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

# Super Yang-Mills on Branched Covers and Weighted Projective Spaces

Roman Mauch 2 and Lorenzo Ruggeri<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we conjecture the Coulomb branch partition function, including
flux and instanton contributions, for the $\mathcal{N}=2$ vector multiplet on
weighted projective space $\mathbb{CP}^{2}_{\boldsymbol{N}}$ for equivariant
Donaldson-Witten and “Pestun-like” theories. More precisely, we claim that
this partition function agrees with the one computed on a certain branched
cover of $\mathbb{CP}^{2}$ upon matching conical deficit angles with
corresponding branch indices. Our conjecture is substantiated by checking that
similar partition functions on spindles agree with their equivalent on certain
branched covers of $\mathbb{CP}^{1}$. We compute the one-loop determinant on
the branched cover of $\mathbb{CP}^{2}$ for all flux sectors via dimensional
reduction from the $\mathcal{N}=1$ vector multiplet on a branched five-sphere
along a free $S^{1}$-action. This work paves the way for obtaining partition
functions on more generic symplectic toric orbifolds.

††preprint: UUITP-10/24

##  1 Introduction

Since the early days of supersymmetric localisation Pestun:2007rz ;
Pestun:2016zxk  the study of observables for supersymmetric quantum field
theories (SQFTs) on spaces with singularities has attracted much attention.
Most notably, the supersymmetric Renyi entropy for $\mathcal{N}=2$
superconformal field theories (SCFTs) has been computed from the partition
function of the SCFT on certain branched covers of $S^{3}$ Nishioka:2013haa .
This result was later generalised to branched covers of spheres $S^{d}$, for
$d=2,4,5$ Crossley:2014oea ; Mori:2015bro ; Alday:2014fsa ; Hama:2014iea .
While these spaces have conical surplus angles at the branch locus, recently
there has been considerable interest in computing observables on spaces with
orbifold singularities (i.e. conical deficit angles) instead. These theories,
which are interesting in their own right, offer the possibility to
holographically reproduce accelerating black hole solutions in supergravity
with orbifold singularities in their near horizon geometry Ferrero:2020laf ;
Ferrero:2020twa ; Ferrero:2021etw . Recently, the authors of Inglese:2023wky ;
Inglese:2023tyc  computed the partition function on
$\mathbb{CP}^{1}_{\boldsymbol{N}}\times S^{1}$ which was then used in
Colombo:2024mts  to reproduce the entropy function of supersymmetric
accelerating black holes in AdS4. The computation was performed for two
different theories denoted twist and anti-twist, which reduce on $S^{2}\times
S^{1}$ to, respectively, the topologically twisted index Benini:2015noa  and
the superconformal index Kim:2009wb ; Imamura:2011su ; Kapustin:2011jm . Their
result was obtained employing an orbifold index theorem
10.1215/S0012-7094-96-08226-5 .

In this work we draw a connection between the aforementioned results on the
branched cover of $\mathbb{CP}^{1}$ and the spindle
$\mathbb{CP}^{1}_{\boldsymbol{N}}$, which we subsequently extend to the
branched cover of $\mathbb{CP}^{2}$ and weighted projective space
$\mathbb{CP}^{2}_{\boldsymbol{N}}$. Our starting point are existing results
for the perturbative partition function of the $\mathcal{N}=2$ vector
multiplet on a branched three-sphere $S^{3}_{\boldsymbol{\alpha}}$
Klebanov:2011uf ; Nishioka:2013haa  and $\mathcal{N}=1$ vector multiplet on a
branched five-sphere $S^{5}_{\boldsymbol{\alpha}}$ Hama:2014iea ;
Alday:2014fsa 111In this work we consider generic integer values for
$\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{r})$, $r=2,3$ (cf. Renyi
entropy).. Both theories are in an $\Omega$-background, i.e. the supercharge
squares (among other symmetries) to translations along a Killing vector field.
The branched spheres can be viewed as principal $S^{1}$-fibrations and our aim
is to compute the corresponding partition function on the base (which we
denote by $B^{1}_{\boldsymbol{\alpha}}$ and $B^{2}_{\boldsymbol{\alpha}}$ in
the following). The method we use for the circle reduction is the one of
Lundin:2021zeb ; Lundin:2023tzw : first, we consider quotients by a finite
subgroup, $S^{2r-1}_{\boldsymbol{\alpha}}/\mathbb{Z}_{h}$ (for $r=2,3$). This
introduces extra topological sectors in the partition function corresponding
to non-trivial flat connections in the localisation locus. The circle
reduction is then obtained in the limit $h\to\infty$, in which these
topological sectors label flux through the two-cycle on the base
$B^{r-1}_{\boldsymbol{\alpha}}$. We consider two different $S^{1}$-fibrations
with respect to a fixed choice of Killing vector: a Hopf-like fibration
leading to an equivariant Donaldson-Witten theory on the base and an anti-
Hopf-like fibration leading to a “Pestun-like” theory. As the main result, we
compute the one-loop determinant and classical part of the full partition
function on $B^{r-1}_{\boldsymbol{\alpha}}$ in (34)-(36), resp. (43)-(45).

We observe that for $r=2$ our result for the one-loop determinant in each flux
sector agrees, for both topological and exotic theories, with, respectively,
the twist and anti-twist theories Inglese:2023wky  on the spindle upon the
identification of weights $N_{1}=\alpha_{1}$, $N_{2}=\alpha_{2}$. For the
special case of $\alpha_{1}=q=\alpha_{2}$ this could already be deduced from
the combined results of Mori:2015bro  and Hosomichi:2015pia ; they showed that
the one-loop determinant on $B^{1}_{(q,q)}$ is equivalent to $q$ copies of the
one on $S^{2}$ in the presence of a gauge defect which in turn is equivalent
to orbifolding the theory on $S^{2}$ by $\mathbb{Z}_{q}$. In 4d our result for
the topologically twisted theory for $r=3$ matches its counterpart on
$\mathbb{CP}^{2}_{\boldsymbol{N}}$ Martelli:2023oqk  upon suitable
identification of weights222A similar relation has been observed for 3d and 4d
SCFTs on orbifolds and branched covers of spheres with a single non-trivial
branch index in Nishioka:2016guu .. Guided by these observations, we make the
subsection 7.1 that also the one-loop determinant for the exotically twisted
$\mathcal{N}=2$ vector multiplet on $\mathbb{CP}^{2}_{\boldsymbol{N}}$ is
simply given by our result on $B^{2}_{\boldsymbol{\alpha}}$ for each flux
sector. Moreover, we believe that this matching for spaces with singular and
branch locus works more generally. It would therefore be interesting to
compute the one-loop determinant also for the $S^{1}$-reduction from branched
covers of other toric Sasakian 5-manifolds. We expect the results to match
with one-loop determinants of the corresponding symplectic toric 4-orbifolds
Martelli:2023oqk ; Colombo:2023fhu .

Finally, capitalising on equivariance of our setup under a $T^{2}$-action, for
the theory on $B^{2}_{\boldsymbol{\alpha}}$ we factorise the one-loop
determinant in each flux sector into individual contributions from the torus
fixed points. In each factor, the Coulomb branch parameter is shifted in a
certain way by the flux through the two-cycle. The instanton part of the
partition function is commonly obtained as a product of Nekrasov partition
functions over the fixed points Nekrasov:2003vi ; Bershtein:2015xfa ;
Festuccia:2018rew ; Festuccia:2019akm ; Lundin:2023tzw , where in each factor
the Coulomb branch parameter is shifted in precisely the way obtained from
factorisation of the one-loop determinant. Provided the equivalence of one-
loop determinants on $B^{2}_{\boldsymbol{\alpha}}$ and
$\mathbb{CP}^{2}_{\boldsymbol{N}}$ in each flux sector, their shifts also
agree. With the Nekrasov partition function on a neighbourhood
$\mathbb{C}^{2}/\mathbb{Z}_{N_{i}}$ of the $i$th fixed point being known
Fucito:2004ry ; Bonelli:2011jx ; Bonelli:2011kv ; Bonelli:2012ny ;
Bruzzo:2013nba ; Bruzzo:2013daa , we conclude our work by providing the
(integrand of the) full partition function (60), including classical, one-loop
and instanton part at all flux sectors for the $\mathcal{N}=2$ vector
multiplet on weighted projective space $\mathbb{CP}^{2}_{\boldsymbol{N}}$
both, for topological and exotic theories.

The outline is as follows. In section 2 we introduce the geometric setup of
branched spheres $S^{2r-1}_{\boldsymbol{\alpha}}$ and discuss the localisation
computation of the vector multiplet on a branched three- and five-sphere.
Section 3 describes the dimensional reduction of the theory along two
different, free $S^{1}$-directions which yield the one-loop determinant on the
respective quotient $S^{2r-1}_{\boldsymbol{\alpha}}/S^{1}$ for a fixed flux
sector. For $r=2,3$ we compute these explicitly in section 4 and 5. Using
equivariance of the setup, we factorise the one-loop determinants into
contributions from each torus fixed point in section 6. Based on our
observation that the determinants on $S^{3}_{\boldsymbol{\alpha}}/S^{1}$ match
the ones on weighted projective space and this also holds for the topological
twist on $S^{5}_{\boldsymbol{\alpha}}/S^{1}$, in section 7 we conjecture and
give some arguments in favour of such a match also for the exotic theory.
Finally, in Appendix A we provide more details on the geometric setup and in
Appendix B we compute the small radius limit of the spindle index.

##  2 Branched Spheres

In this section we introduce our geometric setup and field theoretic starting
point, the $\mathcal{N}=2$ (resp. $\mathcal{N}=1$) vector multiplet partition
function on branched spheres in three (resp. five) dimensions. These are
existing results from Klebanov:2011uf ; Nishioka:2013haa  (resp. Hama:2014iea
; Alday:2014fsa ) to which we refer the reader for more detail333In 3d the
partition function was obtained by first computing it on a background where
the singularities are resolved (which is a $Q$-exact deformation of the theory
on $S^{3}_{\boldsymbol{\alpha}}$) and then taking the singular limit. The
result agrees with the one on a squashed sphere with squashings reciprocal to
the $\\{\alpha_{i}\\}$. A similar relation has been found in 5d for
$\boldsymbol{\alpha}=(\alpha_{1},1,1)$ (which is easily extended to
$\alpha_{2},\alpha_{3}\neq 1$) and also for a squashed
$S^{5}_{\boldsymbol{\alpha}}$..

Throughout this article, we always consider cohomologically twisted theories
with simply-connected gauge group $G$ (see Kallen:2011ny ; Kallen:2012cs  for
the definition of cohomological fields and the corresponding cohomological
complex in three and five dimensions). For both, the (squashed or unsquashed)
spheres $S^{3},S^{5}$ and (the resolution of) their branched cover, Killing
spinors solving the corresponding rigid supergravity background are known
Nishioka:2013haa ; Alday:2014fsa ; Hama:2014iea  and hence, one can switch
between the physical and twisted theories bijectively.

###  2.1 Geometric Setup

The branched spheres in consideration, which we always denote by
$S^{2r-1}_{\boldsymbol{\alpha}}$ in the following (for $r=2,3$), are obtained
from the ordinary ones simply by extending the periodicity of the azimuthal
angles of $S^{2r-1}$ by an integer multiple of $2\pi$. If we denote said
angles by $\theta_{i}$, $i=1,\dots,r$ with $\theta_{i}\sim\theta_{i}+2\pi$
then we extend these to new angles $\tilde{\theta}_{i}$ with
$\tilde{\theta}_{i}\sim\tilde{\theta}_{i}+2\pi\alpha_{i}$, where
$\alpha_{i}\in\mathbb{Z}$. The integers $\\{\alpha_{i}\\}$ are called the
branch indices and in the notation $S^{2r-1}_{\boldsymbol{\alpha}}$ are
collected in the subscript
$\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{r})$. The branched sphere is an
$\alpha_{1}\dots\alpha_{r}$-fold branched cover of the ordinary sphere (see
Appendix A for more detail on the branching structure). More generally, we
obtain a squashed branched sphere by extending the periodicities as above, but
for a squashed $S^{2r-1}$ instead.

#### Three Dimensions.

The metric on $S^{3}_{\boldsymbol{\alpha}}$ can be obtained from the round one
and in terms of the canonical angles reads

$\displaystyle\dd
s^{2}=\dd\phi^{2}+\alpha_{1}^{2}\sin^{2}\phi\dd\theta_{1}^{2}+\alpha_{2}^{2}\cos^{2}\phi\dd\theta_{2}^{2},$
(1)

with $\phi\in[0,\pi/2]$. It was shown in Nishioka:2013haa  that this space has
conical singularities in the curvature on the Hopf-link comprised of the two
$S^{1}$-subspaces $\phi=0,\pi/2$ with surplus angles $\alpha_{1},\alpha_{2}$.
There are two free $S^{1}$-actions (up to $\SL(2,\mathbb{Z})$-transformations)
which are Killing and given by

$\X=\partial_{\tilde{\theta}_{1}}\pm\partial_{\tilde{\theta}_{2}}=\frac{1}{\alpha_{1}}\partial_{\theta_{1}}\pm\frac{1}{\alpha_{2}}\partial_{\theta_{2}}.$
(2)

$S^{3}_{\boldsymbol{\alpha}}$ can be viewed as a “branched” (anti-)Hopf
fibration

$\tikzcd[columnsep=20pt]S^{1}\ar[r,hook]&S^{3}_{\boldsymbol{\alpha}}\ar[r]B^{1}_{\boldsymbol{\alpha}}$
(3)

along this action, whose base we denote by
$B^{1}_{\boldsymbol{\alpha}}=S^{3}_{\boldsymbol{\alpha}}/S^{1}$. Since (2)
preserves the branching, $B^{1}_{\boldsymbol{\alpha}}$ in turn is a branched
cover of $S^{2}$. We can also reach this space by first considering finite
quotients444For simplicity, we take $\gcd(h,\alpha_{i})=1$ for all
$i=1,\dots,r$. This can still be satisfied in the limit $h\to\infty$ by virtue
of Euclid’s theorem. $S^{3}_{\boldsymbol{\alpha}}/\mathbb{Z}_{h}$ along (2),
giving branched lens spaces which we denote $L^{3}_{\boldsymbol{\alpha}}(h,\pm
1)$. Note that the finite quotient introduces holonomies classified by
$\mathbb{Z}_{h}$. We arrive at the base space $B^{1}_{\boldsymbol{\alpha}}$ by
taking the limit $h\to\infty$ (see Lundin:2023tzw ).

#### Five Dimensions.

The metric on $S^{5}_{\boldsymbol{\alpha}}$ can be obtained from the round one
and in terms of the canonical angles reads

$\dd
s^{2}=\dd\varphi^{2}+\sin^{2}\\!\varphi\,\dd\phi+\alpha_{3}^{2}\cos^{2}\\!\varphi\dd\theta_{3}^{2}+\alpha_{2}^{2}\sin^{2}\\!\varphi\sin^{2}\\!\phi\,\dd\theta_{2}^{2}+\alpha_{1}^{2}\sin^{2}\\!\varphi\cos^{2}\\!\phi\,\dd\theta_{1}^{2},$
(4)

with $\varphi,\phi\in[0,\pi/2]$. Similar to the three-dimensional case, the
curvature can be shown to have conical singularities on the three (branched)
$S^{3}$-subspaces $\varphi=\pi/2$ (with surplus $2\pi\alpha_{3}$), $\phi=0$
(with surplus $2\pi\alpha_{2}$) and $\phi=\pi/2$ (with surplus
$2\pi\alpha_{1}$). There are two free $S^{1}$-actions (up to
$\SL(3,\mathbb{Z})$-transformations) which are Killing:

$\X=\frac{1}{\alpha_{1}}\partial_{\theta_{1}}+\frac{1}{\alpha_{2}}\partial_{\theta_{2}}\pm\frac{1}{\alpha_{3}}\partial_{\theta_{3}}.$
(5)

Similar to the three-dimensional case, we are interested in the (free)
$S^{1}$-quotient by this action,
$B^{2}_{\boldsymbol{\alpha}}=S^{5}_{\boldsymbol{\alpha}}/S^{1}$ which gives a
branched cover of $\mathbb{CP}^{2}$. Again, as an intermediate step we can
take finite quotientsfn–gcd $S^{5}_{\boldsymbol{\alpha}}/\mathbb{Z}_{h}$
giving (generalised) branched lens spaces $L^{5}_{\boldsymbol{\alpha}}(h,\pm
1)$ with extra topological sectors classified by $\mathbb{Z}_{h}$.

###  2.2 Localisation

The theories we want to place on $S^{3}_{\boldsymbol{\alpha}}$ (resp.
$S^{5}_{\boldsymbol{\alpha}}$) are those of the $\mathcal{N}=2$ (resp.
$\mathcal{N}=1$) vector multiplet. We again refer to Klebanov:2011uf ;
Nishioka:2013haa  (resp. Hama:2014iea ; Alday:2014fsa ) for supersymmetry
backgrounds and actions.

For localisation, note that away from the branch locus, locally the branched
sphere is diffeomorphic to the ordinary sphere and hence the computation is
locally the same. At the branch locus, however,
$S^{2r-1}_{\boldsymbol{\alpha}}$ is singular. This has the following
consequences: (i) the background R-symmetry connection $A_{R}$ is flat except
at the branch locus where it has a singular field strength compensating for
the curvature singularity and (ii) boundary conditions555The term “boundary
conditions” here stems from the fact that there exists a conformal mapping
from the branched $(2r-1)$-dimensional sphere with
$\boldsymbol{\alpha}=(\alpha,1,\dots,1)$ to $\mathbb{H}^{r-1}\times S^{1}$
where the branch locus is mapped to the circle at infinity, i.e. is an actual
boundary. One could, for instance, consider dynamical gauge fields that
asymptotically have non-trivial holonomy around the $S^{1}$ Hosseini:2019and .
However, here we restrict to trivial holonomies for the dynamical fields. have
to be imposed for the fields of the theory along the branch locus.

Computing the one-loop partition function from localisation essentially
reduces to counting (normalisable) holomorphic functions weighted by their
$T^{r}$-action (see e.g. Qiu:2016dyj ). On
$S^{2r-1}=\\{(z_{1},\dots,z_{r})\in\mathbb{C}^{r}|\,|z_{1}|^{2}+\dots+|z_{r}|^{2}=1\\}$
a basis for such maps is given by polynomials in $z_{1},\dots,z_{r}$. When we
move to the branched cover by extending the periodicities as discussed in the
previous subsection, such a basis is instead given by
$z_{1}^{1/\alpha_{1}},\dots,z_{r}^{1/\alpha_{r}}$ (where $\arg
z_{i}^{1/\alpha_{i}}=\tilde{\theta}_{i}/\alpha_{i}$) and the boundary
conditions on the fields are that the holomorphic functions be smooth in
$\\{z_{i}^{1/\alpha_{i}}\\}$ rather than $\\{z_{i}\\}$. Consequently, when
expanding in Fourier modes $\exp(\ii n_{1}\theta_{1}+\dots+\ii
n_{r}\theta_{r})$, the charge under the free $S^{1}$-action (2), resp. (5) is
$\frac{n_{1}}{\alpha_{1}}+\dots+\frac{n_{r-1}}{\alpha_{r-1}}\pm\frac{n_{r}}{\alpha_{r}}$
with $n_{1},\dots,n_{r}\in\mathbb{Z}$.

The free $S^{1}$-action can be parametrised by an angle $\xi$ with
$\xi\sim\xi+2\pi\lcm\boldsymbol{\alpha}$. Hence, locally, smooth functions on
the branched cover can be expanded in terms of modes $\exp(\ii
t\xi/\lcm\boldsymbol{\alpha})$ along the $S^{1}$ and the charge
$t\in\mathbb{Z}$ is related to the ones corresponding to the canonical angles
by

$\frac{t}{\lcm\boldsymbol{\alpha}}=\frac{n_{1}}{\alpha_{1}}+\dots+\frac{n_{r-1}}{\alpha_{r-1}}\pm\frac{n_{r}}{\alpha_{r}}.$
(6)

This relation will be applied when reducing along the free $S^{1}$ later on.

#### Perturbative Partition Function.

Our observations above suggest that the result of the one-loop computation on
a squashed $S^{2r-1}_{\boldsymbol{\alpha}}$ with squashings
$\\{\omega_{i}\\}_{i=1,\dots,r}$ is simply that on $S^{2r-1}$ Hama:2011ea ;
Imamura:2011wg ; Kallen:2012va ; Imamura:2012efi ; Lockhart:2012vp  for a
background with Killing vector
$\boldsymbol{\R}=(\frac{\omega_{1}}{\alpha_{1}},\dots,\frac{\omega_{r}}{\alpha_{r}})$
and yields the multiple sine function

$S_{r}(\ii\alpha(a)|\boldsymbol{\R})=\prod_{\boldsymbol{n}\in\mathbb{Z}^{r}_{\geq
0}}(\boldsymbol{n}\cdot\boldsymbol{\R}+\ii\alpha(a))\prod_{\boldsymbol{n}\in\mathbb{Z}^{r}_{>0}}(\boldsymbol{n}\cdot\boldsymbol{\R}-\ii\alpha(a))^{(-1)^{r-1}}$
(7)

for a fixed root $\alpha$ of the gauge group $G$ and Coulomb branch parameter
$a$ in the Cartan subalgebra of $G$. Note that for non-trivial squashing the
supercharge used for localisation does not square to rotations along the free
direction $\X$ (cf. footnote 8). The perturbative partition function is
obtained as

$\mathcal{Z}^{\text{pert}}_{S^{2r-1}_{\boldsymbol{\alpha}}}=\int_{\mathfrak{h}}\dd
a~{}\e{-S^{\text{cl}}_{2r-1}}\det{}^{\prime}_{\text{adj}}~{}S_{r}(\ii\alpha(a)|\boldsymbol{\R}),$
(8)

where $\mathfrak{h}\subset\mathfrak{g}$ denotes the Cartan subalgebra and the
determinant is over the roots of $G$, where the prime signifies that possible
zero-mode contributions have already been canceled by the Vandermonde
determinant (which arises when restricting integration from $\mathfrak{g}$ to
$\mathfrak{h}$). The classical action $S^{\text{cl}}_{2r-1}$ for $r=2,3$ is
given by

$\displaystyle S^{\text{cl}}_{3}=-\frac{\ii\pi k}{\R_{1}\R_{2}}\tr
a^{2},\qquad\qquad
S^{\text{cl}}_{5}=\frac{(2\pi)^{3}}{g^{2}_{\text{YM}}\R_{1}\R_{2}\R_{3}}\tr
a^{2}$ (9)

with Chern-Simons level $k$, Yang-Mills coupling $g^{2}_{\text{YM}}$ and the
weights $\R_{i}=\frac{\omega_{i}}{\alpha_{i}}$. We comment on instanton
contributions to the full partition function in section 7.

Finally, we impose that the squashing leave the free $S^{1}$-direction
invariant, i.e.

top: $\displaystyle\omega_{1}+\dots+\omega_{r}=r,$ (10) ex:
$\displaystyle\omega_{1}+\dots+\omega_{r-1}-\omega_{r}=r-2,$

for the two choices of $\X$ corresponding to a Hopf (resp. anti-Hopf)
fibration. We will refer to the respective $(2r-2)$-dimensional theory on
$S^{2r-1}_{\boldsymbol{\alpha}}/S^{1}$ as the topological (resp. exotic)
theory666We adopt this terminology from Festuccia:2019akm . See also the
comment at the end of subsection 3.2., hence the labels top and ex in (10).

##  3 Partition Function on $S^{2r-1}_{\boldsymbol{\alpha}}/S^{1}$

Having the result for the perturbative partition function on
$S^{2r-1}_{\boldsymbol{\alpha}}$ at hand, in this section we will perform the
dimensional reduction along the two777The two fibres differ by the relative
direction with respect to the choice of Killing vector. However, in this
section we discuss the general procedure of the reduction and will mostly not
distinguish between the two choices of free $S^{1}$ explicitly (cf. end of
subsection 3.2). free $S^{1}$-directions identified in (2), resp. (5). In
particular, this will produce the one-loop contribution of the vector
multiplet on the quotient space $B^{r-1}_{\boldsymbol{\alpha}}$ in all flux
sectors (note $H^{2}(B^{r-1}_{\boldsymbol{\alpha}})\simeq\mathbb{Z}$).

The quotient procedure follows the method explained in Lundin:2023tzw  where
we first consider the theory on finite quotients
$S^{2r-1}_{\boldsymbol{\alpha}}/\mathbb{Z}_{h}$ introduced in the previous
section. This requires to extend the localisation locus to include flat
connections with non-trivial holonomy around the free $S^{1}$ and yields the
one-loop contribution around such flat connections to the partition function
on generalised lens spaces $L^{2r-1}_{\boldsymbol{\alpha}}(h,\pm 1)$. In the
limit $h\to\infty$ we reach $B^{r-1}_{\boldsymbol{\alpha}}$ and the
contributions from non-trivial flat connections turn into those with non-
trivial flux. Note that, unlike standard Kaluza-Klein reduction along a
trivially fibred $S^{1}$, in our case all higher KK modes on the non-trivially
fibred $S^{1}$ are not integrated out, but rather distribute themselves into
different topological sectors on the base.

###  3.1 Finite Quotients

In order to understand how the partition function (8) has to be modified for a
finite quotient of the underlying space it suffices to analyse how the fields
contributing to the one-loop determinant change. As was mentioned in
subsection 2.2, these are precisely the holomorphic functions on the space.

Holomorphic functions on $S^{2r-1}_{\boldsymbol{\alpha}}$ are in one-to-one
correspondence with points $\boldsymbol{n}$ in the integral cone
$\mathbb{Z}^{r}_{\geq 0}$ labelling their $T^{r}$-charge. Clearly, functions
invariant under the free $S^{1}$ still contribute to the one-loop determinant
in the same way. It is therefore natural to decompose the cone into slices
corresponding to functions with fixed charge $t$ under $\X$, using the
relation (6):

$\mathcal{B}_{t,\boldsymbol{\alpha}}=\\{(n_{1},\dots,n_{r-1})\in\mathbb{Z}^{r-1}_{\geq
0}|\exists\,n_{r}\in\mathbb{Z}_{\geq
0}:\tfrac{t}{\lcm\boldsymbol{\alpha}}=\tfrac{n_{1}}{\alpha_{1}}+\dots+\tfrac{n_{r-1}}{\alpha_{r-1}}\pm\tfrac{n_{r}}{\alpha_{r}}\\}.$
(11)

Note that in the exotic theory $t\in\mathbb{Z}$ and the slices can be non-
compact while in the topological one $t\in\mathbb{Z}_{\geq 0}$ and the slices
are always compact. Now we can rewrite the product over the cone
$\mathbb{Z}^{r}_{\geq 0}$ in the multiple sine function as

$S_{r}(\ii\alpha(a)|\boldsymbol{\R})=\prod_{t}\prod_{\boldsymbol{n}\in\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(\boldsymbol{n}\cdot\tilde{\boldsymbol{\R}}\pm\frac{\omega_{r}t}{\lcm\boldsymbol{\alpha}}+\ii\alpha(a)\right)\prod_{\boldsymbol{n}\in\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}}\left(\boldsymbol{n}\cdot\tilde{\boldsymbol{\R}}\pm\frac{\omega_{r}t}{\lcm\boldsymbol{\alpha}}-\ii\alpha(a)\right)^{(-1)^{r-1}},$
(12)

where $\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}$ denotes the interior of
the slice (11) and
$\tilde{\boldsymbol{\R}}=(\frac{\epsilon_{1}}{\alpha_{1}},\dots,\frac{\epsilon_{r-1}}{\alpha_{r-1}})$
the weights of the residual $T^{r-1}$-action on the slice, where

$\begin{split}\text{top:}&\quad\epsilon_{i}=\omega_{i}-\omega_{r},\\\
\text{ex:}&\quad\epsilon_{i}=\omega_{i}+\omega_{r},\end{split}$ (13)

where $i=1,\dots,r-1$. Note that in the unsquashed limit $\omega_{i}\to 1$ for
all $i=1,\dots,r$ the equivariance parameters $\\{\epsilon_{i}\\}$
vanish888This is expected since in the unsquashed limit the supercharge in the
$(2r-1)$-dimensional theory squares precisely to rotations along the free
$S^{1}$-direction, $Q^{2}=\mathcal{L}_{\X}$. Accordingly, once we take the
quotient the supercharge squares to zero and equivariance is lost. in the
topological case, while they stay finite for the exotic case. Finally, we can
write $\omega_{r}=\omega_{r}(\epsilon_{1},\dots,\epsilon_{r-1})$ by virtue of
(10). We point out that (12) for topological and exotic case are simply two
equivalent rewritings of (7).

#### Taking the Quotient.

Naively, only holomorphic functions of charge $t=0\mmod h$ along $\X$ descend
to the quotient. However, note that $L^{2r-1}_{\boldsymbol{\alpha}}(h,\pm 1)$
has extra topological sectors labelled by $\mathbb{Z}_{h}$ in the fundamental
group. These are in bijection with flat connections of line bundles over
$L^{2r-1}_{\boldsymbol{\alpha}}(h,\pm 1)$ (up to gauge transformations). By
standard arguments, on the localisation locus the gauge group is broken to its
maximal torus $U(1)^{\rk G}$ Blau:1994rk  and flat connections in the locus
are characterised (up to gauge transformations) by representations
$\Hom(\mathbb{Z}_{h},U(1)^{\rk G})$ (up to conjugation). These can be labelled
by certain elements $\mathfrak{m}$ in the Cartan subalgebra of $G$ (e.g., for
$G=U(N)$ we have
$\mathfrak{m}=\diag(m_{1},\dots,m_{N})\in\mathbb{Z}_{h}^{N\times N}$,
$m_{i}\leq m_{i+1}$). Hence, flat line bundles become part of the locus and
apart from functions with $t=0\mmod h$ (which descend to functions on
$L^{2r-1}_{\boldsymbol{\alpha}}(h,\pm 1)$), for fixed $\mathfrak{m}$ we should
also allow for ones of charge

$t=\alpha(\mathfrak{m})\mmod h$ (14)

which descend to sections of the corresponding flat line bundle on
$L^{2r-1}_{\boldsymbol{\alpha}}(h,\pm 1)$. From these observations we conclude
that the finite quotient along $\X$ can be implemented at the level of the
one-loop determinant simply by restricting the product in (12) to $t$ such
that (14) is satisfied. The partition function then becomes

$\mathcal{Z}_{L^{2r-1}_{\boldsymbol{\alpha}}(h,\pm
1)}=\sum_{\mathfrak{m}}\int_{\mathfrak{h}}\dd
a~{}\e{-S^{\text{cl}}_{2r-1}/h}\det{}^{\prime}_{\text{adj}}~{}S_{r}(\ii\alpha(a)|\boldsymbol{\R})|_{t=\alpha(\mathfrak{m})\mmod
h}$ (15)

with $S^{\text{cl}}_{5}$ as in (9) but

$S^{\text{cl}}_{3}=-\frac{\ii\pi
k}{\R_{1}\R_{2}}\tr(a^{2}\pm\mathfrak{m}^{2})$ (16)

due to the Chern-Simons term Guadagnini:2017lcz . Note that now the
topological and exotic choice describe different theories for $h>1$.

###  3.2 Reduction to the Base

We arrive at the quotient space
$B^{r-1}_{\boldsymbol{\alpha}}=S^{2r-1}_{\boldsymbol{\alpha}}/S^{1}$ by taking
the limit $h\to\infty$ where the $S^{1}$-fibre shrinks to a point (a formal
treatment of this limit for 5-manifolds can be found in Lundin:2023tzw ,
appendix A). For $r=2$ the theory for the $\mathcal{N}=2$ vector multiplet on
$S^{3}_{\boldsymbol{\alpha}}$ descends to the one for an $\mathcal{N}=(2,2)$
vector multiplet on $B^{1}_{\boldsymbol{\alpha}}$. Similarly, for $r=3$ the
$\mathcal{N}=1$ vector multiplet on $S^{5}_{\boldsymbol{\alpha}}$ descends to
an $\mathcal{N}=2$ vector multiplet on $B^{2}_{\boldsymbol{\alpha}}$.
Depending on whether we reduce along the topological or exotic direction, we
obtain two differently twisted theories on $B^{r-1}_{\boldsymbol{\alpha}}$; we
comment on this at the end of the subsection.

In the limit $h\to\infty$, for a fixed root $\alpha$ and topological sector
$\mathfrak{m}$ there is only a single slice
$\mathcal{B}_{t,\boldsymbol{\alpha}}$ satisfying (14) which corresponds to the
holomorphic sections of charge

$t=\alpha(\mathfrak{m}).$ (17)

Moreover, it can be shown (in analogy to Lundin:2023tzw ) from dimensional
reduction of the fields that the flat connections in the locus of the theory
on $L^{2r-1}_{\boldsymbol{\alpha}}(h,\pm 1)$, in the limit $h\to\infty$ give
rise to gauge connections with flux on the two-cycle in
$B^{r-1}_{\boldsymbol{\alpha}}$. This flux is of course labelled by
$\mathfrak{m}$ whose components are now integral (e.g. for $G=U(N)$ we have
$\mathfrak{m}=\diag(m_{1},\dots,m_{N})\in\mathbb{Z}^{N\times N}$). Hence, this
method of reduction not only produces the one-loop determinant for the
topologically trivial sector of the $(2r-2)$-dimensional theory but in fact
for all possible topological sectors (cf. Kaluza-Klein reduction).

The partition function on $B^{r-1}_{\boldsymbol{\alpha}}$ is then obtained by

$\mathcal{Z}_{B^{r-1}_{\boldsymbol{\alpha}}}=\sum_{\mathfrak{m}}\int_{\mathfrak{h}}\dd
a~{}\e{-S^{\text{cl}}_{2r-2}}\det{}^{\prime}_{\text{adj}}~{}S_{r}(\ii\alpha(a)|\boldsymbol{\R})|_{t=\alpha(\mathfrak{m})}.$
(18)

We note at this point that many previous results for the topologically twisted
theory on closed four-manifolds are expressed in terms of a sum over
equivariant fluxes on which certain stability conditions are imposed (e.g.,
Nekrasov:2003vi ; Bershtein:2015xfa ; Bershtein:2016mxz ; Hosseini:2018uzp ;
Crichigno:2018adf ; Bonelli:2020xps ). Here instead, we sum over the
“physical” flux on the two-cycle in $B^{r-1}_{\boldsymbol{\alpha}}$. Explicit
expressions for the one-loop determinants in each flux sector and classical
actions $S^{\text{cl}}_{2r-1}$ on $B^{r-1}_{\boldsymbol{\alpha}}$ are
presented in section 4 and 5. Instanton contributions to
$\mathcal{Z}_{B^{r-1}_{\boldsymbol{\alpha}}}$ will be discussed in section 7.

#### Topologically Twisted and Exotic Theories.

Before we move on to explicit results, let us briefly comment on the different
features of the topologically twisted and exotic theory whose perturbative
partition function we obtained by reducing along the two different directions
in (2), resp. (5). In 4d they are instances of a general cohomological
twisting called Pestunization Festuccia:2018rew ; Festuccia:2019akm . The
topological twist is simply equivariant Donaldson-Witten theory. The canonical
(and first) example of an exotic theory is Pestun’s theory on $S^{4}$
Pestun:2007rz .

In cohomologically twisted theories the one-loop determinant is commonly
computed from the equivariant index of a complex. Depending on whether we have
a topological or an exotic theory, this complex will be either elliptic or
transversally elliptic. This is reflected by the slices
$\mathcal{B}^{t,\boldsymbol{\alpha}}$ being either compact or non-compact.

In 4d, both for topological and exotic theories the partition function
includes additional non-perturbative contributions from (anti-)instantons
localised to the fixed points of a $U(1)$-isometry. While for the topological
twist we have only instantons, the exotic theory allows for more generic
distributions of instantons and anti-instantons at the fixed points. For
instance, on $B^{2}_{\boldsymbol{\alpha}}$ we have anti-instantons at one of
the three fixed points.

##  4 Partition Function on $S^{3}_{\boldsymbol{\alpha}}/S^{1}$

We now set out to compute the integrand of
$\mathcal{Z}_{B^{1}_{\boldsymbol{\alpha}}}$ explicitly. Before considering the
reduction, however, we first perform some simplifications that arise from the
specific form of the double sine function.

###  4.1 Cancellations on $S^{3}_{\boldsymbol{\alpha}}$

We start computing the integrand of
$\mathcal{Z}_{S^{3}_{\boldsymbol{\alpha}}}$ explicitly. Expressions will be
different for the topological and exotic theories which we signify by a
superscript. Before we implement the reduction procedure, on
$S^{3}_{\boldsymbol{\alpha}}$, note that cancellations occur in

$Z_{S^{3}_{\boldsymbol{\alpha}}}(a,\boldsymbol{\R}):=\det{}^{\prime}_{\text{adj}}S_{2}(\ii\alpha(a)|\boldsymbol{\R})$
(19)

which we take care of first. We start by rewriting the product over roots from
the determinant as a product only over the positive roots:

$Z^{\text{top}}_{S^{3}_{\boldsymbol{\alpha}}}=\prod_{\alpha\geq 0}\prod_{t\geq
0}\frac{\prod\limits_{n_{1}\in\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(\left(1-\frac{\epsilon_{1}}{2}\right)\frac{t}{\alpha_{1}\alpha_{2}}+\frac{n_{1}}{\alpha_{1}}\epsilon_{1}+\ii\alpha(a)\right)}{\prod\limits_{n_{1}\in\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}}\left(\left(1-\frac{\epsilon_{1}}{2}\right)\frac{t}{\alpha_{1}\alpha_{2}}+\frac{n_{1}}{\alpha_{1}}\epsilon_{1}+\ii\alpha(a)\right)}\prod_{t^{\prime}\leq
0}\frac{\prod\limits_{n_{1}\in\mathcal{B}_{t^{\prime},\boldsymbol{\alpha}}}\left(\left(1-\frac{\epsilon_{1}}{2}\right)\frac{t^{\prime}}{\alpha_{1}\alpha_{2}}-\frac{n_{1}}{\alpha_{1}}\epsilon_{1}+\ii\alpha(a)\right)}{\prod\limits_{n_{1}\in\mathring{\mathcal{B}}_{t^{\prime},\boldsymbol{\alpha}}}\left(\left(1-\frac{\epsilon_{1}}{2}\right)\frac{t^{\prime}}{\alpha_{1}\alpha_{2}}-\frac{n_{1}}{\alpha_{1}}\epsilon_{1}+\ii\alpha(a)\right)},$
(20) $Z^{\text{ex}}_{S^{3}_{\boldsymbol{\alpha}}}=\prod_{\alpha\geq
0}\prod_{t\in\mathbb{Z}}\frac{\prod\limits_{n_{1}\in\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(-\frac{\epsilon_{1}}{2}\frac{t}{\alpha_{1}\alpha_{2}}+\frac{n_{1}}{\alpha_{1}}\epsilon_{1}+\ii\alpha(a)\right)}{\prod\limits_{n_{1}\in\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}}\left(-\frac{\epsilon_{1}}{2}\frac{t}{\alpha_{1}\alpha_{2}}+\frac{n_{1}}{\alpha_{1}}\epsilon_{1}+\ii\alpha(a)\right)}\prod_{t^{\prime}\in\mathbb{Z}}\frac{\prod\limits_{n_{1}\in\mathcal{B}_{t^{\prime},\boldsymbol{\alpha}}}\left(-\frac{\epsilon_{1}}{2}\frac{t^{\prime}}{\alpha_{1}\alpha_{2}}-\frac{n_{1}}{\alpha_{1}}\epsilon_{1}+\ii\alpha(a)\right)}{\prod\limits_{n_{1}\in\mathring{\mathcal{B}}_{t^{\prime},\boldsymbol{\alpha}}}\left(-\frac{\epsilon_{1}}{2}\frac{t^{\prime}}{\alpha_{1}\alpha_{2}}-\frac{n_{1}}{\alpha_{1}}\epsilon_{1}+\ii\alpha(a)\right)},$
(21)

where we used (13) and (10) to introduce the equivariance parameter
$\epsilon_{1}$. Note that one product in
$Z^{\text{top}}_{\boldsymbol{\alpha}}$ is over $t\geq 0$ and one over
$t^{\prime}\leq 0$. Let us also introduce $a_{\pm}\in\mathbb{Z}$ satisfying

$a_{-}\alpha_{1}-a_{+}\alpha_{2}=1,$ (22)

which always exist (by virtue of Bezout’s lemma). In particular,

$\gcd(a_{-},\alpha_{2})=1=\gcd(a_{+},\alpha_{1}).$ (23)

In the following, we treat the topological and exotic case separately.

#### Topological Theory.

Remember that the charge for rotations along the free $S^{1}$ is
$t^{\text{top}}=\alpha_{2}n_{1}+\alpha_{1}n_{2}$. In order to identify the
cancellations occurring it is useful to plot the slices
$\mathcal{B}_{t,\boldsymbol{\alpha}}$ and
$\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}$ of the cone
$\mathbb{Z}^{2}_{\geq 0}$, see Figure 1. We notice that simplifications occur
for points in the interior of the cone. Unpaired points are the ones on the
$n_{1}$-and $n_{2}$-axis. Note that, while for $\alpha_{1},\alpha_{2}=1$ two
points survive the cancellations in each slice, for $\alpha_{1}\neq\alpha_{2}$
there might only be one or no such point at all.

$n_{1}$$n_{2}$$\vec{\X}^{\text{top}}$$t=3$$t=5$

$n_{1}$$n_{2}$$\vec{\X}^{\text{top}}$$t=3$$t=9$$t=12$

$n_{1}$$n_{2}$$\vec{\X}^{\text{top}}$$t=5$$t=6$$t=20$

Figure 1: We plot the cone for $S^{3}_{\boldsymbol{\alpha}}$ which is simply
the lattice spanned by integers $n_{1},n_{2}\geq 0$. For various values of
$\alpha_{1},\alpha_{2}$ we plot the regions
$\mathcal{B}_{t,\boldsymbol{\alpha}}$ and
$\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}$ as dashed lines. These two
regions appear, respectively, at the numerator and the denominator in the
perturbative partition function. Thus, after cancellations, only light blue
modes survive. These contribute at different values of $t^{\text{top}}$
depending on the values of $\alpha_{1},\alpha_{2}$. From left to right: (i)
$\alpha_{1}=\alpha_{2}=1$, (ii) $\alpha_{1}=3,\alpha_{2}=2$, (iii)
$\alpha_{1}=5,\alpha_{2}=3$.

More explicitly, one can check that the bounds on
$\tilde{n}_{1}=\frac{n_{1}}{\alpha_{1}}\in\mathbb{Z}_{\geq 0}$ are the
following:

$t\geq
0:\quad\begin{cases}\mathcal{B}_{t,\boldsymbol{\alpha}}:&\rmd{a_{+}t}{\alpha_{1}}/\alpha_{1}\leq\tilde{n}_{1}\leq\frac{t}{\alpha_{1}\alpha_{2}}-\rmd{-a_{-}t}{\alpha_{2}}/\alpha_{2}\\\
\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}:&1-\rmd{-a_{+}t}{\alpha_{1}}/\alpha_{1}\leq\tilde{n}_{1}\leq\frac{t}{\alpha_{1}\alpha_{2}}-1+\rmd{a_{-}t}{\alpha_{2}}/\alpha_{2},\end{cases}$
(24)

where $\rmd{x}{y}$ is the remainder of $x/y$. Similarly, the bounds for
negative $t^{\prime}$ are the following:

$t^{\prime}<0:\quad\begin{cases}\mathcal{B}_{t^{\prime},\boldsymbol{\alpha}}:&\frac{t^{\prime}}{\alpha_{1}\alpha_{2}}+\rmd{a_{-}t^{\prime}}{\alpha_{2}}/\alpha_{2}\leq\tilde{n}_{1}\leq-\rmd{-a_{+}t^{\prime}}{\alpha_{1}}/\alpha_{1}\\\
\mathring{\mathcal{B}}_{t^{\prime},\boldsymbol{\alpha}}:&\frac{t^{\prime}}{\alpha_{1}\alpha_{2}}+1-\rmd{-a_{-}t^{\prime}}{\alpha_{2}}/\alpha_{2}\leq\tilde{n}_{1}\leq-1+\rmd{a_{+}t^{\prime}}{\alpha_{1}}/\alpha_{1}.\end{cases}$
(25)

For $\alpha_{1},\alpha_{2}=1$ these reduce to bounds appearing in
Lundin:2021zeb  for the reduction from $S^{3}$ to $S^{2}$. In order to see how
cancellations work in the general case, note that

$\rmd{a_{+}t}{\alpha_{1}}\neq
0~{}\Longrightarrow~{}1-\rmd{-a_{+}t}{\alpha_{1}}/\alpha_{1}=\rmd{a_{+}t}{\alpha_{1}}/\alpha_{1},$
(26)

and similarly for $\rmd{a_{-}t}{\alpha_{2}}\neq 0$. Therefore, depending on
whether $\rmd{a_{+}t}{\alpha_{1}}=0$ and/or $\rmd{a_{-}t}{\alpha_{2}}=0$, only
the term for $\tilde{n}_{1}=0$ and/or $\tilde{n}_{1}=t/\alpha_{1}\alpha_{2}$
survives. Thus, employing the condition in (23), we find999We henceforth
combine $t\geq 0$ and $t^{\prime}\leq 0$ into a single $t\in\mathbb{Z}$.

$\begin{split}t\in\alpha_{1}\mathbb{Z}:\quad&\left(\left(1-\frac{\epsilon_{1}}{2}\right)\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right),\\\
t\in\alpha_{2}\mathbb{Z}:\quad&\left(\left(1+\frac{\epsilon_{1}}{2}\right)\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right).\end{split}$
(27)

We can now substitute in (20):

$Z^{\text{top}}_{S^{3}_{\boldsymbol{\alpha}}}=\prod_{\alpha>0}\prod_{t\in\alpha_{1}\mathbb{Z}}\left(\left(1-\frac{\epsilon_{1}}{2}\right)\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right)\prod_{t\in\alpha_{2}\mathbb{Z}}\left(\left(1+\frac{\epsilon_{1}}{2}\right)\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right).$
(28)

For $\alpha_{1},\alpha_{2}=1$ this expression reproduces the perturbative
partition function for $S^{3}$. For generic values of $\alpha_{1},\alpha_{2}$
the slicing along $t^{\text{top}}$ only allows functions whose charge $t$ is a
multiple of $\alpha_{1}$ or $\alpha_{2}$.

#### Exotic Theory.

We follow a similar procedure here, where
$t^{\text{ex}}=\alpha_{2}n_{1}-\alpha_{1}n_{2}$. Plots of the slices
$\mathcal{B}_{t,\boldsymbol{\alpha}},\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}$
of $\mathbb{Z}^{2}_{\geq 0}$ are shown in Figure 2. Again, only modes on the
$n_{1},n_{2}$-axis survive after cancellations.

$n_{1}$$n_{2}$$\vec{\X}^{\text{ex}}$$t=0$$t=3$$t=-6$

$n_{1}$$n_{2}$$\vec{\X}^{\text{ex}}$$t=0$$t=8$$t=-12$

Figure 2: We plot the cone for $S^{3}_{\boldsymbol{\alpha}}$ which is the
lattice spanned by integers $n_{1},n_{2}\geq 0$. For various values of
$\alpha_{1},\alpha_{2}$ we plot the regions
$\mathcal{B}_{t,\boldsymbol{\alpha}}$ and
$\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}$ as dashed lines. These two
regions appear, respectively, at the numerator and the denominator in the
perturbative partition function. Thus, after cancellations, only light blue
modes survive. These contribute at different values of $t^{\text{ex}}$
depending on the values of $\alpha_{1},\alpha_{2}$. From left to right: (i)
$\alpha_{1}=3,\alpha_{2}=2$, (ii) $\alpha_{1}=5,\alpha_{2}=3$.

Explicitly, the bounds on
$\tilde{n}_{1}=\frac{n_{1}}{\alpha_{1}}\in\mathbb{Z}_{\geq 0}$ are the
following:

$\displaystyle t\geq 0:$
$\displaystyle~{}\begin{cases}\mathcal{B}_{t,\boldsymbol{\alpha}}:&\tilde{n}_{1}\geq\rmd{a_{+}t}{\alpha_{1}}/\alpha_{2},\\\
\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}:&\tilde{n}_{1}\geq
1-\rmd{-a_{+}t}{\alpha_{1}}/\alpha_{2},\end{cases}$ (29) $\displaystyle
t^{\prime}<0:$
$\displaystyle~{}\begin{cases}\mathcal{B}_{t,\boldsymbol{\alpha}}:&\tilde{n}_{1}\geq-\frac{t}{\alpha_{1}\alpha_{2}}+\rmd{-a_{-}t}{\alpha_{1}}/\alpha_{2},\\\
\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}:&\tilde{n}_{1}\geq-\frac{t}{\alpha_{1}\alpha_{2}}+1-\rmd{a_{-}t}{\alpha_{2}}/\alpha_{2}.\end{cases}$
(30)

Similarly to the previous case, due to (26), a mode survives cancellations if
either $\rmd{a_{+}t}{\alpha_{1}}=0$ or $\rmd{a_{-}t}{\alpha_{2}}=0$. The same
bounds hold for $t^{\prime}=-t$. Combining the contributions from both $t$ and
$t^{\prime}$ in (21) and employing (23), we find

$Z^{\text{ex}}_{S^{3}_{\boldsymbol{\alpha}}}=\prod_{\alpha>0}\prod_{t\in\alpha_{1}\mathbb{Z}}\left(-\frac{\epsilon_{1}}{2}\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right)\prod_{t\in\alpha_{2}\mathbb{Z}}\left(+\frac{\epsilon_{1}}{2}\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right).$
(31)

Again, for $\alpha_{1},\alpha_{2}=1$ we reproduce the result for a smooth
$S^{3}$. The difference to the topological twist, besides the shift in $t$,
are the different relations between the squashing parameters and the
equivariance parameters (13).

###  4.2 Finite Quotient and Reduction

In section 3 we explained that the partition function on the branched lens
space $L^{3}_{\boldsymbol{\alpha}}(h,\pm 1)$ can be obtained simply by
imposing the condition (14) on $t$. Doing so for
$Z^{\text{top}}_{S^{3}_{\boldsymbol{\alpha}}}$ and
$Z^{\text{ex}}_{S^{3}_{\boldsymbol{\alpha}}}$ at fixed $\mathfrak{m}$
gives101010See Inglese:2023tyc  for related work.

$Z^{\text{top}}_{L^{3}_{\boldsymbol{\alpha}}(h,1)}=\prod_{\alpha>0}\prod_{t=\alpha(\mathfrak{m})\mmod
h\atop
t\in\alpha_{1}\mathbb{Z}}\left(\left(1-\frac{\epsilon_{1}}{2}\right)\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right)\prod_{t=\alpha(\mathfrak{m})\mmod
h\atop
t\in\alpha_{2}\mathbb{Z}}\left(\left(1+\frac{\epsilon_{1}}{2}\right)\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right),$
(32)
$Z^{\text{ex}}_{L^{3}_{\boldsymbol{\alpha}}(h,-1)}=\prod_{\alpha>0}\prod_{t=\alpha(\mathfrak{m})\mmod
h\atop
t\in\alpha_{1}\mathbb{Z}}\left(-\frac{\epsilon_{1}}{2}\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right)\prod_{t=\alpha(\mathfrak{m})\mmod
h\atop
t\in\alpha_{2}\mathbb{Z}}\left(+\frac{\epsilon_{1}}{2}\frac{t}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right).$
(33)

The partition function on the quotient space $B^{1}_{\boldsymbol{\alpha}}$ is
obtained by taking the limit $h\to\infty$ as explained in subsection 3.2. This
gives

$Z^{\text{top}}_{B^{1}_{\boldsymbol{\alpha}}}=\prod_{\alpha>0}\left(\left(1-\frac{\epsilon_{1}}{2}\right)\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right)\left(\left(1+\frac{\epsilon_{1}}{2}\right)\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right),$
(34)
$Z^{\text{ex}}_{B^{1}_{\boldsymbol{\alpha}}}=\prod_{\alpha>0}\left(-\frac{\epsilon_{1}}{2}\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right)\left(+\frac{\epsilon_{1}}{2}\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right).$
(35)

It is important to note that the first and second factor only contribute at
$\alpha(\mathfrak{m})\in\alpha_{1}\mathbb{Z}$, resp.
$\alpha(\mathfrak{m})\in\alpha_{2}\mathbb{Z}$. For $\alpha_{1},\alpha_{2}=1$
and $\epsilon_{1}=1$ one recovers111111To match with the expression on $S^{2}$
we need to perform a constant shift of the Coulomb branch parameter
$a\rightarrow a-\ii\mathfrak{m}$. Such shift has also been discussed in
Lundin:2021zeb ; Lundin:2023tzw . It will also appear in Appendix B in order
to match with the result for the spindle in Inglese:2023wky . the result for a
smooth $S^{2}$ Benini:2015noa ; Benini:2012ui  while for
$\alpha_{1}=\alpha_{2}$ we reproduce Mori:2015bro . Note that $\epsilon_{1}$
is different in (34) and (35) according to (13).

The classical action $S^{\text{cl}}_{2}$ on $B^{1}_{\boldsymbol{\alpha}}$ is
obtained from the one on $L^{3}_{\boldsymbol{\alpha}}(h,\pm 1)$ (16) by taking
the limit $h\to\infty$ such that the ratio $\ell:=k/h$ is kept fixed:

$S^{\text{cl}}_{2}=-\ii\pi\ell\rho_{3}\tr(a^{2}\pm\mathfrak{m}),$ (36)

where we have introduced the shorthand
$\rho_{3}:=\frac{\alpha_{1}\alpha_{2}}{\omega_{1}\omega_{2}(\epsilon_{1})}$.

At the end of this section we want to point out an interesting observation. In
Inglese:2023wky  the partition function on
$\mathbb{CP}^{1}_{(N_{1},N_{2})}\times S^{1}$ has been computed for two
different backgrounds, called twist and anti-twist (here
$\mathbb{CP}^{1}_{(N_{1},N_{2})}$ denotes $(N_{1},N_{2})$-weighted projective
space, also known as a spindle; see Appendix A). In Appendix B we compute its
small radius-limit with respect to the $S^{1}$ and find that the resulting
one-loop determinant on $\mathbb{CP}^{1}_{(N_{1},N_{2})}$ is in perfect
agreement with (34) and (35) for twist, resp. anti-twist under the
identification $\alpha_{i}=N_{i}$, $i=1,2$. We will have more to say about
this matching in section 7.

##  5 Partition Function on $S^{5}_{\boldsymbol{\alpha}}/S^{1}$

In this section we compute the partition function for an $\mathcal{N}=2$
vector multiplet, at each flux sector but at the trivial instanton sector, on
spaces $B^{2}_{\boldsymbol{\alpha}}$ with surplus conical angles. Also in this
case, we will show how topologically twisted and exotic theories are obtained
from the same theory of an $\mathcal{N}=1$ vector multiplet on the branched
cover of $S^{5}$. Unlike $r=2$, the perturbative partition function on
$S^{5}_{\boldsymbol{\alpha}}$ does not admit any cancellations, as both terms
in the triple sine function appear at the numerator. This both simplifies the
computations and gives a richer dependence on the parameters $\alpha_{i}$.

###  5.1 Finite Quotient and Reduction

Contrary to the case of $r=2$, here both products in the triple sine function
(12) are in the numerator and no cancellations occur. Thus, we can start with
the reduction procedure right away. Similar to $r=2$ we denote the one-loop
piece of the partition function by

$Z_{S^{5}_{\boldsymbol{\alpha}}}(a,\boldsymbol{\R}):=\det{}^{\prime}_{\text{adj}}S_{3}(\ii\alpha(a)|\boldsymbol{\R})$
(37)

and also introduce the special function Pestun:2007rz ; Festuccia:2018rew ;
Hama:2012bg ; Lundin:2023tzw

$\Upsilon^{\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(z|x,y\right)=\prod_{(n_{1},n_{2})\in\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(xn_{1}+yn_{2}+z\right)\prod_{(n_{1},n_{2})\in\mathring{\mathcal{B}}_{t,\boldsymbol{\alpha}}}\left(xn_{1}+yn_{2}+\bar{z}\right),$
(38)

in terms of which we can write the triple sine function as

$S_{3}(\ii\alpha(a)|\boldsymbol{\R})=\prod_{t}\Upsilon^{\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(\ii\alpha(a)\pm\omega_{3}\frac{t}{\lcm\boldsymbol{\alpha}}\bigg{|}\frac{\epsilon_{1}}{\alpha_{1}},\frac{\epsilon_{2}}{\alpha_{2}}\right).$
(39)

Here, $\omega_{3}$ is obtained from (10) as

$\begin{split}\text{top:}&\quad\omega_{3}=+1-\frac{\epsilon_{1}+\epsilon_{2}}{3},\\\
\text{ex:}&\quad\omega_{3}=-\frac{1}{3}+\frac{\epsilon_{1}+\epsilon_{2}}{3}.\end{split}$
(40)

In section 3 we explained that the partition function on
$L^{5}_{\boldsymbol{\alpha}}(h,\pm 1)$ can be obtained from (37) simply by
imposing the condition (14) on $t$. For fixed $\mathfrak{m}$ this gives

$Z^{\text{top}}_{L^{5}_{\boldsymbol{\alpha}}(h,1)}=\prod_{\alpha\in\Delta}\prod_{t=\alpha(\mathfrak{m})\mmod
h}\Upsilon^{\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(\ii\alpha(a)+\left(1-\frac{\epsilon_{1}+\epsilon_{2}}{3}\right)\frac{t}{\lcm\boldsymbol{\alpha}}\bigg{|}\frac{\epsilon_{1}}{\alpha_{1}},\frac{\epsilon_{2}}{\alpha_{2}}\right),$
(41)
$Z^{\text{ex}}_{L^{5}_{\boldsymbol{\alpha}}(h,-1)}=\prod_{\alpha\in\Delta}\prod_{t=\alpha(\mathfrak{m})\mmod
h}\Upsilon^{\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(\ii\alpha(a)\left(\frac{1}{3}-\frac{\epsilon_{1}+\epsilon_{2}}{3}\right)\frac{t}{\lcm\boldsymbol{\alpha}}\bigg{|}\frac{\epsilon_{1}}{\alpha_{1}},\frac{\epsilon_{2}}{\alpha_{2}}\right),$
(42)

where $\Delta$ denotes the root set of $G$. The partition function on the
quotient space $B^{1}_{\boldsymbol{\alpha}}$ is obtained by taking the limit
$h\to\infty$ as explained in subsection 3.2. This gives

$Z^{\text{top}}_{B^{2}_{\boldsymbol{\alpha}}}=\prod_{\alpha\in\Delta}\Upsilon^{\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}}\left(\ii\alpha(a)+\left(1-\frac{\epsilon_{1}+\epsilon_{2}}{3}\right)\frac{\alpha(\mathfrak{m})}{\lcm\boldsymbol{\alpha}}\bigg{|}\frac{\epsilon_{1}}{\alpha_{1}},\frac{\epsilon_{2}}{\alpha_{2}}\right),$
(43)
$Z^{\text{ex}}_{B^{2}_{\boldsymbol{\alpha}}}=\prod_{\alpha\in\Delta}\Upsilon^{\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}}\left(\ii\alpha(a)+\left(\frac{1}{3}-\frac{\epsilon_{1}+\epsilon_{2}}{3}\right)\frac{\alpha(\mathfrak{m})}{\lcm\boldsymbol{\alpha}}\bigg{|}\frac{\epsilon_{1}}{\alpha_{1}},\frac{\epsilon_{2}}{\alpha_{2}}\right).$
(44)

Note that the slices $\mathcal{B}_{t,\boldsymbol{\alpha}}$ in the exotic case
are non-compact and the products infinite. Thus, in order to make sense of
(44) they need to be suitably regulated, e.g. using zeta-function
regularisation.

The classical action $S^{\text{cl}}_{4}$ on $B^{2}_{\boldsymbol{\alpha}}$ is
obtained from the one on $L^{5}_{\boldsymbol{\alpha}}(h,\pm 1)$ by taking the
limit $h\to\infty$ such that the product
$g^{2}_{\text{YM,4d}}:=g^{2}_{\text{YM}}\cdot h$ is kept fixed:

$S^{\text{cl}}_{4}=\frac{(2\pi)^{3}}{g^{2}_{\text{YM,4d}}}\rho_{5}\tr a^{2},$
(45)

where we have introduced the shorthand
$\rho_{5}=\frac{\alpha_{1}\alpha_{2}\alpha_{3}}{\omega_{1}\omega_{2}\omega_{3}(\epsilon_{1},\epsilon_{2})}$.
Note that, contrary to the $r=2$ case, the classical action has no flux-
dependence121212As discussed in Lundin:2023tzw , such dependence can be
achieved by a constant shift of the Coulomb branch parameter proportional to
$\mathfrak{m}$..

###  5.2 Examples

The essential information needed to compute (43), resp. (44) for explicit
examples are the slices $\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$. In
this subsection we consider two different types of slicings corresponding to
choices where

(i) $\displaystyle\quad\gcd(\alpha_{i},\alpha_{j})=1$ (ii)
$\displaystyle\quad\gcd(\alpha_{1},\alpha_{2})=k_{1},~{}\gcd(\alpha_{2},\alpha_{3})=k_{3},~{}\gcd(\alpha_{1},\alpha_{3})=k_{2}$

and illustrate them for specific examples131313The constraint
$\gcd(\alpha_{1},\alpha_{2},\alpha_{3})=1$ implies $\gcd(k_{i},k_{j})=1$ for
all $i,j=1,2,3$.. Note that the slices will always be compact (resp. non-
compact) for the topological (resp. exotic) twist.

#### Choice (i).

For definiteness let us start by looking at
$\alpha_{1}=1,\alpha_{2}=2,\alpha_{3}=3$. Slices
$\mathcal{B}_{t,\boldsymbol{\alpha}}\subset\mathbb{Z}^{3}_{\geq 0}$ are
depicted in Figure 3 for topological and exotic theories.

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

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

Figure 3: Cone for $S^{5}_{1,2,3}$. Left side: sliced along
$\vec{\X}^{\text{top}}$ for $t=5$ (blue) and $t=10$ (red). At $t=0$ the slice
only contains the origin. Right side: sliced along $\vec{\X}^{\text{ex}}$ for
$t=6$ (blue), $t=0$ (green) and $t=-2$ (red). The slices are compact for
$\vec{\X}^{\text{top}}$ and non-compact for $\vec{\X}^{\text{ex}}$.

For $h\to\infty$, only a single slice survives and contributes at the flux
sector $t=\alpha(\mathfrak{m})$. We show the $(n_{1},n_{2})$-projection of
such slices for the topological and exotic theories for two different choices
of $\mathfrak{m}$ in Figure 4 and 5.

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

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

Figure 4: Slices $\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$ for
$B^{2}_{1,2,3}$ and $\alpha(\mathfrak{m})=12$. Left side: topologically
twisted theory. Right side: Exotic theory. Only points in light blue
contribute to the slices.

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

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

Figure 5: Slices $\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$ for
$B^{2}_{1,2,3}$ and $\alpha(\mathfrak{m})=15$. Left side: topologically
twisted theory. Right side: Exotic theory. Only points in light blue
contribute to the slices.

Looking at these slices we notice that $n_{2}$ only comes in
multiples141414Note that a similar behaviour occurs for $r=2$ in section 4. of
$\alpha_{2}=2$. For generic values of $\alpha_{1},\alpha_{2},\alpha_{3}$ one
can check that $n_{i}$ comes in multiples of $\alpha_{i}$. Let us introduce
$\tilde{n}_{i}:=\frac{n_{i}}{\alpha_{i}}\in\mathbb{Z}$ and rewrite the
partition functions as follows:

$Z^{\text{top}}_{B^{2}_{\boldsymbol{\alpha}}}=\prod_{\alpha\in\Delta}\Upsilon^{\widetilde{\mathcal{B}}_{\mathfrak{m},\boldsymbol{\alpha}}}\left(\ii\alpha(a)+\left(1-\frac{\epsilon_{1}+\epsilon_{2}}{3}\right)\frac{\alpha(\mathfrak{m})}{\lcm\boldsymbol{\alpha}}\bigg{|}\epsilon_{1},\epsilon_{2}\right),$
(46)
$Z^{\text{ex}}_{B^{2}_{\boldsymbol{\alpha}}}=\prod_{\alpha\in\Delta}\Upsilon^{\widetilde{\mathcal{B}}_{\mathfrak{m},\boldsymbol{\alpha}}}\left(\ii\alpha(a)+\left(\frac{1}{3}-\frac{\epsilon_{1}+\epsilon_{2}}{3}\right)\frac{\alpha(\mathfrak{m})}{\lcm\boldsymbol{\alpha}}\bigg{|}\epsilon_{1},\epsilon_{2}\right),$
(47)

where, the new slices are defined for
$\rmd{\alpha(\mathfrak{m})}{\alpha_{i}}=0,\,i=1,2$, as

$\begin{split}\text{top:}\quad&\widetilde{\mathcal{B}}_{\mathfrak{m},\boldsymbol{\alpha}}=\\{(\tilde{n}_{1},\tilde{n}_{2})\in\mathbb{Z}^{2}_{\geq
0}\;|\;\tilde{n}_{1}+\tilde{n}_{2}\leq\alpha(\mathfrak{m})/(\alpha_{1}\alpha_{2}\alpha_{3})\\},\\\
\text{ex:}\quad&\widetilde{\mathcal{B}}_{\mathfrak{m},\boldsymbol{\alpha}}=\\{(\tilde{n}_{1},\tilde{n}_{2})\in\mathbb{Z}^{2}_{\geq
0}\;|\;\tilde{n}_{1}+\tilde{n}_{2}\geq\alpha(\mathfrak{m})/(\alpha_{1}\alpha_{2}\alpha_{3})\\}.\end{split}$
(48)

The slices $\widetilde{\mathcal{B}}_{\mathfrak{m},\boldsymbol{\alpha}}$ are
shown in Figure 6 for $\alpha(\mathfrak{m})=12$. Note that whenever no light
blue points appear in the plots, then all points contribute to the slice.

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

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

Figure 6: Slices $\widetilde{\mathcal{B}}_{\mathfrak{m},\boldsymbol{\alpha}}$
for $B^{2}_{1,2,3}$ and $\alpha(\mathfrak{m})=12$. Left side: topologically
twisted theory. Right side: Exotic theory. All points contribute to the slice.

For generic values of $\alpha(\mathfrak{m})$ the slices do not necessarily
start from the origin and then certain shifts depending on
$\rmd{\alpha(\mathfrak{m})}{\alpha_{1}}$ and
$\rmd{\alpha(\mathfrak{m})}{\alpha_{2}}$ need to be included in the definition
of $\widetilde{\mathcal{B}}_{\mathfrak{m},\boldsymbol{\alpha}}$.

In the case $\alpha_{1},\alpha_{2},\alpha_{3}=1$ the result for both
topological and exotic theories agree with those on $\mathbb{CP}^{2}$ in
Lundin:2021zeb . More interestingly,
$Z^{\text{top}}_{B^{2}_{\boldsymbol{\alpha}}}$ is in agreement with the
equivariant index on weighted projective space
$\mathbb{CP}^{2}_{(N_{1},N_{2},N_{3})}$ in Martelli:2023oqk  under the
identification

$N_{1}=\alpha_{2}\alpha_{3},\qquad N_{2}=\alpha_{1}\alpha_{3},\qquad
N_{3}=\alpha_{1}\alpha_{2}.$ (49)

Specifically, for any value of $\alpha(\mathfrak{m})$ we can match151515Note
that in Martelli:2023oqk  the flux
$\alpha(\mathfrak{m})=T(\Lambda_{1},\Lambda_{2},\Lambda_{3})$ is expressed in
terms of equivariant fluxes $\Lambda_{1},\Lambda_{2},\Lambda_{3}$. both, the
size and the shape of
$\widetilde{\mathcal{B}}_{\mathfrak{m},\boldsymbol{\alpha}}$ with the slices
of Martelli:2023oqk . However, in Martelli:2023oqk  topological sectors are
labelled by equivariant fluxes rather than the “physical” flux in our results
which obscures a direct comparison of the final expressions.

#### Choice (ii).

Let us for the moment set $k_{1}=1$ and return to the general case $k\neq 1$
momentarily. Slices
$\mathcal{B}_{t,\boldsymbol{\alpha}}\subset\mathbb{Z}^{3}_{\geq 0}$ for
$k_{2}=3,k_{3}=5$ and different values for $t$ are shown in Figure 7.

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

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

Figure 7: Cone for $S^{5}_{3,5,15}$. Left side: sliced along
$\vec{\X}^{\text{top}}$ for $t=3$ (blue) and $t=5$ (red). At $t=0$ the slice
only contains the origin. Right side: sliced along $\vec{\X}^{\text{ex}}$ for
$t=5$ (blue), $t=0$ (green) and $t=-2$ (red). The slices are compact for
$\vec{\X}^{\text{top}}$ and non-compact for $\vec{\X}^{\text{ex}}$.

For $h\to\infty$, only a single slice survives and contributes at the flux
sector $t=\alpha(\mathfrak{m})$. We show the $(n_{1},n_{2})$-projection of
these slices in 8.

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

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

Figure 8: Slices for $B^{2}_{3,5,15}$. Left side:
$\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$ of the topologically twisted
theory for $\alpha(\mathfrak{m})=10$. For $\alpha(\mathfrak{m})=0$ only the
origin contributes. Right side:
$\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$ of the exotic theory for
$\alpha(\mathfrak{m})=10$. For $\alpha(\mathfrak{m})\leq 0$ the entire
quadrant contributes. All points contribute to the slice.

We can now understand the dependence on $k_{1}$. Let us recall that the slices
$\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$ are defined to be the pairs
$(n_{1},n_{2})\in\mathbb{Z}^{2}_{\geq 0}$ satisfying:

$k_{3}n_{1}+k_{2}n_{2}=t^{\text{top,ex}}\mp k_{1}n_{3}.$ (50)

If $k_{1}=1$ this condition translates into:

$\begin{split}\text{top}:\;&\quad k_{3}n_{1}+k_{2}n_{2}\leq t,\\\
\text{ex}:\;&\quad k_{3}n_{1}+k_{2}n_{2}\geq t.\end{split}$ (51)

Instead, if $k_{1}\neq 1$, only pairs $(n_{1},n_{2})$, for which there exists
$n_{3}\in\mathbb{Z}_{\geq 0}$ such that (50) holds, contribute. We present
slices for $B^{2}_{2,2,1}$ in Figure 9.

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

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

Figure 9: Slices for $B^{2}_{2,2,1}$. Left side:
$\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$ of the topologically twisted
theory for $\alpha(\mathfrak{m})=3$. Right side:
$\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$ of the exotic theory for
$\alpha(\mathfrak{m})=3$. Only points in light blue contribute to the slices.
The shape of the slice is as for $\mathbb{CP}^{2}$ Lundin:2021zeb  but, here,
only certain pairs $(n_{1},n_{2})$ contribute.

Unlike the previous case, one can check that $n_{1}$ (resp. $n_{2}$) jumps by
$k_{1}$ at a fixed value of $n_{2}$ (resp. $n_{1}$) and thus the shape of the
slice is not affected. More precisely, only pairs belonging to the sublattice
of $\mathbb{Z}^{2}_{\geq 0}$ defined by

$\\{(n_{1},n_{2})\in\mathbb{Z}^{2}_{\geq 0}\;|\;\exists c\in\mathbb{Z}_{\geq
0}:n_{1}+n_{2}=ck_{1}+\rmd{\alpha(\mathfrak{m})}{k_{1}}\\}.$ (52)

and satisfying (51) contribute to the slice
$\mathcal{B}_{\mathfrak{m},\boldsymbol{\alpha}}$. This explains, for example,
why the origin does not contribute on the left-hand side in Figure 9.

We have previously observed that both, $Z_{B^{1}_{\boldsymbol{\alpha}}}$ and
$Z_{B^{2}_{\boldsymbol{\alpha}}}$ agree with the respective one-loop
determinant on $\mathbb{CP}^{1}_{(N_{1},N_{2})}$ and
$\mathbb{CP}^{2}_{(N_{1},N_{2},N_{3})}$ for each flux sector if we choose the
weights $\\{N_{i}\\}$ such that the resulting degrees of singularity match the
branch indices on $B^{1}_{\boldsymbol{\alpha}}$, resp.
$B^{2}_{\boldsymbol{\alpha}}$. Therefore, we propose that also for choice (ii)
the one-loop determinant matches the one on an orbifold whose locus and degree
of singularities agrees with the branch locus and -indices of
$B^{2}_{\boldsymbol{\alpha}}$. Rather than weighted projective space, for
choice (ii) the corresponding orbifold is an orbifold projective space Abreu01
.

##  6 Factorised Expressions

In the previous section we have computed the one-loop determinant for fixed
$\mathfrak{m}$ and the classical part of the partition function on
$B^{2}_{\boldsymbol{\alpha}}$. In this section we use equivariance under the
respective torus-action to factorise the one-loop determinant into
contributions from each torus fixed point (see e.g. Lundin:2023tzw  for more
detail in the manifold case). The factorisation data can then be used to
conjecture the instanton part of the partition function.

It turns out that the local equivariance parameters obtained from
factorisation for choice (i) match those on $\mathbb{CP}^{2}$ Lundin:2021zeb
with the only difference being an overall rescaling by
$1/(\alpha_{1}\alpha_{2}\alpha_{3})$ in front of the shift proportional to
$\alpha(\mathfrak{m})$ and some shifts depending on
$\rmd{\alpha(\mathfrak{m})}{\alpha_{1}}$ and
$\rmd{\alpha(\mathfrak{m})}{\alpha_{2}}$. Thus, we focus on the factorisation
of choice (ii) here. Let us start by expressing the five-dimensional
perturbative partition function (12) as a product of contributions from
neighbourhoods $\hat{\mathbb{C}}^{2}_{\epsilon_{1},\epsilon_{2}}\times S^{1}$
around the fixed fibres (i.e. the vertices of the toric diagram;
$\hat{\mathbb{C}}^{2}$ denotes the respective branched cover of
$\mathbb{C}^{2}$). As for an ordinary $S^{5}$, the factoised expression for
the perturbative partition function on $S^{5}_{\boldsymbol{\alpha}}$ is
determined by a choice of imaginary part of the equivariance
parameters161616Each individual contribution is affected by the choice of
imaginary part. However, the partition function is independent of this choice.
$\epsilon_{1},\epsilon_{2}$, and thus of the vector $\X$. Then, we have the
following factorisation property:

$\prod_{\alpha\in\Delta}\prod_{t}\Upsilon^{\mathcal{B}_{t,\boldsymbol{\alpha}}}\left(\ii\alpha(a)\pm\omega_{3}t\bigg{|}\frac{\epsilon_{1}}{\alpha_{1}},\frac{\epsilon_{2}}{\alpha_{2}}\right)=\prod_{i=1}^{3}\prod_{\alpha\in\Delta}\prod_{t}\Upsilon_{i}(\ii\alpha(a)+\beta_{i}^{-1}t|\epsilon^{i}_{1},\epsilon^{i}_{2})^{s_{i}},$
(53)

where $i$ labels the three fixed fibres and the local equivariance parameters
for the $T^{3}$-action are $\epsilon_{1}^{i},\epsilon_{2}^{i},\beta_{i}^{-1}$.
These, together with the parameter $s_{i}$ can be read off from Table 1.

$i$ | 1 | 2 | 3
---|---|---|---
$\epsilon_{1}^{\text{top},i}$ | $\frac{\epsilon_{1}}{\alpha_{1}}$ | $-\frac{\epsilon_{1}}{\alpha_{1}}$ | $\frac{k_{3}\epsilon_{1}}{\alpha_{1}}-\frac{k_{2}\epsilon_{2}}{\alpha_{2}}$
$\epsilon_{2}^{\text{top},i}$ | $\frac{\epsilon_{2}}{\alpha_{2}}$ | $\frac{k_{2}\epsilon_{2}}{\alpha_{2}}-\frac{k_{3}\epsilon_{1}}{\alpha_{1}}$ | $-\frac{\epsilon_{2}}{\alpha_{2}}$
$\tilde{\beta}_{i}^{-1,\text{top}}$ | $0$ | $\frac{\epsilon_{1}}{\alpha_{1}k_{3}}$ | $\frac{\epsilon_{2}}{\alpha_{2}k_{2}}$
$s_{i}$ | 1 | 1 | -1

$i$ | 1 | 2 | 3
---|---|---|---
$\epsilon_{1}^{\text{ex},i}$ | $\frac{\epsilon_{1}}{\alpha_{1}}$ | $\frac{\epsilon_{1}}{\alpha_{1}}$ | $\frac{k_{3}\epsilon_{1}}{\alpha_{1}}-\frac{k_{2}\epsilon_{2}}{\alpha_{2}}$
$\epsilon_{2}^{\text{ex},i}$ | $\frac{\epsilon_{2}}{\alpha_{2}}$ | $\frac{k_{2}\epsilon_{2}}{\alpha_{2}}-\frac{k_{3}\epsilon_{1}}{\alpha_{1}}$ | $\frac{\epsilon_{2}}{\alpha_{2}}$
$\tilde{\beta}_{i}^{-1,\text{ex}}$ | $0$ | $\frac{\epsilon_{1}}{\alpha_{1}k_{3}}$ | $\frac{\epsilon_{2}}{\alpha_{2}k_{2}}$
$s_{i}$ | 1 | -1 | 1

Table 1: Local equivariance parameters for the topological (left) and exotic
(right) theories. For ease of notation we relabeled
$\tilde{\beta}_{i}^{-1,\text{top}}\equiv\beta_{i}^{-1,\text{top}}-(k_{1}k_{2}k_{3})^{-1}(1-1/3(\epsilon_{1}+\epsilon_{2}))$
in the topological and
$\tilde{\beta}_{i}^{-1,\text{ex}}\equiv\beta_{i}^{-1,\text{ex}}-(3k_{1}k_{2}k_{3})^{-1}(1-(\epsilon_{1}+\epsilon_{2}))$
in the exotic case.

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

$\Upsilon_{i}(z|\epsilon^{i}_{1},\epsilon^{i}_{2})=\prod_{(j,k)\in\mathcal{D}_{i}}(\epsilon^{i}_{1}j+\epsilon^{i}_{2}k+z)\prod_{(j,k)\in\mathcal{D}_{i}^{\prime}}(\epsilon^{i}_{1}j+\epsilon^{i}_{2}k+\bar{z}).$
(54)

The lattices $\mathcal{D}_{i},\mathcal{D}^{\prime}_{i}\subset\mathbb{Z}^{2}$
depend on the imaginary part of $\epsilon_{1},\epsilon_{2}$ and are closely
related to those appearing in Lundin:2023tzw  for toric Sasakian manifolds.
However, there are two main differences:

  * •

When $t$ is not a multiple of either $k_{3}$ or $k_{2}$, the shifts,
respectively, by $\beta_{2}^{-1},\beta_{3}^{-1}$ do not lead to points on the
lattice. This happens because the distance from the shift by $\beta_{1}^{-1}$
is not integer.

  * •

When $k_{1}\neq 1$ there is an extra complications as not all points in the
regions $\mathcal{D}_{i},\mathcal{D}^{\prime}_{i}$ have to be considered, as
it is obvious from (52).

Thus, we define the regions $\mathcal{D}_{i},\mathcal{D}^{\prime}_{i}$ as in
eq. (6.3) Lundin:2023tzw  but consider all values of171717As in Lundin:2023tzw
, depending on the regularisation, $i,j$ can be greater than 0 or 1.
$(i,j)\in\mathbb{R}^{2}_{\geq 0}$ such that when they are shifted by
$\beta_{2}^{-1},\beta_{3}^{-1}$ they belong to the sublattice defined in (52).

Once the factorisation of the perturbative partition function on
$S^{5}_{\boldsymbol{\alpha}}$ is determined, the one-loop contributions at
each topological sector for $L^{5}_{\boldsymbol{\alpha}}(h,\pm 1)$ and
$B^{2}_{\boldsymbol{\alpha}}$ follow simply by restricting $t$:

$Z_{L^{5}_{\boldsymbol{\alpha}}(h,\pm
1)}=\prod_{\alpha\in\Delta}\prod_{i=1}^{3}\prod_{t=\alpha(\mathfrak{m})\mmod
h}\Upsilon_{i}(\ii\alpha(a)+\beta_{i}^{-1}t|\epsilon^{i}_{1},\epsilon^{i}_{2})^{s_{i}}$
(55)
$Z_{B^{2}_{\boldsymbol{\alpha}}}=\prod_{\alpha\in\Delta}\prod_{i=1}^{3}\Upsilon_{i}(\ii\alpha(a)+\beta_{i}^{-1}\alpha(\mathfrak{m})|\epsilon^{i}_{1},\epsilon^{i}_{2})^{s_{i}}.$
(56)

Note that, as for quasi-toric four-manifolds Lundin:2023tzw , the flux-
dependence enters via a shift of the Coulomb branch parameter $a$ which can be
read out from Table 1.

Finally, given that one-loop determinants agree between the branched covers
$B^{1,2}_{\boldsymbol{\alpha}}$ and weighted projective space
$\mathbb{CP}^{1,2}_{\boldsymbol{N}}$, the factorisation also agrees.

##  7 Discussion

So far we have computed the one-loop determinants at all flux sectors and
classical action for the 2d $\mathcal{N}=(2,2)$ vector multiplet on a branched
cover $B^{1}_{\boldsymbol{\alpha}}$ of $S^{2}$, (34)-(36) and the 4d
$\mathcal{N}=2$ vector multiplet on a branched cover
$B^{2}_{\boldsymbol{\alpha}}$ of $\mathbb{CP}^{2}$, (43)-(45). We obtain these
results from dimensional reduction of the 3d $\mathcal{N}=2$ and 5d
$\mathcal{N}=1$ vector multiplet on branched spheres $S^{3}$, resp. $S^{5}$.
These are principal $S^{1}$-fibrations over $B^{1,2}_{\boldsymbol{\alpha}}$
and the reduction is implemented, according to Lundin:2023tzw , by taking
finite quotients along the $S^{1}$-fibre first. This introduces extra
topological sectors which, in the limit where the order of the quotient group
becomes infinite, label different flux sectors in the 2d (resp. 4d) theory.
Depending on the choice of fibration we obtain a topologically twisted or
exotic theory on the base space.

###  7.1 Weighted Projective Space $\mathbb{CP}^{r-1}_{\boldsymbol{N}}$

Let us now draw the connection to weighted projective spaces. First,
Nishioka:2016guu  showed that for the case of only a single non-trivial branch
index $\alpha_{1}=q$, the one-loop determinant $Z(S^{d}_{q})$ on the branched
cover of $S^{d}$ can actually be factorised into $q$ separate contributions.
Each of these are equivalent to the theory on a single sheet $S^{d}$ in the
presence of a gauge defect supported on the branch locus, whose effect is to
implement the monodromy $\e{2\pi\ii n/q}$ ($n=0,\dots,q-1$) of the fields
associated to the covering. On the other hand, in 2d Hosomichi:2015pia  showed
that the theory on $S^{2}$ with codimension-two gauge defect is equivalent to
one without the defect, but orbifolded; this implements the same twisted
periodicity condition on the fields. We summarise this schematically as

$Z(S^{2}_{q})\simeq\prod_{n=0}^{q-1}Z(S^{2}_{\mathrm{defect}})\simeq\prod_{n=0}^{q-1}Z(S^{2}/\mathbb{Z}_{q}).$
(57)

Note that the product on the right-hand side is over all twisted periodicity
conditions and thus yields the one-loop determinant for the theory on the
orbifold $S^{2}/\mathbb{Z}_{q}$.

In section 4 we have observed that both, the one-loop determinants for
topological and exotic theories on $B^{1}_{(\alpha_{1},\alpha_{2})}$ agree
with those on weighted projective space
$\mathbb{CP}^{1}_{(\alpha_{1},\alpha_{2})}$ for every flux sector. This proves
(57) also for the case $\alpha_{1}\neq\alpha_{2}$. Note that the factorisation
can be traced back to the following resummation property Inglese:2023wky  of
the index for a corresponding twisted Dolbeault complex on
$\mathbb{CP}^{1}_{\boldsymbol{\alpha}}$ (see Appendix B for notation):

$\frac{1}{\alpha_{1}}\sum_{j=0}^{\alpha_{1}-1}\frac{\omega_{+}^{-jp_{+}}q_{+}^{-p_{+}}}{1-\omega_{+}^{j\sigma}q_{+}^{\sigma}}+\frac{1}{\alpha_{2}}\sum_{j=0}^{\alpha_{2}-1}\frac{\omega_{-}^{-jp_{-}}q_{-}^{-p_{-}}}{1-\omega_{-}^{j\sigma}q_{-}^{\sigma}}~{}=~{}\frac{q^{-\sigma\lfloor\sigma
p_{+}/\alpha_{1}\rfloor}}{1-q^{\sigma}}+\frac{q^{\lfloor-
p_{-}/\alpha_{2}\rfloor}}{1-q^{-1}}.$ (58)

The one-loop determinant on $\mathbb{CP}^{1}_{\boldsymbol{\alpha}}$ is
extracted from this index. For $\alpha_{1}=q,\alpha_{2}=1$ the left-hand side
precisely yields the product in (57). A similar resummation occurs for the
complex of the topological twist on $\mathbb{CP}^{2}_{\boldsymbol{N}}$
Martelli:2023oqk  and, indeed, we observed in section 5 that for choice (i)
the one-loop determinant for the topological twist on
$B^{2}_{(\alpha_{1},\alpha_{2},\alpha_{3})}$ agrees with the one on
$\mathbb{CP}^{2}_{(\alpha_{2}\alpha_{3},\alpha_{1}\alpha_{3},\alpha_{1}\alpha_{2})}$.
We therefore make the following For a fixed flux sector $\mathfrak{m}$, the
one-loop determinant for the 4d $\mathcal{N}=2$ vector multiplet for exotic
theories agrees on the two spaces $B^{2}_{\boldsymbol{\alpha}}$ and
$\mathbb{CP}^{2}_{\boldsymbol{N}}$,

$Z^{\text{ex}}_{B^{2}_{\boldsymbol{\alpha}}}(a,\mathfrak{m};\epsilon_{1},\epsilon_{2})=Z^{\text{ex}}_{\mathbb{CP}^{2}_{\boldsymbol{N}}}(a,\mathfrak{m};\epsilon_{1},\epsilon_{2}),$

for $\gcd(\alpha_{i},\alpha_{j})=1$, $i=1,2,3$ and the weight vector
$\boldsymbol{N}$ as in (49). Note that we require $\boldsymbol{N}$ in terms of
the branch indices $\boldsymbol{\alpha}$ to be such that the singular locus on
$\mathbb{CP}^{2}_{\boldsymbol{N}}$ matches precisely the branch locus on
$B^{2}_{\boldsymbol{\alpha}}$ (e.g. viewed as embedded in $S^{5}$). The degree
of singularity for each component of the former matches the branch index of
the respective component on the latter. Hence, the spaces simply differ in the
type of conical singularities: one has deficit angles, the other has surplus
angles.

Finally, we expect that the resummation property of the index (and thus the
factorisation of one-loop determinants) might hold more generally for theories
on branched covers and orbifolds with matching branch/singular locus. In
particular, we expect the conjecture to extend to topological and exotic
theories on $B^{2}_{\boldsymbol{\alpha}}$ for choice (ii) and orbifold
projective spaces.

#### Full Partition Function on $\mathbb{CP}^{2}_{\boldsymbol{N}}$.

Let us now comment on instantons in the theory on
$\mathbb{CP}^{2}_{\boldsymbol{N}}$. By a standard argument Nekrasov:2003vi ;
Festuccia:2018rew , the instanton part of the partition function is given by a
product of Nekrasov partition functions on
$\mathbb{C}^{2}_{\epsilon_{1}^{i},\epsilon_{2}^{i}}/\mathbb{Z}_{N_{i}}$
Fucito:2004ry ; Bonelli:2011jx ; Bonelli:2011kv ; Bonelli:2012ny ;
Bruzzo:2013nba ; Bruzzo:2013daa  around the fixed points $i=1,2,3$. For each
fixed point contribution, the Coulomb branch parameter is shifted by a
corresponding flux contribution obtained from the factorisation. The instanton
partition function is then obtained as

$Z_{\mathbb{CP}^{2}_{\boldsymbol{N}}}^{\text{inst}}=\prod_{i=1}^{\ell}Z^{Nek}_{\mathbb{C}^{2}_{\epsilon_{1}^{i},\epsilon_{2}^{i}}/\mathbb{Z}_{N_{i}}}(\ii
a+\beta_{i}^{-1}\mathfrak{m}|\epsilon^{i}_{1},\epsilon^{i}_{2},q)\prod_{i=\ell+1}^{3}Z^{Nek}_{\mathbb{C}^{2}_{\epsilon_{1}^{i},\epsilon_{2}^{i}}/\mathbb{Z}_{N_{i}}}(\ii
a+\beta_{i}^{-1}\mathfrak{m}|\epsilon^{i}_{1},\epsilon^{i}_{2},\bar{q}).$ (59)

The shifts in $a$ and the local equivariance parameters are discussed in
section 6. For the topological twist we have $\ell=3$ (i.e. instantons at all
three fixed points) while for the exotic theory we have $\ell=2$ (i.e. anti-
instantons at one fixed point). Consequently, the full partition function on
weighted projective space $\mathbb{CP}^{2}_{\boldsymbol{N}}$ reads

$\mathcal{Z}_{\mathbb{CP}^{2}_{\boldsymbol{\alpha}}}=\sum_{\mathfrak{m}}\int_{\mathfrak{h}}\dd
a~{}e^{-S^{\text{cl}}_{4}/(\alpha_{1}\alpha_{2}\alpha_{3})^{2}}\cdot
Z_{B^{2}_{\boldsymbol{\alpha}}}\cdot
Z_{\mathbb{CP}^{2}_{\boldsymbol{\alpha}}}^{\text{inst}},$ (60)

with $S^{\text{cl}}_{4}$ in (45) and $Z_{B^{2}_{\boldsymbol{\alpha}}}$ in
(43), resp. (44). We believe a similar expression holds for orbifold
projective spaces employing the the shifts in $a$ and the equivariance
parameters in Table 1.

###  7.2 Future Directions

#### Branched Coverings of Toric Sasakian Manifolds.

An natural extension of this work is to consider the $\mathcal{N}=1$ vector
multiplet on branched covers of other five-dimensional toric Sasakian
manifolds admitting a globally free $S^{1}$-action. From dimensional reduction
we would obtain four-dimensional branched covers of quasi-toric manifolds
Festuccia:2019akm ; Lundin:2023tzw . Also in this case, we expect the one-loop
determinant for a fixed flux sector to be equal to its orbifold pendant. The
simplest examples to study are branched coverings of $Y^{p,q}$ Martelli:2004wu
which are $S^{1}$-principal bundles over branched covers
$B^{1}_{\boldsymbol{\alpha}}\times B^{1}_{\boldsymbol{\alpha}^{\prime}}$.
According to our argument, it should be possible to reproduce the one-loop
determinant including flux contributions on
$\mathbb{CP}^{1}_{\boldsymbol{\alpha}}\times\mathbb{CP}^{1}_{\boldsymbol{\alpha}^{\prime}}$.

#### Locally Free $S^{3}$-Action.

Weighted projective spaces are obtained quotienting $S^{2r-1}$ along a locally
free $S^{1}$-action. Hence, orbifold singularities on the base space arise
from a $\mathbb{Z}_{\alpha_{i}}$-subgroup of $U(1)$ acting trivially.
Consequently, the singularities are of the form
$\mathbb{C}^{r-1}/\mathbb{Z}_{\alpha_{i}}$. If we were instead to consider
quotients by locally free $S^{3}$-actions, elements acting trivially can be
given by finite subgroups of $SU(2)$. The simplest choice to investigate this
is $S^{7}$. Here, the quotient along a locally free $S^{3}$-action yields
weighted quaternionic projective space
$\mathbb{HP}^{1}_{\boldsymbol{\alpha}}$, which is an $S^{4}$ with orbifold
singularities at the two poles of the form $\mathbb{C}^{2}/\Gamma$ for a
finite subgroup $\Gamma\subset SU(2)$. One could envision to obtain instantons
in 4d from flat $SU(2)$-connections on finite quotients of $S^{7}$.

#### Acknowledgments

We are grateful to Guido Festuccia, Dario Martelli, Leonardo Santilli, Itamar
Yaakov and Alberto Zaffaroni for stimulating discussions and correspondence on
the subject. We thank Guido Festuccia for comments on the manuscript. RM
acknowledges support from the Centre for Interdisciplinary Mathematics at
Uppsala University. LR acknowledges support from the Shuimu Tsinghua Scholar
Program.

##  Appendix A Branched Coverings and Weighted Projective Spaces

In this appendix we briefly recall the definition of branched coverings and
discuss the spaces $S^{3}_{\boldsymbol{\alpha}}$ and
$S^{5}_{\boldsymbol{\alpha}}$ and their free $S^{1}$-quotient. We also give
the basic definition and properties of weighted projective space and point out
similarities between the two spaces in support of subsection 7.1.

###  A.1 Branched Coverings of Spheres

A map $p:\tilde{M}\rightarrow M$ into a $d$-dimensional manifold $M$ is called
an $n$-fold branched covering if there exists a codimension-2 subcomplex
$L\subset M$, called the branch locus, such that $p^{-1}(L)\subset\tilde{M}$
is a codimension-2 subcomplex and $p|_{\tilde{M}-p^{-1}(L)}$ is an $n$-fold
covering. A generic point $\tilde{y}\in p^{-1}(L)$ has an open neighbourhood
homeomorphic to $D^{2}\times I^{d-2}$ on which $p$ takes the form $D^{2}\times
I^{d-2}\rightarrow D^{2}\times I^{d-2},(z,x)\mapsto(z^{\alpha},x)$. The
positive integer $\alpha$ is called the branch index of $\tilde{y}$ 181818The
sum of the branch indices for all points in the preimage of a point in $y\in
L$ is the degree of the cover, i.e. $\sum_{\tilde{y}\in
p^{-1}(y)}\alpha(\tilde{y})=n$. This is generally true for any point $y\in M$,
but only for $y\in L$ is the resulting partition of $n$ non-trivial..

For our case, we take $M=S^{2r-1}$ and $L$ to be a codimension 2 subspace

$L\subset\bigcup_{i=1}^{r}\\{(z_{1},\dots,z_{r})\in S^{2r-1}|z_{i}=0\\}.$ (61)

From the toric viewpoint which we take frequently, $S^{2r-1}$ can be seen as a
Lagrangian $T^{r}$-fibration over the moment polytope which is an
$(r-1)$-simplex. Then $L$ describes a subset of its facets (which are
themselves $S^{2r-3}$). At intersections of two facets the corresponding
spheres are glued along an $S^{2r-5}$ and so on. For $r>3$ this geometry
becomes fairly complicated and since we focus on computing partition functions
for $r=2,3$ later on, we henceforth restrict to these low-dimensional cases.
However, everything that follows can be extended beyond $r=3$.

#### Three Dimensions.

Consider a branched covering over the round $S^{3}$. We parametrise a point in
$S^{3}$ (which we mostly view as a Hopf fibration over $S^{2}$ in the
following) by

$z_{1}=\sin\phi\e{\ii\theta_{1}},\qquad z_{2}=\cos\phi\e{\ii\theta_{2}}$ (62)

with angles $\phi\in[0,\pi/2]$ and $\Delta\theta_{i}=2\pi$. In order to move
to a branched covering $S^{3}_{\boldsymbol{\alpha}}$, we can simply extend the
periodicity of the azimuthal angles to
$\Delta\tilde{\theta}_{i}=2\pi\alpha_{i}$, which we now write as
$\tilde{\theta}_{i}$, to distinguish from the canonical periodicity of
$\theta_{i}$. The branch locus is comprised of the two $S^{1}$-fibres where
$\phi=0$ and $\phi=\pi/2$ which we denote by $C_{1},C_{2}$, respectively (note
that the two fibres form a Hopf link). At $C_{1}$ (resp. $C_{2}$),
$S^{3}_{\boldsymbol{\alpha}}$ is an $\alpha_{2}$-fold (resp.
$\alpha_{1}$-fold) covering of $S^{3}$. Then $C_{1}\subset L$ has a closed
neighbourhood homeomorphic to $D^{2}\times S^{1}$ on which the covering takes
the form
$(z,e^{\ii\theta_{2}})\mapsto(z^{\alpha_{1}},e^{\ii\alpha_{2}\theta_{2}})$,
i.e. points on $p^{-1}(C_{1})$ have branch index $\alpha_{1}$; similarly,
points on $p^{-1}(C_{2})$ have branch index $\alpha_{2}$ (which we record in
the subscript $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2})$). Note that in
order for functions to be smooth on the preimage of such a neighbourhood it
has to be smooth already on a single sheet, i.e. in $\theta_{i}$.

For the round metric on $S^{3}$, the corresponding metric on
$S^{3}_{\boldsymbol{\alpha}}$ is given (in terms of the $2\pi$-periodic
angles) by

$\dd
s^{2}=\dd\phi^{2}+\alpha_{1}^{2}\sin^{2}\\!\phi\,\dd\theta_{1}^{2}+\alpha_{2}^{2}\cos^{2}\\!\phi\,\dd\theta_{2}^{2}.$
(63)

Also from this metric we see that close to the fibres at $\phi=0$ (resp.
$\phi=\pi/2$) there is a conical singularity with surplus angle
$2\pi\alpha_{1}$ (resp. $2\pi\alpha_{2}$). Clearly, the $T^{2}$-action on
$S^{3}$ lifts to $S^{3}_{\boldsymbol{\alpha}}$ and it is straightforward to
identify free $S^{1}$-actions that preserve the branching structure transverse
to $L$. They are generated (up to $\SL(2,\mathbb{Z})$-transformations) by the
vector field

$\X=\partial_{\tilde{\theta}_{1}}\pm\partial_{\tilde{\theta}_{2}}=\frac{1}{\alpha_{1}}\partial_{\theta_{1}}\pm\frac{1}{\alpha_{2}}\partial_{\theta_{2}}.$
(64)

#### Five Dimensions.

Let us start with parametrising the round $S^{5}$ by

$z_{1}=\sin\varphi\cos\phi\e{\ii\theta_{1}},\qquad
z_{2}=\sin\varphi\sin\phi\e{\ii\theta_{2}},\qquad
z_{3}=\cos\varphi\e{\ii\theta_{3}},$ (65)

with angles $\varphi,\phi\in[0,\pi/2]$ and $\Delta\theta_{i}=2\pi$ for
$i=1,2,3$. Similarly to the three-dimensional case, in order to move to a
branched cover $S^{5}_{\boldsymbol{\alpha}}$ we simply extend the periodicity
of the azimuthal angles to $\Delta\tilde{\theta}_{i}=2\pi\alpha_{i}$. The
branch locus is comprised of three $S^{3}$ where $\varphi=\pi/2$, $\phi=0$ and
$\phi=\pi/2$ which we denote by $C_{1},C_{2}$ and $C_{3}$, respectively and
view as the facets of the moment polytope of $S^{5}$ (viewed as a Sasakian
space). At $\mathring{C}_{1}$ (resp. $\mathring{C}_{2}$, $\mathring{C}_{3}$),
$S^{5}_{\boldsymbol{\alpha}}$ is an $\alpha_{2}\alpha_{3}$-fold (resp.
$\alpha_{1}\alpha_{3}$-fold, $\alpha_{1}\alpha_{2}$-fold) cover of $S^{5}$. On
a tubular neighbourhood $D^{2}\times I^{3}$ around a point in
$\mathring{C}_{i}$ the covering map takes the form
$(z,x)\mapsto(z^{\alpha_{i}},x)$, i.e. points on $p^{-1}(\mathring{C}_{i})$
have branch index $\alpha_{i}$. However, in contrast to the 3d case, the
preimage $p^{-1}(C_{i})$ is itself a branched cover of $C_{i}$, precisely of
the form discussed for $S^{3}_{\boldsymbol{\alpha}}$ in the previous
paragraph. Hence, e.g. a point in the intersection $p^{-1}(C_{1}\cap C_{2})$
has branch index $\alpha_{1}\alpha_{2}$. The branching structure is displayed
in Figure 10. Again, note that in order for functions to be smooth on the
preimage of respective neighbourhoods around points in $L$, they have to be
smooth already in the $2\pi$-periodic angles $\theta_{i}$.

For the round metric on $S^{5}$, the corresponding metric on
$S^{5}_{\boldsymbol{\alpha}}$ is given (in terms of the $2\pi$-periodic
angles) by

$\dd
s^{2}=\dd\varphi^{2}+\sin^{2}\\!\varphi\,\dd\phi+\alpha_{3}^{2}\cos^{2}\\!\varphi\dd\theta_{3}^{2}+\alpha_{2}^{2}\sin^{2}\\!\varphi\sin^{2}\\!\phi\,\dd\theta_{2}^{2}+\alpha_{1}^{2}\sin^{2}\\!\varphi\cos^{2}\\!\phi\,\dd\theta_{1}^{2}.$
(66)

We see that, e.g. at $p^{-1}(\mathring{C}_{i})$ there is a conical singularity
with surplus angle $2\pi\alpha_{i}$. The $T^{3}$-action on $S^{5}$ lifts to
$S^{5}_{\boldsymbol{\alpha}}$ and free $S^{1}$-actions preserving the
branching structure transverse to $L$ are generated (up to
$\SL(3,\mathbb{Z})$-transformations) by the vector field

$\X=\frac{1}{\alpha_{1}}\partial_{\theta_{1}}+\frac{1}{\alpha_{2}}\partial_{\theta_{2}}\pm\frac{1}{\alpha_{3}}\partial_{\theta_{3}}.$
(67)

###  A.2 Weighted Projective Space from Spheres

Hereinafter, we consider odd-dimensional spheres $S^{2r-1}$ embedded into
$\mathbb{C}^{r}$ in the standard way. In particular, we consider $S^{2r-1}$ as
an $S^{1}$-fibration for the following locally free $S^{1}$-action:

$(z_{1},\dots,z_{r})\mapsto(\e{\ii N_{1}\xi}z_{1},\dots,\e{\ii
N_{r}\xi}z_{r}),\qquad\xi\sim\xi+2\pi,$ (68)

where $N_{1},\dots,N_{r}\in\mathbb{N}$ with $\gcd(N_{1},\dots,N_{r})=1$ (this
guarantees an effective action). We commonly refer to these as weights and
define the weight vector $\boldsymbol{N}=(N_{1},\dots,N_{r})$. The locally
free nature of this action can be seen easily by restricting to the
$S^{1}$-fibres $|z_{i}|=1$ ($i=1,\dots,r$). There, group elements of the form
$\xi=2\pi k/N_{i}$ for $k\in\mathbb{Z}_{N_{i}}$ act trivially. Consequently,
the base space of the fibration contains orbifold singularities and, more
precisely, we obtain an $S^{1}$-orbibundle

$\tikzcd[columnsep=small]S^{1}\ar[r]&S^{2r-1}\ar[d,"\pi"]\\\
\mathbb{CP}^{r-1}_{\boldsymbol{N}}$ (69)

over a weighted projective space191919This definition of complex weighted
projective space might be unfamiliar to some readers who instead have in mind
the definition as the symplectic quotient $\mathbb{C}^{r}/\\!/S^{1}$ by (68).
Topologically, the two procedures agree. In this article, we always understand
$\mathbb{CP}^{r-1}_{\boldsymbol{N}}$ in the sense of (69).
$\mathbb{CP}^{r-1}_{\boldsymbol{N}}$. Note that
$\mathbb{CP}^{r-1}_{(1,\dots,1)}=\mathbb{CP}^{r-1}$ and (69) is simply the
Hopf fibration. This case has been studied in Lundin:2021zeb .

The orbifold structure of $\mathbb{CP}^{r-1}_{\boldsymbol{N}}$ can be inferred
from (68) as follows: denote by $[z]$ a point in
$\mathbb{CP}^{r-1}_{\boldsymbol{N}}$ with $z_{j}\neq 0$ for some
$j\in\\{1,\dots,r\\}$ and let $m$ be the greatest common divisor of the
corresponding set of weights $N_{j}$. Then the isotropy group $\Gamma_{[z]}$
of $[z]$ is isomorphic to $\mathbb{Z}_{m}$. In particular, we have
$\Gamma_{[z]}\simeq\mathbb{Z}_{N_{i}}$ for $[z]=[0:\ldots:0:z_{i}:0:\ldots:0]$
with $z_{i}\neq 0$ and, due to $\gcd\boldsymbol{N}=1$, at points where
$z_{i}\neq 0$ for all $i\in\\{1,\dots,r\\}$, $\Gamma_{[z]}$ is trivial.

#### Three Dimensions.

In this case, (69) simply corresponds to a Seifert fibration of $S^{3}$ with
two exceptional fibres $(N_{1},\beta_{1})$, $(N_{2},\beta_{2})$ located at
$|z_{1}|=1$ and $|z_{2}|=1$, respectively. In order for the fundamental group
of the Seifert fibration to be trivial, we impose202020This can always be
arranged for given $N_{1},N_{2}$ with $\gcd(N_{1},N_{2})=1$ by virtue of
Bezout’s lemma. $N_{1}\beta_{2}+N_{2}\beta_{1}=1$. The base space, which
topologically is $S^{2}$, has conical singularities at the two poles with
deficit angle $2\pi/N_{1}$ and $2\pi/N_{2}$, respectively. This space is
commonly referred to as a spindle.

#### Five Dimensions.

In this case, the singularity structure of (69) can be more interesting.
Viewing $\mathbb{CP}^{2}_{(N_{1},N_{2},N_{3})}$ as a toric orbifold for the
moment, the $i$th facet $F_{i}$ of its polytope is located at $z_{i}=0$.
Hence, points in the interior $\mathring{F}_{i}$ have isotropy group
isomorphic to $\mathbb{Z}_{\gcd(N_{j},N_{k})}$, where $j,k\neq i$. Moreover,
the facet $F_{i}$ itself corresponds to a spindle
$\mathbb{CP}^{1}_{(N_{j},N_{k})}$. At a vertex of the polytope where facets
$F_{i},F_{j}$ intersect, we have isotropy group isomorphic to
$\mathbb{Z}_{N_{k}}$, where $i,j\neq k$. The singularity structure is
displayed in Figure 10.

$N_{3}$$N_{1}$$N_{2}$$\gcd(N_{1},N_{3})$$F_{2}$$\gcd(N_{2},N_{3})$$F_{1}$$\gcd(N_{1},N_{2})$$F_{3}$$\alpha_{1}\alpha_{2}$$\alpha_{2}\alpha_{3}$$\alpha_{1}\alpha_{3}$$\alpha_{2}$$C_{2}$$\alpha_{1}$$C_{1}$$\alpha_{3}$$C_{3}$
Figure 10: Left-hand side: singularity structure of
$\mathbb{CP}^{2}_{\boldsymbol{N}}$ in terms of the $S^{5}$\- moment polytope.
The labels in black on vertices and edges denote the degree of the singularity
while the grey labels denote the facets. Right-hand side: branching structure
of $S^{5}_{\boldsymbol{\alpha}}$ in terms of the $S^{5}$-moment polytope. The
labels in black on vertices and edges denote the branch index while the grey
labels denote the facets.

As an example, let us consider the weight vector
$\boldsymbol{N}=(k_{1}k_{2},k_{1}k_{3},k_{2}k_{3})$, for some positive
integers $k_{1},k_{2},k_{3}$ with $\gcd(k_{i},k_{j})=1$ for $i,j=1,2,3$. Then
we have

$\Gamma_{[z]\in\mathring{F}_{1}}\simeq\mathbb{Z}_{k_{3}},\qquad\Gamma_{[z]\in\mathring{F}_{2}}\simeq\mathbb{Z}_{k_{2}},\qquad\Gamma_{[z]\in\mathring{F}_{3}}\simeq\mathbb{Z}_{k_{1}}.$
(70)

For the facets themselves we have
$F_{1}\simeq\mathbb{CP}^{1}_{(k_{1},k_{2})}$,
$F_{2}\simeq\mathbb{CP}^{1}_{(k_{3},k_{1})}$,
$F_{3}\simeq\mathbb{CP}^{1}_{(k_{3},k_{2})}$ and hence, at the vertices:

$\Gamma_{F_{1}\cap F_{2}}\simeq\mathbb{Z}_{k_{2}k_{3}},\qquad\Gamma_{F_{2}\cap
F_{3}}\simeq\mathbb{Z}_{k_{1}k_{2}},\qquad\Gamma_{F_{1}\cap
F_{3}}\simeq\mathbb{Z}_{k_{1}k_{3}}.$ (71)

Hence, for $r\geq 3$ we have, in general, already codimension 2 singularities.

###  A.3 Free Quotient of Branched Spheres

All partition functions in this work will be computed on the free quotient
$B^{r-1}_{\boldsymbol{\alpha}}:=S^{2r-1}_{\boldsymbol{\alpha}}/S^{1}$ defined
by (2), respectively (5). Depending on the choice of sign in the last
component of the vector field we call the fibration topological or exotic (see
e.g. Festuccia:2019akm  for terminology). The branching structure descends to
$B^{r-1}_{\boldsymbol{\alpha}}$ and can be read off from Figure 10 for $r=3$.
Here we provide some more detail about this space.

#### Three Dimensions.

Let us define new angles $\tilde{\xi},\tilde{\psi}$ such that

$\tilde{\theta}_{1}=\tilde{\xi}+\tilde{\psi},\qquad\tilde{\theta}_{2}=\pm\tilde{\xi}\mp\tilde{\psi}.$
(72)

It is easy to see that in terms of the new angles the $S^{1}$-action is
generated by

$\X^{\mathrm{top,ex}}=\partial_{\tilde{\theta}_{1}}\pm\partial_{\tilde{\theta}_{2}}=\partial_{\tilde{\xi}}$
(73)

A naive rewriting of the metric (63) simply produces a fibration over the
round $S^{2}$. However, note that the periodicities of
$\tilde{\xi},\tilde{\psi}$ are not the canonical ones and change
discontinuously over the $S^{3}_{\boldsymbol{\alpha}}$. Instead, it can be
shown that the branched covering, as an $S^{1}$-bundle, can be patched over
$B^{1}_{\boldsymbol{\alpha}}$ (which is topologically an $S^{2}$) by two
charts $\hat{D}^{2}_{N,S}\times S^{1}$ around the north and south pole. Here,
$\hat{D}^{2}_{N,S}$ denotes the $\alpha_{1},\alpha_{2}$-fold branched cover of
the open disc $D^{2}$ with radius $0\leq r_{N,S}<\pi/2$ and angles
$\Delta\psi_{N,S}=2\pi$. The metric on each patch reads

$\displaystyle\dd s_{N}^{2}=$ $\displaystyle\dd
r_{N}^{2}+\alpha_{1}^{2}\sin^{2}(2r_{N})\dd\psi_{N}^{2}+(\dd\xi_{N}\mp\alpha_{1}\sin(r_{N})^{2}\dd\psi_{N})^{2},$
(74a) $\displaystyle\dd s_{S}^{2}=$ $\displaystyle\dd
r_{S}^{2}+\alpha_{2}^{2}\sin^{2}(2r_{S})\dd\psi_{S}^{2}+(\dd\xi_{S}\pm\alpha_{2}\sin(r_{S})^{2}\dd\psi_{S})^{2},$
(74b)

where $\Delta\xi_{N,S}=2\pi$. Note that the connection one-forms are well-
defined on the whole patch. We glue the two patches along an $S^{1}$ for fixed
$r_{N,S}>0$ by reversing one of the patches’ orientation.

We can match the singularity structure of
$\mathbb{CP}^{1}_{\boldsymbol{\alpha}}$ with the branching structure of
$B^{1}_{\boldsymbol{\alpha}}$ simply by choosing
$\boldsymbol{\alpha}=\boldsymbol{N}$.

#### Five Dimensions.

Introducing new angles $\tilde{\xi},\tilde{\psi},\tilde{\phi}$ (see _e.g._
Lundin:2021zeb ) such that

$\tilde{\theta}_{1}=\tilde{\xi}-\frac{1}{2}(\tilde{\psi}-\tilde{\phi}),\quad\tilde{\theta}_{2}=\tilde{\xi}-\frac{1}{2}(\tilde{\psi}+\tilde{\phi}),\quad\tilde{\theta}_{3}=\pm\tilde{\xi},$
(75)

we find:

$\X^{\mathrm{top,ex}}=\partial_{\tilde{\theta}_{1}}+\partial_{\tilde{\theta}_{2}}\pm\partial_{\tilde{\theta}_{3}}=\partial_{\tilde{\xi}}.$
(76)

In complete analogy with the previous case we can patch the fibred metric for
the branched five-sphere $S^{5}_{\boldsymbol{\alpha}}$ and we omit a detailed
analysis at this point.

Note that, unless $\alpha_{i}=1$ for all $i=1,2,3$ (in which case
$S^{5}_{\boldsymbol{\alpha}}$ simply becomes the round $S^{5}$), the space
$B^{2}_{\boldsymbol{\alpha}}$ always has codimension 2 singularities (cf.
Figure 10). In contrast, on $\mathbb{CP}^{2}_{\boldsymbol{\alpha}}$ we only
obtain codimension 2 singularities if $\gcd(\alpha_{i},\alpha_{j})>1$ for at
least one pair of weights. Hence, in order to match singularity degrees of
$\mathbb{CP}^{2}_{\boldsymbol{N}}$ and branch indices of
$B^{2}_{\boldsymbol{\alpha}}$ we can only consider weighted projective spaces
with weights $\boldsymbol{N}=(N_{1},N_{2},N_{3})$ such that at least one pair
has non-trivial greatest common divisor.

Therefore, unless otherwise specified, in the following we always restrict to
weight vectors of the form

$\boldsymbol{N}=(\alpha_{2}\alpha_{3},\alpha_{1}\alpha_{3},\alpha_{1}\alpha_{2})$
(77)

with $\gcd(\alpha_{i},\alpha_{j})=1$ for all $i,j=1,2,3$.

##  Appendix B Dimensional Reduction of the Spindle Index

In this appendix we show how to match our result for the partition function of
the 2d vector multiplet on the branched cover $B^{1}_{\boldsymbol{\alpha}}$
with that on a spindle $\mathbb{CP}^{1}_{\boldsymbol{\alpha}}$. The latter is
obtained from dimensional reduction of the spindle index computed in
Inglese:2023wky , to which we refer for details. Note that our results on the
spindle are obtained reducing from a single theory for an $\mathcal{N}=2$
vector multiplet on the branched cover of the sphere
$S^{3}_{\boldsymbol{\alpha}}$. Instead, the dimensional reduction of the
spindle index, as it is performed along a trivially fibred $S^{1}$, it is such
that from each $\mathcal{N}=2$ theory on
$\mathbb{CP}^{1}_{\boldsymbol{\alpha}}\times S^{1}$ one obtains a unique
theory on $\mathbb{CP}^{1}_{\boldsymbol{\alpha}}$.

###  B.1 Spindle Index

Following Inglese:2023wky , the partition function on
$\mathbb{CP}^{1}_{\boldsymbol{\alpha}}\times S^{1}$ for an $\mathcal{N}=2$
vector multiplet can be written as:

$\mathcal{Z}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}\times
S^{1}}^{\sigma}=\sum_{\mathfrak{m}}\oint\dd u\,e^{-S_{cl}}\cdot
Z^{vm,\sigma}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}\times S^{1}}.$ (78)

The sum is over magnetic charges $\mathfrak{m}$ that label configurations with
flux for the field strength on the spindle. The classical part is given by a
Chern-Simons term evaluated on the BPS locus,

$S_{\text{cl}}=\frac{2\pi\ii k}{\alpha_{1}\alpha_{2}}\tr(\mathfrak{m}u),$ (79)

and the integral is performed over the holonomies $u$ for the gauge
connections. Topologically twisted and exotic theories correspond,
respectively, to $\sigma=1$ and $\sigma=-1$. For convenience, let us start by
introducing some notation from Inglese:2023wky :

$\begin{split}&\mathfrak{m}=\alpha_{1}m_{-}-\alpha_{2}m_{+},\qquad
p_{+}=\alpha(m_{+})-\sigma,\qquad p_{-}=\alpha(m_{-})+1,\\\
&\mathfrak{b}=1+\sigma\lfloor\sigma p_{+}/\alpha_{1}\rfloor+\lfloor-
p_{-}/\alpha_{2}\rfloor,\qquad\mathfrak{c}=\rmd{-p_{-}}{\alpha_{2}}/\alpha_{2}-\sigma\,\rmd{p_{+}}{\alpha_{1}}/\alpha_{1},\\\
&\gamma=-\frac{n}{2}+\omega\frac{\chi_{-\sigma}}{4},\qquad\frac{\chi_{\sigma}}{2}=\frac{1}{2}\left(\frac{1}{\alpha_{2}}+\frac{\sigma}{\alpha_{1}}\right)=\frac{1}{2\pi}\int_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}\dd
A_{R},\\\ &q=e^{2\pi\ii\omega},\qquad
y=q^{\mathfrak{c}/2}e^{2\pi\ii(2\gamma-\alpha(u))},\end{split}$ (80)

where $m_{\pm}\in\mathbb{Z}$, $n\in\mathbb{Z}$ and $\alpha$ denote the roots
of the gauge algebra. The notation $\lfloor x\rfloor$ denotes the floor
function and $\sigma=\pm 1$ for the topologically twisted and exotic theory,
respectively (or the twist and anti-twist in the language of Inglese:2023wky
).

The one-loop contribution to the partition function for the 3d vector
multiplet on $\mathbb{CP}^{1}_{\boldsymbol{\alpha}}\times S^{1}$ can be
obtained from an index computation of a twisted Dolbeault operator and the
index reads

$\begin{split}I^{\sigma}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}=&\frac{1}{\alpha_{1}}\sum_{j=0}^{\alpha_{1}-1}\frac{\omega_{+}^{-jp_{+}}q_{+}^{-p_{+}}}{1-\omega_{+}^{j\sigma}q_{+}^{\sigma}}+\frac{1}{\alpha_{2}}\sum_{j=0}^{\alpha_{2}-1}\frac{\omega_{-}^{-jp_{-}}q_{-}^{-p_{-}}}{1-\omega_{-}^{j\sigma}q_{-}^{\sigma}}\\\
=&\frac{q^{-\sigma\lfloor\sigma
p_{+}/\alpha_{1}\rfloor}}{1-q^{\sigma}}+\frac{q^{\lfloor-
p_{-}/\alpha_{2}\rfloor}}{1-q^{-1}}.\end{split}$ (81)

Here, $q_{\pm}=q^{1/\alpha_{1,2}}$ and
$\omega_{\pm}=e^{2\pi\ii/\alpha_{1,2}}$. This result can be combined with the
one on the $S^{1}$ and the fugacities for gauge and flavour symmetries and
then converted into the one-loop determinant:

$Z^{\sigma}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}\times
S^{1}}=\prod_{\alpha}(-y)^{\frac{1}{4}(1-\sigma-2\mathfrak{b})}q^{\frac{1}{8}(1-\sigma)(\mathfrak{b}-1)}\frac{\left(q^{\frac{1}{2}(1+\mathfrak{b})}y^{-1};q\right)_{\infty}}{\left(q^{\frac{\sigma}{2}(1-\mathfrak{b})}y^{-\sigma};q\right)_{\infty}},$
(82)

where $(z;q)_{n}$ is the $n$-th $q$-Pochhammer symbol

$(a;q)_{n}=\prod_{k=0}^{n-1}\left(1-aq^{k}\right).$ (83)

###  B.2 Dimensional Reduction

In order to match with our results in 2d we compute the small radius-limit
where the $S^{1}$-factor shrinks. Our computation follows the dimensional
reduction for the superconformal index in Benini:2012ui . We set

$q=e^{-\beta},\quad u=\frac{\beta a}{2\pi},$ (84)

where $\beta$ is the circumference of the $S^{1}$ and $a$ the 2d scalar
arising from the component of the gauge connection along the $S^{1}$. This
gives:

$y=e^{-\beta\left(\frac{\mathfrak{c}}{2}+r\frac{\chi_{-\sigma}}{4}+\ii\alpha(a)\right)},$
(85)

as $r\in 2\mathbb{Z}$ and $n\in\mathbb{Z}$. In the limit $\beta\rightarrow 0$
the following relation holds:

$\lim_{z\rightarrow
1}\frac{(z^{s};z)_{\infty}}{(z^{t};z)_{\infty}}=\frac{\Gamma(t)}{\Gamma(s)}(1-z)^{t-s},$
(86)

where

$\Gamma(q)=\frac{\sqrt{2\pi}}{\prod_{n=0}^{\infty}(q+n)}.$ (87)

In our case $z=e^{-\beta}$ and

$\begin{split}s=&\frac{1}{2}\left(1+\mathfrak{b}-\mathfrak{c}-r\chi_{-\sigma}/2-\ii\alpha(a)\right),\\\
t=&\frac{\sigma}{2}\left(1-\mathfrak{b}-\mathfrak{c}-r\chi_{-\sigma}/2-\ii\alpha(a)\right).\end{split}$
(88)

Hence, simply using $\lim_{\beta\to 0}(e^{-\beta})=1+\mathcal{O}(\beta)$, all
contributions to the left of the Pochhammer symbols in (82) simplify and we
find:

$Z^{\sigma}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}=\lim_{\beta\rightarrow
0}Z^{\sigma}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}\times
S^{1}}\beta^{\sum_{\alpha}(s-t)}=\prod_{\alpha}\prod_{k=0}^{\infty}\frac{(s+k)}{(t+k)}.$
(89)

This result is the one-loop at each flux sector on
$\mathbb{CP}^{1}_{\boldsymbol{\alpha}}$. The extra power of $\beta$, which is
needed to recover the result on $S^{2}$ starting from the superconformal index
with $\sigma=-1$ for $\alpha_{1}=\alpha_{2}=1$, is discussed in section 8.1 of
Benini:2012ui .

###  B.3 Spindle

First we rewrite

$\begin{split}\mathfrak{b}=&1+\sigma\lfloor\sigma
p_{+}/\alpha_{1}\rfloor+\lfloor-p_{-}/\alpha_{2}\rfloor\\\
=&1+\frac{\alpha(m_{+})}{\alpha_{1}}-\frac{\alpha(m_{-})}{\alpha_{2}}-\sigma\,\frac{\rmd{\sigma
p_{+}}{\alpha_{1}}}{\alpha_{1}}-\frac{\rmd{-p_{-}}{\alpha_{2}}}{\alpha_{2}}-\left(\frac{1}{\alpha_{2}}+\frac{\sigma}{\alpha_{1}}\right)\\\
=&1-\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}-\sigma\,\frac{\rmd{\sigma
p_{+}}{\alpha_{1}}}{\alpha_{1}}-\frac{\rmd{-p_{-}}{\alpha_{2}}}{\alpha_{2}}-\left(\frac{1}{\alpha_{2}}+\frac{\sigma}{\alpha_{1}}\right).\end{split}$
(90)

Also recalling

$\mathfrak{c}=\frac{\rmd{-p_{-}}{\alpha_{2}}}{\alpha_{2}}-\sigma\,\frac{\rmd{p_{+}}{\alpha_{1}}}{\alpha_{1}},$
(91)

we find that

$\begin{split}\frac{1}{2}\left(1+\mathfrak{b}-\mathfrak{c}-\chi_{-\sigma}\right)&=\frac{1}{2}\left(2-\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}-2\,\frac{\rmd{-p_{-}}{\alpha_{2}}}{\alpha_{2}}-\left(\frac{1}{\alpha_{2}}+\frac{\sigma}{\alpha_{1}}\right)-\left(\frac{1}{\alpha_{2}}-\frac{\sigma}{\alpha_{1}}\right)\right)\\\
&=1-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-p_{-}}{\alpha_{2}}}{\alpha_{2}}-\frac{1}{\alpha_{2}},\end{split}$
(92)

and

$\begin{split}\frac{\sigma}{2}\left(1-\mathfrak{b}-\mathfrak{c}-\chi_{-\sigma}\right)&=\frac{\sigma}{2}\left(\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}+2\sigma\,\frac{\rmd{\sigma
p_{+}}{\alpha_{1}}}{\alpha_{1}}+\left(\frac{1}{\alpha_{2}}+\frac{\sigma}{\alpha_{1}}\right)-\left(\frac{1}{\alpha_{2}}-\frac{\sigma}{\alpha_{1}}\right)\right)\\\
&=\sigma\left(\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\sigma\,\frac{\rmd{\sigma
p_{+}}{\alpha_{1}}}{\alpha_{1}}\right)+\frac{1}{\alpha_{1}}.\end{split}$ (93)

The two expressions above simplify noting that

$\begin{split}&-\frac{\rmd{-p_{-}}{\alpha_{2}}}{\alpha_{2}}=-1+\frac{1}{\alpha_{2}}+\frac{\rmd{\alpha(m_{-})}{\alpha_{2}}}{\alpha_{2}},\\\
&\sigma\,\frac{\rmd{\sigma
p_{+}}{\alpha_{1}}}{\alpha_{1}}=\sigma\left(1-\frac{1}{\alpha_{1}}-\frac{\rmd{-\sigma\alpha(m_{+})}{\alpha_{1}}}{\alpha_{1}}\right).\end{split}$
(94)

Furthermore, one can always find integers $a_{\pm}\in\mathbb{Z}$ such that
$a_{-}\alpha_{1}-a_{+}\alpha_{2}=1$ (by virtue of Bezout’s lemma). Then we can
rewrite

$m_{\pm}=(a_{\pm}+t\alpha_{1,2})\mathfrak{m}.$ (95)

Putting all this together, we arrive at

$s=\frac{1}{2}\left(1+\mathfrak{b}-\mathfrak{c}-\chi_{1}-2\ii\alpha(a)\right)=-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}-\ii\alpha(a),$
(96)
$t=\frac{\sigma}{2}\left(1-\mathfrak{b}-\mathfrak{c}-\chi_{1}-2\ii\alpha(a)\right)=1+\sigma\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-\sigma
a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}-\ii\sigma\alpha(a).$ (97)

We are now ready to substitute into the one-loop determinant (89).

#### Topological Twist.

This is the case of $\sigma=1$ and we find

$Z^{\text{top}}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}=\prod_{\alpha}\prod_{k=0}^{\infty}\frac{k-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}-\ii\alpha(a)}{k+1+\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}-\ii\alpha(a)}.$
(98)

We can rewrite this expression as follows

$\begin{split}\alpha(\mathfrak{m})\geq
0:\;Z^{\text{top}}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}=&\prod_{\alpha}\prod_{k=0}^{\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}-\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}}\left(k-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}-\ii\alpha(a)\right)\\\
=&\prod_{\alpha}\prod_{j=-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}^{\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}}\left(j-\ii\alpha(a)\right),\end{split}$
(99)
$\begin{split}\alpha(\mathfrak{m})<0:\;Z^{\text{top}}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}=&\prod_{\alpha}\prod_{k=0}^{-\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}-2+\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}\left(k+1+\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}-\ii\alpha(a)\right)^{-1}\\\
=&\prod_{\alpha}\prod_{l=\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}+1}^{-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-1+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}\left(l-\ii\alpha(a)\right)^{-1}.\end{split}$
(100)

where
$j=k-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}$
and
$l=k+1+\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}$.
A further rewriting yields

$\begin{split}Z^{\text{top}}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}=&\prod_{\alpha>0}\prod_{j=-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}^{\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}}\left(j-\ii\alpha(a)\right)\prod_{l=-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{+a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}+1}^{\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-1+\frac{\rmd{-a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}\left(l+\ii\alpha(a)\right)^{-1}\\\
=&\prod_{\alpha>0}\prod_{j=-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}^{\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}}\left(j-\ii\alpha(a)\right)\prod_{l=-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}+1}^{\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-1+\frac{\rmd{+a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}}\left(l-\ii\alpha(a)\right)^{-1}\\\
=&\prod_{\alpha>0}\left(\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right)\left(-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right).\end{split}$
(101)

Here we used the relation (26) and, between the first and the second line, we
neglected an overall minus sign in the second factor while also sending
$l\rightarrow-l$. The first factor in the bottom line appears only if
$\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}=\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}=0$
and the second only if
$\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}=\rmd{a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}=0$.
Employing (23) these two conditions are equivalent, respectively, to
$\alpha(\mathfrak{m})\in\alpha_{1}\mathbb{Z}$ and
$\alpha(\mathfrak{m})\in\alpha_{2}\mathbb{Z}$. Finally, this agrees with our
result (34) on the branched cover upon setting $\epsilon_{1}=1$ and a constant
shift in the Coulomb branch parameter in (34),

$a\rightarrow a-\ii\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}.$ (102)

This constant shift is such that also the classical contribution to the
partition function matches Lundin:2021zeb .

#### Exotic.

This is the case of $\sigma=-1$ and we find

$\begin{split}Z^{\text{ex}}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}=&\prod_{\alpha}\prod_{k=0}^{\infty}\frac{k-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}-\ii\alpha(a)}{k+1-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}+\ii\alpha(a)}\\\
=&\prod_{\alpha>0}\prod_{k=0}^{\infty}\frac{k-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}-\ii\alpha(a)}{k+1-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}+\ii\alpha(a)}\frac{k+\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{-a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}+\ii\alpha(a)}{k+1+\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}-\ii\alpha(a)}\\\
=&\prod_{\alpha>0}\prod_{k=0}^{\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}-\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}}\left(k-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}-\ii\alpha(a)\right)\\\
\quad\quad&\prod_{k=0}^{\frac{\alpha(\mathfrak{m})}{\alpha_{1}\alpha_{2}}-2+\frac{\rmd{+a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}+\frac{\rmd{-a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}\left(k+1-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{+a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}+\ii\alpha(a)\right)^{-1}\\\
=&\prod_{\alpha>0}\prod_{j=-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\frac{\rmd{a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}^{\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-\frac{\rmd{-a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}}\left(j-\ii\alpha(a)\right)\prod_{l=-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+1-\frac{\rmd{+a_{+}\alpha(\mathfrak{m})}{\alpha_{1}}}{\alpha_{1}}}^{\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}-1+\frac{\rmd{-a_{-}\alpha(\mathfrak{m})}{\alpha_{2}}}{\alpha_{2}}}\left(l+\ii\alpha(a)\right)^{-1}\end{split}$
(103)

This is the expression we encountered in the previous case, which reduces to:

$Z^{\text{ex}}_{\mathbb{CP}^{1}_{\boldsymbol{\alpha}}}=\prod_{\alpha>0}\left(\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right)\left(-\frac{\alpha(\mathfrak{m})}{2\alpha_{1}\alpha_{2}}+\ii\alpha(a)\right),$
(104)

where the first term contributes only if
$\alpha(\mathfrak{m})\in\alpha_{1}\mathbb{Z}$ and the second only if
$\alpha(\mathfrak{m})\in\alpha_{2}\mathbb{Z}$. This agrees with (35) for
$\epsilon_{1}=1$ and replacing $a\leftrightarrow\eta$. Also in this case the
classical contributions can be shown to match what we found reducing from the
branched cover.

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