11institutetext: Department of Physics and Astronomy, Uppsala Universitet, 752
37 Uppsala, Sweden22institutetext: Yau Mathematical Sciences Center, Tsinghua
University, Beijing, 100084, China

# From 5d Flat Connections to 4d Fluxes
(the Art of Slicing the Cone)

Jim Lundin 1 Roman Mauch 2 and Lorenzo Ruggeri<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the Coulomb branch partition function of the 4d $\mathcal{N}=2$
vector multiplet on closed simply-connected quasi-toric manifolds $B$. This
includes a large class of theories, localising to either instantons or anti-
instantons at the torus fixed points (including Donaldson-Witten and Pestun-
like theories as examples). The main difficulty is obtaining flux
contributions from the localisation procedure. We achieve this by taking a
detour via the 5d $\mathcal{N}=1$ vector multiplet on closed simply-connected
toric Sasaki-manifolds $M$ which are principal $S^{1}$-bundles over $B$. The
perturbative partition function can be expressed as a product over slices of
the toric cone. By taking finite quotients $M/\mathbb{Z}_{h}$ along the
$S^{1}$, the locus picks up non-trivial flat connections which, in the limit
$h\to\infty$, provide the sought-after fluxes on $B$. We compute the one-loop
partition functions around each topological sector on $M/\mathbb{Z}_{h}$ and
$B$ explicitly, and then factorise them into contributions from the torus
fixed points. This enables us to also write down the conjectured instanton
part of the partition function on $B$.

††preprint: UUITP-12/23

##  1 Introduction

Four-dimensional $\mathcal{N}=2$ supersymmetric quantum field theories (SQFTs)
have proven, over the years, to be an excellent laboratory for studying
connections between QFT and mathematics and for enhancing our insight into
more realistic QFTs. The work of Witten Witten:1988ze  showed how to interpret
Donaldson invariants of four-manifolds Donaldson:1985zz  in terms of
observables in a topological subsector of the full $\mathcal{N}=2$ SQFT
counting instantons. Later, the derivation of the Seiberg-Witten exact
prepotential of an $SU(2)$ $\mathcal{N}=2$ gauge theory in the IR
Seiberg:1994rs  greatly improved our understanding of strongly coupled QFTs.
This result relied on a version of electric-magnetic duality and on the
holomorphicity of the moduli space of vacua. It was later reinterpreted by
Nekrasov Nekrasov:2002qd , as the non-equivariant limit of the equivariant
volume of the instanton moduli space in the $\Omega$-background. This
computation employed supersymmetric localisation (see Pestun:2016zxk  for a
comprehensive review) to reduce the integral over the entire instanton moduli
space to a discrete set of fixed points under a $T^{2}$-isometry. The Nekrasov
partition function is the building block of Pestun’s result Pestun:2007rz  for
the exact partition function of the $\mathcal{N}=2^{\ast}$ theory on $S^{4}$:
on this background, the field strength localises to point-like instantons at
the north pole and point-like anti-instantons at the south pole. The partition
function is obtained by gluing two copies of the Nekrasov partition function,
one for instantons and one for anti-instantons. Subsequently, a plethora of
exact results for SQFTs have been obtained. Among these, related to our work
are studies of $\mathcal{N}=2$ topologically twisted theories on compact toric
manifolds Bawane:2014uka ; Rodriguez-Gomez:2014eza ; Bershtein:2015xfa ;
Bershtein:2016mxz ; Bonelli:2020xps . See also Hosseini:2018uzp  for
topologically twisted theories on $M_{4}\times S^{1}$, where $M_{4}$ is a
compact 4d toric Kähler manifold.

More recently, in Festuccia:2018rew ; Festuccia:2019akm ; Festuccia:2020yff  a
framework was developed to generalise Pestun’s result to any compact simply-
connected four-manifold with a $T^{2}$-isometry with isolated fixed points and
an arbitrary distribution of (anti-)instantons at those points. The partition
function is conjecturally obtained by gluing (anti-)instanton Nekrasov
partition functions. However, this is not the full story, as four-manifolds
$B$ with non-trivial $H^{2}(B;\mathbb{Z})$ also admit gauge configurations
where the field strength has flux on non-contractible cycles. It was
conjectured in Nekrasov:2003vi ; Festuccia:2018rew  that the flux
contributions enter as a shift of the Coulomb branch parameter in each copy of
the Nekrasov partition function. The full partition function is then obtained
as a sum over these flux sectors. In the literature Nekrasov:2003vi ;
Nakajima:2003pg ; Nakajima:2003uh ; Gottsche:2006tn ; Bawane:2014uka ;
Bershtein:2015xfa ; Bershtein:2016mxz ; Hosseini:2018uzp ; Crichigno:2018adf ;
Festuccia:2018rew ; Festuccia:2019akm ; Bonelli:2020xps , the sum is over
equivariant fluxes which are then constrained by stability equations.

The main objective of this work is to compute the one-loop partition function
around fluxes, for the 4d $\mathcal{N}=2$ vector multiplet on compact simply-
connected quasi-toric manifolds. In particular, we allow the theories to
localise to different distributions of (anti-)instantons at the torus fixed
points (also known as Pestunization). We explicitly compute the one-loop
partition function of these theories around each flux sector, whose factorised
form111By “factorised” we mean an expression for the partition function in
terms of a product over its contributions from individual torus fixed points
on the manifold. This notion was introduced in Pasquetti:2011fj ; Beem:2012mb
. agrees with the form conjectured in Nekrasov:2003vi ; Festuccia:2018rew .
From this form we can read out the shifts of the Coulomb branch parameter at
each torus fixed point by the fluxes and write down the Coulomb branch
partition function222This is the holomorphic “integrand” of the full partition
function and we make this distinction since we will not concern ourselves with
the remaining integral over the Coulomb branch parameter in this work.,
including instanton and classical parts.

The main difficulty is obtaining flux solutions from the 4d localisation
equations. We circumvent this issue by taking a detour via the 5d
$\mathcal{N}=1$ vector multiplet on closed toric Sasakian manifolds. This
setup is well-studied Kallen:2012cs ; Kallen:2012va ; Imamura:2012efi ;
Lockhart:2012vp ; Kim:2012qf ; Qiu:2013pta ; Qiu:2013aga ; Qiu:2014oqa ;
Alday:2015lta ; Festuccia:2016gul , in particular for $S^{5}$ and $Y^{p,q}$
spaces (see also Qiu:2016dyj  for a review). In order to, ultimately, return
to 4d, we assume the five-dimensional manifold $M$ to be a (non-trivial)
principal $S^{1}$-bundle

$\tikzcd[columnsep=small]S^{1}\ar[r,hook]&M\ar[r,"p"]B$ (1)

over $B$. We can usually have multiple different such bundles over $B$, which
lead to different (anti-)instanton distributions (depending on the relative
orientation of the free $S^{1}$-direction with respect to the Reeb vector).
For simply-connected $M$, the perturbative partition function was computed in
Qiu:2014oqa  and is given by

$Z_{M}=\prod_{\alpha\in\Delta}\prod_{\vec{n}\in\mathcal{C}\cap\mathbb{Z}^{3}}(\vec{n}\cdot\vec{\R}+\ii\alpha(a))\prod_{\vec{n}\in\mathcal{C}^{\circ}\cap\mathbb{Z}^{3}}(\vec{n}\cdot\vec{\R}-\ii\alpha(a)).$

with $\vec{\R}$ the Reeb vector and $\mathcal{C}$ the three-dimensional (dual)
moment map cone of the $T^{3}$-action on $M$. We first rewrite this result as
a product over two-dimensional slices $\mathcal{B}_{t}$ of the cone along the
free $S^{1}$-direction (as displayed in Figure 2), labelled by their integral
charge $t$ under the $S^{1}$:

$Z_{M}=\prod_{\alpha\in\Delta}\prod_{t}\prod_{(n_{1},n_{2})\in\mathcal{B}_{t}}(n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+t\tfrac{\R_{3}}{l_{3}}+\ii\alpha(a))\prod_{(n_{1},n_{2})\in\mathcal{B}_{t}^{\circ}}(n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+t\tfrac{\R_{3}}{l_{3}}-\ii\alpha(a)).$

Here, $\epsilon_{1},\epsilon_{2}$ are the equivariance parameters
corresponding to the remaining $T^{2}$-action and $l_{3}$ some constant.

As a first step towards $B$, we introduce a quotient of the simply-connected
$M$ along the free $S^{1}$ by a finite subgroup, $X=M/\mathbb{Z}_{h}$. The
quotient has $\pi_{1}(X)=\mathbb{Z}_{h}$ and the localisation locus must be
extended to include topologically non-trivial flat connections, valued in the
Cartan of the gauge algebra. The effect of the quotient on the partition
function is to discard most slices and only keep the ones for which
$t=c_{1}\mmod h$, controlled by the first Chern class $c_{1}$ of the
corresponding $U(1)$-bundles333Here and below, when we write $c_{1}\mmod h$ we
are implicitly using the isomorphism
$H^{2}(X)\simeq\mathbb{Z}_{h}\oplus\mathbb{Z}^{b_{2}(X)}$ under which $c_{1}$
takes values in $\mathbb{Z}_{h}$ (see Appendix A).. The partition function on
$X$ is then given by a sum over these topological sectors. We suspect that
this procedure can be interpreted as gauging a $\mathbb{Z}_{h}$ 0-form
symmetry on $M$.

Finally, we return to $B$ by taking the limit of large $h$. The non-trivial
connections in the 5d locus on $X$, upon reduction, become connections in the
4d locus on $B$ with flux on a two-cycle. In terms of the partition function,
instead of a product over different slices, for large $h$ only a single slice
of the cone survives, labelled by the aforementioned flux $\mathfrak{m}$:

$Z_{B}=\prod_{\alpha\in\Delta}\prod_{(n_{1},n_{2})\in\mathcal{B}_{\mathfrak{m}}}(n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+\alpha(\mathfrak{m})\tfrac{\R_{3}}{l_{3}}+\ii\alpha(a))\prod_{(n_{1},n_{2})\in\mathcal{B}_{\mathfrak{m}}^{\circ}}(n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+\alpha(\mathfrak{m})\tfrac{\R_{3}}{l_{3}}-\ii\alpha(a)).$

Note that one can think of $Z_{M}$ as counting holomorphic functions444Since
$C(M)$ is a toric manifold, the integral lattice of the toric cone
$\mathcal{C}$ is in 1:1 correspondence to holomorphic functions on $C(M)$. See
Qiu:2016dyj  for precise relations to the one-loop calculation. on the metric
cone $C(M)$. Then $Z_{B}$ counts precisely the holomorphic functions on $C(M)$
that have charge $c_{1}$ under $\X$. For the full partition function, the
topological sectors are summed over as before:

$\mathcal{Z}_{B}=\sum_{\mathfrak{m}}\int_{\mathfrak{h}}da\;e^{-S_{\text{cl}}}\cdot
Z_{B}\cdot Z_{B}^{\text{inst}}.$

The one-loop partition function $Z_{B}$ for flux $\mathfrak{m}$ can then be
factorised using 5d techniques. Note that, depending on which fibration (1) we
choose, the remaining slice $\mathcal{B}_{\mathfrak{m}}$ can be either compact
or non-compact. The theory on $B$ corresponding to the former is commonly
known as the topologically twisted theory, localising to instantons on all
torus fixed points (resulting in an elliptic complex; e.g. Donaldson-Witten
theory). The latter case appears for more general distributions of
(anti-)instantons (resulting in a transversally elliptic complex; e.g.
Pestun’s theory on $S^{4}$).

The 5d detour described above makes the inclusion of fluxes into the 4d locus
a simple consequence of the non-trivial nature of the fibration (1). Given
that we obtain flux contributions on $B$ as the base of an $S^{1}$-fibration,
one might wonder whether a similar technique could be applied to analyse non-
Abelian features of the locus from non-Abelian fibrations over the four-
manifold.

Finally, as a disclaimer, whenever $H^{2}(M;\mathbb{Z})\neq 0$, our procedure
might only produce part of the sum over all possible fluxes in the full
partition function on $B$. This is because our starting point in 5d is a
localisation result Qiu:2014oqa , whose locus does not include any flux. As a
consequence, our result for the 4d partition function might miss out on these
fluxes. In order to include them, it would be necessary to allow for a more
general 5d localisation locus (possibly by relaxing reality conditons or
including singular configurations, cf. Benini:2015noa ; Closset:2015rna  in 3d
and 2d).

The outline is as follows. In section 2 we introduce some basic facts about
toric Sasakian geometry and the condition under which these manifolds admit a
free $S^{1}$-action. As an example, we consider $Y^{p,q}$. Then, in section 3,
we explain the supersymmetric setup on these manifolds and on their quotients.
We also discuss how flat connections in $X$ lead to flux on $B$. In section 4
we perform in detail the steps of slicing and reducing outlined above and
compute the one-loop partition function around flat connections on the
quotient $X$ and around fluxes on the four-dimensional base $B$. In
particular, we show how each flux sector arises as a single slice of the
three-dimensional cone. The construction is illustrated by explicit examples
in section 5, considering $S^{5}$, $Y^{p,q}$ and $A^{p,q}$. The factorisation
properties of the partition functions are studied in section 6 where,
employing the shifts derived from the one-loop part around fluxes, we can
finally write the full partition function, including instanton and classical
contributions. We summarise our results in section 7 and comment on possible
future directions.

This work is a generalisation of earlier work Lundin:2021zeb  on
$\mathbb{CP}^{2}$ by two of the authors.

##  2 Toric Sasaki-Manifolds

In this section, we briefly review some facts about toric Sasaki-manifolds on
which we later place our theory. A detailed account of contact structures and
Sasakian geometry can be found in Boyer:2008era , for toric geometry see
Fulton:1993 . Throughout the article, we assume all manifolds as being smooth
and connected.

###  2.1 Toric Sasakian Geometry

#### Contact Structures.

The basic underlying structure of Sasakian manifolds is a contact structure.
It can be viewed as the odd-dimensional cousin of a symplectic structure on
even-dimensional manifolds. More precisely, on a $(2n-1)$-dimensional manifold
$M$, consider a field $\xi\subset TM$ of hyperplanes on $M$. This field can be
expressed as $\xi=\ker\kappa$, where $\kappa$ is a one-form555In fact,
$\kappa$ is globally defined only if $\xi$ is co-orientable, i.e. the line
bundle $TM/\xi$ is trivial. Equivalently, one can always find a global
$\kappa$ with $\xi=\ker\kappa$ if both, $M$ and $\xi$ are orientable. on $M$.
If $\kappa\wedge(\dd\kappa)^{n-1}\neq 0$ everywhere666An equivalent
formulation of this condition is $(\dd\kappa)^{n-1}|_{\xi}\neq 0$. Hence, we
anticipate that $\dd\kappa$ provides a symplectic structure for a suitable
codimension-one submanifold. on $M$, then $\xi$ is called a contact structure,
$\kappa$ a contact form and $(M,\xi)$ a contact manifold. Note that in this
case $\kappa\wedge(\dd\kappa)^{n-1}$ defines a volume form of $M$ (i.e., in
particular, $M$ must be orientable).

To a contact form $\kappa$ we can associate a vector field $\R$ defined by the
equations

$\iota_{\SR}\kappa=1,\qquad\iota_{\SR}\dd\kappa=0.$ (2)

This is called the Reeb vector field. Since we consider $M$ to be Riemannian,
we demand some compatibility conditions between the metric and contact
structure: the metric $g$ should be preserved, i.e. $\mathcal{L}_{\SR}g=0$ and
there should exist an almost complex structure $J$ on $\ker\kappa$ which can
be extended to $TM$ via $J(\R)=0$ such that for vector fields $X,Y$ on $M$,
$g(X,Y)=\frac{1}{2}\dd\kappa(X,J(Y))+\kappa(X)\kappa(Y)$ (note that
$(\ker\kappa,J|_{\ker\kappa})$ provides $M$ with an almost CR-structure). If
these conditions are satisfied, $(M,g,\kappa,J)$ is called a K-contact
manifold. Note that a consequence of the second condition and (2) is
$\kappa=g(\R,\us)$.

The Reeb vector field generates a flow on $M$ and its orbits (which are
geodesics) can be viewed as the leaves of a one-dimensional foliation, the so-
called Reeb foliation $\mathcal{F}_{\SR}$. Note that the tangent bundle $TM$
can be orthogonally decomposed as $TM\simeq\ker\kappa\oplus L_{\SR}$ with
respect to $g$, where $L_{\SR}$ is the (trivial) line bundle consisting of
vectors tangent to the leaves of $\mathcal{F}_{\SR}$.

One way to characterise $M$ is by the regularity property of the Reeb
foliation. $\mathcal{F}_{\SR}$ is called quasi-regular if there is an integer
$k$ such that for every point $x\in M$ there is a foliated chart through which
each leaf passes at most $k$-times. $\mathcal{F}_{\SR}$ is called regular if
$k=1$ and irregular if it is not quasi-regular777Note that in the case of a
regular (quasi-regular, irregular) foliation we also talk about a regular
(quasi-regular, irregular) Reeb vector field or contact form. However,
different contact forms in a contact structure can produce drastically
different foliations, so it does not make sense to talk about a regular
(quasi-regular, irregular) contact structure.. Note that if $M$ is compact and
$\mathcal{F}_{\SR}$ quasi-regular, the orbits are circles and we get a locally
free $S^{1}$-action on $M$ (which is isometric since ${\R}$ is Killing). If
$\mathcal{F}_{\SR}$ is regular, the $S^{1}$-action is free. Hence, in this
case, we have a principal $S^{1}$-bundle $\pi:M\rightarrow B$ over the space
of leaves $B=M/\mathcal{F}_{\SR}$ which is a compact, symplectic manifold with
symplectic form $\omega$ such that $\pi^{\ast}\omega=\dd\kappa$. In the
locally free case instead, $B$ is a symplectic orbifold. In contrast, in the
irregular case, ${\R}$ generates a group action with orbits whose closure is
isomorphic to a torus.

#### Sasakian Manifolds.

As mentioned above, a Sasakian manifold is a special case of a K-contact
manifold and can be viewed as the odd-dimensional cousin of a Kähler manifold
in even dimensions. More precisely, a K-contact manifold $(M,g)$ is Sasakian
if its metric cone $(C(M),\bar{g})$ with $C(M)=\mathbb{R}_{>0}\times M$ and
$\bar{g}=\dd r^{2}+r^{2}g$ is Kähler. The corresponding Kähler form and
complex structure are

$\bar{\omega}=\frac{1}{2}\dd(r^{2}\kappa),\qquad\bar{J}=J+\frac{1}{r}\zeta\otimes\dd
r-r\partial_{r}\otimes\kappa.$ (3)

Here, $J$ and $\kappa$ are trivially extended to $C(M)$ and
$\zeta=\bar{J}(r\partial_{r})$ is the characteristic vector field. The Kähler
condition is equivalent to $\bar{J}$ being covariantly constant,
$\bar{\nabla}\bar{J}=0$, with respect to the Levi-Civita connection (see e.g.
McDuff:2017 , lemma 4.2.5). It follows that $\|\zeta\|=r^{2}$,
$\mathcal{L}_{\zeta}\bar{g}=0$ and $\zeta$ is tangent to hypersurfaces of
constant $r$. Hence, $M$ can be naturally embedded into $C(M)$ as the
hypersurface with $r=1$ and $\zeta$ can be identified with the Reeb vector
field of $M$.

As described above, we can characterise a Sasakian manifold as regular, quasi-
regular, or irregular, depending on its Reeb foliation. For a compact regular
(quasi-regular) Sasakian manifold, the space of leaves $B$ is a compact Kähler
manifold (orbifold) (see Boyer:2008era , theorem 7.1.3). In turn,
$\mathcal{F}_{\SR}$ is a transverse Kähler foliation. Note that regular
Sasakian manifolds are rare (e.g. $S^{5}$ with $B=\mathbb{CP}^{2}$) and most
Sasakian manifolds are either quasi-regular or irregular (e.g. $Y^{p,q}$ which
we discuss in more detail below). Since the aim of this work is to start from
a five-dimensional Sasakian manifold and reduce, along a free direction, to
four dimensions, it is clear that this direction, in general, has to be
different from the Reeb vector field.

#### Toric Sasakian Geometry.

A $(2n-1)$-dimensional Sasakian manifold $M$ is called toric if, on its cone
$(C(M),\bar{g},\bar{\omega},\bar{J})$, there exists an effective, holomorphic
$T^{n}$-action that is Hamiltonian and such that $\zeta\in\mathfrak{t}_{n}$ is
an element888Note the abuse of notation by identifying an element in
$\mathfrak{t}_{n}$ with its induced vector field. of the Lie algebra of
$T^{n}$. The Hamiltonian property implies existence of a $T^{n}$-equivariant
moment map

$\mu:C(M)\longrightarrow\mathfrak{t}_{n}^{\vee}\simeq\mathbb{R}^{n},$ (4)

where $\mathfrak{t}_{n}^{\vee}$ denotes the dual, such that

$\forall v\in\mathfrak{t}_{n}:\;\dd\langle\mu,v\rangle=\iota_{v}\bar{\omega}.$
(5)

From (3) it follows that (up to an additive constant) the moment map is
determined by

$\langle\mu,v\rangle=\frac{1}{2}r^{2}\kappa(v).$ (6)

Moreover, the image $\mu(C(M))\cup\\{0\\}$ is a strictly convex, rational,
polyhedral cone $\mathcal{C}^{\vee}$ in
$\mathfrak{t}_{n}^{\vee}\simeq\mathbb{R}^{n}$ (see Lerman:2001zua , theorem
4.2), i.e. $\mathcal{C}^{\vee}$ can be presented as

$\mathcal{C}^{\vee}=\\{u\in\mathfrak{t}_{n}^{\vee}|\forall i=1,\dots,m:\langle
u,v_{i}\rangle\geq 0\\}.$ (7)

Here, $v_{i}\in\mathfrak{t}_{n}$, $i=1,\dots,m$ are the “inward-pointing
normals” of the facets (codimension-one faces) of the cone and $m$ the number
of facets. The rationality means that the $v_{i}$ are elements of the lattice
of circle subgroups of $T^{n}$, i.e. upon
$\mathfrak{t}_{n}\simeq\mathbb{R}^{n}$ we have $v_{i}\in\mathbb{Z}^{n}$.
Furthermore, we can assume $\\{v_{i}\\}_{i=1,\dots,m}$ to be minimal and
primitive. Since the cone $\mathcal{C}^{\vee}$ is strictly convex, its base is
a compact convex $(n-1)$-dimensional polytope; hence, $m\geq n$. An intuitive
way of thinking about the moment map (4) is as a $T^{n}$-fibration over
$\mathcal{C}^{\vee}$ where at the $i$th facet of the cone the circle subgroup
specified by $v_{i}$ degenerates. We can also define the dual cone
$\mathcal{C}\subset\mathfrak{t}_{n}$ as

$\mathcal{C}=\\{v\in\mathfrak{t}_{n}|\forall u\in\mathcal{C}^{\vee}:\langle
u,v\rangle\geq 0\\}=\\{v\in\mathfrak{t}_{n}|\forall i:v\cdot v_{i}\geq 0\\},$
(8)

which is again a convex, rational, polyhedral cone Sparks:2010sn . Note that,
for the characteristic vector field, $\zeta\in\mathcal{C}^{\circ}$ since
$\langle\mu,\zeta\rangle=\frac{1}{2}r^{2}>0$ (here and in the following,
$\mathcal{C}^{\circ}$ denotes the interior of $\mathcal{C}$). Note that, once
we pick a basis $\\{e_{i}\\}\subset T^{n}$, we are free to perform
$SL(n,\mathbb{Z})$-transformations which in turn generate
$SL(n,\mathbb{Z})$-transformations of $\im\mu$. Hence, the cones and polytopes
are unique only up to such transformations.

Recall that the toric Sasakian manifold $M$ is naturally embedded into $C(M)$
as $\\{1\\}\times M$. Hence, we can define a moment map on $M$ by restriction
$\mu|_{\\{1\\}\times M}$ (which we again denote by $\mu$) and, using (6), the
image of $M$ under the moment map is given by

$\mu(M)=\\{u\in\mathcal{C}^{\vee}|\langle u,\R\rangle=\tfrac{1}{2}\\},$ (9)

i.e. the intersection of $\mathcal{C}^{\vee}$ with the characteristic plane
$\\{u\in\mathfrak{t}^{\vee}_{n}|\langle u,\R\rangle=\frac{1}{2}\\}$. This
image is an $(n-1)$-dimensional compact convex polytope Boyer:2000 . Similarly
to $C(M)$, we can view $M$ as a $T^{n}$-fibration where the base is now
$\mu(M)$, i.e. $(n-1)$-dimensional. This is illustrated in Figure 1 for the
five-dimensional case ($n=3$) which is the relevant one for us. In particular,
at the vertices of the polytope, only an $S^{1}$ survives whose orbit is a
closed Reeb orbit ($\R\neq 0$ everywhere on $M$). Hence, locally around the
vertices, $M$ looks like $\mathbb{C}^{2}\times S^{1}$.

$\mathcal{C}^{\vee}$$\mu(M)$$T^{3}$$T^{2}$$S^{1}$ Figure 1: On the left, the
image $\mu(M)$ of $M$ is depicted for $n=3$ and $m=4$ as the section of
$\mathcal{C}^{\vee}$ with the characteristic plane. On the right, we show
$\mu(M)$ with the inward-pointing normals and the $T^{3}$-fibration
degenerating to $T^{2}$ at the edges and $S^{1}$ at the vertices.

The cone structure encodes some topology of the toric Sasakian manifold $M$,
namely, we have
$\pi_{1}(M)\simeq\mathbb{Z}^{n}/\sspan_{\mathbb{Z}}\\{v_{i}\\}$ and
$\pi_{2}(M)\simeq\mathbb{Z}^{m-n}$. Hence, if $\\{v_{i}\\}$ with
$\mathbb{Z}$-coefficients span all of $\mathbb{Z}^{n}$ then $M$ is simply-
connected. Moreover, if $n=3$ and $\pi_{1}(M)=0$, then $M$ is diffeomorphic to
$(S^{2}\times S^{3})^{\\#k}$, where $k=m-n$ (see Sparks:2010sn , prop. 5.3 and
cor. 5.4).

Finally, note that the metric cone $C(M)$ with its symplectic structure and
Kähler metric can be completely recovered from the moment map cone
$\mathcal{C}^{\vee}$ via Kähler reduction Lerman:2001zua : if
$\\{v_{i}\\}_{i=1,\dots,m}\subset\mathbb{Z}^{n}$ denotes the set of normals
for $\mathcal{C}^{\vee}$ and $\Lambda\subset\mathbb{Z}^{n}$ the lattice they
span, then there is a map of tori

$T^{m}\simeq\mathbb{R}^{m}/2\pi\mathbb{Z}^{m}\longrightarrow\mathbb{R}^{n}/2\pi\Lambda$
(10)

induced by the linear map
$\mathbb{R}^{m}\rightarrow\mathbb{R}^{n},\,E_{i}\mapsto v_{i}$ taking the
$i$th canonical basis vector $E_{i}\in\mathbb{R}^{m}$ to the normal $v_{i}$.
The kernel $\mathcal{K}$ of (10) is isomorphic to $T^{m-n}\times\Gamma$
($\Gamma$ some finite Abelian group) and we get

$C(M)=\mathbb{C}^{m}/\\!/\mathcal{K}.$ (11)

The group $T^{m}/\mathcal{K}\simeq T^{n}$ acts symplectically on $C(M)$ and
the image of the moment map is precisely $\mathcal{C}^{\vee}$ (see
Lerman:2001zua , theorem 2.18). This way, we can express the coefficients
$\\{\R_{i}\\}$ of the Reeb vector field $\R=\sum_{a=1}^{n}\R_{a}e_{a}$,
$\\{e_{a}\\}\subset\mathcal{C}$ in terms of the charges
$\\{\omega^{(j)}_{i}\\}^{j=1,\dots,m-n}_{i=1,\dots,m}$ with which the $j$th
$S^{1}\subset\mathcal{K}$ acts on $\mathbb{C}^{m}$. Moreover, note that, for a
fixed toric Sasakian manifold $M$, the smooth Sasakian metrics are
parametrised by the Reeb $\R\in\mathcal{C}^{\circ}$ Martelli:2005tp . Hence,
given a cone $\mathcal{C}$, deforming $\R$ in a way that leaves it inside the
cone gives again a Sasakian structure on $M$, with a squashed metric. This
deformation of the Reeb will be necessary for regular Sasakian manifolds (cf.
$S^{5}$ in subsection 5.1) in order to obtain an equivariant theory after
dimensional reduction to the 4d topologically twisted theory. The deformation
enables us to tune the equivariance parameters in the 4d theory.

#### Free $S^{1}$-Action.

Henceforward, we focus on the five-dimensional case, i.e. $n=3$ and the cones
$\mathcal{C}^{\vee},\mathcal{C}$ are three-dimensional. It was mentioned above
already that $\R$ is, in general, not generating a free action on $M$.
Therefore, in order to be able to quotient or reduce along a free $S^{1}$, we
have to find some element $\vec{\X}=\sum_{i=1}^{3}\X_{i}e_{i}$ in
$\mathfrak{t}_{n}$ whose orbits are of some period $L$ and form a regular
foliation $\mathcal{F}_{\vec{\X}}$ of $M$. Its components $\X_{i}$ are
obtained as the solution to the following set of equations (where
$v_{m+1}=v_{1}$):

$\forall
i=1,\dots,m:\vec{\X}\cdot(\vec{v}_{i}\times\vec{v}_{i+1})=\pm\frac{L}{2\pi},$
(12)

whereby the period $L$ has to be chosen as the smallest such for which the
coefficients of $\vec{\X}$ are integral. This ensures that the
$\vec{\X}$-orbits close and pass through each foliated chart of
$\mathcal{F}_{\vec{\X}}$ only once (i.e. the stabiliser is trivial everywhere
on $M$). It is not obvious whether (12) always has a solution and, in fact,
there are well known examples for which no regular foliation exists
Cvetic:2005ft . However, since we are ultimately interested in 4d theories
obtained as the quotient of $M$ by a free $S^{1}$, we limit our considerations
to manifolds for which (12) has solutions.

Note that, since $\vec{\R}$ lies within the dual cone, we can write
$\vec{\R}=\sum_{i=1}^{m}\lambda_{i}\vec{v}_{i}$ with $\lambda_{i}>0$ (cf.
(8)). Hence, $\vec{\R}\cdot(\vec{v}_{i}\times\vec{v}_{i+1})>0$ for all $i$ and
therefore (12) determines whether $\vec{\X}$ and $\vec{\R}$ are parallel or
anti-parallel at the vertices of the polytope $\mu(M)$.

###  2.2 An Example: $Y^{p,q}$

Let us now consider the concrete example $Y^{p,q}$. These are an infinite
family of closed, five-dimensional toric Sasakian999In fact, the $Y^{p,q}$ are
toric Sasaki-Einstein manifolds. These can be viewed as odd-dimensional
cousins of Calabi-Yau manifolds, i.e. $C(Y^{p,q})$ is CY. This additional
property places some restrictions on the cone $\mathcal{C}^{\vee}$ which we
will mention en passant. manifolds with $p,q\in\mathbb{Z}$ such that $p>1$,
$p>q$ for which explicit metrics are known Gauntlett:2004yd ; Martelli:2004wu
. For $p$, $q$ coprime, i.e. $\gcd(p,q)=1$, $Y^{p,q}$ is simply-connected and
topologically $S^{2}\times S^{3}$. Geometrically, it is a principal
$S^{1}$-bundle $Y^{p,q}\rightarrow B$ over the product of an axially squashed
and a round sphere $B=S^{2}\times S^{2}$, such that the first Chern number on
the standard two-cycles are $p$ and $q$.

Locally, we can choose coordinates $\theta\in[0,\pi],\phi\in[0,2\pi)$ and
$y\in[y_{1},y_{2}],\psi\in[0,2\pi)$ as polar, azimuthal angles on the two
spheres and $\gamma\in[0,2\pi)$ parametrising the $S^{1}$-fibre. In these
coordinates, a metric can be given explicitly and the Reeb vector field is

$\R=3\partial_{\psi}-\tfrac{1}{2\ell}\partial_{\gamma}.$ (13)

$Y^{p,q}$ has effectively acting isometry group $SO(3)\times U(1)^{2}$ for
$p,q$ odd and $U(2)\times U(1)$ else Martelli:2004wu . In both cases, we have
an effective Hamiltonian $T^{3}$-action on $Y^{p,q}$ for the following choice
of basis:

$\displaystyle e_{1}=$ $\displaystyle-\partial_{\phi}-\partial_{\psi},$ (14)
$\displaystyle e_{2}=$
$\displaystyle\,\partial_{\phi}-\tfrac{l}{2}\partial_{\gamma},$ $\displaystyle
e_{3}=$ $\displaystyle\,\partial_{\gamma}.$

Note that the choice (14) is different from the standard Killing vectors
$\partial_{\phi},\partial_{\psi},\partial_{\gamma}$ which, in general, do not
generate an effective action with closed orbits. This is due to the
submanifolds $y=y_{1},y_{2}$ being lens spaces $S^{3}/\mathbb{Z}_{k}$,
$S^{3}/\mathbb{Z}_{l}$ ($k=p+q$, $l=p-q$) on which the standard basis
$\partial_{\phi},\partial_{\gamma}$ does not generate an effective action with
closed orbits101010See Martelli:2004wu , section 4 for a discussion on the
subtleties related to effective torus actions on lens spaces..

In the basis above, we obtain the following moment map for $C(Y^{p,q})$
Martelli:2004wu :

$\mu=(-\tfrac{r^{2}}{6}(1-y)(\cos\theta-1),\tfrac{r^{2}}{6}(1-y)\cos\theta-\tfrac{r^{2}}{2}l\ell
y,\ell r^{2}y);$ (15)

here, $\ell=q(3q^{2}-2p^{2}+p(4p^{2}-3q^{2})^{1/2})^{-1}$ and $r$ is the cone
direction. At the edges of the cone, $T^{3}$ collapses into an $S^{1}$; on
$Y^{p,q}$ this is the case at the poles of the two spheres ($\theta=0,\pi$ and
$y=y_{1},y_{2}$), i.e. the image of (15) is spanned by four edges. By fixing,
for example, $r=1$ in the cone direction and evaluating $\mu$ at the poles
above, we obtain the edge vectors

$\vec{u}_{1}=[0,0,1],\quad\vec{u}_{2}=[0,p,-1],\quad\vec{u}_{3}=[p+q,-q,-1],\quad\vec{u}_{4}=[p-q,q-p,1]$
(16)

and inward-pointing primitive normals111111We point out that there is a vector
$\vec{\xi}=[1,0,0]$ such that $\vec{\xi}\cdot\vec{v}_{i}=1$ for all $i$. Cones
satisfying this condition for some $\vec{\xi}$ are called Gorenstein, which is
the case for all Sasaki-Einstein manifolds (see e.g. Festuccia:2016gul  for a
discussion).

$\vec{v}_{1}=[1,0,0],\quad\vec{v}_{2}=[1,1,0],\quad\vec{v}_{3}=[1,2,p-q],\quad\vec{v}_{4}=[1,1,p].$
(17)

The cone spanned by (16) is depicted in Figure 5 for $p=2,q=1$.

The Kähler cone of $Y^{p,q}$ can also be obtained by symplectic reduction of
$\mathbb{C}^{4}$ under a $U(1)$ Boyer:2011ia ; Qiu:2013pta . If we denote by
$\\{E_{j}\\}_{j=1,\dots,4}$ the generators of the $U(1)^{4}$ acting
canonically on $\mathbb{C}^{4}$, then the reduction is performed along the
$U(1)$ with charge $[p+q,p-q,-p,-p]$. The Reeb vector field
$\R=\sum_{i=1}^{3}\R_{i}e_{i}$ can also be written as
$\R=\sum_{j=1}^{4}\omega_{j}E_{j}$ and the $\R_{i}$ related to the general
equivariance parameters $\omega_{j}$:

$\R_{1}=-\omega_{1}-\omega_{2}-\omega_{3}-\omega_{4},\qquad\R_{2}=-\omega_{1}-\omega_{2}-2\omega_{4},\qquad\R_{3}=-p\omega_{2}+(q-p)\omega_{4}.$
(18)

For the Reeb vector field (13) we find

$\omega_{1}=0,\qquad\omega_{2}=\frac{1}{(p+q)\ell},\qquad\omega_{3}=\omega_{4}=\left(\frac{3}{2}-\frac{1}{2\ell(p+q)}\right).$
(19)

The Reeb vector field here is quasi-regular if $\ell$ is rational and
irregular else Martelli:2004wu . In contrast, solving (12) we find that free
actions are generated by

$\vec{\X}^{\text{top}}=\begin{cases}[1,1,\tfrac{p}{2}],&p\text{ even}\\\
[2,2,p],&p\text{ odd}\end{cases},\qquad\vec{\X}^{\text{ex}}=[0,0,1].$ (20)

Here, $\vec{\X}^{\text{top}}$ is parallel to $\R$ at all four vertices of the
polytope while $\vec{\X}^{\text{ex}}$ is parallel to $\R$ only at the first
two vertices and anti-parallel at the last two (“top”, “ex” for topological,
exotic; see discussion at the end of subsection 3.3 or Festuccia:2019akm ).

##  3 Super Yang-Mills on Sasakian 5-Manifolds

In this section, we discuss $\mathcal{N}=1$ super Yang-Mills (SYM) theory on
the five-dimensional toric Sasakian manifold $M$. Furthermore, we always
assume that $M$ has a free direction (i.e. (12) has solutions). We start by
introducing the $\mathcal{N}=1$ field content and supersymmetric actions for
compact simply-connected $M$ in a suitably twisted way. Then we discuss the
theory on finite quotients $M/\mathbb{Z}_{h}$ along a free direction and
finally take $h\to\infty$ to reduce to four-dimensional $\mathcal{N}=2$ SYM.

Let us stress right from the beginning that all results in this work are
computed for the cohomologically twisted theories both, in 5d and 4d. If we
start on a spin manifold with the physical theory, spinors solving the rigid
SUGRA background Festuccia:2011ws  can be used to twist the theory. This has
been done in 5d $\mathcal{N}=1$ for Sasaki-Einstein manifolds Kallen:2012va
(for which existence of Killing spinors is guaranteed, see e.g. Sparks:2010sn
) and in 4d $\mathcal{N}=2$ for any compact spin manifold Festuccia:2018rew .
The twisting can be viewed as a “change of variables” and the partition
functions of physical and twisted theory are expected to agree. However, not
all manifolds we consider in this work are spin so there might be global
obstructions to writing the physical theory. On the other hand, the twisted
field content are just differential forms and therefore the twisted theory
still makes sense globally on all manifolds considered.

###  3.1 Simply-Connected Case

#### Cohomological Complex.

The 5d $\mathcal{N}=1$ vector multiplet contains as bosonic fields a gauge
field $A$, a real scalar $\sigma$, and a two-form $H$; the gauginos are given
in terms of a one-form $\Psi$ and a two-form $\chi$; all fields (except $A$)
are valued in the adjoint of the gauge group121212For simplicity, we assume
$G$ to be simply-connected. $G$. The two-forms $\chi,H$ satisfy the following
projection condition:

$P^{+}\chi=\chi,\qquad P^{+}H=H.$ (21)

The projector $P_{\pm}=\frac{1}{2}(1\pm\iota_{\SR}\star)$ maps two-forms in
$\Omega^{2}(M)$ to their horizontal131313We can decompose a form
$\omega\in\Omega^{\bullet}(M)$ as
$\omega={\R}\wedge\iota_{\SR}\omega+\tilde{\omega}$, with
$\iota_{\SR}\tilde{\omega}=0$. The component $\tilde{\omega}$ is called
horizontal and the corresponding subspace is denoted
$\Omega^{\bullet}_{H}(M)$., (anti-)self-dual component in
$\Omega^{2\pm}_{H}(M)$. This is the five-dimensional analogue of the 4d
projector onto (anti-)self-dual two-forms. The cohomological complex reads

$\begin{array}[]{lll@{\hskip 4em}lll}QA&=&i\Psi,\hfil\hskip
40.00006pt&Q\Psi&=&-\iota_{\SR}F+\DD\sigma,\\\ Q\chi&=&H,\hfil\hskip
40.00006pt&QH&=&-iL_{\SR}^{A}\chi-[\sigma,\chi],\\\
Q\sigma&=&-i\iota_{\SR}\Psi\hfil\hskip 40.00006pt&&&\\\ \end{array}$ (22)

and the supercharge squares to

$Q^{2}=-iL_{\SR}^{A}+G_{\sigma}.$ (23)

Here, $\DD\sigma$ and $F$ denote the covariant derivative and field strength
with respect to $A$ (locally, $\DD\sigma=\dd\sigma-\ii[A,\sigma]$ and $F=\dd
A-\ii A^{2}$), $L_{\SR}^{A}=L_{\SR}-\ii[\iota_{\SR}A,\us]$ is the covariant
Lie derivative along the Reeb and $G_{\sigma}$ is a gauge transformation with
parameter $\sigma$:

$G_{\sigma}A=\DD\sigma,\qquad G_{\sigma}\Phi=\ii[\sigma,\Phi]$ (24)

with $\Phi$ any field in the adjoint of the gauge group.

Finally, we want to remark that a deformation of $\R$ in the cohomological
formalism can be implemented easily; one simply replaces $\R$ in the complex
(22) with its deformed version. This is another advantage compared to the
ordinary formulation, where more background fields would have to be introduced
in the (rigid) supergravity multiplet in order to preserve supersymmetry.

#### Action and BPS Locus.

In terms of the twisted fields introduced above, the SYM action can be written
as

$S_{\text{YM}}=-\left(CS_{3,2}(A+\sigma\kappa)+\ii\tr\int\frac{1}{g_{\text{YM}}^{2}}(\kappa\wedge
d\kappa\wedge\Psi\wedge\Psi)\right)+QW_{\text{vec}},$ (25)

where

$\displaystyle CS_{3,2}(A)=\tr\int\frac{1}{g_{\text{YM}}^{2}}\kappa\wedge
F\wedge F,$ (26) $\displaystyle
W_{\text{vec}}=\tr\int\frac{1}{g_{\text{YM}}^{2}}\left(\Psi\wedge\star(-\iota_{\SR}F-\DD\sigma)-\frac{1}{2}\chi\wedge\star
H+2\chi\wedge\star F+\sigma\kappa\wedge d\kappa\wedge\chi\right).$ (27)

Here, $g^{2}_{\text{YM}}$ denotes the 5d YM coupling constant. Note that the
terms in (25) written in parentheses are $Q$-closed but not $Q$-exact and can
be viewed as a supersymmetric observable.

The partition function for this theory can be computed using localisation
techniques141414An extensive review of such techniques can be found in
Pestun:2016zxk .. The localisation term $t\cdot QV$ which we add to (25) uses
Qiu:2013pta

$V=\tr\int\frac{1}{2}\Psi\wedge\star(-\iota_{\SR}F-\DD\sigma)-\chi\wedge\star
H+2\chi\wedge\star F.$ (28)

Upon Wick rotation of $\sigma,H$ we arrive at the following localisation
locus:

$F^{+}_{H}=0,\quad\iota_{\SR}F=0,\quad\DD\sigma=0.$ (29)

The first two equations can equivalently be written as the single
equation151515In analogy to 4d, solutions to this equation automatically
satisfy the YM equation. $\star F=-\kappa\wedge F$ and its solutions are
called contact instantons. Since we are only concerned with the trivial
instanton sector in this work, we restrict to zero contact instanton-number
for which the localisation locus reads Qiu:2016dyj

$A=0,\quad\sigma=a\in i\mathfrak{g},$ (30)

i.e. $A$ is the trivial connection and $\sigma$ a constant valued in the Lie
algebra $\mathfrak{g}$ of $G$, up to gauge transformations. The classical
piece in the localised partition function is obtained by evaluating (25) at
the locus above and only receives a contribution from
$CS_{3,2}(A+\sigma\kappa)$:

$S_{\text{cl}}=-\tr\int\frac{1}{g_{\text{YM}}^{2}}a^{2}\kappa\wedge
d\kappa\wedge d\kappa=-\frac{8\pi^{3}}{g_{\text{YM}}^{2}}\varrho\tr(a^{2}),$
(31)

where $\varrho\equiv\vol_{M}\\!/\\!\vol_{S^{5}}$ and volume form of $M$ is
$\frac{1}{8}\kappa\wedge(\dd\kappa)^{2}$. Optionally, we could multiply the
right-hand side by a parameter controlling the size of $M$.

###  3.2 Taking Finite Quotients

The main player in this work is finite quotients of the simply-connected toric
Sasakian manifolds $M$ discussed above. More specifically, we always
assume161616Note that there always is a locally free $S^{1}\subset T^{3}$;
precisely the one that agrees with the remaining, non-degenerate $S^{1}$ at
the vertices of the polygon, cf. Figure 1. that there is a free $S^{1}\subset
T^{3}$ and take a quotient by a finite Abelian subgroup $\mathbb{Z}_{h}$
($h\in\mathbb{N}_{>1}$ fixed) along this $S^{1}$. So we have a principal
$\mathbb{Z}_{h}$-bundle

$\tikzcd[columnsep=small]\mathbb{Z}_{h}\ar[r,hook]&M\ar[r,"\pi"]X$ (32)

with $X=M/\mathbb{Z}_{h}$ the finite quotient. Note that $X$ is no longer
simply-connected but $\pi_{1}(X)\simeq\mathbb{Z}_{h}$ (see Appendix A) and we
can view $M$ as the $h$-sheeted connected (universal) cover of $X$. This
means, intuitively, that $X$ and $M$ are “the same” locally and differences
are only noticeable on a global level. In particular, since the vector field
$\X$ generating the free $S^{1}$ is Killing, the metric on $M$ induces a
metric on $X$ such that the covering is Riemannian. Moreover, since
$S^{1}/\mathbb{Z}_{h}\simeq S^{1}$ (via $s\mapsto s^{h}$) we still have a
$T^{3}$-action on $X$.

In order for the theory in subsection 3.1 to descend to $X$ we first restrict
to bundles over $M$ that are equivariant under deck transformations171717Note
that $\gamma\in\Aut(\pi)$ is an isometry since the covering $\pi$ is
Riemannian. $\gamma\in\Aut(\pi)\simeq\pi_{1}(X)$ of the covering $\pi$; these
bundles descend to $X$. We mainly take the view of equivariant bundles on $M$
instead of bundles on $X$ given that we have good control over the theory on
$M$. All fields (except the connection) in the cohomological formulation are
(endomorphism-valued) differential forms181818Here, $\mathfrak{g}_{P}$ denotes
the associated bundle $\mathfrak{g}_{P}=P\times_{\text{ad}}\mathfrak{g}$ of
the gauge bundle $P$. $\omega\in\Omega^{r}(M;\mathfrak{g}_{P})$ and they
descend to $X$ if they are $\Aut(\pi)$-invariant. Namely, let $\omega$ be such
a form. Then $\omega$ descends to a differential form on $X$ if

$\forall\gamma\in\Aut(\pi):\gamma^{\ast}\omega=\omega.$ (33)

If we restrict our theory on $M$ to fields satisfying (33) then also the
cohomological complex (22) is invariant under deck transformations191919This
follows from various properties of the pullback,
$\gamma^{\ast}\dd\omega=\dd\gamma^{\ast}\omega$,
$\gamma^{\ast}(\omega\wedge\eta)=\gamma^{\ast}\omega\wedge\gamma^{\ast}\eta$,
$\gamma^{\ast}\iota_{\SR}\omega=\iota_{(\gamma^{-1})_{\ast}{\R}}\gamma^{\ast}\omega$
as well as the fact that $\gamma_{\ast}{\R}={\R}$ since ${\R}$ and $\X$
commute (remember ${\R}\in\mathfrak{t}_{3}$). and descends to $X$. Similarly,
one can show that $S_{\text{YM}}$ in (25) descends to the action on $X$, i.e.
the complex is again a symmetry of the action. Hence, as we expect, local
expressions can be “pushed down” and we obtain the supersymmetric theory on
$X$.

However, since $X$ is not simply-connected, we should also account for field
configurations arising from this non-trivial topology. In particular, there
are $h$ topologically inequivalent $U(1)$-bundles over $X$, labelled by their
first Chern class $c_{1}$ valued in $H^{2}(X;\mathbb{Z})=H^{2}(X)$
(henceforth, if no coefficients are specified, we always assume $\mathbb{Z}$),
each admitting a flat connection. Thus, for functions on $M$, condition (33)
is too strong and we should relax it such that functions on $M$ are pushed
down to sections over aforementioned line bundles on $X$. Specifically, given
a function $f:M\rightarrow\mathbb{C}$ on $M$, by virtue of the periodicity in
$\X$-direction, we can expand

$f(x,\alpha)=\sum_{t\in\mathbb{Z}}f_{t}(x)\e{2\pi\ii t\alpha/L},$ (34)

where $\alpha\sim\alpha+L$ parametrises the $S^{1}$ and $x$ the other four
dimensions. Since $\X$ is a linear combination of the three $S^{1}$-directions
such that (12) is satisfied, $L$ can be different from $2\pi$. Clearly,
$\gamma^{\ast}f=f$ implies $t=0\mmod h$. Instead of restricting to such
functions only, we impose the more general condition

$t=c_{1}\mmod h,$ (35)

where we use
$H^{2}(X;\mathbb{Z})\simeq\mathbb{Z}^{b_{2}(X)}\oplus\mathbb{Z}_{h}$ ($b_{2}$
the second Betti number of $X$) and the fact that $c_{1}$ takes values in the
$\mathbb{Z}_{h}$-part (see Appendix A). In this way, we can view $f$ as a
section of a flat line bundle over $X$ that acquires a phase $\exp(2\pi\ii
t/h)$ around a loop $\gamma\in\pi_{1}(X)$, which is simply the holonomy around
$\gamma$ for the flat connection of the bundle. Hence, the theory still
descends from $M$ to $X$ for an $S^{1}$-action with weights satisfying (35).
In terms of the moment map cone $\mathcal{C}$ introduced in (8) this means
restricting to slices

$\mathcal{C}_{t}=\\{\vec{v}\in\mathcal{C}\;|\;\vec{v}\cdot\vec{\X}=t\\}$ (36)

of $\mathcal{C}$, as depicted in Figure 2. The orientation of
$\mathcal{C}_{t}$ inside $\mathcal{C}$ is determined by the free direction
$\vec{\X}$. Notice that this is a generalisation of the slice (9) along the
Reeb. In particular, when
$\vec{\X}\notin\mathcal{C}^{\circ}\cup(-\mathcal{C}^{\circ})$, the resulting
polytope is no longer compact (see, e.g., Figure 3 or Figure 5).

$\mathcal{C}$$\mathcal{C}_{c_{1}+5h}$$\mathcal{C}$$\mathcal{C}_{c_{1}+2h^{\prime}}$
Figure 2: Different slices $\mathcal{C}_{t}$ of the cone $\mathcal{C}$ for
(left) $t=c_{1}+n\cdot h$ ($n=0,\dots,5$) and (right) $t=c_{1}+n\cdot
h^{\prime}$ ($n=0,1,2$), where $h^{\prime}=2h$. Hence, as $h$ becomes larger,
the slices space out. Note that here we have chosen $\X$ inside $\mathcal{C}$,
hence the slices are compact (cf. Figure 3 and 5 with non-compact slices).

Another consequence of $X$ not being simply-connected is the existence of non-
trivial flat connections $A$ for the gauge bundle $G\hookrightarrow
P\rightarrow X$ which now contribute to the locus (30). It is well known that
(up to gauge transformations) flat connections are in one-to-one
correspondence with representations of $\pi_{1}(X)$ in $G$ (up to conjugation)
via their holonomy $\text{h}_{A}:\pi_{1}(X)\rightarrow G$ around loops
$[\gamma]\in\pi_{1}(X)$ (see e.g. Taubes:2011 ). For example, $G=U(N)$ yields

$\Hom(\pi_{1}(X),U(N))\simeq\\{\diag(\e{2\pi\ii m_{1}/h},\dots,\e{2\pi\ii
m_{N}/h})|m_{i}\in\mathbb{N}_{\leq h}\\}$ (37)

and conjugation acts simply by permuting the exponents. Hence, flat
connections for $G=U(N)$ are labelled by an array of integers

$\mathfrak{m}\equiv\diag(m_{1},\dots,m_{N})\in\mathbb{N}^{N\times N}_{\leq
h},\qquad m_{i}\leq m_{i+1}.$ (38)

Note that, since $\pi_{1}(X)$ is Abelian, all representations in $G$ will be
contained in its maximal Abelian (i.e. Cartan) subgroup. We henceforth use the
symbol $\mathfrak{m}$ to denote the elements in (the Cartan subalgebra of)
$\mathfrak{g}$ labelling flat connections also for generic $G$.

Next, we revisit the BPS locus (30). On $X$, in addition to the trivial
connection, the contact instanton equation in the zero-instanton sector is
solved by any flat connection. In particular, the topologically non-trivial
flat connections $A_{\mathfrak{m}}$ discussed above are part of the BPS locus
now. Then, for fixed $\mathfrak{m}$, the solution to $\DD\sigma=0$ is again
given by a constant,

$\sigma=a\in\ii\mathfrak{g},\qquad[\mathfrak{m},a]=0,$ (39)

up to gauge transformations. Therefore, in general, $G$ is broken to its
Cartan subgroup $U(1)^{\rk G}$ and the path integral over gauge bundles $P$
reduces to principal $U(1)^{\rk G}$-bundles202020These can effectively be seen
as a product of $\rk G$ $U(1)$-bundles via the pullback along the diagonal map
$X\rightarrow X\times\dots\times X$, on sections of which the free $S^{1}$
acts with infinitesimal weight (35). which are classified precisely by
$\mathfrak{m}$. Therefore, the free $S^{1}$ acts on fluctuations of the
adjoint-valued fields around the BPS locus with infinitesimal weight

$t=\alpha(\mathfrak{m})\mmod h$ (40)

according to (35) and the discussion thereafter. Note that the complex (22)
does not mix topological sectors and hence, the form of the one-loop partition
function obtained on $M$ is unchanged in each topological sector212121Since
fluctuations around the locus do not jump between sectors and the connections
are flat, in particular, the index computation (cf. Qiu:2014oqa ;
Festuccia:2019akm ) that yields the one-loop determinant is unchanged., of
course, up to imposing the descendance conditions on the fields and
$S^{1}$-action discussed above. However, on $X$ we now have a sum of these
one-loop pieces over all topological sectors, labelled by $\mathfrak{m}$. This
will be further discussed in subsection 4.2.

Similar to the simply-connected case, the classical piece only receives a
contribution from the Chern-Simons action $CS_{3,2}(A_{\mathfrak{m}}+a\kappa)$
on $X$:

$S_{\text{cl}}(a,\mathfrak{m})=-\frac{8\pi^{3}}{g_{\text{YM}}^{2}}\frac{\varrho}{h}\tr(a^{2}),$
(41)

where we have used (39) and $CS_{3,2}(A_{\mathfrak{m}})=0$ since
$A_{\mathfrak{m}}$ is flat. The quotient introduces a factor of $\frac{1}{h}$.
The independence of $S_{\text{cl}}$ from $\mathfrak{m}$ suggests that all
topological sectors are weighted equally in the partition function.

###  3.3 Reduction to Four Dimensions

While the theory on finite quotients of $M$ is interesting in its own right,
in this work we ultimately want to reduce to the 4d $\mathcal{N}=2$ theory in
order to see how fluxes feature in the partition function. We achieve this by
taking the order of the finite group $\mathbb{Z}_{h}$ we quotient by to be
very large. This makes the free $S^{1}$ in $M$ shrink more and more until, in
the limit where $h\to\infty$, the $S^{1}$ shrinks to a point and we are left
with the base manifold $M/S^{1}$ which we denote by $B$ in the following (a
more formal treatment of the limit can be found in Appendix A). Note that, in
general, $B$ is not toric Kähler if $\X$ is not aligned with the Reeb $\R$.

#### Flux.

The bundles on $M$ that descend to $B$ are the ones equipped with connections
whose holonomy around the free $S^{1}$ is trivial222222See Festuccia:2016gul ,
appendix A for a detailed discussion. and fields in our theory on $M$ that
descend to $B$ are the forms $\omega$ on $M$ that are invariant along $\X$,
i.e. $\mathcal{L}_{\SX}\omega=0$ (the infinitesimal version of invariance
under deck transformations discussed in subsection 3.2). The reduction of the
cohomological complex (22) and the action (25) is straightforward; it has been
performed in Festuccia:2016gul ; Festuccia:2019akm  and matches the four-
dimensional cohomological complex and Yang-Mills action in Festuccia:2018rew ,
except for the peculiarity that the four-dimensional Yang-Mills coupling is
now position-dependent232323Hence, the resulting 4d theory can be viewed as a
slight generalisation of the one discussed in Festuccia:2018rew . whenever
$\X\neq\pm\R$ (which is expected since the radius of the free $S^{1}$ is
position-dependent in these cases). Moreover, we can rewrite the locus
equations (29) in terms of fields pulled back from 4d via
$\tikzcd[columnsep=small]S^{1}\ar[r,hook]&X\ar[r,"p"]B$ using

$\displaystyle A$ $\displaystyle=p^{\ast}A_{4}+\varphi\,b$ (42) $\displaystyle
F$ $\displaystyle=p^{\ast}F_{4}+\DD\varphi\wedge b+\varphi\,\dd b,$ (43)

where $b=g(\X/\|\X\|^{2},\us)$ and the covariant derivative $\DD\varphi$ is
with respect to $p^{\ast}A_{4}$ (cf. Festuccia:2019akm , section 4). In
particular, $\iota_{\SR}F=0$ produces two equations:

$\iota_{v}F_{4}-\DD(\iota_{\SR}b\,\varphi)=0,\qquad\iota_{v}\DD\varphi=0,$
(44)

where $v=p_{\ast}{\R}$ is Killing with respect to the 4d metric and represents
the remaining torus action. The two 4d scalars $\sigma,\varphi$ are
simultaneously diagonalisable, breaking the gauge group to its Cartan subgroup
$U(1)^{\rk G}$ as usual. The first equation in (44) then has solutions
corresponding to line bundles242424They are equivariant under the remaining
torus action on $B$ represented by $v$. $F$ provides a symplectic form and
$\iota_{\SR}b\,\varphi$ a moment map for the action by $v$. on $B$
characterised by their first Chern class $c_{1}$.

Explicitly, the flat connections in the 5d locus on $X$ have non-trivial
holonomy along the generator $[\gamma]\in\pi_{1}(X)$, being the loop along the
$\X$-direction. Using (42), we have $\varphi\propto\mathfrak{m}$ on the locus.
But since $A$ is flat, using (43) yields

$p^{\ast}F_{4}+\varphi\dd b=0.$ (45)

Note that, while $\dd b$ is basic with respect to $\X$, it is not an exact
form on $B$ and hence, the field strength $F_{4}$ satisfying (45) carries
flux, determined by $\varphi$. In this way, flat connections in the locus on
$X$ yield $F_{4}$ with flux in the locus of the 4d theory on $B$.

More geometrically, remember that the flat connections on $X$ are
characterised by $c_{1}$ with $\im c_{1}\simeq\mathbb{Z}_{h}\subset H^{2}(X)$.
Note that $H^{2}(X)$ is generated by the (Poincaré duals of the) faces of the
moment map polytope252525To be precise, the generators are the torus fibration
given by the moment map, restricted such that its image is an edge of the
polytope. $\mu(M)$, modulo some relations and the $\mathbb{Z}_{h}$-subgroup is
generated by a particular linear combination. Consequently, quotienting along
the free $S^{1}$ by taking $h\to\infty$ yields the generating set for
$H^{2}(B)$; the generator for the $\mathbb{Z}_{h}$-subgroup yields a generator
$[c]$ for a $\mathbb{Z}$-subgroup in $H^{2}(B)$.

In terms of the projection condition (40), for $h\to\infty$ only a single mode

$t=\alpha(\mathfrak{m})\in\mathbb{Z}$ (46)

is allowed along the free $S^{1}$, i.e. a single slice $\mathcal{C}_{t}$ of
the moment map cone $\mathcal{C}$ (cf. Figure 2); for larger values of $h$ the
additional slices move further up and hide at infinity in the limit.
Therefore, the sum over flat connections labelled by $\mathfrak{m}$ in (38)
for the 5d partition function on $X$, in the limit, becomes a sum over fluxes
on the two-cycle $c$,

$\frac{1}{2\pi}\int_{c}F_{4}=\mathfrak{m}=\diag(m_{1},\dots,m_{\rk
G})\in\mathbb{Z}^{\rk G\times\rk G},$ (47)

for the 4d partition function on $B$.

#### Instantons.

Another interesting feature of the 4d theory concerns instanton contributions.
It was shown in Festuccia:2019akm  that the space of horizontal (with respect
to $\R$), self-dual two-forms $\Omega^{2+}_{H}(M;\mathfrak{g}_{P})$ is
isomorphic to another three-dimensional subbundle of
$\Omega^{2}_{H}(M;\mathfrak{g})$ which is transverse to $\X$ and given by the
image of the following projector:

$P=\frac{1}{1+g(\hat{\X},\R)^{2}}(1+g(\hat{\X},\R)\star_{4}-g_{4}(v,\us)\wedge\iota_{v}),$
(48)

where $g$ and $g_{4}$ are the metrics on $M$ and $B$ and $\hat{\X}=\X/\|\X\|$.
Note that, at the vertices of the polytope $\mu(M)$, we must have
$g(\hat{\X},\R)=\pm 1$ (cf. discussion succeeding (12)) and $v=0$ (since the
vertices are precisely the fixed points with respect to the remaining torus
action). Then $P$ coincides with $P^{\pm}$; hence, on $B$ this happens
precisely at the torus fixed points. Therefore, depending on our choice of
free direction $\X$, the 4d theory localises to self-dual (SD) or anti-self-
dual (ASD) connections at the fixed points and gives either instanton or anti-
instanton contributions. This is a generalisation of Pestun’s theory on
$S^{4}$ and is known as Pestunization Festuccia:2018rew . For the examples in
section 5 different choices of $\X$ are allowed and, consequently, we obtain
different SD/ASD contributions at the 4d torus fixed points. The choice for
which we have SD (or ASD) contributions at all fixed points is usually called
the (anti)-topological theory, while we call theories with mixed distributions
exotic.

It should be mentioned at this point that, in general, we cannot reach 4d
theories with arbitrary SD/ASD distributions from 5d because there might not
always be enough toric Sasakian $S^{1}$-fibrations over $B$ to start with
(e.g., there is no such fibration for Pestun’s theory on $S^{4}$).
Nevertheless, for all cases obtainable in the above fashion, we are able to
write down the full partition function of the 4d theory, as we show in the
next section.

#### 4d Geometries.

Given the reduction procedure described above for a large class of five-
dimensional manifolds $M$, it is natural to ask about a classification of the
possible 4d base manifolds $B$. Indeed, such a classification was provided in
Festuccia:2016gul  for the special case of toric Sasaki-Einstein manifolds
whose spin structure can be pushed down to $B$. Here, it was possible to write
the normals of the moment map cone $\mathcal{C}^{\vee}$ and the free direction
$\X$ in a standard form and, by computing the intersection forms of elements
in $H_{2}(B)$, it was proved that $B$ is homeomorphic to connected sums
$(S^{2}\times S^{2})^{\\#k}$, $k=m/2-1$ ($m$ the number of vertices of the
polytope, which in this case is always even and $m\geq 4$).

Since we do not have to impose a spin structure on $B$ (by virtue of the
cohomological formulation) and the moment map cone is not required to be
Gorenstein, we should be able to reach all quasi-toric 4d manifolds

$(S^{2}\times
S^{2})^{\\#a}\\#(\mathbb{CP}^{2})^{\\#b}\\#(\overline{\mathbb{CP}^{2}})^{\\#c},\qquad
a,b,c\in\mathbb{Z}_{\geq 0}.$ (49)

However, we are not currently able to provide a proof of this statement.

##  4 Partition Functions

In this section, we derive our main result: the one-loop partition function
around fluxes of an $\mathcal{N}=2$ vector multiplet on a large class of four-
dimensional manifolds and for all distributions of SD/ASD complexes at the
fixed points that can be obtained from 5d. As a starting point, we take the
partition function for the 5d $\mathcal{N}=1$ vector multiplet on the simply-
connected manifold $M$. A detailed exposition of this result can be found in
Qiu:2016dyj . Our focus will be on the perturbative partition function $Z_{M}$
which involves a product over charges $\vec{n}=(n_{1},n_{2},n_{3})$ of the
various fields under the $T^{3}$-action. These charges are determined by the
integer-valued vectors inside the three-dimensional dual cone $\mathcal{C}$ of
$M$ (cf. (8)). We can build up $\mathcal{C}\cap\mathbb{Z}^{3}$ as a collection
of 2d slices (cf. Figure 2) labelled by the charge $t$ under the free $S^{1}$
instead and replace the product over $\vec{n}$ by $(n_{1},n_{2},t)$. Then, at
fixed $t$, the perturbative partition function receives contributions from the
two-dimensional slice $\mathcal{C}_{t}\subset\mathcal{C}$. The orientation of
$\mathcal{C}_{t}$ inside $\mathcal{C}$ depends on the choice of free $S^{1}$.

When moving to the quotient $X=M/\mathbb{Z}_{h}$ the rewriting of $Z_{M}$ in
terms of a product over $(n_{1},n_{2},t)$ makes it easy to implement the
projection condition (40). In particular, only slices
$...,\mathcal{C}_{\alpha(\mathfrak{m})-h},\mathcal{C}_{\alpha(\mathfrak{m})},\mathcal{C}_{\alpha(\mathfrak{m})+h},...$
contribute to a single topological sector $\mathfrak{m}$. The partition
function on $X$ then involves a sum over all these sectors.

In the limit where $h\to\infty$, we reduce to the four-dimensional base
$B=M/S^{1}$ and the flat connections on $X$ give rise to configurations with
flux on $B$. Correspondingly, we arrive at the zero-instanton one-loop
partition function on $B$ which, at a given flux sector, is a product over
charges $(n_{1},n_{2})$ under the remaining $T^{2}$-action. These charges
belong to the projection of ${C}_{\alpha(\mathfrak{m})}$ to its first two
components.

We point out that all expressions for $Z$ below are unfactorised and we leave
for section 6 a discussion of their factorisation properties.

###  4.1 Simply-Connected Case

The cohomological complex for an $\mathcal{N}=1$ vector multiplet on a simply-
connected toric Sasakian manifold $M$ with gauge group $G$ has been introduced
in (22) and the action in (25). The partition function is given by Qiu:2016dyj

$\begin{split}\mathcal{Z}_{M}&=\int_{\mathfrak{h}}da\;e^{-S_{\text{cl}}}\cdot
Z_{M}\cdot Z_{M}^{\text{inst}}\\\
&=\int_{\mathfrak{h}}da\;e^{-\frac{(2\pi)^{3}}{g_{\text{YM}}^{2}}\varrho\tr(a^{2})}\cdot\mbox{det}_{\text{adj}}^{\prime}S_{3}^{\mathcal{C}}(\ii
a|\R)\cdot\prod_{i=1}^{m}Z^{\text{Nek}}_{\mathbb{C}^{2}_{\epsilon^{i}_{1},\epsilon^{i}_{2}}\times
S^{1}}(a|\epsilon^{i}_{1},\epsilon^{i}_{2},\beta_{i}^{-1}).\end{split}$ (50)

The integral with respect to the Coulomb branch parameter $a$ is over the
Cartan subalgebra $\mathfrak{h}\subseteq\mathfrak{g}$ of the gauge group.
Contact instantons contribute to the non-perturbative part of the partition
function and it is conjectured in Qiu:2016dyj  that $Z^{\text{inst}}_{M}$ is
obtained by gluing Nekrasov partition functions at each of the $m$ fixed
fibres. In the expression above,
$\epsilon^{i}_{1},\epsilon^{i}_{2},\beta_{i}^{-1}$ are the local equivariance
parameters for the $T^{3}$-action on each neighbourhood
$\mathbb{C}^{2}_{\epsilon^{i}_{1},\epsilon^{i}_{2}}\times S^{1}$ around the
fixed fibres; in particular, $\beta_{i}$ is the radius of the fixed fibre.

We focus on the perturbative partition function262626Most results on the
perturbative partition function in the literature assume that $M$ is Einstein
or, equivalently, that the cone $C(M)$ is Calabi-Yau. However, even without
this condition, $Z_{M}$ is still given by the triple sine function Qiu:2016dyj
., written in terms of the triple sine function
$S_{3}^{\mathcal{C}}(\ii\alpha(a)|\R)$ (see, e.g., Kurokawa2003MultipleSF ;
narukawa2004modular ) which explicitly reads

$Z_{M}=\prod_{\alpha\in\Delta}\prod_{\vec{n}\in\mathcal{C}\cap\mathbb{Z}^{3}}(\vec{n}\cdot\vec{\R}+\ii\alpha(a))\prod_{\vec{n}\in\mathcal{C}^{\circ}\cap\mathbb{Z}^{3}}(\vec{n}\cdot\vec{\R}-\ii\alpha(a)).$
(51)

Here, $(n_{1},n_{2},n_{3})$ are the charges of the modes under rotations along
the $T^{3}$-action and $\R$ the Reeb, as usual. The dual moment map cone
$\mathcal{C}$ of $M$ and its interior $\mathcal{C}^{\circ}$ have been
introduced in (8). Finally, $\Delta$ denotes the root set of the gauge algebra
$\mathfrak{g}$. Instead of using the $\det_{\text{adj}}^{\prime}$-notation
from (51) we henceforth write the one-loop part of the partition function
explicitly as a product over the roots, omitting possible factors arising from
fermionic zero-modes that would cancel a Vandermonde determinant in the
integral over $a$. This way, shifts of $a$ via flat connections will be more
apparent in the notation.

Before we proceed to non-simply-connected spaces, we have to point out that
the partition function (51) is not the full story yet. Indeed, whenever
$H^{2}(M)\neq 0$ there might be an additional sum over flux configurations in
the path integral. The fact that these configurations are not part of the
localisation locus (29) points to the fact that more general loci should be
allowed in this case. These are possibly complex or singular field
configurations (see, e.g. Benini:2015noa ; Closset:2015rna  in 2 and 3d). Such
an analysis is outside the scope of this work, where we consider (51) as our
starting point. However, we expect our analysis to go through in the exact
same way in the presence of these additional fluxes.

###  4.2 Finite Quotients and Reduction

As stated above, we consider toric Sasakian manifolds $M$ admitting a free
$S^{1}$-action. In terms of a basis
$\\{e_{i}\\}_{i=1,2,3}\subset\mathfrak{t}_{3}$ for an effective
$T^{3}$-action, the vector field generating rotations along the free $S^{1}$
is

$\vec{\X}=l_{1}e_{1}+l_{2}e_{2}+l_{3}e_{3},$ (52)

where the $\mathbb{Z}$-valued coefficients $l_{1},l_{2},l_{3}$ are obtained by
solving (12); they depend on the geometry of $\mathcal{C}$ and will be
discussed in concrete examples in section 5.

#### Slicing the Cone.

The first step towards deriving the one-loop partition function around fluxes
on $B=M/S^{1}$ consists in rewriting (51) as a product over the slices
$\mathcal{C}_{t}$. Here, $t$ is the integer-valued charge under the free
$S^{1}$ and can be written in terms of the charges $\\{n_{i}\\}$ corresponding
to the initial choice of basis $\\{e_{i}\\}$ as follows:

$t=l_{1}n_{1}+l_{2}n_{2}+l_{3}n_{3}.$ (53)

Thus, substituting272727W.l.o.g. we assume $l_{3}\neq 0$ here. Substituting
for any of $n_{1},n_{2},n_{3}$ leads to the same result.
$n_{3}=l_{3}^{-1}(t-l_{1}n_{1}-l_{2}n_{2})$ in (51) and defining

$\epsilon_{1}\equiv\R_{1}-\frac{l_{1}}{l_{3}}\R_{3},\quad\epsilon_{2}\equiv\R_{2}-\frac{l_{2}}{l_{3}}\R_{3},$
(54)

we find

$Z_{M}=\prod_{\alpha\in\Delta}\prod_{t}\prod_{(n_{1},n_{2})\in\mathcal{B}_{t}}(n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+t\tfrac{\R_{3}}{l_{3}}+\ii\alpha(a))\prod_{(n_{1},n_{2})\in\mathcal{B}_{t}^{\circ}}(n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+t\tfrac{\R_{3}}{l_{3}}-\ii\alpha(a)).$
(55)

Here, the region $\mathcal{B}_{t}$ is defined as

$\mathcal{B}_{t}=(\proj_{12}\mathcal{C}_{t})\cap\mathbb{Z}^{2}=\proj_{12}\\{\vec{v}\in\mathcal{C}\;|\;\vec{v}\cdot\vec{\X}=t\\}\cap\mathbb{Z}^{2},$
(56)

where $\proj_{12}[v_{1},v_{2},v_{3}]=[v_{1},v_{2}]$ projects onto the first
two components and $\mathcal{B}_{t}^{\circ}$ denotes the interior of
$\mathcal{B}_{t}$. In order to write (55) more compactly, we define a slight
generalisation of $\Upsilon$-functions (see, e.g., Pestun:2007rz ; Hama:2012bg
; Festuccia:2018rew ):

$\Upsilon^{\mathcal{B}_{t}}(z|\epsilon_{1},\epsilon_{2})=\prod_{(n_{1},n_{2})\in\mathcal{B}_{t}}(\epsilon_{1}n_{1}+\epsilon_{2}n_{2}+z)\prod_{(n_{1},n_{2})\in\mathcal{B}^{\circ}_{t}}(\epsilon_{1}n_{1}+\epsilon_{2}n_{2}+\bar{z}),$
(57)

Using this definition, we can rewrite (55) as

$Z_{M}=\prod_{\alpha\in\Delta}\prod_{t}\Upsilon^{\mathcal{B}_{t}}(\ii\alpha(a)+\tfrac{\R_{3}}{l_{3}}t|\epsilon_{1},\epsilon_{2}).$
(58)

As discussed above, we can also deform the Reeb by introducing a squashing of
$M$. Let us recall that $M$ is obtained via symplectic reduction (11) of
$\mathbb{C}^{m}$, and that the $\R_{i}$ ($i=1,2,3$) can be expressed in terms
of equivariance parameters $\omega_{j}$, ($j=1,...,m)$. These, in turn, can be
squashed without spoiling the supersymmetric theory (cf. discussion in
subsection 3.1, Cohomological Complex). Hence, we impose that the squashing
along the fibre vanishes so that the fibre employed for dimensional reduction
is invariant under deformations of the $\R_{i}$. This gives a constraint on
the sum:

$l_{1}\R_{1}+l_{2}\R_{2}+l_{3}\R_{3}=C,$ (59)

where $C$ is a constant to be determined. We will use this constraint in the
examples below to express $\R_{3}$ in terms of the equivariance parameters
$\epsilon_{1},\epsilon_{2}$.

#### Taking the Quotient.

The next step in the construction is to pass to the quotient
$X=M/\mathbb{Z}_{h}$ introduced in subsection 3.2. The action of the free
$S^{1}$ on $M$ descends to $X$ if the projection condition

$t=\alpha(\mathfrak{m})\mmod h$ (60)

is satisfied, which is the restriction we need to impose on the modes
considered in the superdeterminant. Thanks to our rewriting of $Z_{M}$ as (58)
this is achieved easily for fixed $\mathfrak{m}$:

$Z_{X}=\prod_{\alpha\in\Delta}\prod_{t=\alpha(\mathfrak{m})\mmod
h}\Upsilon^{\mathcal{B}_{t}}(\ii\alpha(a)+\tfrac{\R_{3}}{l_{3}}t|\epsilon_{1},\epsilon_{2}).$
(61)

This contribution is the one-loop superdeterminant around flat connections.
The full partition function on $X$ is then a sum over the topological sectors:

$\mathcal{Z}_{X}=\sum_{\mathfrak{m}}\int_{\mathfrak{h}}da\;e^{-S_{\text{cl}}}\cdot
Z_{X}\cdot Z_{X}^{\text{inst}}.$ (62)

The classical piece of the partition function is given in (41). We see that
the slices $\mathcal{B}_{t}$ contributing to $Z_{M}$ now distribute over
different topological sectors282828We expect a similar treatment to be
possible for the contributions from contact instantons. according to their
value of $t$.

#### Reduction to Base.

In order to compute the one-loop partition function around fluxes on the four-
dimensional base manifold $B$, we take the limit of $h\to\infty$ as discussed
in subsection 3.3. For fixed $\mathfrak{m}$, only a single mode (46) survives
and the one-loop contributions become

$Z_{B}=\prod_{\alpha\in\Delta}\Upsilon^{\mathcal{B}_{\mathfrak{m}}}(\ii\alpha(a)+\tfrac{\R_{3}}{l_{3}}\alpha(\mathfrak{m})|\epsilon_{1},\epsilon_{2}),$
(63)

where, using the fact that $t$ only takes the single value
$t=\alpha(\mathfrak{m})$, we write
$\mathcal{B}_{\mathfrak{m}}\equiv\mathcal{B}_{t}$ here. Clearly, the infinite
products in the $\Upsilon$-functions need to be regulated appropriately, for
instance, using zeta function regularisation. The full partition function on
$B$ is a sum over flux sectors:

$\mathcal{Z}_{B}=\sum_{\mathfrak{m}}\int_{\mathfrak{h}}da\;e^{-S_{\text{cl}}}\cdot
Z_{B}\cdot Z_{B}^{\text{inst}}.$ (64)

The classical and instanton parts are determined in section 6.

Let us stress again that the sum over fluxes in (64) might be incomplete,
since our starting point was (51). If $H^{2}(M)\neq 0$, there might be
additional flux configurations that should already be included at the level of
$\mathcal{Z}_{M}$.

##  5 Examples

In this section, we present concrete examples of the reduction procedure
explained above. First, we consider the simplest toric Sasakian manifold,
$S^{5}$, which is also Einstein and regular. Here, one of the two free
directions we find is the Reeb itself, which yields the topologically twisted
theory on $\mathbb{CP}^{2}$. The other direction results in an exotic theory.
The second example, $Y^{p,q}$, is Einstein but not regular and reduces to $B$
homeomorphic to $S^{2}\times S^{2}$. We find again two free directions (one
“top” and one “ex”), but none of the two coincides with the Reeb. Finally, we
present the manifolds $A^{p,q}$, which are neither Einstein nor regular and
reduce to $(\overline{\mathbb{CP}^{2}})^{\\#2}$. Here, we only find an
“exotic” direction.

###  5.1 $S^{5}$

This example has been discussed in detail in Lundin:2021zeb  and we include it
here for completeness. The metric cone is $C(S^{5})=\mathbb{C}^{3}$. Moreover,
as stated above, $S^{5}$ is a regular Sasaki-Einstein manifold and thus, the
Reeb vector generates a free $S^{1}$-action which can be used to dimensionally
reduce to $S^{5}/S^{1}=\mathbb{CP}^{2}$. In general, reducing along the Reeb
results in a topologically twisted theory. This can be understood from the
discussion following (12). Additionally, we find a second fibre which yields
an exotic theory.

Choosing a basis for an effectively acting $T^{3}$, the edge vectors of the
three-faceted dual cone $\mathcal{C}$ are:

$\vec{u}_{1}=[0,1,0],\quad\vec{u}_{2}=[0,0,1],\quad\vec{u}_{3}=[1,0,0]$ (65)

and the inward-pointing normals are given by:

$\vec{v}_{1}=[1,0,0],\quad\vec{v}_{2}=[0,1,0],\quad\vec{v}_{3}=[0,0,1].$ (66)

The Reeb vector field can be written as follows:

$\R=\omega_{1}\vec{v}_{1}+\omega_{2}\vec{v}_{2}+\omega_{3}\vec{v}_{3},$ (67)

where, for $S^{5}$, $\R_{i}=\omega_{i}\equiv 1+a_{i}\in\mathbb{R}$ are the
equivariance parameters deformed by some parameters $a_{i}$. The integer-
valued vectors $\vec{n}$ in the dual cone $\mathcal{C}$ (see (8)) are found
solving $\vec{n}\cdot\vec{v}_{i}\geq 0$. They are given by:

$\vec{n}\in\mathcal{C}\cap\mathbb{Z}^{3}=\mathbb{Z}_{\geq 0}^{3}.$ (68)

Solving (12), we find two inequivalent free $S^{1}$-directions:

$\begin{split}\mbox{top:}\;&\vec{\X}^{\text{top}}=\vec{v}_{1}+\vec{v}_{2}+\vec{v}_{3}=[1,1,1]\sim\R,\\\
\mbox{ex:}\;&\vec{\X}^{\text{ex}}=\vec{v}_{1}+\vec{v}_{2}-\vec{v}_{3}=[1,1,-1].\end{split}$
(69)

These choices result in +++ and ++ distributions of SD/ASD complexes292929Note
that in Lundin:2021zeb , rather than ++, the distribution + was studied and
thus certain coefficients differ below.. Correspondingly, the charges of the
modes under the $S^{1}$-rotation along the fibre are:

$\begin{split}\mbox{top:}\;&t^{\text{top}}=n_{1}+n_{2}+n_{3},\\\
\mbox{ex:}\;&t^{\text{ex}}=n_{1}+n_{2}-n_{3},\end{split}$ (70)

where $t^{\text{top}}\geq 0$ while $t^{\text{ex}}$ ranges from $-\infty$ to
$+\infty$. Notice that, since the supercharge (23) squares to a translation
along $\R$, reducing along the Reeb, in the undeformed case, only gives
ordinary Donaldson-Witten theory with $Q^{2}=0$. The deformation of the Reeb
used for dimensional reduction, which makes it tilt away from the fibre, is
introduced here in order to obtain an equivariant four-dimensional theory also
in the case of reducing along $\vec{\X}^{\text{top}}$.

It is crucial to set the $\omega_{i}$ so that the deformation acts only on the
base manifold while the fibre is invariant:

$\begin{split}\mbox{top:}&\quad+a_{1}+a_{2}+a_{3}=0,\\\
\mbox{ex:}&\quad+a_{1}+a_{2}-a_{3}=0,\end{split}$ (71)

Then, we redefine303030Notice that the equivariance parameters
$\epsilon^{\text{top}}_{1,2}$ vanish in the undeformed limit while this is not
the case for $\epsilon^{\text{ex}}_{1,2}$. This again shows that reducing
along the Reeb results in the non-equivariant limit of Donaldson-Witten
theory.:

$\begin{array}[]{rll}\mbox{top:}&\quad\epsilon_{1}^{\text{top}}=\omega_{1}-\omega_{3},&\epsilon_{2}^{\text{top}}=\omega_{2}-\omega_{3},\\\
\mbox{ex:}&\quad\epsilon_{1}^{\text{ex}}=\omega_{1}+\omega_{3},&\epsilon_{2}^{\text{ex}}=\omega_{2}+\omega_{3}.\end{array}$
(72)

Imposing (71), we find:

$\begin{split}\mbox{top:}&\quad\omega_{3}^{\text{top}}=+1-\frac{\epsilon^{\text{top}}_{1}+\epsilon_{2}^{\text{top}}}{3},\\\
\mbox{ex:}&\quad\omega_{3}^{\text{ex}}=-\frac{2}{3}+\frac{\epsilon^{\text{ex}}_{1}+\epsilon_{2}^{\text{ex}}}{3}.\end{split}$
(73)

When confusion cannot arise, we henceforth drop the superscripts “ex” and
“top”. Finally, we substitute $n_{3}=\pm(t-n_{1}-n_{1})$ in (55) and obtain:

$Z_{S^{5}}^{\text{top}}=\prod_{\alpha\in\Delta}\prod_{t\geq
0}\prod_{(n_{1},n_{2})\in\mathcal{B}_{t}}\bigg{(}\epsilon_{1}n_{1}+\epsilon_{2}n_{2}+\ii\alpha(a)+\omega_{3}t\bigg{)}\prod_{(n_{1},n_{2})\in\mathcal{B}^{\circ}_{t}}\bigg{(}\epsilon_{1}n_{1}+\epsilon_{2}n_{2}+\ii\alpha(a)+\omega_{3}t\bigg{)},$
(74)
$Z_{S^{5}}^{\text{ex}}=\prod_{\alpha\in\Delta}\prod_{t\in\mathbb{Z}}\prod_{(n_{1},n_{2})\in\tilde{\mathcal{B}}_{t}}\bigg{(}\epsilon_{1}n_{1}+\epsilon_{2}n_{2}+\ii\alpha(a)-\omega_{3}t\bigg{)}\prod_{(n_{1},n_{2})\in\tilde{\mathcal{B}}^{\circ}_{t}}\bigg{(}\epsilon_{1}n_{1}+\epsilon_{2}n_{2}+\ii\alpha(a)-\omega_{3}t\bigg{)}.$
(75)

The slices (56) of the dual cone $\mathcal{C}$ are given by:

$\begin{split}\mbox{top:}&\quad\mathcal{B}_{t}=\proj_{12}\\{\vec{v}\in\mathcal{C}\;|\;\vec{v}\cdot\vec{\X}^{\text{top}}=t^{\text{top}}\\}\cap\mathbb{Z}^{2},\\\
\mbox{ex:}&\quad\tilde{\mathcal{B}}_{t}=\proj_{12}\\{\vec{v}\in\mathcal{C}\;|\;\vec{v}\cdot\vec{\X}^{\text{ex}}=t^{\text{ex}}\\}\cap\mathbb{Z}^{2}.\end{split}$
(76)

Using the definition of $\Upsilon^{\mathcal{B}_{t}}$-functions in (57) we can
express (74)-(75) as:

$Z_{S^{5}}^{\text{top}}=\prod_{\alpha\in\Delta}\prod_{t\geq
0}\Upsilon^{\mathcal{B}_{t}}(\ii\alpha(a)+(1-\tfrac{\epsilon_{1}+\epsilon_{2}}{3})t|\epsilon_{1},\epsilon_{2}),$
(77)
$Z_{S^{5}}^{\text{ex}}=\prod_{\alpha\in\Delta}\prod_{t\in\mathbb{Z}}\Upsilon^{\tilde{\mathcal{B}}_{t}}(\ii\alpha(a)-(\tfrac{2}{3}-\tfrac{\epsilon_{1}+\epsilon_{2}}{3})t|\epsilon_{1},\epsilon_{2}).$
(78)

The dual cone $\mathcal{C}$ and the two possible slicings
$\mathcal{B}_{t},\tilde{\mathcal{B}}_{t}$ are depicted in Figure 3.

$\vec{u}_{3}$$\vec{u}_{1}$$\vec{u}_{2}$$\vec{\X}^{\text{top}}$

$\vec{u}_{3}$$\vec{u}_{1}$$\vec{u}_{2}$$\vec{\X}^{\text{ex}}$

Figure 3: Dual cone $\mathcal{C}$ of $S^{5}$. Left side: sliced along
$\vec{\X}^{\text{top}}$ for $t=2$ (blue) and $t=4$ (red). At $t=0$ the slice
only contains the origin. Right side: sliced along $\vec{\X}^{\text{ex}}$ for
$t=2$ (blue), $t=0$ (green) and $t=-2$ (red). The slices are compact for
$\vec{\X}^{\text{top}}$ and non-compact for $\vec{\X}^{\text{ex}}$.

Up to this point, (77) and (78) are just rewritings of the perturbative
partition function on $S^{5}$. The difference between the two cases arises
when we introduce a quotient acting along either of the two fibres. The
resulting manifold is a higher-dimensional generalisation of lens spaces
$X=S^{5}/\mathbb{Z}_{h}$ and the charge of the modes under rotations along the
fibre is constrained by the projection condition (40):

$t=\alpha(\mathfrak{m})\mmod h.$ (79)

Thus, the one-loop partition function around flat connections is a sum over
inequivalent topological sectors and, at each of them, only slices
$\mathcal{B}_{t},\tilde{\mathcal{B}}_{t}$ satisfying the projection condition
contribute.

When reducing to the base manifold $\mathbb{CP}^{2}$ by taking the large $h$
limit, we set $t=\alpha(\mathfrak{m})$ and find, for each flux sector:

$Z_{\mathbb{CP}^{2}}^{\text{top}}=\prod_{\alpha\in\Delta}\Upsilon^{\mathcal{B}_{\mathfrak{m}}}(\ii\alpha(a)+(1-\tfrac{\epsilon_{1}+\epsilon_{2}}{3})\alpha(\mathfrak{m})|\epsilon_{1},\epsilon_{2}).$
(80)
$Z_{\mathbb{CP}^{2}}^{\text{ex}}=\prod_{\alpha\in\Delta}\Upsilon^{\tilde{\mathcal{B}}_{\mathfrak{m}}}(\ii\alpha(a)-(\tfrac{2}{3}-\tfrac{\epsilon_{1}+\epsilon_{2}}{3})\alpha(\mathfrak{m})|\epsilon_{1},\epsilon_{2}).$
(81)

The first expression corresponds to a +++ distribution of complexes at all
three fixed points of $\mathbb{CP}^{2}$, and thus to an equivariant
topological twisting, while, for the exotic theory, one fixed point flips to
ASD and the distribution of complexes is ++ instead.

At each flux sector on $\mathbb{CP}^{2}$, the charges of the modes
contributing $\mathcal{B}_{\mathfrak{m}},\tilde{\mathcal{B}}_{\mathfrak{m}}$
can be represented by projecting the slices in Figure 3 to the
$(n_{1},n_{2})$-plane. Explicitly, the integer valued vectors in the dual cone
$\vec{n}\in\mathcal{C}$ are determined solving $\vec{v}_{i}\cdot\vec{n}\geq
0$. Substituting $n_{3}=t^{\text{top}}-n_{1}-n_{2}$, the slice
$\mathcal{B}_{\mathfrak{m}}$ is determined by:

$n_{1}\geq 0,\qquad n_{2}\geq 0,\qquad n_{1}+n_{2}\leq t^{\text{top}}.$ (82)

Similarly, substituting $n_{3}=-t^{\text{ex}}+n_{1}+n_{2}$, the slice
$\tilde{\mathcal{B}}_{\mathfrak{m}}$ is determined by:

$n_{1}\geq 0,\qquad n_{2}\geq 0,\qquad n_{1}+n_{2}\geq t^{\text{ex}}.$ (83)

The slices we obtain are depicted in Figure 4. For the topologically twisted
theory the slices are compact while those for the exotic theory are non-
compact. This property is due to the complexes of the two theories being,
respectively, elliptic and transversally elliptic. In the trivial flux sector,
the results agree with those computed using Mauch:2021fgc .

$n_{1}$$n_{2}$

$n_{1}$$n_{2}$

Figure 4: Slices for $S^{5}$. Left side: $\mathcal{B}_{\mathfrak{m}}$ of the
topologically twisted theory for $\alpha(\mathfrak{m})=2$. For
$\alpha(\mathfrak{m})=0$ only the origin contributes. Right side:
$\tilde{\mathcal{B}}$ of the exotic theory for $\alpha(\mathfrak{m})=2$
(blue). For $\alpha(\mathfrak{m})\leq 0$ the entire quadrant contributes.

Lastly, we point out that the one-loop partition function around fluxes for
the topological theory on $\mathbb{CP}^{2}$ has already been computed in
Bershtein:2015xfa . The result was expressed as a sum over equivariant fluxes,
compared to our partition function summing over “physical” fluxes. This makes
a direct comparison rather difficult and it would be desirable to gain an
understanding of the relation between the two results in the future.

###  5.2 $Y^{p,q}$

As a second example, we consider the infinite class of Sasaki-Einstein
manifolds $Y^{p,q}$, which are either quasi-regular or irregular. They are
homeomorphic to $S^{2}\times S^{3}$ and have been introduced in
Gauntlett:2004yd ; Martelli:2004wu  (see also subsection 2.2). The
perturbative partition function of the $\mathcal{N}=1$ vector multiplet on
$Y^{p,q}$ has been computed in Qiu:2013pta ; Qiu:2013aga ; Schmude:2014lfa
and the full partition function (50), including contact instanton
contributions, is conjecturally obtained by gluing Nekrasov partition
functions at each fixed fibre. Using our reduction procedure we will obtain
the one-loop partition function around fluxes at the zero instanton sector on
$B=Y^{p,q}/S^{1}$ (which is homeomophic to $S^{2}\times S^{2}$). As this is a
new result, we give a more detailed presentation.

A basis for an effective $T^{3}$-action has been introduced in (14). The edge
vectors $\vec{u}_{i}$ and inward-pointing normals of the four-faceted cone
have been computed in (16) and (17). We also recall that the vector fields
generating free $S^{1}$-actions are obtained by solving (12), resulting in the
choices of fibre (20):

$\begin{split}\mbox{top:}&\quad\vec{\X}^{\text{top}}=\begin{cases}[1,1,\frac{p}{2}],&p\text{
even}\\\ [2,2,p],&p\text{ odd}\end{cases},\\\
\mbox{ex:}&\quad\vec{\X}^{\text{ex}}=[0,0,1],\end{split}$ (84)

and, respectively, in ++++ and ++ distributions of SD/ASD complexes on the
base manifold. Correspondingly, the charges (53) under rotations along the
fibres are:

$\begin{split}\mbox{top:}&\quad
t^{\text{top}}=\begin{cases}n_{1}+n_{2}+\frac{p}{2}n_{3},&p\text{ even}\\\
2n_{1}+2n_{2}+pn_{3},&p\text{ odd}\end{cases},\\\ \mbox{ex:}&\quad
t^{\text{ex}}=n_{3}.\end{split}$ (85)

#### Effectively Acting $T^{3}$.

When introducing a quotient along the free $S^{1}$, and thus considering
$X=Y^{p,q}/\mathbb{Z}_{h}=Y^{hp,hq}$, the four fixed points of the base
manifold $B$ do not change. However, the quotient affects the submanifolds
found at $y=y_{1},y_{2}$:

$y=y_{1}:\;S^{3}/\mathbb{Z}_{\lcm(h,k)},\quad
y=y_{2}:\;S^{3}/\mathbb{Z}_{\lcm(h,l)}$ (86)

(remember $k=p+q,\,l=p-q$). Hence, we modify the basis for the effectively
acting $T^{3}$ (14) accordingly:

$\begin{split}&e_{1}=-\partial_{\phi}-\partial_{\psi},\\\
&\tilde{e}_{2}=\partial_{\phi}-\tfrac{\lcm(h,l)}{2}\partial_{\gamma},\\\
&e_{3}=\partial_{\gamma}.\end{split}$ (87)

It is useful to study the matrix which relates the previous basis (14) to
(LABEL:eq.Ypq/Zhbasis):

$A=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&\frac{l-\lcm(h,l)}{2}\\\
0&0&1\end{array}\right),\quad A\cdot\left(\begin{array}[]{c}e_{1}\\\ e_{2}\\\
e_{3}\end{array}\right)=\left(\begin{array}[]{c}e_{1}\\\ \tilde{e}_{2}\\\
e_{3}\end{array}\right),\quad\mbox{det}A=1.$ (88)

For $l$ even, or for both $l$ and $h$ odd, $A\in SL(3,\mathbb{Z})$. In this
case, we can keep the previous basis (14) as we always have the freedom to
rotate by an $SL(3,\mathbb{Z})$-transformation. For $l$ odd and $h$ even, we
can have $A\notin SL(3,\mathbb{Z})$ and need to use (LABEL:eq.Ypq/Zhbasis)
instead. However, we are mainly interested in the dimensional reduction to
$B=Y^{p,q}/S^{1}$. Hence, in the following, to avoid the need of introducing
(LABEL:eq.Ypq/Zhbasis), if $l$ is odd we simply choose $h$ to be odd.

#### Example: $Y^{2,1}$.

In the following, we show explicitly how the one-loop partition function
around fluxes on $B=Y^{2,1}/S^{1}$ arises as different slicings of the cone
for the two choices of fibre above. For $p=2,q=1$ they become:

$\begin{split}\mbox{top:}&\quad\vec{\X}^{\text{top}}=[1,1,1],\\\
\mbox{ex:}&\quad\vec{\X}^{\text{ex}}=[0,0,1]\end{split}$ (89)

and correspond, respectively, to a ++++ and a ++ distribution of SD/ASD
complexes on the base manifold. Accordingly, the charges of the modes under a
rotation along the fibre are:

$\begin{split}\mbox{top:}&\quad t^{\text{top}}=n_{1}+n_{2}+n_{3},\\\
\mbox{ex:}&\quad t^{\text{ex}}=n_{3}.\end{split}$ (90)

Notice that, similar to $S^{5}$, $t^{\text{top}}$ can only be positive while
$t^{\text{ex}}$ can also be negative.

From the symplectic reduction in subsection 2.2, we can express the
coefficients $\R_{i}$ of the Reeb in terms of the general equivariance
parameters:

$\omega_{1}=a_{1},\quad\omega_{2}=\frac{1}{3\ell}+a_{2},\quad\omega_{3}=\left(\frac{3}{2}-\frac{1}{6\ell}\right)+a_{3},\quad\omega_{4}=\left(\frac{3}{2}-\frac{1}{6\ell}\right)+a_{4},$
(91)

where we have included deformations $a_{i}$ that will control the equivariance
parameters of the dimensionally reduced 4d theory. Let us now repeat the
procedure done earlier for $S^{5}$ by redefining

$\begin{array}[]{rll}\mbox{top:}&\quad\epsilon_{1}^{\text{top}}=\R_{1}-\R_{3},&\epsilon_{2}^{\text{top}}=\R_{2}-\R_{3},\\\
\mbox{ex:}&\quad\epsilon_{1}^{\text{ex}}=\R_{1},&\epsilon_{2}^{\text{ex}}=\R_{2},\end{array}$
(92)

and setting the deformation to act only on the base:

$\begin{split}\mbox{top:}&\quad-2a_{1}-4a_{2}-a_{3}-4a_{4}=0,\\\
\mbox{ex:}&\quad-2a_{2}-a_{4}=0.\end{split}$ (93)

This leads to:

$\begin{split}\mbox{top:}&\quad\R_{1}+\R_{2}+\R_{3}=-\frac{1}{2}\left(15+\frac{1}{\ell}\right)\equiv
C^{2,1},\\\
\mbox{ex:}&\quad\R^{\text{ex}}_{3}=-\frac{1}{2}\left(3+\frac{1}{\ell}\right),\end{split}$
(94)

and thus,

$\R_{3}^{\text{top}}=-\frac{1}{3}(\epsilon_{1}^{\text{top}}+\epsilon_{2}^{\text{top}}-C^{2,1}).$
(95)

We are now ready to substitute into the perturbative partition function (55).
In terms of $\Upsilon^{\mathcal{B}_{t}}$-functions (57), we find:

$Z_{Y^{2,1}}^{\text{top}}=\prod_{\alpha\in\Delta}\prod_{t\geq
0}\Upsilon^{\mathcal{B}_{t}}(\ii\alpha(a)+\R_{3}t|\epsilon_{1},\epsilon_{2}),$
(96)
$Z_{Y^{2,1}}^{\text{ex}}=\prod_{\alpha\in\Delta}\prod_{t\in\mathbb{Z}}\Upsilon^{\tilde{\mathcal{B}}_{t}}(\ii\alpha(a)+\R_{3}t|\epsilon_{1},\epsilon_{2}).$
(97)

Here, $\mathcal{B}_{t},\,\tilde{\mathcal{B}}_{t}$ are defined as in (56)
using, respectively, $\vec{\X}^{\text{top}}$ and $\vec{\X}^{\text{ex}}$. We
show the slices $\mathcal{B}_{t}$ and $\tilde{\mathcal{B}}_{t}$ of the dual
cone $\mathcal{C}$ in Figure 5.

$\vec{u}_{1}$$\vec{u}_{2}$$\vec{u}_{3}$$\vec{u}_{4}$$\vec{\X}^{\text{top}}$

$\vec{u}_{1}$$\vec{u}_{2}$$\vec{u}_{3}$$\vec{u}_{4}$$\vec{\X}^{\text{ex}}$

Figure 5: Dual cone $\mathcal{C}$ of $Y^{2,1}$. Left side: sliced along
$\vec{\X}^{\text{top}}$ for $t=1$ (blue) and $t=2$ (red). At $t=0$ the slice
only contains the origin. Right side: sliced along $\vec{\X}^{\text{ex}}$ for
$t=2$ (green), $t=0$ (blue) and $t=-1$ (red). The slices are compact for
$\vec{\X}^{\text{top}}$ and non-compact for $\vec{\X}^{\text{ex}}$.

Let us stress that, up until this point, the two expressions (96) and (97) are
just rewritings of the perturbative partition function on $Y^{2,1}$ for the
two different slicings and hence equal to each other. The difference between
the two cases arises when we consider the quotients
$X=Y^{p,q}/\mathbb{Z}_{h}=Y^{hp,hq}$ along the fibres $\vec{\X}^{\text{top}}$
and $\vec{\X}^{\text{ex}}$. As explained previously, the one-loop partition
function around flat connections is given by a sum over topological sectors
(see (61)). Only those slices $\mathcal{B}_{t}$, $\tilde{\mathcal{B}_{t}}$
satisfying the projection condition (40) contribute to a given topological
sector.

Upon reducing to the base manifold $B\simeq S^{2}\times S^{2}$ by taking the
large $h$ limit, we set $t=\alpha(\mathfrak{m})$ and obtain, for each flux
sector,

$Z_{B}^{\text{top}}=\prod_{\alpha\in\Delta}\Upsilon^{\mathcal{B}_{\mathfrak{m}}}(\ii\alpha(a)+\R_{3}\alpha(\mathfrak{m})|\epsilon_{1},\epsilon_{2}),$
(98)
$Z_{B}^{\text{ex}}=\prod_{\alpha\in\Delta}\Upsilon^{\tilde{\mathcal{B}}_{\mathfrak{m}}}(\ii\alpha(a)+\R_{3}\alpha(\mathfrak{m})|\epsilon_{1},\epsilon_{2}).$
(99)

The first expression corresponds to a distribution ++++ (i.e. all SD
complexes) at the fixed points of $B$ and the second one to ++, where two
fixed points flip to ASD.

The contributions
$\mathcal{B}_{\mathfrak{m}},\tilde{\mathcal{B}}_{\mathfrak{m}}$ for each flux
sector are obtained by projecting the slices in Figure 5 to the
$(n_{1},n_{2})$-plane. Explicitly, the integer-valued vectors in the dual cone
$\vec{n}\in\mathcal{C}$ are determined by solving $\vec{v}_{i}\cdot\vec{n}\geq
0$. Substituting $n_{3}=t^{\text{top}}-n_{1}-n_{2}$, the slice
$\in\mathcal{B}_{\mathfrak{m}}$ is determined by

$n_{1}\geq 0,\qquad n_{1}+n_{2}\geq 0,\qquad n_{2}\geq-t^{\text{top}},\qquad
n_{1}+n_{2}\leq 2t^{\text{top}}.$ (100)

Similarly, substituting $n_{3}=t^{\text{ex}}$, the slice
$\tilde{\mathcal{B}}_{\mathfrak{m}}$ is determined by

$n_{1}\geq 0,\qquad n_{1}+n_{2}\geq 0,\qquad
n_{1}+2n_{2}\geq-t^{\text{ex}},\qquad n_{1}+n_{2}\leq 2t^{\text{ex}}.$ (101)

The slices are shown in Figure 6. As for $S^{5}$, we obtain compact slices for
the topologically twisted theory while those for the exotic theory are non-
compact. Again, this property is due to the complexes of the two theories
being, respectively, elliptic and transversally elliptic. The results for the
trivial flux sector agree with those computed using Festuccia:2016gul ;
Festuccia:2018rew ; Mauch:2021fgc .

$n_{1}$$n_{2}$

$n_{1}$$n_{2}$

Figure 6: Slices for $Y^{2,1}$. Left side: $\mathcal{B}_{\mathfrak{m}}$ of the
topologically twisted theory for $\alpha(\mathfrak{m})=1$ (blue) and
$\alpha(\mathfrak{m})=2$ (red). For $\alpha(\mathfrak{m})=0$ only the origin
contributes. Right side: $\tilde{\mathcal{B}}_{\mathfrak{m}}$ of the exotic
theory for $\alpha(\mathfrak{m})=0$ (blue), $\alpha(\mathfrak{m})=2$ (green)
and $\alpha(\mathfrak{m})=-1$ (red).

#### General Case.

For arbitrary $p,q$ the choices of fiber and the corresponding charges of the
modes have been introduced in (84)-(85). Then, the (squashed) equivariance
parameters read:

$\omega_{1}=a_{1},\quad\omega_{2}=\frac{1}{(p+q)\ell}+a_{2},\quad\omega_{3}=\left(\frac{3}{2}-\frac{1}{2(p+q)\ell}\right)+a_{3},\quad\omega_{4}=\left(\frac{3}{2}-\frac{1}{2(p+q)\ell}\right)+a_{4}.$
(102)

As in the previous examples, we define

$\begin{array}[]{rll}\mbox{top:}&\quad\epsilon_{1}^{\text{top}}=\R_{1}-\frac{2}{p}\R_{3},&\epsilon_{2}^{\text{top}}=\R_{2}-\frac{2}{p}\R_{3},\\\
\mbox{ex:}&\quad\epsilon_{1}^{\text{ex}}=\R_{1},&\epsilon_{2}^{\text{ex}}=\R_{2},\end{array}$
(103)

and set the deformation to act only on the base:

$\begin{split}\mbox{top:}&\quad-2a_{1}-(2+\frac{p^{2}}{2})a_{2}-a_{3}-(3+\frac{p}{2}(p-q))a_{4}=0,\\\
\mbox{ex:}&\quad-2a_{2}-a_{4}=0,\end{split}$ (104)

which is equivalent to

$\begin{split}\mbox{top:}&\quad\R_{1}+\R_{2}+\frac{p}{2}\R_{3}=-6-\frac{p}{4}\left(\frac{1}{\ell}+3(p-q)\right)\equiv
C,\\\
\mbox{ex:}&\quad\R_{3}^{\text{ex}}=-\frac{3}{2}\left(1+\frac{1}{(p+q)\ell}\right).\end{split}$
(105)

After some lengthy but straightforward computations, one finds:

$\R_{3}^{\text{top}}=-\frac{1}{1+8/p^{2}}(\epsilon_{1}^{\text{top}}+\epsilon_{2}^{\text{top}}-C).$
(106)

Finally, substituting into (51) and using the definition of
$\Upsilon^{\mathcal{B}_{t}}$-functions in (57):

$Z_{Y^{p,q}}^{\text{top}}=\prod_{\alpha\in\Delta}\prod_{t\geq
0}\Upsilon^{\mathcal{B}_{t}}(\ii\alpha(a)+\tfrac{2}{p}\R_{3}t|\epsilon_{1},\epsilon_{2}),$
(107)
$Z_{Y^{p,q}}^{\text{ex}}=\prod_{\alpha\in\Delta}\prod_{t\in\mathbb{Z}}\Upsilon^{\tilde{\mathcal{B}}_{t}}(\ii\alpha(a)+\R_{3}t|\epsilon_{1},\epsilon_{2}),$
(108)

where, as for $Y^{2,1}$, $\mathcal{B}_{t},\,\tilde{\mathcal{B}}_{t}$ are
defined in (56).

The one-loop around partition function around flat connections on $Y^{hp,hq}$
for the topologically twisted and exotic theories are obtained simply by
imposing the projection condition (40) and summing over flat connections.
Finally, taking the limit of large $h$, we find the one-loop partition
function around fluxes on $B=Y^{p,q}/S^{1}$:

$Z_{B}^{\text{top}}=\prod_{\alpha\in\Delta}\Upsilon^{\mathcal{B}_{\mathfrak{m}}}(\ii\alpha(a)+\tfrac{2}{p}\R_{3}\alpha(\mathfrak{m})|\epsilon_{1},\epsilon_{2}).$
(109)
$Z_{B}^{\text{ex}}=\prod_{\alpha\in\Delta}\Upsilon^{\tilde{\mathcal{B}}_{\mathfrak{m}}}(\ii\alpha(a)+\R_{3}\alpha(\mathfrak{m})|\epsilon_{1},\epsilon_{2}).$
(110)

Again, we obtain the slicings by studying the conditions on $(n_{1},n_{2})$,
looking for solutions of $\vec{v}_{i}\cdot\vec{n}\geq 0$ at given $t$. For
$\mathcal{B}_{\mathfrak{m}}$ we find:

$n_{1}\geq 0,\qquad n_{1}+n_{2}\geq
0,\qquad(\tfrac{p}{2(p-q)}-1)n_{1}+(\tfrac{p}{(p-q)}-1)n_{2}\geq-\,t^{\text{top}},\qquad
n_{1}+n_{2}\leq 2\,t^{\text{top}}.$ (111)

Similarly, substituting $n_{3}=t^{\text{ex}}$, we find for
$\tilde{\mathcal{B}}_{\mathfrak{m}}$:

$n_{1}\geq 0,\qquad n_{1}+n_{2}\geq 0,\qquad
n_{1}+2n_{2}\geq-(p-q)\,t^{\text{ex}},\qquad
n_{1}+n_{2}\geq-p\,t^{\text{ex}}.$ (112)

As a consistency check, one can see that these reduce to (98)-(99) for
$p=2,\;q=1$. The slices
$\mathcal{B}_{\mathfrak{m}},\tilde{\mathcal{B}}_{\mathfrak{m}}$ depend
explicitly on $p,q$. Although we do not have a proof, we believe that this
dependence cannot be entirely removed. Notice that this is not in disagreement
with Festuccia:2018rew , where it is shown that infinitesimal deformations of
the metric enter the Lagrangian through $\delta$-exact terms. But different
values of $p,q$ correspond to variations of the metric on $B\simeq S^{2}\times
S^{2}$ that are not connected to the identity.

###  5.3 $A^{p,q}$

In this last example we consider a class of manifolds which are not Einstein
(in contrast to $S^{5},Y^{p,q}$). In particular, this means that the moment
map cone $\mathcal{C}^{\vee}$ is not Gorenstein in this case. This property
was exploited in Festuccia:2016gul  in view of retaining a spin structure on
the base $B$ after reduction, which is needed for the ordinary formulation of
the vector multiplet supersymmetry and for both formulations of the
hypermultiplet one. As discussed in subsection 3.1, we can drop this condition
without any consequence.

Apart from the Einstein property, the Sasakian manifolds $A^{p,q}$ with
$p>q>0$ differ from $Y^{p,q}$ by their base space
$A^{p,q}/S^{1}\simeq(\overline{\mathbb{CP}^{2}})^{\\#2}$. The edge vectors of
the moment map cone are given by

$\vec{u}_{1}=[0,0,1],\quad\vec{u}_{2}=[q,0,1],\quad\vec{u}_{3}=[2p-q,p-q,1],\quad\vec{u}_{4}=[0,p,-1],$
(113)

along with the inward-pointing normals:

$\vec{v}_{1}=[1,0,0],\quad\vec{v}_{2}=[0,1,0],\quad\vec{v}_{3}=[-1,2,q],\quad\vec{v}_{4}=[-1,1,p].$
(114)

Furthermore, we identify the fibre that is rotated by a free $S^{1}$-action by
solving (12):

$\begin{split}\mbox{ex:}&\quad\vec{\X}^{\text{ex}}=[0,0,1],\end{split}$ (115)

corresponding to a +++ distribution of SD/ASD complexes on the base manifold.
The charge under rotations along the fibre is given by $t^{\text{ex}}=n_{3}$.
Compared to the previous examples we only find a single solution to (12) and
we are unable to access the topological theory on $B$ using the five-
dimensional procedure.

For simplicity, in the following calculations we do not turn on a deformation
of the Reeb vector and thus we simply relabel
$\epsilon^{\text{ex}}_{1}=\R_{1},\epsilon^{\text{ex}}_{2}=\R_{2}$.
Substituting in (51) gives:

$Z_{A^{p,q}}^{\text{ex}}=\prod_{\alpha\in\Delta}\prod_{t\in\mathbb{Z}}\Upsilon^{\tilde{\mathcal{B}}_{t}}\big{(}\ii\alpha(a)+\R_{3}t|\epsilon_{1},\epsilon_{2}\big{)}.$
(116)

The slices $\tilde{\mathcal{B}}_{t}$ are shown on the left in Figure 7 for the
case of $p=2,q=1$. As in the previous examples, the slices are normal to the
vector $\vec{\X}^{\text{ex}}$.

$\vec{u}_{1}$$\vec{u}_{2}$$\vec{u}_{3}$$\vec{u}_{4}$$\vec{\X}^{\text{ex}}$

$n_{1}$$n_{2}$

Figure 7: $\mathcal{C}$ and $\tilde{\mathcal{B}}_{\mathfrak{m}}$ of $A^{2,1}$.
Left side: dual cone $\mathcal{C}$ sliced along $\vec{\X}^{\text{ex}}$ for
$t=1$ (green), $t=0$ (blue) and $t=-1$ (red). Note that the slices are non-
compact. Right side: $\tilde{\mathcal{B}}_{\mathfrak{m}}$ for
$\alpha(\mathfrak{m})=1$ (green), $\alpha(\mathfrak{m})=0$ (blue) and
$\alpha(\mathfrak{m})=-1$ (red).

We now introduce a quotient by $\mathbb{Z}_{h}$ acting on the fibre. This
introduces a sum over flat connections and demands that we impose the
projection condition on $t^{\text{ex}}$. Taking the large $h$ limit we find
the one-loop partition function around fluxes on the base manifold:

$Z_{B}^{\text{ex}}=\prod_{\alpha\in\Delta}\Upsilon^{\tilde{\mathcal{B}}_{\mathfrak{m}}}\big{(}\ii\alpha(a)+\R_{3}\alpha(\mathfrak{m})|\epsilon_{1},\epsilon_{2}\big{)}.$
(117)

The form of the partition function is identical to (99), however, the slices
$\tilde{\mathcal{B}}_{\mathfrak{m}}$ are different. They are found solving
$\vec{v}_{i}\cdot\vec{n}\geq 0$:

$n_{1}\geq 0,\qquad n_{2}\geq 0,\qquad-
n_{1}+2n_{2}\geq-q\,t^{\text{ex}},\qquad-n_{1}+n_{2}\geq-p\,t^{\text{ex}}$
(118)

and we display them on the right hand side of Figure 7, again for the case of
$p=2,q=1$. Notice that, in this example, we have not expressed $\R_{3}$ in
terms of $\epsilon_{1},\epsilon_{2}$. To achieve this we would need to study
how the Kähler cone of $A^{p,q}$ can be obtained by symplectic reduction of
$\mathbb{C}^{4}$. This would allow us to relate the components of the Reeb
$\R_{i}$ to general equivariance parameters $\omega_{j}$, as in (18) for
$Y^{p,q}$. This, in turn, would enable us to introduce a deformation of $\R$
acting on the base $B$ only and, finally, to write $\R_{3}$ in terms of
$\epsilon_{1},\epsilon_{2}$.

##  6 Factorised Partition Functions

Given the equivariance in our setup, it is natural to ask whether the one-loop
around fluxes can be expressed as a product of local contributions on a
neighbourhood $\mathbb{C}^{2}_{\epsilon^{i}_{1},\epsilon^{i}_{2}}\times S^{1}$
around the fixed fibres in 5d and, correspondingly, on a neighbourhood
$\mathbb{C}^{2}_{\epsilon^{i}_{1},\epsilon^{i}_{2}}$ around the torus fixed
points in 4d. This factorisation property was confirmed for the perturbative
partition function in Qiu:2014oqa  and Festuccia:2019akm , respectively.

While, in the 5d case, the usual way of factorising $Z_{M}$ is by expressing
the triple-sine function in terms of $q$-Pochhammer symbols, in our case we
need to factorise $\Upsilon$-functions instead, due to the slicing of the
cone. Their factorisation has been discussed in Festuccia:2019akm  and
requires giving an imaginary part to the hitherto real vector field $\X$ and
equivariance parameters $\epsilon_{1},\epsilon_{2}$. Then we have the
following factorisation property:

$\prod_{t}\Upsilon^{\mathcal{B}_{t}}(\ii\alpha(a)+\tfrac{\R_{3}}{l_{3}}t|\epsilon_{1},\epsilon_{2})=\prod_{i=1}^{m}\prod_{t\in\mathbb{Z}}\Upsilon_{i}(\ii\alpha(a)+\beta_{i}^{-1}t|\epsilon^{i}_{1},\epsilon^{i}_{2})^{s_{i}},$
(119)

where $i$ labels the $m$ fixed fibres and
$\epsilon^{i}_{1},\epsilon^{i}_{2},\beta_{i}^{-1}$ are the local equivariance
parameters for the $T^{3}$-action on
$\mathbb{C}^{2}_{\epsilon^{i}_{1},\epsilon^{i}_{2}}\times S^{1}$, given by

$\beta_{i}^{-1}=(\vec{v}_{i}\times\vec{v}_{i+1})\cdot\vec{\R},\quad\epsilon^{i}_{1}=(\vec{\R}\times\vec{v}_{i+1})\cdot\vec{\X},\quad\epsilon^{i}_{2}=(\vec{v}_{i}\times\vec{\R})\cdot\vec{\X}.$
(120)

The $\Upsilon_{i}$-functions are defined as follows:

$\Upsilon_{i}(z|\epsilon_{1},\epsilon_{2})=\prod_{(j,k)\in\mathcal{D}_{i}}(\epsilon_{1}j+\epsilon_{2}k+z)\prod_{(j,k)\in\mathcal{D}_{i}^{\prime}}(\epsilon_{1}j+\epsilon_{2}k+\bar{z}).$
(121)

Note that this is essentially a generalisation of (57) where the second
product now is over the region $\mathcal{D}^{\prime}_{i}$ which does not
necessarily coincide with the interior of $\mathcal{D}_{i}$. The regions
$\mathcal{D}_{i},\mathcal{D}^{\prime}_{i}$ depend on the imaginary parts of
$\epsilon_{1}^{i},\epsilon_{2}^{i}$ in the following way:

$\begin{split}\imp(\epsilon^{i}_{1})>0,\;\imp(\epsilon_{2}^{i})>0:\quad&\mathcal{D}=\\{(j,k)\in\mathbb{Z}^{2}\;|\;j\geq
0\mbox{ and }k\geq 0\\},\quad\quad\,\;s_{i}=+\frac{1}{2},\\\
&\mathcal{D}^{\prime}=\\{(j,k)\in\mathbb{Z}^{2}\;|\;j\geq 1\mbox{ and }k\geq
1\\},\\\
\imp(\epsilon^{i}_{1})>0,\;\imp(\epsilon^{i}_{2})<0:\quad&\mathcal{D}=\\{(j,k)\in\mathbb{Z}^{2}\;|\;j\geq
0\mbox{ and }k\leq-1\\},\quad\;\;\,s_{i}=-\frac{1}{2},\\\
&\mathcal{D}^{\prime}=\\{(j,k)\in\mathbb{Z}^{2}\;|\;j\geq 1\mbox{ and }k\leq
0\\},\\\
\imp(\epsilon^{i}_{1})<0,\;\imp(\epsilon^{i}_{2})<0:\quad&\mathcal{D}=\\{(j,k)\in\mathbb{Z}^{2}\;|\;j\leq-1\mbox{
and }k\leq-1\\},\quad s_{i}=+\frac{1}{2},\\\
&\mathcal{D}^{\prime}=\\{(j,k)\in\mathbb{Z}^{2}\;|\;j\leq 0\mbox{ and }k\leq
0\\}\\\
\imp(\epsilon^{i}_{1})<0,\;\imp(\epsilon^{i}_{2})>0:\quad&\mathcal{D}=\\{(j,k)\in\mathbb{Z}^{2}\;|\;j\leq-1\mbox{
and }k\geq 0\\},\quad\;\;\,s_{i}=-\frac{1}{2},\\\
&\mathcal{D}^{\prime}=\\{(j,k)\in\mathbb{Z}^{2}\;|\;j\leq 0\mbox{ and }k\geq
1\\}.\end{split}$ (122)

The different choices in (122) are known as different regularisations at the
fixed fibres. Note, in particular, that the sign of $s_{i}$ can be negative at
the fixed fibres, for which the respective contributions of $\Upsilon_{i}$
appear in the denominator (hence, from the superdeterminant viewpoint, they
can be interpreted as bosonic modes remaining after cancellations, instead of
fermionic ones). We choose an arbitrary sign for the imaginary part of the
equivariance parameters at one fixed fibre, from which the signs at all other
fixed fibres follow. It can be shown Festuccia:2019akm  that the partition
function is independent of this initial choice of sign.

Once we consider the quotient $X=M/\mathbb{Z}_{h}$, one can still glue
contributions on $\mathbb{C}^{2}_{\epsilon^{i}_{1},\epsilon^{i}_{2}}\times
S^{1}$ from the fixed fibres as in (119), but has to impose the projection
condition (40) for the charge under rotations along the free $S^{1}$. Hence,
for each topological sector:

$Z_{X}=\prod_{\alpha\in\Delta}\prod_{i=1}^{m}\prod_{t=\alpha(\mathfrak{m})\mmod
h}\Upsilon_{i}(\ii\alpha(a)+\beta_{i}^{-1}t|\epsilon^{i}_{1},\epsilon^{i}_{2})^{s_{i}}$
(123)

and, upon reduction, we can express the one-loop partition function around
fluxes on $B$ in a factorised form as a product over the torus fixed points:

$Z_{B}=\prod_{\alpha\in\Delta}\prod_{i=1}^{m}\Upsilon_{i}(\ii\alpha(a)+\beta_{i}^{-1}\alpha(\mathfrak{m})|\epsilon^{i}_{1},\epsilon^{i}_{2})^{s_{i}}.$
(124)

Now we can simply read out the shifts of the Coulomb branch parameter by the
flux contributions at each fixed point and write down the classical and
instanton contributions to $\mathcal{Z}_{B}$ explicitly (using equivariant
localisation). The classical piece gives

$\e{S_{\text{cl}}}=\exp\left(-\sum_{i=1}^{m}\frac{4\pi^{2}}{g_{\text{YM}}^{2}(x_{i})}\frac{\phi^{2}(x_{i})}{\epsilon_{1}^{i}\epsilon_{2}^{i}}\right)$
(125)

with $\phi=\ii\sigma+\ii(\iota_{\R}b)\varphi$ (note that $\sigma$ is imaginary
in our convention). The instanton piece is the usual product of Nekrasov
partition functions on $\mathbb{C}^{2}_{\epsilon^{i}_{1},\epsilon^{i}_{2}}$
over the fixed points, applying the appropriate shifts to $a$ Nekrasov:2003vi
; Festuccia:2018rew . For $\X$ such that the cohomologically twisted
background on $B$ localises to instantons at $r$ of the $m$ fixed points and
anti-instantons at the remaining ones we obtain

$Z_{B}^{\text{inst}}=\prod_{i=1}^{r}Z^{Nek}_{\mathbb{C}^{2}}(\ii
a+\beta_{i}^{-1}\mathfrak{m}|\epsilon^{i}_{1},\epsilon^{i}_{2},q)\prod_{i=r+1}^{m}Z^{Nek}_{\mathbb{C}^{2}}(\ii
a+\beta_{i}^{-1}\mathfrak{m}|\epsilon^{i}_{1},\epsilon^{i}_{2},\bar{q}),$
(126)

where $q$ is the usual instanton counting parameter. Hence, we finally have
all the ingredients to write down the full partition function (64) for the
vector multiplet on $B$.

#### Example: $Y^{p,q}$.

Considering $p$ even, the local equivariance parameters for both choices of
free direction are considered in subsection 5.2, and are shown in Table 1.

$i$ | 1 | 2 | 3 | 4
---|---|---|---|---
$\epsilon_{1}^{\text{top},i}$ | $\frac{p}{2}(\epsilon_{1}-\epsilon_{2})$ | $q\left(\epsilon_{1}-\epsilon_{2}\right)+\frac{p}{2}\epsilon_{2}$ | $\frac{p}{2}(\epsilon_{2}-\epsilon_{1})$ | $-\frac{p}{2}\epsilon_{2}$
$\epsilon_{2}^{\text{top},i}$ | $\frac{p}{2}\epsilon_{2}$ | $\frac{p}{2}(\epsilon_{2}-\epsilon_{1})$ | $q\left(\epsilon_{2}-\epsilon_{1}\right)-\frac{p}{2}\epsilon_{2}$ | $\frac{p}{2}(\epsilon_{1}-\epsilon_{2})$
$\beta_{i}^{-1}$ | $\R^{\text{top}}_{3}$ | $\left(p-q\right)\left(\epsilon_{1}-\epsilon_{2}\right)+\R^{\text{top}}_{3}$ | $(p+q)\epsilon_{1}-q\epsilon_{2}+\R^{\text{top}}_{3}$ | $p\epsilon_{2}+\R^{\text{top}}_{3}$
$\epsilon_{1}^{\text{ex},i}$ | $\epsilon_{1}-\epsilon_{2}$ | $2\epsilon_{1}-\epsilon_{2}$ | $\epsilon_{1}-\epsilon_{2}$ | $-\epsilon_{2}$
$\epsilon_{2}^{\text{ex},i}$ | $\epsilon_{2}$ | $\epsilon_{2}-\epsilon_{1}$ | $-2\epsilon_{1}+\epsilon_{2}$ | $\epsilon_{2}-\epsilon_{1}$
$\beta_{i}^{-1}$ | $\R^{\text{ex}}_{3}$ | $\left(p-q\right)\left(\epsilon_{1}-\epsilon_{2}\right)+\R^{\text{ex}}_{3}$ | $\left(p+q\right)\epsilon_{1}-q\epsilon_{2}-\R^{\text{ex}}_{3}$ | $p\epsilon_{2}-\R^{\text{ex}}_{3}$
Table 1: Local equivariance parameters for the topological (top half) and
exotic theory (bottom half).

We recall that the equivariance parameters $\epsilon_{1},\epsilon_{2}$ have
different definitions (103) for topological and exotic theory, and
$\R_{3}^{\text{top}},\R_{3}^{\text{ex}}$ have been defined in (106)-(105).
Assuming, without loss of generality,
$\imp(\epsilon_{1})>\imp(\epsilon_{2})>0$, the perturbative partition
functions (107), (108) become:

$Z^{\text{top}}_{Y^{p,q}}=\prod_{\alpha\in\Delta}\prod_{t\in\mathbb{Z}}\frac{\Upsilon_{1}(\ii\alpha(a)+\beta^{-1,\text{top}}_{1}t|\epsilon_{1}^{\text{top},1},\epsilon_{2}^{\text{top},1})^{1/2}\cdot\Upsilon_{3}(\ii\alpha(a)+\beta^{-1,\text{top}}_{3}t|\epsilon_{1}^{\text{top},3},\epsilon_{2}^{\text{top},3})^{1/2}}{\Upsilon_{2}(\ii\alpha(a)+\beta^{-1,\text{top}}_{2}t|\epsilon_{1}^{\text{top},2},\epsilon_{2}^{\text{top},2})^{1/2}\cdot\Upsilon_{4}(\ii\alpha(a)+\beta^{-1,\text{top}}_{4}t|\epsilon_{1}^{\text{top},4},\epsilon_{2}^{\text{top},4})^{1/2}},$
(127)
$Z^{\text{ex}}_{Y^{p,q}}=\prod_{\alpha\in\Delta}\prod_{t\in\mathbb{Z}}\frac{\Upsilon_{1}(\ii\alpha(a)+\beta^{-1,\text{ex}}_{1}t|\epsilon_{1}^{\text{ex},1},\epsilon_{2}^{\text{ex},1})^{1/2}\cdot\Upsilon_{4}(\ii\alpha(a)+\beta^{-1,\text{ex}}_{3}t|\epsilon_{1}^{\text{ex},4},\epsilon_{2}^{\text{ex},4})^{1/2}}{\Upsilon_{2}(\ii\alpha(a)+\beta^{-1,\text{ex}}_{2}t|\epsilon_{1}^{\text{ex},2},\epsilon_{2}^{\text{ex},2})^{1/2}\cdot\Upsilon_{3}(\ii\alpha(a)+\beta^{-1,\text{ex}}_{4}t|\epsilon_{1}^{\text{ex},3},\epsilon_{2}^{\text{ex},3})^{1/2}}.$
(128)

It is now straightforward to obtain from this $Z_{Y^{p,q}/\mathbb{Z}_{h}}$ and
$Z_{Y^{p,q}/S^{1}}$ in the way described above.

#### Index Computation.

As a byproduct, once the factorised expression for the one-loop part around
fluxes $Z_{B}$ is known, it can be used to compute the equivariant index of a
transversally elliptic complex that would arise in the intrinsically four-
dimensional formulation of the theory on $B$ Festuccia:2018rew . In the
trivial flux sector the index has been computed in Festuccia:2019akm ;
Mauch:2021fgc  and (124) can be used to generalise this result for non-zero
flux (at least for the SD/ASD distributions on $B$ that can be reached from
5d).

##  7 Discussion

In this work, we have computed the Coulomb branch partition function,
including flux contributions, for the 4d $\mathcal{N}=2$ vector multiplet on a
large class of closed simply-connected four-manifolds $B$. We started on a
principal $S^{1}$-bundle over $B$ whose bundle space is a simply-connected
toric Sasakian five-manifold. Making use of existing results for this setup,
we have computed the one-loop partition function on finite quotients
$X=M/\mathbb{Z}_{h}$. This was done by restricting the modes contributing to
the superdeterminant to ones that satisfy the projection condition (40) for
the charge $t$ under the free $S^{1}$. Most importantly, since $\pi_{1}(X)\neq
0$ we had to include non-trivial flat connections into the localisation locus.
Taking the limit of large $h$, we finally obtained the one-loop partition
function on $B$ with flux contributions originating from the 5d flat
connections. Depending on the relative orientation of $\X$ and $\R$, the
theory on $B$ can be either topologically twisted or exotic. Finally, we have
factorised the one-loop partition functions on $M$, $X$, and $B$ and were able
to read out the shifts of the Coulomb branch parameter by the fluxes at each
fixed fibre/point. This enabled us to write down the Coulomb branch partition
function, including instanton and classical parts.

Finally, we point out once again that, whenever $H^{2}(M;\mathbb{Z})\neq 0$,
our procedure might only produce part of the sum over all possible fluxes in
the full partition function on $B$ (e.g. for $Y^{p,q}$,
$H^{2}(Y^{p,q};\mathbb{Z})\simeq\mathbb{Z}$ and $\mathcal{Z}_{Y^{p,q}}$ is
expected to contain a sum over these fluxes). In order to include the
remaining flux contributions, it would be necessary to extend the 5d
localisation locus accordingly, which we leave for future work. Once this is
done, using our procedure, one obtains the full partition function on $B$
(again, for $Y^{p,q}$, $H^{2}(Y^{p,q}/S^{1})\simeq\mathbb{Z}^{2}$: one sector
is carried over from 5d, the other one is introduced by our procedure).

#### BPS Strings.

Let us suggest an interpretation for our procedure to compute the four-
dimensional one-loop partition function around fluxes in terms of the BPS
objects of the 5d theories. Restricting, for simplicity, the gauge group to be
$SU(2)$, it is known that all rank one 5d SCFTs descend from 6d E-string
theory upon circle compactification313131It is conjectured Jefferson:2018irk
that all 5d $\mathcal{N}=1$ SCFTs can be obtained via an RG flow from 6d
$\mathcal{N}=(1,0)$ SCFTs Witten:1995zh ; Strominger:1995ac ; Seiberg:1996qx
on a circle. The degrees of freedom at a generic point in the tensor
branch323232All known interacting $\mathcal{N}=(1,0)$ SCFTs include a tensor
multiplet, whose components are a tensor field $B$, with anti-self-dual field
strength, two fermions, and one scalar. At a generic point of the tensor
branch the scalar acquires a vev. of these 6d $\mathcal{N}=(1,0)$ SCFTs are
tensionful strings. These can descend, taking the radius of the sixth
dimension to be small, to either electrically charged BPS particles in 5d, if
they wrap the $S^{1}$-fibre, or magnetically charged BPS strings coupling to
$F_{D}=\star F$, if they do not wrap the $S^{1}$-fibre. If we further take our
five-dimensional space-time to be a toric Sasakian manifold $M$, we find a 5d
SCFT whose IR description is that of a weakly coupled $SU(2)$ gauge theory on
$M$.

As conjectured in Lockhart:2012vp , the partition function of an
$\mathcal{N}=1$ vector multiplet on $S^{5}$ only has contributions from BPS
particles while BPS strings do not contribute333333Considering toric Sasakian
manifolds with non-trivial $H^{2}(M;\mathbb{Z})$, it is expected that magnetic
strings wrapping two-cycles in $M$ will contribute to flux sectors already in
5d. We limit the discussion in this paragraph to the new flux sectors which
arise dimensionally reducing to 4d.. However, with our dimensional reduction
along a non-trivial fibre $S^{1}\hookrightarrow M\rightarrow B$, we can show
explicitly how BPS strings in 5d give rise to new flux sectors in 4d by
wrapping the fibre which is the orbit of the free $S^{1}$-action generated by
$\vec{\X}$. After taking the limit in which the fibre shrinks, these wrapped
BPS strings result in magnetically charged particles in the 4d $\mathcal{N}=2$
theories. Therefore, at the trivial sector of these new fluxes arising from
dimensional reduction, the partition function $Z_{B}$ receives contributions
only from electrically charged BPS particles. Instead, at a generic flux
sector, the partition function receives contributions from both electrically
and magnetically charged BPS particles. At the intermediate step, on
$X=M/\mathbb{Z}_{h}$, the partition function at a generic, non-trivial, flat
connection sector, receives contributions from magnetically charged BPS
strings. Notice that a string wrapping $h$ times the fibre of $X$ does not
contribute since it behaves similarly to strings not wrapping the fibre at
all.

A similar discussion holds for the factorised expression presented in section
6. Thus, one can place the 6d $\mathcal{N}=(1,0)$ SCFT on
$\mathbb{C}^{2}_{\epsilon^{i}_{1},\epsilon^{i}_{2}}\times T^{2}$ and, after
dimensionally reducing on the $T^{2}$, find the contributions of electric and
magnetic BPS particles to the one-loop partition function in 4d
$\mathcal{N}=2$ at the trivial instanton sector. These contributions arise
from 6d effective strings wrapping, respectively, the sixth and the fifth
direction.

###  7.1 Future Directions

#### Complete Partition Function.

In this work, we have determined the instanton part by computing the shifts of
the Coulomb branch parameter at each fixed point and writing the partition
function as a product of Nekrasov partition functions around the fixed points.
Although this is common practice in 4d, a rigorous proof, to the best of our
knowledge, is still missing.

Moreover, it would be nice to extend our approach to also include
hypermultiplets. However, even in the cohomological formulation this requires
a spin structure on $M$ and the reduction to 4d has to be carried out more
carefully in order for the spin bundle on $M$ to descend to a spin (or at
least spinc) bundle on $B$. Such an analysis was performed in
Festuccia:2016gul  for $M$ Sasaki-Einstein (using the ordinary formulation of
supersymmetry) and it would be interesting to extend it according to our
procedure.

There are existing results for the partition function of the topologically
twisted theory including flux on some compact toric 4-manifolds Bawane:2014uka
; Rodriguez-Gomez:2014eza ; Bershtein:2015xfa ; Bershtein:2016mxz ;
Bonelli:2020xps  where the sum is over equivariant fluxes. It would be
desirable to compare these to our result in terms of physical fluxes, possibly
making use of a suitable notion of S-duality Festuccia:2020xtv .

Finally, in order to explicitly compute the partition function, the contour
integral of the Coulomb branch partition function over $a$ needs to be
understood. Since in 5d SYM there is only one scalar field $\sigma$ and the
fundamental group is finite, we expect there to be a unique choice of
integration contour. If this choice commutes with the $h\to\infty$ limit, then
we would have an easy recipe for the 4d contour whenever the 4d theory arises
from 5d using our procedure.

#### Locally Free $S^{1}$-Actions.

In this work, we assumed the $S^{1}$-action generated by $\vec{\X}$ to be
globally free such that $B=M/S^{1}$ has no singularities. More generally, we
could consider a locally free $\widetilde{S^{1}}$-action343434For Sasakian
manifolds, such actions are discussed in Sparks:2010sn .. In this case, $M$
will contain points with finite isotropy group that descend to orbifold
singularities on $\widetilde{B}=M/\widetilde{S^{1}}$. The intermediate step
would be to consider non-simply-connected orbifolds
$\widetilde{X}=M/\widetilde{\mathbb{Z}_{h}}$, where the quotient acts on the
fibre which is the orbit of the $\widetilde{S^{1}}$-action.

Let us briefly sketch how we think the procedure in section 4 can be adapted
to these cases. For simplicity, let us consider the toric Sasakian manifold
$S^{5}$ discussed in subsection 3.1. We recall that the vectors spanning the
edges of the two-dimensional dual cone are:

$\vec{u}_{1}=[0,1,0],\quad\vec{u}_{2}=[0,0,1],\quad\vec{u}_{3}=[1,0,0]$ (129)

and thus the inward-pointing normals are $\vec{v}_{3}=\vec{u}_{2}$,
$\vec{v}_{2}=\vec{u}_{1}$, $\vec{v}_{1}=\vec{u}_{3}$. The reduction of an
$\mathcal{N}=1$ vector multiplet is performed along the orbits of the free
$S^{1}$-actions, generated by:

$\vec{\X}^{\text{top}}=[1,1,1]\sim\R,\qquad\vec{\X}^{\text{ex}}=[1,1,-1].$
(130)

However, locally free $\widetilde{S^{1}}$-actions are generated by:

$\vec{\tilde{\X}}^{\text{top}}=[l_{1},l_{2},l_{3}],\qquad\vec{\tilde{\X}}^{\text{ex}}=[l_{1},l_{2},-l_{3}],$
(131)

where, as long as $l_{1},l_{2},l_{3}\in\mathbb{N}$ and
$\mbox{gcd}(l_{1},l_{2},l_{3})=1$, the action is effective. Following our
procedure, the next step would be to introduce a quotient by $\mathbb{Z}_{h}$
acting on the fibres (131), sum over non-trivial flat connections and
eventually take the large $h$ limit. At finite $h$, the resulting space is an
orbifold with a conical singularity, representing a deficit angle. The
procedure can be generalized starting from a generic toric Sasakian manifold
$M$ and will appear in future work.

The setup above is related to the study of two-dimensional weighted projective
spaces, also known as spindles Ferrero:2020laf . In Lundin:2021zeb  the
reduction of an $\mathcal{N}=2$ vector multiplet from $S^{3}$ to $S^{2}$ is
considered. As a consistency check, taking an $\mathcal{N}=1$ vector multiplet
$S^{3}\times S^{1}$ as starting point and generalizing the reduction to
locally free $S^{1}$-actions, one should reproduce the result of
Inglese:2023wky  for both topological and exotic theories. Moreover, notice
that on $S^{3}$, as the double sine function has huge cancellations between
numerator and denominator, we expect the dependence on the choice of locally
free $S^{1}$-action to only affect the shift353535This is already what happens
reducing on the two fibres $\vec{\X}^{\text{top}},\vec{\X}^{\text{ex}}$ of
$S^{3}$ Lundin:2021zeb . in $\alpha(\mathfrak{m})$. Instead, reducing from 5d
to 4d, we expect the different slicings to depend on the choice of locally
free $S^{1}$-action as the triple sine function does not have such
cancellations. These results would significantly enlarge the observables for
SQFTs on orbifolds and, in the large-$N$ limit, they would allow an in-depth
study of the gravitational block formulas conjectured in Faedo:2021nub ;
Faedo:2022rqx .

#### 3-Sasakian Reduction

Another possibility to extend our procedure is to consider the dimensional
reduction to 4d from a 7d $\mathcal{N}=1$ vector multiplet on a 3-Sasakian
hypertoric manifold Polydorou:2017jha ; Rocen:2018xwo ; Iakovidis:2020znp .
Seven-dimensional 3-Sasakian manifolds are an $S^{3}$-fibration over a four-
dimensional quaternion Kähler orbifold. However, when the $S^{3}$-action is
free, the fibration is over a four-dimensional quaternion Kähler manifold. An
example is the Hopf fibration $S^{3}\hookrightarrow S^{7}\rightarrow S^{4}$.
For $n=2$, the perturbative partition function is given by a quadruple sine
function and it is shown to factorise in contributions arising from the fixed
point of the $T^{2}$-action on the four-dimensional base Iakovidis:2020znp .
In general, reducing on a sphere breaks half of the supersymmetry and thus,
shrinking the size of the $S^{3}$, one ends up with an $\mathcal{N}=2$ theory
in 4d. For example, we would expect to find both an equivariant version of
Donaldson-Witten theory and Pestun’s theory on $S^{4}$ from the same seven-
dimensional theory. In general, one could consider the reduction to
$\mathcal{N}=2$ theories from a generic quaternion Kähler manifold, thus
enlarging the class of manifolds considered in our setup. Finally, we have
shown how fluxes, whose nature is purely Abelian, arise from dimensional
reductions along non-trivial $S^{1}$-fibrations over $B$. It would be
interesting to understand what configurations arise when dimensionally
reducing along $SU(2)$-fibrations.

#### Acknowledgments

We are grateful to Guido Festuccia, Jian Qiu, and Maxim Zabzine for
stimulating discussions on the subject. We thank them, Matteo Sacchi, Leonardo
Santilli and Itamar Yaakov for comments on the manuscript. We also thank
Seidon Alsaody and Thomas Kragh for mathematical advice. RM acknowledges
support from Vetenskapsrådet under grant 2018-05572 and the Centre for
Interdisciplinary Mathematics at Uppsala University. LR acknowledges support
from the Shuimu Tsinghua Scholar Program.

##  Appendix A Topology of Finite Quotients and the Base

In this appendix, we collect some topological facts about the quotient
manifolds on which we place the $\mathcal{N}=1$ SYM theory in the main part of
this work and the four-dimensional base.

###  A.1 Finite Quotients

Let $M$ be a closed, simply-connected manifold that has a free $S^{1}$-action.
We consider the quotient of $M$ by the discrete subgroup
$\mathbb{Z}_{h}\subset S^{1}$, $h\in\mathbb{N}_{\geq 2}$. This quotient
$X=M/\mathbb{Z}_{h}$ is again a smooth manifold with a free
$(S^{1}/\mathbb{Z}_{h}\simeq S^{1})$-action . However, $X$ is not simply-
connected but has $\pi_{1}(X)\simeq\mathbb{Z}_{h}$. This follows from the fact
that there is a fibration

$\tikzcd[columnsep=small,rowsep=small]\mathbb{Z}_{h}\ar[r]&M\ar[d]\\\ X$ (132)

giving rise to a long exact sequence (LES) in homotopy:

$\tikzcd[columnsep=small,rowsep=tiny]\dots\ar[r]&\pi_{1}(M)\ar[d,equal]\ar[r]\pi_{1}(X)\ar[r]\pi_{0}(\mathbb{Z}_{h})\ar[d,equal]\ar[r]\pi_{0}(M)\ar[d,equal]\ar[r]\dots\\\
0\mathbb{Z}_{h}0$ (133)

Moreover, by continuing the LES above, we find $\pi_{i}(M)\simeq\pi_{i}(X)$
for $i>2$.

In order to make sense of the projection condition (35) we first use the
following363636In the following, we always consider (co)homology with integral
coefficients, unless specified otherwise.

$H^{2}(X)\simeq\mathbb{Z}_{h}\oplus\mathbb{Z}^{b_{2}(X)}$, where $b_{2}(X)$
denotes the second Betti number of $X$. We already know
$\pi_{1}(X)\simeq\mathbb{Z}_{h}$ and thus, by the Hurewicz theorem,
$H_{1}(X)\simeq\mathbb{Z}_{h}$. Now we apply the universal coefficient theorem
(UCT) for integral coefficients:

$\tikzcd[columnsep=small]0\ar[r]&\Ext^{1}_{\mathbb{Z}}(H_{1}(X),\mathbb{Z})\ar[r]H^{2}(X)\ar[r]\Hom_{\mathbb{Z}}(H_{2}(X),\mathbb{Z})\ar[r]0.$
(134)

Since $H_{\bullet}(X)$ is finitely generated, we can decompose
$H_{\bullet}(X)\simeq\mathbb{Z}^{b_{\bullet}(X)}\oplus T_{\bullet}$ with
$b_{\bullet}(X)$ the Betti number and $T_{\bullet}$ the torsion part. But (see
e.g. Hatcher:2002  p.195):

$\displaystyle\Hom_{\mathbb{Z}}(H_{\bullet}(X),\mathbb{Z})$
$\displaystyle\simeq\Hom_{\mathbb{Z}}(\mathbb{Z}^{b_{\bullet}(X)},\mathbb{Z})\oplus\Hom_{\mathbb{Z}}(T_{\bullet},\mathbb{Z})\simeq\mathbb{Z}^{b_{\bullet}(X)},$
(135) $\displaystyle\Ext^{1}_{\mathbb{Z}}(H_{\bullet}(X),\mathbb{Z})$
$\displaystyle\simeq\Ext^{1}_{\mathbb{Z}}(\mathbb{Z}^{b_{\bullet}(X)},\mathbb{Z})\oplus\Ext^{1}_{\mathbb{Z}}(T_{\bullet},\mathbb{Z})\simeq
T_{\bullet},$ (136)

where we have used that $\Hom$ and $\Ext$ preserve limits. Since (134) splits
(although not naturally), we arrive at the result.

Consider the space $Y^{p,q}$ from subsection 2.2 with $\gcd(p,q)\geq 1$. Then
$\pi_{1}(Y^{p,q})\simeq\mathbb{Z}_{\gcd(p,q)}$ (see Gauntlett:2004yd ,
appendix A) and $H^{2}(Y^{p,q})\simeq\mathbb{Z}_{\gcd(p,q)}\oplus\mathbb{Z}$.

subsection A.1 shows that the torsion part of $H^{2}(X)$ originates from
$\pi_{1}(X)$. Since flat line bundles $\mathbb{C}^{\times}\hookrightarrow
L\rightarrow X$ are characterised by the first Chern class $c_{1}(L)$, we need
$c_{1}(L)\in\mathbb{Z}_{h}\subset H^{2}(X)$ for the projection condition (35)
to make sense. This is guaranteed by the following

$\im c_{1}\simeq\Ext^{1}(\pi_{1}(X),\mathbb{Z})$. First, note that
(isomorphism classes of) flat line bundles are precisely373737We have
$H^{1}(X;\mathbb{C}^{\times})\simeq
H^{1}(X;S^{1})\simeq\Hom(\pi_{1}(X),S^{1})$, where $X$ connected. elements of
$H^{1}(X;{\mathbb{C}}^{\times})$. We have a SES of coefficient rings

$\tikzcd[columnsep=scriptsize]0\ar[r]&{\mathbb{Z}}\ar[r]{\mathbb{C}}\ar[r,"\exp"]{\mathbb{C}}^{\times}\ar[r]0,$
(137)

giving rise to a LES in cohomology:

$\tikzcd[columnsep=scriptsize]\dots\ar[r]&H^{1}(X;{\mathbb{C}})\ar[r,"f"]H^{1}(X;{\mathbb{C}}^{\times})\ar[r,"c_{1}"]H^{2}(X;{\mathbb{Z}})\ar[r]\dots$
(138)

with the connecting homomorphism being the first Chern class (Putman:2012 ,
sec. 2.2). Since $X$ is connected, the universal coefficient theorem implies

$H^{1}(X;{\mathbb{C}})\simeq\Hom(H_{1}(X;{\mathbb{Z}}),{\mathbb{C}}),\qquad
H^{1}(X;{\mathbb{C}}^{\times})\simeq\Hom(H_{1}(X;{\mathbb{Z}}),{\mathbb{C}}^{\times}).$
(139)

Under the isomorphisms above, the map $f$ in the LES corresponds to the map

$\tilde{f}:\Hom(H_{1}(X;{\mathbb{Z}}),{\mathbb{C}})\longrightarrow\Hom(H_{1}(X;{\mathbb{Z}}),{\mathbb{C}}^{\times}),\;\varphi\longmapsto\exp\circ\varphi.$
(140)

The maps in the codomain that cannot be reached by $\tilde{f}$ are precisely
those that map torsion elements non-trivially, hence
$\coker\tilde{f}\simeq\Hom(H_{1}(X;{\mathbb{Z}})^{\mathrm{tor}},{\mathbb{C}}^{\times})$.
Using the first isomorphism theorem and the fact that
$H_{1}(X;\mathbb{Z})^{\mathrm{tor}}$ is a finite Abelian group, we obtain

$\im
c_{1}\simeq\Hom(H_{1}(X;{\mathbb{Z}})^{\mathrm{tor}},{\mathbb{C}}^{\times})\simeq\Ext^{1}(H_{1}(X;\mathbb{Z}),\mathbb{Z}).$
(141)

Applying the Hurewicz theorem gives the result.

Hence, $c_{1}$ takes values in $\pi_{1}(X)\simeq\mathbb{Z}_{h}$ and the
projection condition (40) is justified.

###  A.2 The Base

As for the topology of $B$, we have a fibration (in particular, a principal
$S^{1}$-bundle) over the base space $B=M/S^{1}$,

$\tikzcd[columnsep=small,rowsep=small]S^{1}\ar[r]&M\ar[d]\\\ B$ (142)

from which we obtain the LES in homotopy:

$\tikzcd[columnsep=small,rowsep=tiny]\dots\ar[r]&\pi_{2}(S^{1})\ar[r]\ar[d,equal]\pi_{2}(M)\ar[r]\ar[d,equal]\pi_{2}(B)\ar[r,"f"]\pi_{1}(S^{1})\ar[r]\ar[d,equal]\pi_{1}(M)\ar[r]\ar[d,equal]\pi_{1}(B)\ar[r]\pi_{0}(S^{1})\ar[r]\ar[d,equal]\dots\\\
0H^{3}(M)\mathbb{Z}00$ (143)

We conclude that $B$ is again simply-connected and (using the Hurewicz theorem
and Poincaré duality) we have $\mathbb{Z}\subset H^{2}(B)$ from the surjection
$f$. This is the subgroup generated by the two-cycle that is obtained from the
three-cycle in $X$; the image of $c_{1}$ upon reduction takes values in this
subgroup. However, the corresponding domain of $c_{1}$ are no longer flat
connections ($\pi_{1}(B)=0$ and thus $H^{1}(B;\mathbb{C}^{\times})=0$) but the
subspace of connections $A$ with curvature $F_{A}$ satisfying (45) and
$c_{1}(F_{A})\in\mathbb{Z}\subset H^{2}(B)$.

Finally, we present a more formal argument for the dimensional reduction of
$M$ to $B=M/S^{1}$ via the quotient $X=M/\mathbb{Z}_{h}$. Let $X_{n}:=M/Z_{n}$
and $Z_{n}:=\mathbb{Z}/h^{n}\mathbb{Z}$. Then we have the direct system

$\tikzcd[]X_{1}\ar[r,twoheads,"f_{1}"]&X_{2}\ar[r,twoheads,"f_{2}"]X_{3}\ar[r,twoheads,"f_{3}"]X_{4}\ar[r,twoheads,"f_{4}"]\dots$
(144)

with $f_{n}$ the canonical projections. The colimit of (144) (in the category
of compactly generated weakly Hausdorff spaces, CGWH) is given by

$\tikzcd
X_{1}\ar[r,twoheads,"f_{1}"]\ar[drr,"\iota_{1}"^{\prime}]&X_{2}\ar[r,twoheads,"f_{2}"]\ar[dr,"\iota_{2}"]X_{3}\ar[r,twoheads,"f_{3}"]\ar[d,"\iota_{3}"]X_{4}\ar[r,twoheads,"f_{4}"]\ar[dl,"\iota_{4}"]\dots\\\
\varinjlim X_{n}$ (145)

such that the triangles commute. On the other hand, by virtue of the
fibrations

$\tikzcd[columnsep=small]S^{1}\ar[r,hook]&X_{n}\ar[r,"\pi_{n}"]B,$ (146)

we have another co-cone for (144) given by

$\tikzcd
X_{1}\ar[r,twoheads,"f_{1}"]\ar[drr,"\pi_{1}"^{\prime}]&X_{2}\ar[r,twoheads,"f_{2}"]\ar[dr,"\pi_{2}"]X_{3}\ar[r,twoheads,"f_{3}"]\ar[d,"\pi_{3}"]X_{4}\ar[r,twoheads,"f_{4}"]\ar[dl,"\pi_{4}"]\dots\\\
B$ (147)

Then, by the universal property of $\varinjlim X_{n}$, there is a unique,
continuous map $\phi:\varinjlim X_{n}\rightarrow B$ such that
$\pi_{n}=\phi\circ\iota_{n}$ for all $n\in\mathbb{N}$. We now want to find a
map $\psi$ such that the triangles in the following diagram commute:

$\tikzcd\dots\ar[r,twoheads]&X_{n}\ar[rr,twoheads,"f_{n}"]\ar[dr,"\pi_{n}"]\ar[ddr,"\iota_{n}"^{\prime}]X_{n+1}\ar[r,twoheads]\ar[dl,"\pi_{n+1}"^{\prime}]\ar[ddl,"\iota_{n+1}"]\dots\\\
B\ar[d,"\psi"]\\\ \varinjlim X_{n}$ (148)

There is a unique such map $\psi:B\rightarrow\varinjlim X_{n}$,
$b\mapsto\iota_{1}(\tilde{b})$ for some $\tilde{b}\in X_{1}$ such that
$\pi_{1}(\tilde{b})=b$. But then $\psi$ must be the unique isomorphism.
Moreover, since both spaces are compact and Hausdorff383838This essentially
follows from the fact that the weak Hausdorffification of $\varinjlim
S^{1}/Z_{n}$ in Top is a point., $\psi$ is a homeomorphism. In fact, the
smooth structure on $\varinjlim X_{n}$ expected from the quotienting procedure
is precisely the one induced from $B$, which turns $\psi$ into a
diffeomorphism.

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