N}a_{i}$
(87)

  2. 2.

the Ray-Singer torsion with $N\geq 3$ has poles at $\alpha(\phi)=n\pi$ with
$n\neq Pm$ (while for $N\leq 2$ there are no poles)

  3. 3.

the Ray-Singer torsion has zeros at $\alpha(\phi)=m\pi P$

  4. 4.

the Ray-Singer torsion is an even function of its arguments, in the sense that

$\tau_{M}(\phi,\mathbf{n}_{i})=\tau_{M}(-\phi,a_{i}-\mathbf{n}_{i})=\tau_{M}(-\phi,-\mathbf{n}_{i})$
(88)

Using the symmetry (87), we may either ‘compactify’ the field $\phi$ or limit
the range of $\mathbf{n}_{0}$. Either the way, the integral over $\lambda$
leads to a delta function constraint on $\phi$, namely

$\phi=\phi_{C}\equiv-\frac{2\pi iP}{d}\widehat{\mathbf{q}}$ (89)

and so it appears that only these values contribute to the path integral.
These values correspond to reducible connections as we have seen in the past
BT-CCS and so one might conclude that the only flat connections that
contribute to the $BF$ partition function are those that are Abelian. However,
because of the presence of poles, this conclusion is a bit quick, and we will
take a closer look at this issue in section 4.4.

Before moving on, it is worthwhile reminding ourselves that there are
restrictions on $M$ so the analysis we are performing holds. As a counter-
example, presume that $M$ is a product $\Sigma\times S^{1}$, or more generally
a Seifert mapping torus, i.e. with $c_{1}(\mathcal{L}_{M})=0$ (so definitely
not one of the manifolds that we are considering). Then there would be no
symmetry such as (87) for the action and the integral over $\lambda$ would not
lead to (89).

###  4.2 The $\phi_{\mathbf{C}}$ Contribution to $\mathbb{Z}$ Homology Spheres

Here we specialise to the $\phi_{C}$ contribution to the path integral for $M$
either an $\mathbb{Z}$ or a $\mathbb{Q}$ Homology Sphere. We will use the
symmetries available not to limit the range of $\phi$ but rather to set
$\mathbf{n}_{0}\in I(G)/dI(G)$ and we will keep the $\mathbf{n}_{i}$ and so
take the classical field contribution to be

$\phi_{C}=-2\pi i\frac{P}{d}\sum_{i=0}^{N}\frac{\mathbf{n}_{i}}{a_{i}}$ (90)

and any other choice leads to the same conclusions that we arrive at
presently. As we only want the $\phi_{C}$ contribution we must ensure that
$\tau_{M}$ has no poles and this is achieved by passing to the massive Ray-
Singer torsion $\tau_{M}(\phi,\mu)$ which we do.

If $M$ is a $\mathbb{Z}$ homology sphere then $d=1$ and the symmetry allows us
to set $\mathbf{n}_{0}=0$ so that

$\phi_{C}=-2\pi iP\sum_{i=1}^{N}\frac{\mathbf{n}_{i}}{a_{i}}\in I(G)$ (91)

Note that these are precisely the (potentially problematic) integral values of
$\phi$ discussed at the end of section 3.3. We will (have to) come back to
this below. A tiny bit of gymnastics333By (33)
$Pb_{j}/a_{j}=1-a_{j}t_{j},\;t_{j}\in\mathbb{Z}$ and
$Pb_{j}r_{j}/a_{j}=r_{j}-r_{j}a_{j}t_{j}=(1-a_{j}s_{j})P/a_{j},\;t_{j}\in\mathbb{Z}$
whence $r_{j}-P/a_{j}=0\mod a_{j}\mathbb{Z}$. Consequently, $\phi_{C}+2\pi
i\mathbf{n}_{j}r_{j}=-2\pi iP\sum_{i\neq j}\mathbf{n}_{i}/a_{i}+2\pi
i\mathbf{n}_{j}.(r_{j}-P/a_{j})=0\mod 2\pi ia_{j}I(G)$. shows that

$\phi_{C}+2\pi i\mathbf{n}_{j}r_{j}=0\mod 2\pi ia_{j}I(G)$ (92)

One immediately sees that the Ray-Singer Torsion (without a mass) on the
circle vanishes and the overall ratio in (82) is zero. The poles and the zeros
of the finite dimensional integral arise as the zeros of the (inverse)
determinant of the $D_{\phi}$ operator. The poles are there when $D_{\phi}$
acts on 1-forms and the zeros arise when acting on 0-forms at the special
values (89) of $\phi$.

In order to understand the contributions we turn on the mass regulator, as in
(84). Then we find

$\tau_{M}(\phi,\mu)=\mathrm{det}_{\mathfrak{g}}\left(I-\mbox{e}^{\,\textstyle-2\pi\mu}\right)^{2-N}\prod_{i=1}^{N}\mathrm{\det}_{\mathfrak{g}}\left(I-\mbox{e}^{\,\textstyle-2\pi\mu/a_{i}}\right)$
(93)

The small mass limit gives us, as in (28),

$\tau_{M}(\phi,\mu)=(2\pi\mu)^{2\dim{G}}.(a_{1}\dots a_{N})^{-\dim{G}}+\dots$
(94)

The exponent of $\mu$ is understood as follows: the torsion of $M$ has been
pushed to that on $S^{1}$ and as on $S^{1}$ the twisted cohomology groups
satisfy $\mathrm{H}_{\omega}^{0}=\mathrm{H}_{\omega}^{1}$ the combination
$3\dim{\mathrm{H}_{\omega}^{0}}-\dim{\mathrm{H}_{\omega}^{1}}=2\dim{\mathrm{H}_{\omega}^{0}}$
and $\dim{\mathrm{H}_{\omega}^{0}}=\dim{G}$ for the trivial connection which
is the maximally reducible connection.

This result is quite far from the known Reidemeister Torsion for regular
elements of a Brieskorn Sphere $\Sigma(a_{1},a_{2},a_{3})$ Freed-Brieskorn ,

$(a_{1}a_{2}a_{3})^{-1}\prod_{i=1}^{3}4\sin^{2}{\left(\frac{n_{i}r_{i}\pi}{a_{i}}\right)}$
(95)

in the case of $SU(2)$. The semi-classical analysis of Witten would have
suggested

$Z_{BF}[\Sigma(a_{1},a_{2},a_{3})\,SU(2)]=(a_{1}a_{2}a_{3})^{-1}\prod_{i=1}^{3}\sum_{n_{i}=1}^{a_{i}-1}4\sin^{2}{\left(\frac{n_{i}r_{i}\pi}{a_{i}}\right)}+\mathrm{reducible\,terms}$
(96)

rather than the result that we obtained.

From (94) we understand that the $\phi_{C}$ contributions that we are looking
at come from reducible connections, for these manifolds the trivial connection
as $G=SU(2)$, and so it appears that we see only the non acyclic terms in
(96). For example for $S^{3}$ the only flat connection is the trivial
connection and, as it has maximal reducibility, one sees that a formula of the
type (94) correctly captures this fact. Indeed, for any (trivial) $G$ bundle
over a $\mathbb{Z}$ Homology sphere the trivial connection is flat and
isolated and would give a contribution as above. What are missing are the
contributions of non-reducible or, for higher rank groups, less reducible
connections.

###  4.3 The $\phi_{\mathbf{C}}$ Contribution to $\mathbb{Q}$ Homology Spheres

The zeros that were faced when $M$ is an $\mathbb{Z}$ Homology Sphere are
avoided to some extent when the manifold is a $\mathbb{Q}$ Homology Sphere.
The first advantage is that the points at which we wish to evaluate the
torsion are no longer necessarily at integer values but rather at

$\phi_{C}=-2\pi i\frac{P}{d}\sum_{i=0}^{N}\frac{\mathbf{n}_{i}}{a_{i}}$ (97)

Even if $\phi_{C}$ does at some points take integer values, the mass
regularised torsion remains well defined and that is what we will use. We will
give an example of this later on. Recall that here we fix the symmetry (87) by
insisting that $\mathbf{n}_{0}\in I(G)/d.I(G)$.

The $\phi_{C}$ contribution to the partition function is, with the masses re-
instated so that we may follow the reducibility of the connections,

$\displaystyle\left.Z_{BF}[M,G,\mu]\right|_{\phi_{C}}$
$\displaystyle\;\;\;=\sum_{\mathbf{n}_{0}}\dots\sum_{\mathbf{n_{N}}}\tau_{S^{1}}(-2\pi
i(P/d)\widehat{\mathbf{q}}+2\pi
i\mathbf{n}_{0},\mu)^{2-N}.\prod_{i=1}^{N}\tau_{S^{1}}((-2\pi
i(P/d)\widehat{\mathbf{q}}+2\pi ir_{i}\mathbf{n}_{i})/a_{i},\mu/a_{i})$ (98)

This result can be understood in terms of the homotopy representations for
flat connections on the Seifert $\mathbb{Q}$ Homology Sphere to which we turn
shortly. Notice that the contribution from these connections is precisely of
the form that was anticipated in (26). However, for present purposes the most
useful form for us is where we have set the $\mathbf{n}_{i}$ for $i\geq 1$ to
zero and restrict $\mathbf{n}_{0}\in I(G)/d.I(G)$

$\left.Z_{BF}[M,G,\mu]\right|_{\phi_{C}}=\sum_{\mathbf{n}_{0}\in
I(G)/d.I(G)}\tau_{S^{1}}(-2\pi
i(P/d)\mathbf{n}_{0},\mu)^{2-N}.\prod_{i=1}^{N}\tau_{S^{1}}(-2\pi
i(P/d)\mathbf{n}_{0}/a_{i},\mu/a_{i})$ (99)

The classical examples of $\mathbb{Q}$ Homology Spheres are of course the Lens
spaces $L(p,q)$. The simplest Seifert presentation of the Lens space $L(p,q)$
is with $N=1$, $d=p$ and $a=q^{*}$ where $qq^{*}=1\mod p$,
$p=b_{0}q^{*}+b_{1}$. Substituting these values into (99)

$Z_{BF}[L(p,q),G,\mu]=\sum_{\mathbf{n}\in I(G)/p.I(G)}\tau_{S^{1}}(2\pi
iq^{*}\mathbf{n}/p+\mu).\tau_{S^{1}}(2\pi i\mathbf{n}/p+\mu/q^{*})$ (100)

To specialise to $G=SU(2)$ we note that the massive Ray-Singer Torsion (84)
takes the form

$\tau_{S^{1}}(\phi,\mu)=(1-\mbox{e}^{\,\textstyle-2\pi\mu}).(1+\mbox{e}^{\,\textstyle-4\pi\mu}-\mbox{e}^{\,\textstyle-2\pi\mu}2\cos{2\phi})$
(101)

so that on specialising (100) to $G=SU(2)$ one obtains

$\displaystyle Z_{BF}[L(p,q),SU(2),\mu]$
$\displaystyle=\left(\frac{\mu^{2}}{q^{*}}\right)^{3}+\dots$
$\displaystyle\;+\left(\frac{\mu^{2}}{q^{*}}\right)\sum_{n=1}^{p-1}\sin^{2}{(2\pi
iq^{*}n/p)}.\sin^{2}{(2\pi in/p)}+\dots$ (102)

We note that happily (100) and (102) have the form that we expected from the
covariant analysis in Section 2 and in particular agree with the general form
(28).

The first term in (102) corresponds to the trivial connection with
reducibility of order 3 while the second term is the sum over the Abelian
connections with reducibility of order 1 and the torsions agree with the
formulae obtained by Ray Ray for Lens spaces. This should also be compared
with the Chern-Simons large $k$ limit formula (5.7) in Jeffrey and we see that
there is complete agreement with the expression for the Ray-Singer Torsion of
the Lens spaces for the Abelian connections (the only difference with
Jeffrey’s formula is that we are summing over the Ray-Singer Torsion not its
square root).

We note that $\phi_{C}$ in this case is the complete contribution to the
integral (86) for, as noted in item 2 after (87), when $N=1$ there are no
poles in the integrand and the integral is well defined as is.

Now consider quotients of the Poincare sphere
$\Sigma(2,3,5)=M[-1,g=0,(2,1),(3,1),(5,1)]$. The fundamental group of
$\Sigma(2,3,5)$ is $\pi_{1}(\Sigma(2,3,5))=I^{*}$ the binary icosahedral group
of order 120 while the quotient manifold
$\Sigma(2,3,5)_{d}=S(\mathcal{L}_{\Sigma(2,3,5)}^{\otimes d})$ has fundamental
group $\mathbb{Z}_{d}\times I^{*}$ (Theorem 2 (v) page 112 in Orlik ). This
means that $\Sigma(2,3,5)_{d}$ comes equipped with the non-Abelian flat
connections of $\Sigma(2,3,5)$ through the $I^{*}$ factor of the fundamental
group and with Abelian connections through the $\mathbb{Z}_{d}$ factor of
$\pi_{1}$. In this case we have

$\displaystyle\left.Z_{BF}[\Sigma(2,3,5)_{d},G,\mu]\right|_{\phi_{C}}\simeq$
$\displaystyle\sum_{0\leq\mathbf{n}\leq d-1}\tau_{S^{1}}(60\pi
i\mathbf{n}/d+\mu)^{-1}.\tau_{S^{1}}(30\pi
i\mathbf{n}/d+\mu/2).\tau_{S^{1}}(20\pi
i\mathbf{n}/d+\mu/3).\tau_{S^{1}}(12\pi i\mathbf{n}/d+\mu/5)$ (103)

Apart from having the standard form we note that we can expect, at least in
the case of $SU(2)$ that this formula correctly captures the contributions of
the reducible connections.

Indeed all the so called small Seifert manifolds with genus zero and
orientable base have finite fundamental groups with cyclic factors. Hence
quite generally the space of flat connections for these manifolds naturally
splits into the non-Abelian and Abelian parts. The cyclic group is generated
by the fibre generator $h$. For large Seifert manifolds the fibre generator
also generates the cyclic group as a normal subgroup of the infinite order
fundamental group. Then $\mathbb{Z}_{d}\subset\pi_{1}(M_{d})$ and
$\mathbb{Z}_{d}=(\pi_{1}/[\pi_{1},\pi_{1}])(M_{d})$.

The interpretation of the formulae at $\phi_{C}$ that we arrived at above in
terms of homotopy representations has been given in BT-CCS . At these
reducible connections the Chern-Simons invariant agrees with that given by
Kirk and Klassen in Kirk-Klassen , Auckly Auckly and Nishi Nishi and is
consistent with the identification of the connections being reducible and
flat. Also the known relationship between the Ray-Singer Torsion for flat
connections and Reidemeister torsion corroborates the identification of the
contributions that we find.

###  4.4 A Residue Formula for $SU(2)$

As we saw in Section 4.3 that Abelian reducible connections are accounted for
by $\phi_{C}$ we now take another and closer look at the finite-dimensional
integral to see where the contributions from non-Abelian (ir)reducibles could
come from.

The integrand of (86) has poles and zeros and so we must give meaning to the
integral. A similar situation arises in Chern-Simons theory BT-CS where one
explicitly removes the poles with the addition of a mass term. There is a good
reason for this in Chern-Simons theory as, say on $\Sigma\times S^{1}$, one is
counting the finite number of states in the theory. Here, if we use a mass
regularisation then there is the same effect, namely the poles do not
contribute. However, this then excludes important contributions to the $BF$
partition function. Also, it should be said, that for $BF$ theory one does not
have a finite dimensional Hilbert space of states and so there is no
‘directive’ to ensure that the poles should not be counted in some way.

One could rotate the $\phi$ integral to say
$\exp{\left(-i\epsilon\right)}\times\mathfrak{t}$ which would ensure that one
avoids the poles altogether. However, one would need to simultaneously require
a different path of integration for $\lambda$ to ensure convergence of the
integrals. Passing to the massive Ray-Singer Torsion is another way of
avoiding the poles as the mass pushes the poles off the real $\mathfrak{t}$
axis and into the complex plane $\mathbb{C}\times\mathfrak{t}$.

Short of a guiding principle we use the first symmetry in the Properties of
the Integrand to restrict the range of each component of the $\phi$ field to
lie in $(-\pi P,\pi P)$. In this way we do not have to worry about convergence
issues with respect to integration over $\phi$. However, there are still the
poles to contend with and the integration over $\lambda$.

In order to be concrete and to fix ideas we now focus on $G=SU(2)$. From the
outset we are tasked with having to make sense of

$Z_{BF}[M,SU(2)]\sim\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}\times\mathbb{R}}\tau_{M}(\phi)\,.\exp{\left(i\lambda(\phi\frac{d}{P}-2\pi
n)\right)}$ (104)

We firstly sum over $n$ to exchange $\lambda$ with the integers and exchange
the integral with the sum

$Z_{BF}[M,SU(2)]\sim\int_{\mathbb{R}}\tau_{M}(\phi)\,.\sum_{n\in\mathbb{Z}}\exp{\left(in\phi\frac{d}{P}\right)}$
(105)

As $\tau_{M}(\phi)$ is even in $\phi$ we have

$\displaystyle Z_{BF}[M,SU(2)]$ $\displaystyle\sim\int_{-\pi P}^{\pi
P}\tau_{M}(\phi)\,.\frac{\left(1+\exp{\left(id\phi/P\right)}\right)}{\left(1-\exp{\left(id\phi/P\right)}\right)}$
$\displaystyle\sim\int_{-\pi P}^{\pi
P}\tau_{M}(\phi)\,.\frac{f(\phi)}{\left(1-\exp{\left(id\phi/P\right)}\right)}$
(106)

and we have used the symmetries available to limit the range of $\phi$ to lie
in the range $(-\pi P,\pi P)$. Here the function $f$ is

$f(\phi)=\left(1+\exp{\left(id\phi/P\right)}\right)$ (107)

There are now poles on the real axis coming both from the Ray-Singer Torsion
as well as from the result of the geometric sum. We push all of these, in the
complex plane, to lie above the real line (say by use of an $i\epsilon$
prescription).

The contour that we choose in the complex plane in order to evaluate the
integral is as follows: travel along the real axis from $-\pi P$ to $\pi P$
then straight up along the imaginary axis to $\pi P+iR$ followed by an arc to
$-\pi P+iR$ then straight down to $-\pi P$ and finally take the
$R\rightarrow\infty$ limit. For the integral over the arc to make sense one
regularises the integrand by multiplying by $\exp{(i\epsilon\phi^{2})}$ with
$\epsilon>0$. The integrals over the other parts of the contour are convergent
as we will see below. The fact that the prescription for the poles on the
interval $(-\pi P,\pi P)$ are now such that they all lie inside the contour
(in the upper half plane) means that the contour integral is just given by the
residue of all of those poles.

Denote the contribution of each segment of the integral by the start and end
points as $I(x,y)$ so that we have, with $R\rightarrow\infty$

$I(-\pi P,\pi P)+I(\pi P,\pi P+iR)-I(-\pi P,-\pi P+iR)=2\pi
i\mathrm{Res}\left(\tau_{M}(\phi)\,.\frac{\left(1+\exp{\left(id\phi/P\right)}\right)}{\left(1-\exp{\left(id\phi/P\right)}\right)}\right)$
(108)

as the integral over the arc is zero. Had we preserved the symmetry
$\phi\rightarrow\phi+2\pi P$ with our regulator then we would have been able
to declare that $I(\pi P,\pi P+iR)=I(-\pi P,-\pi P+iR)$. However as $\phi$
becomes large the integrand of $I(\pm\pi P,\pm\pi P+iR)$ goes as

$\exp{\left(i\epsilon(i\phi\pm\pi
P)^{2}-2\phi(N-2-\sum_{i=1}^{N}1/a_{i})\right)}\,+\dots$ (109)

and for $N\geq 3$ (but not for $\Sigma(2,3,5)$ the Poincare sphere) there is
convergence without the need for the Gaussian oscillatory behaviour so that
one may take the $\epsilon\rightarrow 0$ limit and the integrals over the two
vertical parts of the contour cancel each other in the limit. In short, we
have that with this prescription for handling the poles that

$Z_{BF}[M,SU(2)]=2\pi
i\mathrm{Res}\left(\tau_{M}(\phi)\,.\frac{\left(1+\exp{\left(id\phi/P\right)}\right)}{\left(1-\exp{\left(id\phi/P\right)}\right)}\right)$
(110)

One would also like to understand a little better what the residue formula
implies. To that end let

$\displaystyle\tau_{M}(\phi)$
$\displaystyle=\prod_{\alpha>0}\left(\sin{(\alpha(\phi))}\right)^{4-2N}\,.\,\prod_{\alpha>0}\prod_{i=1}^{N}\left(\sin{(\alpha(\phi)/a_{i})}\right)^{2}$
$\displaystyle=\prod_{\alpha>0}\left(\sin{(\alpha(\phi))}\right)^{4-2N}\,.\,\widetilde{\tau}_{M}(\phi)$
(111)

The residue formula in the case of $SU(2)$ takes the simple form

$\displaystyle Z_{BF}[M,SU(2)]$ $\displaystyle=Z_{BF}[M,SU(2)]|_{\phi_{C}}$
$\displaystyle\,\;+\sum_{r=0}^{2N-5}\left(\begin{array}[]{c}2N-5-r\\\
r\end{array}\right)\sum_{m=1-P}^{P-1}\left(f(m\pi).\,\frac{1}{1-\mbox{e}^{\,\textstyle
im\pi d/P}}\right)^{(2N-5-r)}\widetilde{\tau}_{M}(m\pi)^{(r)}$ (114)

where the sum over $m$ indicates the poles of
$\left(\sin{(\alpha(\phi))}\right)^{4-2N}$. One sees that this is essentially
a sum over derivatives of the Reidemeister torsion $\widetilde{\tau}_{M}$ (up
to normalisation).

###  4.5 $SU(2)$ $BF$ Theory on Brieskorn Spheres $\Sigma(a_{1},a_{2},a_{3})$

We now look concretely at $SU(2)$ $BF$ Theory on Brieskorn Spheres
$\Sigma(a_{1},a_{2},a_{3})$, in order to see if the suggested contour
reproduces the expectations from perturbation theory as derived by Witten WBF
. Even though the result for the path integral is not exactly that predicted
by Witten, we will see that this choice does have contributions from
irreducible flat connections which is quite pleasing in its own right though,
as we explain below, not completely unexpected.

With $N=3$ and $d=1$ the path integral becomes, apart from the $\phi_{C}$
contributions of the previous sections

$\displaystyle Z_{BF}[\Sigma(a_{1},a_{2},a_{3}),SU(2)]$
$\displaystyle\simeq\sum_{m=1}^{P-1}\left(\frac{1}{P}\csc^{2}{(m\pi/2P)}-4\cot{(m\pi/P)}\sum_{i=1}^{3}\frac{1}{a_{i}}\cot{(m\pi/a_{i})}\right)\widetilde{\tau}_{\Sigma(a_{1},a_{2},a_{3})}(m\pi)$
$\displaystyle\simeq\frac{1}{a_{1}a_{2}a_{3}}\sum_{m=1}^{P-1}h(m)\widetilde{\tau}_{\Sigma(a_{1},a_{2},a_{3})}(m\pi)$
(115)

To compare with (96) we note that the integer $m$ can be written in various
ways in particular

$m=a_{1}\alpha_{1}+n_{1}r_{1}=a_{2}\alpha_{2}+n_{2}r_{2}=a_{3}\alpha_{3}+n_{3}r_{3},\quad\alpha_{i},\,n_{i}\in\mathbb{Z}$
(116)

to arrive at

$Z_{BF}[\Sigma(a_{1},a_{2},a_{3}),SU(2)]\simeq\frac{1}{a_{1}a_{2}a_{3}}\sum_{n_{i}}g(n_{i})\,\prod_{i=1}^{3}4\sin^{2}{\left(\frac{\pi
n_{i}r_{i}}{a_{i}}\right)}$ (117)

which would agree with (96) if the function $g$ is $n_{i}$ independent.

As our first example (even though as we noted the derivation provided fails)
consider the Poincaré homology sphere $\Sigma(2,3,5)$. The Ray-Singer-Torsion
is the same for the values of $(n_{1},n_{2},n_{3})$ corresponding to
$(1,1,r)$, $(1,1,5-r)$ and $(1,2,r)$. In this way we have, apart from the
trivial connection, the two possible non-Abelian connections $(1,1,1)$ and
$(1,1,2)$. By Proposition 2.8 of Fintushel-Stern , this count is correct for
the flat connections on $\Sigma(2,3,5)$.

As our next example let us fix on $\Sigma(2,3,7)$ for which there are 2
irreducible isolated flat connections. The Ray-Singer-Torsion is the same for
the values of $(n_{1},n_{2},n_{3})$ corresponding to $(1,1,r)$, $(1,1,7-r)$
and $(1,2,r)$. If we identify those contributions as corresponding to the same
isolated and irreducible connection then we get 4 copies each of three
different connections, say of $(1,1,1)$, $(1,1,2)$ and $(1,1,3)$. In that case
the sum in (117) is over the 3 different connections each of which corresponds
to 4 different values of $m$ as tabulated below (the values of $g$ are
approximate)

$\begin{array}[]{|c|c|c|}\hline\cr m&(n_{1},n_{2},n_{3})&g/P=\sum_{m}h(m)/P\\\
\hline\cr 1,13,29,41&(1,1,1)&-34.45\\\ \hline\cr 5,19,23,37&(1,1,2)&6.96\\\
\hline\cr 11,17,25,31&(1,1,3)&3.35\\\ \hline\cr\end{array}$ (118)

Hence the partition function does not just give the sum of these connections
with equal weight.

The situation is somewhat less clear than just indicated and rather more
puzzling as the third contribution $m=11,17,25,31$ does not correspond to a
flat $SU(2)$ connection on $\Sigma(2,3,7)$. Rather this connection is one
identified in GMP-resurge as a flat $SL(2,\mathbb{C})$ connection that is
conjugate to $SL(2,\mathbb{R})$ and not to $SU(2)$ (cf. also Kitano-SL2R ).

The holonomies of the isolated irreducible flat connections are elements of
the group and the holonomies can, independently, be conjugated into a
preferred maximal torus (even though they are not reducible). The contribution
to the Reidemeister Torsion of these holonomies is then determined by the
integers $\mathbf{n}$ as described in Freed-Brieskorn equation 2.8 for
Brieskorn spheres. To apply this to an actual flat connection there are
conditions on the integers arising from the presentation of the fundamental
group. What we have in our situation is that all such integers arise as the
poles of the Ray-Singer Torsion and, unfortunately, our residue formula (110)
does not restrict only to the ones that correspond to honest flat $SU(2)$
connections.

###  4.6 A Residue Formula For Higher Rank and Outlook

Here we would like to briefly give a formula in terms of residues for the
partition function (86) to use as a definition for general $G$.

A possible definition, is

$Z_{BF}[M,G]\simeq 2\pi
i\mathrm{Res}\left(f(\phi)\tau_{M}(\phi)\,.\,\prod_{j=1}^{\dim\mathfrak{t}}\frac{1}{1-\mbox{e}^{\,\textstyle
i\phi_{j}d/P}}\right)$ (119)

where $\phi=\phi_{j}\alpha^{j}$ and the function $f$ is such that
$f(2\pi\mathbf{n}P/d)=1$ for all $\mathbf{n}\in I(G)$,

$f(\phi)=\sum_{\mathbf{n}\in\Lambda}\mbox{e}^{\,\textstyle
in^{j}\phi_{j}d/P}\,f_{\mathbf{n}}$ (120)

The poles of $\prod_{j}(1-\exp{(i\phi_{j}d/P)})^{-1}$ are at $\phi_{C}$, (89),
so summing over its residues at these poles is equivalent to performing the
$\lambda$ integral. The properties of $f(\phi)$ are so that at those poles $f$
is unity (whence the contribution is exactly the same as performing the
$\lambda$ integral). The function $f$ is obtained, as in the $SU(2)$ case, by
first performing the sum over $\mathbf{n}$ to restrict the form of $\lambda$
and symmetry properties within the integral

$\sum_{\mathbf{n}}\int_{\mathfrak{t}}d\lambda\,\exp{\left(i\mathop{\mathrm{Tr}}\nolimits{\lambda\left(d\phi/P+2\pi
i\mathbf{n}\right)}\right)}\rightarrow\prod_{j=1}^{\dim\mathfrak{t}}\frac{f(\phi)}{1-\mbox{e}^{\,\textstyle
i\phi_{j}d/P}}$ (121)

It must also be remembered that the range of $\phi$ is constrained.

The poles of $\tau_{M}$ may also contribute to the residue. If one takes the
attitude that one should use the massive Ray-Singer Torsion directly then
there are no extra poles and (119) agrees with the naive evaluation of the
path integral. Alternatively, (119) allows one to take the poles of $\tau_{M}$
into account after which one may reinstate the mass.

We have not shown that there is a contour that leads to (119) but one may
reasonably hope that a generalisation of that used in the $SU(2)$ case will be
available.

Before ending this section we would like to suggest a possible apriori reason
for why the definition adopted here has shortcomings. We start by observing
that it is somewhat surprising that the possible isolated irreducible flat
connections are determined by poles in the Ray-Singer Torsion. This is
certainly counterintuitive as the poles of the Ray-Singer Torsion are
understood to arise when one has moduli, i.e. 1-form zero modes. A possible
reason for this is a shortcoming in the gauge-fixing choice that has been
made. As already noted in order to gauge fix $\lambda$ to be constant on the
fibre we require that $\phi$ be generic ‘enough’. To see that this may be the
cause of concern, imagine that one wishes, as we did in Section 2, to expand
around an isolated irreducible flat connection. We do this but we still insist
on gauge fixing the quantum fields as we did in Section 3 so that $\lambda$ is
constant along the fibre. Then we would need to solve the equivalent of (53)

$\lambda+D_{\phi_{f}+\phi}\Lambda=\lambda_{0}$ (122)

where $\phi_{f}$ is the fibre component of the background flat connection and
$\phi$ is the fibre component of the quantum field. If one thinks of the
quantum field as having a $\hbar$ in front of it then to lowest order the
equation to be solved is

$\lambda+D_{\phi_{f}}\Lambda\simeq\lambda_{0}$ (123)

and the field $\phi_{f}$ is certainly not generic. The discussion in Section 3
tells us that precisely for these $\phi_{f}$, as explained around (62-63), the
operator $D_{\phi_{f}}$ is not invertible and the gauge
$\partial_{t}\lambda=0$ cannot be achieved.

Given this question about the gauge choice and the appearance of a complex
flat connection in the $SU(2)$ theory one may wonder if resurgence is an
approach that may demystify the situation. The $BF$ theory, as we have
approached it here, arises as one possible limit of $G_{\mathbb{C}}$ Chern-
Simons theory (see the next section). In the $SU(2)$ case the reducible
connections are Abelian and one may hope that the resurgence programme will
yield the correct non-Abelian connections as they do in GMP-resurge with the
correct contribution to the partition function.

##  5 Large $\boldsymbol{k}$ Asymptotics of $\boldsymbol{G\times G}$ and
$\boldsymbol{G_{\mathbb{C}}}$ Chern-Simons Theory

Our aim in this section is to see if embedding $BF$ theory in a ‘bigger’
theory may act to give a suitable definition which can be used in case $M$ is
an $\mathbb{Z}$ Homology Sphere. To that end we consider some naive large $k$
limits of Chern-Simons theory with the compact gauge group $G\times G$ (with
levels $k$ and $-k$ for the two factors) and the complexified group
$G_{\mathbb{C}}$. The ‘fattened’ groups $G_{-k}\times G_{k}$ and
$G_{\mathbb{C}}$, as has already been explained (see Appendix A), have been
chosen as in the limits to be discussed they both contract to $TG$.

The advantage of such an approach is that we avoid the possible issues we had
with the generic nature (or not) of $\phi$ as the values of $\phi$ signalled
out prior to taking the large $k$ limit allow for the gauge where $\lambda$ is
constant along the fibre.

In the case of $G_{-k}\times G_{k}$ we find that the theory does not appear to
‘decompactify’ enough in the large $k$ limit to capture all the features of
the $TG$ theory. However, in examples, we show that it does actually pick up
those flat connections which are both isolated and irreducible correctly. The
$G_{\mathbb{C}}$ Chern-Simons theory, on the otherhand, has all the features
of the $TG$ theory from the outset. We show that the partition function has
the formal properties that would allow an application of the large $k$
analysis that have worked in the case of the compact groups.

###  5.1 Definition 2: $BF$ and the Large $k$ limit of $G_{k}\times$ $G_{-k}$
Chern-Simons Theory

Here we consider the Chern-Simons action for two connections on two copies of
the same bundle for the product gauge group $G\times G$ and for the following
gymnastics we require that the bundle in question is trivial. The action under
consideration is

$CS(A_{+},A_{-})=\frac{1}{4\pi}\int_{M}\left[\mathop{\mathrm{Tr}}\nolimits(A_{+}\wedge
dA_{+}+\frac{2}{3}A_{+}\wedge A_{+}\wedge
A_{+})-\mathop{\mathrm{Tr}}\nolimits(A_{-}\wedge dA_{-}+\frac{2}{3}A_{-}\wedge
A_{-}\wedge A_{-})\right]$ (124)

The relative sign means that the two gauge theories have opposite levels and
we denote such a theory, with action $k\,CS(A_{+},A_{-})$ as a $G_{k}\times
G_{-k}$ Chern-Simons theory. If we denote the level $k$ Chern-Simons theory
partition function for group $G$ by $Z_{CS}[M,G_{k}]$ then the partition
function for the $G_{k}\times$ $G_{-k}$ theory is

$Z_{CS}[M,G_{k}\times G_{-k}]=Z_{CS}[M,G_{k}].Z_{CS}[M,G_{-k}]$ (125)

Now to show how this is related to $BF$ theory make the substitutions

$A_{\pm}=A\pm\frac{\pi}{k}B,\quad A=\frac{1}{2}(A_{+}+A_{-}),\quad
B=\frac{k}{2\pi}(A_{+}-A_{-})$ (126)

to arrive at

$CS(A_{+},A_{-})=\int_{M}\mathop{\mathrm{Tr}}\nolimits\left(B\wedge
F_{A}+\frac{\pi^{2}}{k^{2}}B\wedge B\wedge B\right)$ (127)

There may be a Jacobian $J$, depending on $k$ in passing to the new variables
in the path integral, so the relationship that we expect is finally

$Z_{BF}[M,G]=\lim_{k\rightarrow\infty}J(k)\,Z_{CS}[M,G_{k}\times G_{-k}]$
(128)

where the limit as $k\rightarrow\infty$ formally ensures that the cubic term
in $B$ may be neglected. We allow $J$ to also soak up any other factors of $k$
that may be present due to zero modes and so on.

This formal result says that the $k\rightarrow\infty$ limit of a $G_{k}\times$
$G_{-k}$ Chern-Simons theory is equivalent to a pure BF theory. However, this
statement certainly requires some elaboration as adding and subtracting
connections at will as in (126) is not usually an operation that makes sense
in the theory of bundles. This is mirrored in the gauge transformations that
one obtains for $A$ and $B$ which are also not what one might call ‘standard’.
To emphasise that the limit in question is by no means obvious consider the
3-manifold $\Sigma\times S^{1}$. Chern-Simons theory on such a manifold is
perfectly sensible and the large $k$ limit is well understood, the partition
function having leading term $k^{n/2}\mathrm{Vol}(\mathfrak{M}[\Sigma,G])$ the
symplectic volume of the moduli space of $G$ connections on $\Sigma$ where
$n=\dim{(\mathfrak{M}[\Sigma,G])}$ Witten-2d . The large $k$ limit of the
$G_{k}\times$ $G_{-k}$ partition function is then proportional to
$k^{n}\mathrm{Vol}(\mathfrak{M}[\Sigma,G])^{2}$ and not what one would expect
from $BF$ theory, say $k^{n}\mathrm{Vol}(T\mathfrak{M}[\Sigma,G])$ (even
though this is ill defined).

So we need to better understand when the limit will indeed correspond to $BF$
theory. At large $k$, the stationary phase approximation tells us that the
functional integral Witten-CS localises around the critical points of the
action $CS(A)$, which in this case are precisely the flat connections. We then
have, for isolated flat connections, as k$\rightarrow$ $\infty$,

$Z_{k}[M]\sim\sum_{\alpha}\mbox{e}^{\,\textstyle 2\pi
ikCS(\alpha)}\,e^{-\frac{i\pi}{2}I({\alpha})}\,\sqrt{\tau_{M}({\alpha})}$
(129)

where ${\alpha}$, $\tau_{M}(\alpha)$ and $I({\alpha})$ are the flat
connections, the Ray-Singer torsion at the flat connection and the spectral
flow to the flat connection respectively.

The $G_{k}$ $\times$ $G_{-k}$ theory then has the asymptotic form

$Z_{k}[M]Z_{-k}[M]\sim\sum_{\alpha,\beta}\mbox{e}^{\,\textstyle
ik\left(CS({\alpha})-CS({\beta})\right)}f_{\alpha\beta}\sqrt{\tau_{M}({\alpha})}\sqrt{\tau_{M}({\beta})}$
(130)

where

$f_{\alpha\beta}=\exp{\left(-\frac{i\pi}{2}[I(A_{\alpha})-I(A_{\beta})]\right)}$
(131)

One would expect therefore, that in the limit as $k\rightarrow\infty$, that
providing that flat connections $\alpha$ and $\beta$ having the same Chern-
Simons invariant implies they are the same flat connection,

$CS(\alpha)=CS(\beta)\,\mod 2\pi\mathbb{Z}\quad\implies\alpha=\beta$ (132)

then the oscillations in (130) would ensure that we would indeed only need to
sum over $\alpha=\beta$. In such a situation then one could reasonably expect
that (128) holds. If this condition does not hold then there may be ‘non-
diagonal’ contributions to the sum (130) and so the relationship with $BF$
theory becomes more tenuous.

Indeed for any connected component of non-isolated flat connections all the
flat connections in that component have the same Chern-Simons invariant and
fail our test (132). The flat connections on $\Sigma\times S^{1}$ that we
discussed previously are of this type and this, to some extent, explains why
they do not match the expectations for $BF$ theory. The lesson here is that we
must concentrate on isolated flat connections. Fortunately for us Fintushil
and Stern Fintushel-Stern have shown that the Brieskorn spheres (the Seifert
$\mathbb{Z}$ homology spheres with 3 exceptional fibres) have moduli spaces of
flat $SU(2)$ connections made up of a finite number of discrete points.

We shall now illustrate when (132) holds with a few examples in the
literature. Consider firstly the Poincaré Homology Sphere, $M=\Sigma(2,3,5)$.
The Poincaré $\mathbb{Z}$ homology sphere has three flat $SU(2)$ connections,
one of which is the reducible trivial connection and two which are non-
Abelian. Freed and Gompf Freed-Gompf determine the large $k$ behaviour for the
Chern-Simons partition function and find

$Z_{k}[\Sigma(2,3,5)]\sim\sqrt{\frac{2}{5}}e^{-3\pi
i/4}\Big{[}\sin\Big{(}\frac{\pi}{5}\Big{)}e^{-\pi
i(k+2)/60}+\sin\Big{(}\frac{2\pi}{5}\Big{)}e^{-49\pi i(k+2)/60}\Big{]}$ (133)

Taking the modulus square, and presuming the limit of the norm agrees with the
norm of the limit, one gets

$\lim_{k\rightarrow\infty}Z_{k}[\Sigma(2,3,5)]Z_{-k}[\Sigma(2,3,5)]\sim\frac{2}{5}\Big{[}\sin^{2}\Big{(}\frac{\pi}{5}\Big{)}+\sin^{2}\Big{(}\frac{2\pi}{5}\Big{)}\Big{]}$
(134)

this corresponds to the sum of the Ray-Singer torsions of the irreducible
connections for $\Sigma(2,3,5)$ up to a finite normalisation. Similarly Freed-
Gompf provides us with the example of $\Sigma(2,3,17)$ and, with the same
caveat on exchanging limits and norms, we find that the norm square of the
Chern-Simons partition function is the sum of the Ray-Singer Torsion over the
6 irreducible flat $SU(2)$ connections.

There are manifolds, however, where you find two or more isolated flat
connections giving the same Chern-Simons invariant so that (132) does not
hold. Lens spaces yield examples of this phenomena. As we have seen for Lens
spaces $L(p,q)$ the Ray-Singer Torsion is given by

$\tau_{L(p,q)}=\frac{16}{p}\,\sin^{2}\Big{(}\frac{2\pi
n}{p}\Big{)}\quad\sin^{2}\Big{(}\frac{2\pi q^{*}n}{p}\Big{)}$ (135)

Here one integer is enough to describe the flat (Abelian) connections.

A typical example where two different connections have the same Chern-Simons
invariant is afforded by the Lens space $L(12,5)$. For $L(12,5)$ we have
$q^{*}=5$ and one can see for example that $n=1$ and $n=5$ give the same
Chern-Simons invariant (mod $2\pi$), namely

$CS(n)=q^{*}n^{2}/p=\frac{5}{12}$ (136)

so we are not free to use the norm of the Chern-Simons theory in this case as
a way to define the $BF$ theory.

The discussion of this section shows that this definition of $BF$ theory as a
limit of the $G_{-k}\times G_{k}$ Chern-Simons theory needs to be handled with
care and we might have to know more about the moduli space itself than we
would care to. However, for the part of the moduli space where the connections
are isolated and irreducible this appears to be an appropriate definition.

###  5.2 Definition 3: $BF$ and the Large $k$ or $s$ Limit of $G_{\mathbb{C}}$
Chern-Simons Theory

The final definition that we give involves the gauge group $G_{\mathbb{C}}$.
Note that the classical groups $TG$ and $G_{\mathbb{C}}$ are diffeomorphic (as
spaces). The Inönü-Wigner contraction that establishes the Lie algebra
relationship is given in Appendix A. Furthermore, the $G_{\mathbb{C}}$
connection is

$A_{\mathbb{C}}=A+iB$ (137)

and as $G_{\mathbb{C}}$ contracts to the compact group $G$ the connection $A$
may be considered a $G$ connection while $B$ is then a Lie algebra
$\mathfrak{g}$ valued one-form. This is a much clearer decomposition than the
mixing of objects that appeared in the $G_{-k}\times G_{k}$ theory of the
previous section.

The $G_{\mathbb{C}}$ Chern-Simons action is

$\displaystyle I(k,s)$
$\displaystyle=\frac{k}{4\pi}\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(A\wedge
dA+\frac{2}{3}A\wedge A\wedge A-B\wedge d_{A}B\right)}$
$\displaystyle\quad-\frac{s}{2\pi}\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B\wedge
F_{A}-\frac{1}{3}B\wedge B\wedge B\right)}$ (138)

with $k\in\mathbb{Z}$ and we take $s\in\mathbb{R}$ Witten-CCS . One can see
that the complex Chern-Simons action has a number of limits which formally
lead to the $BF$ action. The first, and most straightforward way, to arrive at
the $BF$ theory is to set $k=0$ and consider the $s\rightarrow\infty$ limit
where one sends $B\rightarrow B/s$ to formally arrive at

$I(k,s)\rightarrow-\frac{1}{2\pi}\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B\wedge
F_{A}\right)}$ (139)

Alternatively one can consider the large $k$ limit, for finite $s$, and this
time by scaling $B\rightarrow B/\sqrt{k}$. One does not get the $BF$ action
directly, but rather

$I(k,s)\rightarrow\frac{k}{4\pi}\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(A\wedge
dA+\frac{2}{3}A\wedge A\wedge
A\right)}-\frac{1}{4\pi}\int_{M}\mathop{\mathrm{Tr}}\nolimits{\left(B\wedge
d_{A}B\right)}$ (140)

About an isolated flat connection this would be the same as the large $k$
limit of Chern-Simons theory except that the integral over the field $B$ has,
formally the effect of replacing the square root of the Ray-Singer torsion of
the flat connection with the Ray-Singer torsion itself and so would, in the
limit, reproduce the partition function of $BF$ theory with a Chern-Simons
action contribution of the flat connection. Both limits (and indeed any other
limits of $k$ and $s$) need to be carefully taken as in principle one would
most likely land on the moduli space of flat $G_{\mathbb{C}}$ connections and
not just the flat $G$ connections.

For finite $k$ and with $s=0$ Gukov and Pei Gukov-Pei have evaluated the
partition function on $M=\Sigma\times S^{1}$. Even though the Hilbert space is
also infinite dimensional in this case, they ‘filter’ it by giving the $B$
field a mass and so are able to extract meaningful results. Unlike the
situation with the $G_{-k}\times G_{k}$, where one gets essentially the square
of the compact case, here one sees that the results come from a non-compact
group and formally diverges as $k\rightarrow\infty$ as we would expect for the
$BF$ theory.

Returning to the case at hand note that the corresponding finite dimensional
integral to be performed in the case of complex Chern-Simons after
Abelianisation BT-CCS has as its action

$\displaystyle I(k,s)$
$\displaystyle=-\frac{k}{4\pi}\mathop{\mathrm{Tr}}\nolimits(\phi^{2}-\lambda^{2})c_{1}\left(\mathcal{L}_{M}\right)+k\sum_{i=1}^{N}\frac{1}{a_{i}}\mathop{\mathrm{Tr}}\nolimits(-i\phi\mathbf{n}_{i}+r_{i}\pi\mathbf{n}_{i}^{2})+k\mathop{\mathrm{Tr}}\nolimits(-i\phi\mathbf{n}_{0})$
$\displaystyle+\frac{s}{2\pi}\mathop{\mathrm{Tr}}\nolimits\Big{(}\lambda\left(\phi
c_{1}\left(\mathcal{L}_{M}\right)+2\pi
i\sum_{i=1}^{N}\frac{1}{a_{i}}\mathbf{n}_{i}+2\pi
i\mathbf{n}_{0}\right)\Big{)}$ (141)

while the Ray-Singer Torsion goes over to the complex Ray-Singer Torsion

$\tau_{M}(\phi)\rightarrow\sqrt{\tau_{M}(\phi+i\lambda)\tau_{M}(\phi-i\lambda)}$
(142)

Formally the complex Ray-Singer Torsion under either scaling goes back to the
‘real’ Ray-Singer Torsion. The $k=0$ and $s\rightarrow\infty$ limit lands us
on the finite dimensional action that we have been using for $BF$ theory. Here
we would like to investigate the other limit.

The limit of interest then is large $k$ and for
$\lambda\rightarrow\lambda/\sqrt{k}$. As $\lambda$ now only appears in the
action quadratically one may integrate it out. In this limit the finite action
goes over to

$I(k,s)\rightarrow-\frac{k}{4\pi}\mathop{\mathrm{Tr}}\nolimits(\phi^{2})c_{1}\left(\mathcal{L}_{M}\right)+k\sum_{i=1}^{N}\frac{1}{a_{i}}\mathop{\mathrm{Tr}}\nolimits(-i\phi\mathbf{n}_{i}+r_{i}\pi\mathbf{n}_{i}^{2})+k\mathop{\mathrm{Tr}}\nolimits(-i\phi\mathbf{n}_{0})$
(143)

which is just the standard Chern-Simons action for compact group $G$. The
partition function, however, includes an extra factor of the square root of
the Ray-Singer Torsion (arising from the integral over the field $B$ in (140))
in the integral over $\mathfrak{t}$ so this is not the partition function of
compact Chern-Simons theory.

As the action agrees with that of $G$ Chern-Simons theory then all the
properties of the integrand (see Appendix B for the symmetry properties) and
hence the considerations that appear in LR and Marino apply here too. This
includes the choice of contour to take the large $k$ limit, the only
difference being that of considering a slightly different function that takes
into account that we have the Ray-Singer Torsion not its square root in the
integrand. The resulting large $k$ asymptotics are, for Chern-Simons theory,
just as predicted by Witten in Witten-CS , contributions around flat
connections. Consequently, in the limit that we are considering in the
$G_{\mathbb{C}}$ theory, this means that the asymptotic form will have
contributions from flat connections in $BF$ theory as anticipated in Section
2.

Specifically in the case of $SU(2)$ the large $k$ limit is that of the flat
connection contributions on page 302 of LR but with $F(y)$ there replaced with
$F(y)^{2}$ in order to pass to the $BF$ theory (and one should not include the
framing as it ought not to arise in complex Chern-Simons theory) while for a
general compact group one can refer to (4.17) in Marino from which one can
deduce the large $k$ behaviour. At this point this particular definition of
$BF$ theory is the most complete that we have.

If one would like to obtain more explicit formulae for the $BF$ theory one
could follow the explanation of LR on how to arrive at Rozansky’s formula for
the large $k$ expansion of $SU(2)$ Chern-Simons theory Rozansky-large-k which
reproduces Witten’s expansion. Of course this also encodes some of the large
$k$ structure of the $G_{\mathbb{C}}$ theory itself so that by following
different parts of the asymptotic expansion we would be able to obtain parts
of the perturabation theory. In particular we also have in mind the
contributions about the trivial connection. We leave these issues, and the
question of how and whether complex connections contribute to the $BF$ limit
for the future.

## Acknowledgement

M. Kakona thanks the External Activities Unit of the ICTP for a Ph.D. grant at
EAIFR in the University of Rwanda and the HECAP group at the ICTP for
supporting a visit to complete this work. He is also indebted to OFID and the
STEP programme of the ICTP for funding and support in the initial stages of
his studies. The group of M. Blau is supported by the NCCR SwissMAP (The
Mathematics of Physics) of the Swiss Science Foundation.

##  Appendix A The Tangent Bundle Group $TG$ and its Lie Algebra
$\mathfrak{tg}$

###  A.1 Basic Properties of $TG$ and $\mathfrak{tg}$

let $G$ be a compact Lie group and (to not unnecessarily complicate things)
assume that $G$ is connected, simply-connected and semi-simple. Via a right-
invariant trivialisation of the tangent bundle, the group $TG$ can be
identified with the semi-direct product of $G$ with its Lie algebra
$\mathfrak{g}$,

$TG\simeq G\ltimes\mathfrak{g}\equiv G\times_{\mathsf{Ad}}\mathfrak{g}\;\;,$
(144)

with multiplication

$(g,v)(h,w)=(gh,v+\mathsf{Ad}_{g}w)\\}\;\;,$ (145)

where the adjoint action of $g\in G$ on $w\in\mathfrak{g}$ is
$\mathsf{Ad}_{g}w=gwg^{-1}$. Correspondingly, its Lie algebra has the form

$\mathsf{Lie}(TG)\equiv\mathfrak{tg}\simeq\mathfrak{g}\oplus_{\mathsf{ad}}\mathfrak{g}_{Ab}$
(146)

where $\mathfrak{g}_{Ab}$ is the Abelian Lie algebra based on the underlying
vector space $\mathfrak{g}$, i.e. the commutator is

$[(x,v),(y,w)]=([x,y],[x,w]+[v,y])=([x,y],\mathsf{ad}_{x}w-\mathsf{ad}_{y}v)$
(147)

and the (inverse) adjoint action of $TG$ on its Lie algebra is

$(g,v)^{-1}(y,w)(g,v)=(\mathsf{Ad}_{g^{-1}}y,\mathsf{Ad}_{g^{-1}}(w+[y,v]))$
(148)

Let $\ell_{a}$ be a (real) basis of the (real) Lie algebra $\mathfrak{g}$,
with

$[\ell_{a},\ell_{b}]=f_{ab}^{c}\ell_{c}\;\;.$ (149)

Then

$j_{a}=(\ell_{a},0)\quad,\quad p_{a}=(0,\ell_{a})$ (150)

are a basis of $\mathfrak{tg}$, and the Lie algebra commutation relations take
the form

$[j_{a},j_{b}]=f_{ab}^{c}j_{c}\quad,\quad[j_{a},p_{b}]=[p_{a},j_{b}]=f_{ab}^{c}p_{c}\quad,\quad[p_{a},p_{b}]=0\;\;.$
(151)

Here are two useful properties of $\mathfrak{tg}$:

  1. 1.

The Lie algebra $\mathfrak{tg}$ of $TG$ can be obtained as a contraction of
the Lie algebra of the group $G\times G$

     * •

Consider the $G\times G$ Lie algebra

$\mathfrak{g}_{+}\oplus\mathfrak{g}_{-}=\mathfrak{g}\oplus\mathfrak{g}$ (152)

with generators

$[j_{a}^{\pm},j_{b}^{\pm}]=f_{ab}^{c}j_{c}^{\pm}\quad,\quad[j_{a}^{+},j_{b}^{-}]=0\;\;.$
(153)

     * •

Perform the redefinition

$j_{a}^{\pm}=\tfrac{1}{2}(j_{a}\pm p_{a}/\epsilon)\quad\Leftrightarrow\quad
j_{a}=j_{a}^{+}+j_{a}^{-}\quad,\quad p_{a}=\epsilon(j_{a}^{+}-j_{a}^{-})$
(154)

     * •

Then the algebra $\mathfrak{g}\oplus\mathfrak{g}$ takes the form

$[j_{a},j_{b}]=f_{ab}^{c}j_{c}\quad,\quad[j_{a},p_{b}]=[p_{a},j_{b}]=f_{ab}^{c}p_{c}\quad,\quad[p_{a},p_{b}]=\epsilon^{2}f_{ab}^{c}j_{c}\;\;.$
(155)

     * •

In the limit $\epsilon\rightarrow 0$, this reduces to (151), the Lie algebra
of $TG$.

     * •

If one sets $\epsilon=i\delta$, the same reasoning shows that $\mathfrak{tg}$
can be obtained as the contraction of the complexification
$\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}\oplus i\mathfrak{g}$ of
$\mathfrak{g}$.

  2. 2.

Existence of Invariant Scalar Products on $\mathfrak{tg}$

Let us consider the case where the Lie algebra $\mathfrak{g}$ is simple. In
that case, $\mathfrak{g}$ has a preferred (and up to a choice of scale unique)
non-degenerate and $\mathsf{ad}$-invariant metric scalar product, namely the
Killing-Cartan form
$\mathop{\mathrm{Tr}}\nolimits\mathsf{ad}_{x}\mathsf{ad}_{y}$ (the trace in
the adjoint representation). With respect to the basis $\ell_{a}$ of
generators, this metric has the components

$g_{ab}\equiv<\ell_{a},\ell_{b}>=\mathop{\mathrm{Tr}}\nolimits\mathsf{ad}_{\ell_{a}}\mathsf{ad}_{\ell_{b}}=f^{d}_{ac}f^{c}_{bd}$
(156)

Turning to the Lie algebra

$\mathfrak{tg}\simeq\mathfrak{g}\oplus_{\mathsf{ad}}\mathfrak{g}_{Ab}\;\;,$
(157)

as it is not semi-simple, its Killing-Cartan form will be
$\mathfrak{tg}$-invariant (by construction, i.e. by the Jacobi identity) but
degenerate. Indeed, in terms of the generators $(j_{a},p_{a})$ one has

$\mathop{\mathrm{Tr}}\nolimits\mathsf{ad}_{j_{a}}\mathsf{ad}_{j_{b}}=2g_{ab}=2f^{d}_{ac}f^{c}_{bd}$
(158)

(from the adjoint action of $\mathsf{ad}_{j_{a}}\mathsf{ad}_{j_{b}}$ on
$j_{c}$ and on $p_{c}$), and

$\mathop{\mathrm{Tr}}\nolimits\mathsf{ad}_{j_{a}}\mathsf{ad}_{p_{b}}=\mathop{\mathrm{Tr}}\nolimits\mathsf{ad}_{p_{a}}\mathsf{ad}_{p_{b}}=0$
(159)

(because both $\mathsf{ad}_{j_{a}}\mathsf{ad}_{p_{b}}$ and
$\mathsf{ad}_{p_{a}}\mathsf{ad}_{p_{b}}$ only act non-trivially on a $j_{c}$
and take it to a linear combination of the $p_{c}$). This scalar product can
therefore also be written as

$<j_{a},j_{b}>=2g_{ab}\quad,\quad<j_{a},p_{b}>=<p_{a},p_{b}>=0\;\;.$ (160)

In addition to the (degenerate) Killing-Cartan form, $\mathfrak{tg}$
exceptionally also possesses a non-degenerate $\mathsf{ad}$-invariant scalar
product given by

$\ll j_{a},j_{b}\gg=\ll p_{a},p_{b}\gg=0\quad,\quad\ll
j_{a},p_{b}\gg=g_{ab}\;\;.$ (161)

It is easily verified that this is indeed both invariant and non-degenerate.

The existence of this second invariant scalar product, or of the overall two-
parameter family of invariant scalar products, can be understood in terms of
the contraction of the Lie algebra $\mathfrak{g}\oplus\mathfrak{g}$ to
$\mathfrak{tg}$ mentioned above. Indeed, starting with the non-degenerate
Killing-Cartan metric on the Lie algebra $\mathfrak{g}\oplus\mathfrak{g}$
(153) of $G\times G$, with coefficients $c^{\pm}$,

$<j_{a}^{\pm},j_{b}^{\pm}>=c^{\pm}g_{ab}\quad,\quad<j_{a}^{+},j_{b}^{-}>=0\;\;,$
(162)

and performing the redefinition (154), one finds

$<j_{a},j_{b}>=(c^{+}+c^{-})g_{ab}\quad,\quad<j_{a},p_{b}>=\epsilon(c^{+}-c^{-})g_{ab}\quad,\quad<p_{a},p_{b}>=\epsilon^{2}(c^{+}+c^{-})g_{ab}$
(163)

Taking the contraction $\epsilon\rightarrow 0$ with $c^{+}=c^{-}=1$, one finds
(160), while taking $c^{+}=-c^{-}=1/2\epsilon$ one finds (161),

$\displaystyle c^{+}=c^{-}=1$ $\displaystyle\Rightarrow\quad<.,.>$ (164)
$\displaystyle c^{+}=-c^{-}=1/2\epsilon$
$\displaystyle\Rightarrow\quad\ll.,.\gg$

###  A.2 Diagonalisation / Abelianisation in $TG$ and $\mathfrak{tg}$

Let $G$ be compact, $T_{G}$ be a maximal torus, $\mathfrak{t}_{G}$ its Lie
algebra, a Cartan subalgebra. Then the following assertions are true:

  1. 1.

For any $h\in G$ one can find a $g\in G$ such that

$h\in G\quad\Rightarrow\quad\exists g\in G:\quad\mathsf{Ad}_{g}h=ghg^{-1}\in
T_{G}\;\;.$ (165)

  2. 2.

For any $y\in\mathfrak{g}$ one can find a $g\in G$ such that

$y\in\mathfrak{g}\quad\Rightarrow\quad\exists g\in
G:\quad\mathsf{Ad}_{g}y=gyg^{-1}\in\mathfrak{t}_{G}\;\;.$ (166)

As $TG$ is not compact and not semi-simple, a priori the corresponding
statements do not necessarily hold (and would usually be far from true for a
generic such group). Nevertheless, it turns out that the above statements
carry over literally to the case of the group $TG$, provided that we replace
the maximal torus $T_{G}$ of $G$ and the Cartan subalgebra $\mathfrak{t}_{G}$
of $\mathfrak{g}$ by

$\displaystyle T_{TG}$ $\displaystyle=T(T_{G})\simeq
T_{G}\times_{\mathsf{Ad}}\mathfrak{t}_{G}=T_{G}\times\mathfrak{t}_{G}$ (167)
$\displaystyle\mathfrak{t}_{TG}$
$\displaystyle=\mathsf{Lie}(T_{G}\times\mathfrak{t}_{G})=\mathfrak{t}_{G}\oplus\mathfrak{t}_{G}$
(168)

(since $\mathfrak{t}_{G}$ is already an Abelian Lie algebra, here it is not
necessary to write $(\mathfrak{t}_{G})_{Ab}$ in the second factor / summand).
Indeed, we can now prove the following two statements:

  1. 1.

Diagonalisation / Abelianisation in $TG$

$\forall(h,w)\in TG\simeq
G\times_{\mathsf{Ad}}\mathfrak{g}\quad\exists(g,v)\in
TG:\quad(g,v)^{-1}(h,w)(g,v)=(t,\tau_{2})\in T_{G}\times\mathfrak{t}_{G}$
(169)

  2. 2.

Diagonalisation / Abelianisation in $\mathfrak{tg}$

$\forall(y,w)\in\mathfrak{tg}\simeq\mathfrak{g}\oplus_{\mathsf{ad}}\mathfrak{g}\quad\exists(g,v)\in
TG:\quad(g,v)^{-1}(y,w)(g,v)=(\tau_{1},\tau_{2})\in\mathfrak{t}_{G}\oplus\mathfrak{t}_{G}$
(170)

Here are the proofs of these assertions:

  1. 1.

Diagonalisation / Abelianisation in $TG$

     * •

The (inverse) Ad-action of $TG$ on itself is

$(g,v)^{-1}(h,w)(g,v)=(\mathsf{Ad}_{g^{-1}}h,\mathsf{Ad}_{g^{-1}}(w+\mathsf{Ad}_{h}v-v))$
(171)

In particular, for the action of $(g,0)$ one finds

$(g,0):\quad h\mapsto\mathsf{Ad}_{g^{-1}}h\quad,\quad
w\mapsto\mathsf{Ad}_{g^{-1}}w$ (172)

and for that of $(e,v)$ one has

$(e,v):\quad h\mapsto h\quad,\quad w\mapsto w+\mathsf{Ad}_{h}v-v\;\;.$ (173)

     * •

First of all, we can and will choose $g\in G$ such that

$h\mapsto g^{-1}hg=t\in T_{G}\;\;.$ (174)

     * •

Next we define $S$ to be the stabiliser of $t$ under the adjoint action of
$G$,

$S:=\mathsf{Stab}_{G}(t)=\\{g\in G:\;g^{-1}tg=t\\}\;\;.$ (175)

One has

$S\supseteq T_{G}\;\;,$ (176)

with equality if $t$ is generic (regular). In the non-generic case, $S$ will
be a product of simple factors and $U(1)$s. We use the convention that
$T_{S}=T_{G}$, and likewise at the Lie algebra level,

$T_{S}=T_{G}\quad,\quad\mathfrak{t}_{S}=\mathfrak{t}_{G}\;\;.$ (177)

At the Lie algebra level one has the (reductive, because $\mathfrak{g}$ is
compact) decomposition

$\mathfrak{g}=\mathfrak{s}\oplus\mathfrak{m}\quad\text{with}\quad[\mathfrak{s},\mathfrak{m}]\subset\mathfrak{m}\;\;.$
(178)

     * •

Now let us turn to the second entry in (171),

$w\mapsto\mathsf{Ad}_{g^{-1}}(w+\mathsf{Ad}_{h}v-v)\;\;.$ (179)

With $h=t\in T_{G}$ and thus $g=s\in S$, this becomes

$w\mapsto\mathsf{Ad}_{s^{-1}}(w+\mathsf{Ad}_{t}v-v)\;\;.$ (180)

Decomposing $v=v^{\mathfrak{s}}+v^{\mathfrak{m}}$ into its components in
$\mathfrak{g}=\mathfrak{s}\oplus\mathfrak{m}$, we see that only the
$\mathfrak{m}$-component of $v$ contributes, since
$\mathsf{Ad}_{t}v^{\mathfrak{s}}=v^{\mathfrak{s}}$,

$\mathsf{Ad}_{t}v-v=(\mathsf{Ad}_{t}-1)v^{\mathfrak{m}}\in\mathfrak{m}\;\;,$
(181)

and $v^{\mathfrak{m}}$ can be chosen to cancel the $\mathfrak{m}$-component
$w^{\mathfrak{m}}$ of $w$ (because by definition of $\mathfrak{m}$ the
operator $(\mathsf{Ad}_{t}-1)$ is invertible on $\mathfrak{m}$).

     * •

We are then left with

$w\mapsto\mathsf{Ad}_{s^{-1}}w^{\mathfrak{s}}\;\;,$ (182)

and we can now choose $s\in S$ such that

$w\mapsto\mathsf{Ad}_{s^{-1}}w^{\mathfrak{s}}=\tau_{2}\in\mathfrak{t}_{S}=\mathfrak{t}_{G}\;\;.$
(183)

     * •

Altogether, we have thus managed to conjugate

$(h,w)\mapsto(t,\tau_{2})\in T_{G}\times\mathfrak{t}_{G}\;\;,$ (184)

as announced.

  2. 2.

Diagonalisation / Abelianisation in $\mathfrak{tg}$

     * •

The inverse adjoint action of $TG$ on its Lie algebra is

$(g,v)^{-1}(y,w)(g,v)=(\mathsf{Ad}_{g^{-1}}y,\mathsf{Ad}_{g^{-1}}(w+[y,v]))$
(185)

In particular, for the action of $(g,0)$ one finds

$(g,0):\quad y\mapsto\mathsf{Ad}_{g^{-1}}y\quad,\quad
w\mapsto\mathsf{Ad}_{g^{-1}}w$ (186)

and for that of $(e,v)$ one has

$(e,v):\quad y\mapsto y\quad,\quad w\mapsto w+[y,v]\;\;.$ (187)

     * •

First of all, we can and will choose $g\in G$ such that

$y\mapsto g^{-1}yg=\tau_{1}\in\mathfrak{t}_{G}\;\;.$ (188)

     * •

Next we define $S$ to be the stabiliser of $\tau_{1}$ under the adjoint action
of $G$,

$S:=\mathsf{Stab}_{G}(\tau_{1})=\\{g\in G:\;g^{-1}\tau_{1}g=\tau_{1}\\}\;\;.$
(189)

One has

$S\supseteq T_{G}\;\;,$ (190)

with equality if $\tau_{1}$ is generic (regular). As above, we set
$T_{S}=T_{G}$ and $\mathfrak{t}_{S}=\mathfrak{t}_{G}$, with
$\mathfrak{g}=\mathfrak{s}\oplus\mathfrak{m}$ and
$[\mathfrak{s},\mathfrak{m}]\subset\mathfrak{m}$.

     * •

Now let us turn to the second entry in (185),

$w\mapsto\mathsf{Ad}_{g^{-1}}(w+[y,v])$ (191)

With $y=\tau_{1}\in\mathfrak{t}_{G}$, and thus $g=s\in S$, this becomes

$w\mapsto\mathsf{Ad}_{s^{-1}}(w+[\tau_{1},v])$ (192)

Decomposing $v=v^{\mathfrak{s}}+v^{\mathfrak{m}}$, only $v^{\mathfrak{m}}$
contributes to the commutator,

$[\tau_{1},v]=[\tau_{1},v^{\mathfrak{m}}]\in\mathfrak{m}\;\;,$ (193)

and $v^{\mathfrak{m}}$ can be chosen to cancel $w^{\mathfrak{m}}$ (because by
definition of $\mathfrak{m}$ the operator $\mathsf{ad}_{\tau_{1}}$ is
invertible on $\mathfrak{m}$).

     * •

We are then left with

$w\mapsto\mathsf{Ad}_{s^{-1}}w^{\mathfrak{s}}\;\;,$ (194)

and as in the proof above we can now choose $s\in S$ such that

$w\mapsto\mathsf{Ad}_{s^{-1}}w^{\mathfrak{s}}=\tau_{2}\in\mathfrak{t}_{S}=\mathfrak{t}_{G}\;\;.$
(195)

     * •

Altogether, we have thus managed to conjugate

$(y,w)\mapsto(\tau_{1},\tau_{2})\in\mathfrak{t}_{G}\oplus\mathfrak{t}_{G}\;\;,$
(196)

as announced.

Remarks:

  1. 1.

We see that for any $(h,w)\in TG$ or $(y,w)\in\mathfrak{tg}$

     * •

one needs $g\in G/S$ (meaning $g\in G$ modulo elements in $S$) to conjugate
$h$ or $y$ into $T_{G}$

     * •

one needs $v^{m}\in\mathfrak{m}\subset\mathfrak{g}$ to map $w\mapsto
w^{\mathfrak{s}}\in\mathfrak{s}$

     * •

one then needs $s\in S/T_{G}$ to conjugate
$w^{\mathfrak{s}}\mapsto\tau_{2}\in\mathfrak{t}_{G}$

Thus in total, for any $g$ or $y$ (regardless of regular or not) the
parameters $(g,v)$ lying in $TT_{G}\simeq T_{G}\times\mathfrak{t}_{G}\subset
TG$ are not needed to accomplish the Abelianisation / diagonalisation.

  2. 2.

In particular, in the generic regular case ($g$ or $y$ regular), one has
$S=T_{G}$ and $\mathfrak{s}=\mathfrak{t}_{G}$. Thus the last step in the proof
(consisting of conjugating $w^{\mathfrak{s}}$ into
$\mathfrak{t}_{S}=\mathfrak{t}_{G}$) is empty. The entire ambiguity in the
construction then lies in the first step, the conjugation of $h$ into $T_{G}$
(or $y$ into $\mathfrak{t}_{G}$). The possible choices of $t$ or $\tau_{1}$
are related by the action of the Weyl Group $W$,

$W=N_{G}(T_{G})/T_{G}\;\;,$ (197)

where $N_{G}(T_{G})$ is the normaliser of $T_{G}$ in $G$.

  3. 3.

In the opposite extreme case of $g=e$ or $y=0$, say, evidently the first step
of the argument is empty, the stabiliser of $g=e$ or $y=0$ is of course all of
$G$, $S=G$, and then in the last step there is then a $W$-fold degeneracy in
the choice $\tau_{2}\in\mathfrak{t}_{G}$.

##  Appendix B A Symmetry of Complex Chern-Simons Theory on Seifert
3-Manifolds

Here we establish that the symmetry that we need in section 3.4 to pass from
having a background field and summing over integers at each orbifold point to
having a background field with just one integer summation to perform exists at
the level of Complex Chern-Simons theory. $BF$ theory is then just a special
limit.444It is straightforward to show that the symmetry is available directly
in $BF$ theory but we wanted to establish the more general result here. One,
possibly surprising, outcome is that the two approaches do not agree at the
level of holomorphic factorisation.

We should also point out that these symmetries are a variation of those found
by Lawrence and Rozansky LR and by Mariño Marino . However, here the
symmetries have a clear geometric meaning which we explain.

###  B.1 Discrete Symmetries

The exponent in the finite dimensional integral that one arrives at in the
case of complex Chern-Simons theory is

$\displaystyle I(k,s)$
$\displaystyle=-\frac{k}{4\pi}\mathop{\mathrm{Tr}}\nolimits(\phi^{2}-\lambda^{2})c_{1}\left(\mathcal{L}_{M}\right)+k\sum_{1}^{N}\frac{1}{a_{i}}\mathop{\mathrm{Tr}}\nolimits(-i\phi\mathbf{n}_{i}+r_{i}\pi\mathbf{n}_{i}^{2})+k\mathop{\mathrm{Tr}}\nolimits(-i\phi\mathbf{n}_{0})$
$\displaystyle+\frac{s}{2\pi}\mathop{\mathrm{Tr}}\nolimits\Big{(}\lambda\left(\phi
c_{1}\left(\mathcal{L}_{M}\right)+2\pi
i\sum_{i=1}^{N}\frac{1}{a_{i}}\mathbf{n}_{i}+2\pi
i\mathbf{n}_{0}\right)\Big{)}$ (198)

(just use this instead of the exponent in (86)).

The exponential of this action as well as the complex Ray-Singer torsion enjoy
a number of symmetries. We will present them here, but before doing that we
note that the transformations below do not depend on the coupling constants
$k$ and $s$ and so those parts of the action are independently invariant. Also
one can see that the $k$-dependent part of the action (198), apart from the
$\lambda^{2}$ term, is just what one gets from Chern-Simons theory on $M$ with
compact gauge group $G$. Furthermore, the field $\lambda$ does not transform,
so that the $k$-dependent part has the same invariance as for the $G$ Chern-
Simons theory.

First, we have the symmetry

$\mathbf{n}_{i}\rightarrow\mathbf{n}_{i}+a_{i}\mathbf{m}_{i},\;\;\mathbf{n}_{0}\rightarrow\mathbf{n}_{0}-\sum_{i=1}^{N}\mathbf{m}_{i}$
(199)

It is clear that (198) is not changed by these transformations (and it is
consistent with our limit on the range of summation over the
$\mathbf{n}_{i}$).

Remark: Note that geometrically this symmetry is the statement that, as the
appropriate powers of the V-line bundle $\mathcal{L}_{i}$ of degree $1/a_{i}$
at the $i$’th orbifold point is an honest line bundle

$\mathcal{L}_{i}^{\otimes m_{i}a_{i}}\simeq\mathcal{L}_{0}^{\otimes m_{i}}$
(200)

any orbifold line bundle $\mathcal{L}$ satisfies

$\displaystyle\mathcal{L}$ $\displaystyle=\mathcal{L}_{0}^{\otimes
n_{0}}\otimes\mathcal{L}_{1}^{\otimes
n_{1}}\otimes\dots\otimes\mathcal{L}_{N}^{\otimes n_{N}}$
$\displaystyle\simeq\mathcal{L}_{0}^{\otimes(n_{0}-\sum_{i}m_{i})}\otimes\mathcal{L}_{1}^{\otimes(n_{1}+a_{1}m_{1})}\otimes\dots\otimes\mathcal{L}_{N}^{\otimes(n_{N}+a_{N}m_{N})}$
(201)

so one has not changed the bundle but just expressed it in a different way.

Second, we also have the transformations

$\mathbf{n}_{i}\rightarrow\mathbf{n}_{i}+b_{i}\mathbf{v},\;\;\phi\rightarrow\phi-2\pi
i\mathbf{v},\;\;\;\mathbf{n}_{0}\rightarrow\mathbf{n}_{0}+b_{0}\mathbf{v}$
(202)

which form a symmetry of the exponential of the action.

Remark: Geometrically this says that the pair $(\mathcal{L},\,\phi)$ of an
orbifold bundle $\mathcal{L}$ with curvature $\phi\,d\kappa$ satisfy

$(\mathcal{L},\,\phi)\simeq(\mathcal{L}\otimes\mathcal{L}_{M}^{\otimes
v},\,\phi-2\pi iv)$ (203)

The first Chern class of $\mathcal{L}_{M}^{\otimes v}$ is

$c_{1}(\mathcal{L}_{M}^{\otimes v})=v.c_{1}(\mathcal{L}_{M})$ (204)

while

$\frac{i}{2\pi}.\int_{\Sigma_{V}}-2\pi
iv\,d\kappa=v.\int_{\Sigma_{V}}d\kappa=-v.c_{1}(\mathcal{L}_{M})$ (205)

It is now possible to combine the symmetries and consider

$\mathbf{n}_{i}\rightarrow\mathbf{n}_{i}+a_{i}\mathbf{u}+b_{i}\mathbf{v},\;\;\phi\rightarrow\phi-2\pi
i\mathbf{v},\;\;\mathbf{n}_{0}\rightarrow\mathbf{n}_{0}+b_{0}\mathbf{v}-N\mathbf{u}$
(206)

One then has a constructive proof that this symmetry allows one to set all the
$\mathbf{n}_{i}$ for $i\neq 0$ to zero. Recall that if the $\gcd(a,\,b)=1$
then any integer $n$ can be expressed as

$n=au+bv$ (207)

for some integers $u$. Furthermore, if $\gcd{(a,\,b)}=1$ and $\gcd{(a,\,c)}=1$
then $\gcd{(a,\,bc)}=\gcd{(a,\,b)}.\gcd{(a,\,c)}=1$.

As a first step let
$(\mathbf{u},\,\mathbf{v})=(\mathbf{u}_{1},\,\mathbf{v}_{1})$ so that

$\mathbf{n}_{1}+a_{1}\mathbf{u}_{1}+b_{1}\mathbf{v}_{1}=0$ (208)

which is guaranteed to have a solution by (207). All the other
$\mathbf{n}_{i}$ are changed by this but are brought back into their
appropriate ranges by the use of (199). Now we want to use (206) again to set
$\mathbf{n}_{2}=0$ while keeping $\mathbf{n}_{1}=0$. If the transformation
$\mathbf{v}$ is proportional to $a_{1}$

$\displaystyle\mathbf{n}_{1}$ $\displaystyle\rightarrow
0+a_{1}\mathbf{u}_{2}+b_{1}a_{1}\mathbf{v}_{2}$ $\displaystyle\mathbf{n}_{2}$
$\displaystyle\rightarrow\mathbf{n}_{2}+a_{2}\mathbf{u}_{2}+b_{2}a_{1}\mathbf{v}_{2}$
$\displaystyle\mathbf{n}_{3}$
$\displaystyle\rightarrow\mathbf{n}_{3}+a_{3}\mathbf{u}_{2}+b_{3}a_{1}\mathbf{v}_{2}$
$\displaystyle\dots$ $\displaystyle\dots\;\;\;\;\dots\;\;\;\dots$
$\displaystyle\mathbf{n}_{N}$
$\displaystyle\rightarrow\mathbf{n}_{N}+a_{N}\mathbf{u}_{2}+b_{N}a_{1}\mathbf{v}_{2}$
(209)

then the change in $\mathbf{n}_{1}$ is proportional to $a_{1}$ and by (199) is
zero. We note that $\gcd(a_{2},\,b_{2}a_{1})=1$ so by (207) we can choose
$(\mathbf{u}_{2},\,\mathbf{v}_{2})$ so the $\mathbf{n}_{2}$ maps to zero. Now
we would like to set $\mathbf{n}_{3}=0$ without changing $\mathbf{n}_{1}$ and
$\mathbf{n}_{2}$. To not change $\mathbf{n}_{1}$ the vector $\mathbf{v}$ must
be proportional to $a_{1}$, as we saw before, and not to change
$\mathbf{n}_{2}$ it must also be proportional to $a_{2}$ so we perform a
transformation

$\displaystyle\mathbf{n}_{1}$ $\displaystyle\rightarrow
0+a_{1}\mathbf{u}_{3}+b_{1}a_{1}a_{2}\mathbf{v}_{3}\simeq 0$
$\displaystyle\mathbf{n}_{2}$ $\displaystyle\rightarrow
0+a_{2}\mathbf{u}_{3}+b_{2}a_{1}a_{2}\mathbf{v}_{3}\simeq 0$
$\displaystyle\mathbf{n}_{3}$
$\displaystyle\rightarrow\mathbf{n}_{3}+a_{3}\mathbf{u}_{3}+b_{3}a_{1}a_{2}\mathbf{v}_{3}$
$\displaystyle\dots$ $\displaystyle\dots\;\;\;\;\dots\;\;\;\dots\quad\dots$
$\displaystyle\mathbf{n}_{N}$
$\displaystyle\rightarrow\mathbf{n}_{N}+a_{N}\mathbf{u}_{2}+b_{N}a_{1}a_{2}\mathbf{v}_{2}$
(210)

Now $\gcd(a_{3},\,b_{3}a_{1}a_{2})=1$ so we can set $\mathbf{n}_{3}=0$ by a
suitable choice of $(\mathbf{u}_{3},\,\mathbf{v}_{3})$.

Clearly the procedure can be repeated till we have set all the
$\mathbf{n}_{i}$ to zero and the parameters we use are

$\mathbf{u}=\sum_{i=1}^{N}\mathbf{u}_{i},\;\;\;\mathbf{v}=\mathbf{v}_{1}+a_{1}\mathbf{v}_{2}+\dots+a_{1}\dots
a_{N-1}\mathbf{v}_{N}$ (211)

###  B.2 Holomorphic Factorisation

The finite dimensional action for complex Chern-Simons theory (198) can be
expressed in terms of complex fields $\phi+i\lambda$ and a complex coupling
constant $t=k+is$ as

$\displaystyle I(t,\bar{t})$
$\displaystyle=-\frac{t}{8\pi}c_{1}(\mathcal{L}_{M})\mathop{\mathrm{Tr}}\nolimits\Phi^{2}-i\frac{t}{2}\mathop{\mathrm{Tr}}\nolimits(\Phi\widehat{\mathbf{q}})+\frac{t\pi}{2}\sum_{i=1}^{N}\mathop{\mathrm{Tr}}\nolimits(\mathbf{n}_{i}^{2})$
(212)
$\displaystyle-\frac{\bar{{t}}}{8\pi}c_{1}(\mathcal{L}_{M})\mathop{\mathrm{Tr}}\nolimits\bar{\Phi}^{2}+i\frac{\bar{t}}{2}\mathop{\mathrm{Tr}}\nolimits(\bar{\Phi}\widehat{\mathbf{q}})+\frac{\bar{t}\pi}{2}\sum_{i=1}^{N}\mathop{\mathrm{Tr}}\nolimits(\mathbf{n}_{i}^{2})$

Now we note that the holomorphic part of the action

$I(t)=-\frac{t}{8\pi}c_{1}(\mathcal{L}_{M})\mathop{\mathrm{Tr}}\nolimits\Phi^{2}-i\frac{t}{2}\mathop{\mathrm{Tr}}\nolimits(\Phi\widehat{\mathbf{q}})+\frac{t\pi}{2}\sum_{i=1}^{N}\mathop{\mathrm{Tr}}\nolimits(\mathbf{n}_{i}^{2})$
(213)

is not invariant under the first symmetry namely under

$\mathbf{n}_{i}\rightarrow\mathbf{n}_{i}+a_{i}\mathbf{m}_{i},\;\;\mathbf{n}_{0}\rightarrow\mathbf{n}_{0}-\sum_{i=1}^{N}\mathbf{m}_{i}$
(214)

even though both $\Phi$ and $\widehat{\mathbf{q}}$ are invariant. The
quadratic term $\mathbf{n}_{i}^{2}$ is not invariant. Furthermore, the
exponential $\exp{\left(iI(t)\right)}$ is also not invariant, unless $s=0$, as
the coupling constant is complex and so one does not just get a phase.

This implies that we can have two inequivalent holomorphic factorizations. The
first is to take the partition function with all the $\mathbf{n}_{i}$ switched
on and then factorise with holomorphic action (213) while the second is to set
all the $\mathbf{n}_{i}=0$ for $i\neq 0$ and then get the factorisation in BT-
CCS .

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