Super Yang-Mills on Branched Covers and Weighted Projective Spaces

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.

Since the early days of supersymmetric localisation [1,2] 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 N = 2 superconformal field theories (SCFTs) has been computed from the partition function of the SCFT on certain branched covers of S 3 [3].This result was later generalised to branched covers of spheres S d , for d = 2, 4, 5 [4][5][6][7].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 [8][9][10].
Recently, the authors of [11,12] computed the partition function on CP1 N × S 1 which was then used in [13] to reproduce the entropy function of supersymmetric accelerating black holes in AdS 4 .The computation was performed for two different theories denoted twist and anti-twist, which reduce on S2 × S 1 to, respectively, the topologically twisted index [14] and the superconformal index [15][16][17].Their result was obtained employing an orbifold index theorem [18].
In this work we draw a connection between the aforementioned results on the branched cover of CP 1 and the spindle CP 1  N , which we subsequently extend to the branched cover of CP 2 and weighted projective space CP 2 N .Our starting point are existing results for the perturbative partition function of the N = 2 vector multiplet on a branched three-sphere S3 α [3,19] and N = 1 vector multiplet on a branched five-sphere S 5 α [6,7] 1 .Both theories are in an Ω-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 α and B 2 α in the following).The method we use for the circle reduction is the one of [20,21]: first, we consider quotients by a finite subgroup, S 2r−1 α /Z h (for r = 2, 3).This introduces extra topological sectors in the partition function corresponding to nontrivial flat connections in the localisation locus.The circle reduction is then obtained in the limit h → ∞, in which these topological sectors label flux through the two-cycle on the base B r−1 α .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 α in (4.16)-(4.18),resp.(5.7)-(5.9).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 [11] on the spindle upon the identification of weights N 1 = α 1 , N 2 = α 2 .For the special case of α 1 = q = α 2 this could already be deduced from the combined results of [5] and [22]; 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 Z q .In 4d our result for the topologically twisted theory for r = 3 matches its counterpart on CP 2 N [23] upon suitable identification of weights 2 .Guided by these observations, we make the Conjecture 7.1 that also the one-loop determinant for the exotically twisted N = 2 vector multiplet on CP 2  N is simply given by our result on B 2 α 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 [23,25].
Finally, capitalising on equivariance of our setup under a T 2 -action, for the theory on B 2 α 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 [21,[26][27][28][29], 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 α and CP 2 N in each flux sector, their shifts also agree.With the Nekrasov partition function on a neighbourhood C 2 /Z N i of the ith fixed point being known [30][31][32][33][34][35], we conclude our work by providing the (integrand of the) full partition function (7.4), including classical, one-loop and instanton part at all flux sectors for the N = 2 vector multiplet on weighted projective space CP 2 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 α 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 α /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 α /S 1 match the ones on weighted projective space and this also holds for the topological twist on S 5 α /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.

Branched Spheres
In this section we introduce our geometric setup and field theoretic starting point, the N = 2 (resp.N = 1) vector multiplet partition function on branched spheres in three (resp.five) dimensions.These are existing results from [3,19] (resp.[6,7]) to which we refer the reader for more detail 3 .
Throughout this article, we always consider cohomologically twisted theories with simply-connected gauge group G (see [36,37] 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 [3,6,7] and hence, one can switch between the physical and twisted theories bijectively. 3In 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 α ) and then taking the singular limit.The result agrees with the one on a squashed sphere with squashings reciprocal to the {αi}.A similar relation has been found in 5d for α = (α1, 1, 1) (which is easily extended to α2, α3 ̸ = 1) and also for a squashed S 5 α .

Geometric Setup
The branched spheres in consideration, which we always denote by S 2r−1 α 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 2π.If we denote said angles by θ i , i = 1, . . ., r with θ i ∼ θ i + 2π then we extend these to new angles θi with θi ∼ θi + 2πα i , where α i ∈ Z.The integers {α i } are called the branch indices and in the notation S 2r−1 α are collected in the subscript α = (α 1 , . . ., α r ).The branched sphere is an α 1 . . .α 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 α can be obtained from the round one and in terms of the canonical angles reads with ϕ ∈ [0, π/2].It was shown in [3] that this space has conical singularities in the curvature on the Hopf-link comprised of the two S 1 -subspaces ϕ = 0, π/2 with surplus angles α 1 , α 2 .There are two free S 1 -actions (up to SL(2, Z)-transformations) which are Killing and given by S 3 α can be viewed as a "branched" (anti-)Hopf fibration along this action, whose base we denote by B 1 α = S 3 α /S 1 .Since (2.2) preserves the branching, B 1 α in turn is a branched cover of S 2 .We can also reach this space by first considering finite quotients4 S 3 α /Z h along (2.2), giving branched lens spaces which we denote L 3 α (h, ±1).Note that the finite quotient introduces holonomies classified by Z h .We arrive at the base space B 1 α by taking the limit h → ∞ (see [21]).
Five Dimensions.The metric on S 5 α can be obtained from the round one and in terms of the canonical angles reads with φ, ϕ ∈ [0, π/2].Similar to the three-dimensional case, the curvature can be shown to have conical singularities on the three (branched) S 3 -subspaces φ = π/2 (with surplus 2πα 3 ), ϕ = 0 (with surplus 2πα 2 ) and ϕ = π/2 (with surplus 2πα 1 ).There are two free S 1 -actions (up to SL(3, Z)-transformations) which are Killing: Similar to the three-dimensional case, we are interested in the (free) S 1 -quotient by this action, B 2 α = S5 α /S 1 which gives a branched cover of CP 2 .Again, as an intermediate step we can take finite quotients 4 S 5 α /Z h giving (generalised) branched lens spaces L 5 α (h, ±1) with extra topological sectors classified by Z h .
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 α 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 conditions 5 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.[39]).
1} a basis for such maps is given by polynomials in z 1 , . . ., 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/α 1 1 , . . ., z 1/αr r (where arg z 1/α i i = θi /α i ) and the boundary conditions on the fields are that the holomorphic functions be smooth in {z 1/α i i } rather than {z i }.Consequently, when expanding in Fourier modes exp(in nr αr with n 1 , . . ., n r ∈ Z.The free S 1 -action can be parametrised by an angle ξ with ξ ∼ ξ + 2π lcm α.Hence, locally, smooth functions on the branched cover can be expanded in terms of modes exp(itξ/ lcm α) along the S 1 and the charge t ∈ Z is related to the ones corresponding to the canonical angles by 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 α with squashings {ω i } i=1,...,r is simply that on S 2r−1 [40][41][42][43][44] for a background with Killing vector r = ( ω 1 α 1 , . . ., ωr αr ) and yields the multiple sine function for a fixed root α 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 where h ⊂ 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 g to h).
The classical action S cl 2r−1 for r = 2, 3 is given by (2.9) with Chern-Simons level k, Yang-Mills coupling g 2 YM and the weights r i = ω i α 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: 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 α /S 1 as the topological (resp.exotic) theory 6 , hence the labels top and ex in (2.10).

Partition Function on
Having the result for the perturbative partition function on S 2r−1 α at hand, in this section we will perform the dimensional reduction along the two7 free S 1 -directions identified in (2.2), resp.(2.5).In particular, this will produce the one-loop contribution of the vector multiplet on the quotient space B r−1 α in all flux sectors (note H 2 (B r−1 α ) ≃ Z).The quotient procedure follows the method explained in [21] where we first consider the theory on finite quotients S 2r−1 α /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 α (h, ±1).In the limit h → ∞ we reach B r−1 α 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.

Finite Quotients
In order to understand how the partition function (2.8) 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 section 2.2, these are precisely the holomorphic functions on the space.
Holomorphic functions on S 2r−1 α are in one-to-one correspondence with points n in the integral cone Z r ≥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 (2.6): Note that in the exotic theory t ∈ Z and the slices can be non-compact while in the topological one t ∈ Z ≥0 and the slices are always compact.Now we can rewrite the product over the cone Z r ≥0 in the multiple sine function as ) where Bt,α denotes the interior of the slice (3.1) and r = ( ϵ 1 α 1 , . . ., ϵ r−1 α r−1 ) the weights of the residual T r−1 -action on the slice, where top: where i = 1, . . ., r − 1.Note that in the unsquashed limit ω i → 1 for all i = 1, . . ., r the equivariance parameters {ϵ i } vanish8 in the topological case, while they stay finite for the exotic case.Finally, we can write ω r = ω r (ϵ 1 , . . ., ϵ r−1 ) by virtue of (2.10).We point out that (3.2) for topological and exotic case are simply two equivalent rewritings of (2.7).
Taking the Quotient.Naively, only holomorphic functions of charge t = 0 mod h along x descend to the quotient.However, note that L 2r−1 α (h, ±1) has extra topological sectors labelled by Z h in the fundamental group.These are in bijection with flat connections of line bundles over L 2r−1 α (h, ±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 [45] and flat connections in the locus are characterised (up to gauge transformations) by representations Hom(Z h , U (1) rk G ) (up to conjugation).These can be labelled by certain elements m in the Cartan subalgebra of G (e.g., for ).Hence, flat line bundles become part of the locus and apart from functions with t = 0 mod h (which descend to functions on L 2r−1 α (h, ±1)), for fixed m we should also allow for ones of charge which descend to sections of the corresponding flat line bundle on L 2r−1 α (h, ±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 (3.2) to t such that (3.4) is satisfied.The partition function then becomes with S cl 5 as in (2.9) but due to the Chern-Simons term [46].Note that now the topological and exotic choice describe different theories for h > 1.

Reduction to the Base
We arrive at the quotient space B r−1 α = S 2r−1 α /S 1 by taking the limit h → ∞ where the S 1 -fibre shrinks to a point (a formal treatment of this limit for 5-manifolds can be found in [21], appendix A).For r = 2 the theory for the N = 2 vector multiplet on S 3 α descends to the one for an N = (2, 2) vector multiplet on B 1 α .Similarly, for r = 3 the N = 1 vector multiplet on S 5 α descends to an N = 2 vector multiplet on B 2 α .Depending on whether we reduce along the topological or exotic direction, we obtain two differently twisted theories on B r−1 α ; we comment on this at the end of the subsection.
In the limit h → ∞, for a fixed root α and topological sector m there is only a single slice B t,α satisfying (3.4) which corresponds to the holomorphic sections of charge Moreover, it can be shown (in analogy to [21]) from dimensional reduction of the fields that the flat connections in the locus of the theory on L 2r−1 α (h, ±1), in the limit h → ∞ give rise to gauge connections with flux on the two-cycle in B r−1 α .This flux is of course labelled by m whose components are now integral (e.g. for 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 α is then obtained by 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., [26,27,[47][48][49][50]).Here instead, we sum over the "physical" flux on the two-cycle in B r−1 α .Explicit expressions for the one-loop determinants in each flux sector and classical actions S cl 2r−1 on B r−1 α are presented in section 4 and 5. Instanton contributions to Z B r−1 α 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 (2.2), resp.(2.5).In 4d they are instances of a general cohomological twisting called Pestunization [28,29].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 [1].
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 B t,α 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 α we have anti-instantons at one of the three fixed points.
4 Partition Function on S 3 α /S 1 We now set out to compute the integrand of Z B 1 α explicitly.Before considering the reduction, however, we first perform some simplifications that arise from the specific form of the double sine function.

Cancellations on S 3 α
We start computing the integrand of Z S 3 α 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 α , note that cancellations occur in 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: where we used (3.3) and (2.10) to introduce the equivariance parameter ϵ 1 .Note that one product in Z top α is over t ≥ 0 and one over t ′ ≤ 0. Let us also introduce a ± ∈ Z satisfying which always exist (by virtue of Bezout's lemma).In particular, gcd(a − , α 2 ) = 1 = gcd(a + , α 1 ).(4.5) 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 In order to identify the cancellations occurring it is useful to plot the slices B t,α and Bt,α of the cone Z 2 ≥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 α 1 , α 2 = 1 two points survive the cancellations in each slice, for α 1 ̸ = α 2 there might only be one or no such point at all.More explicitly, one can check that the bounds Figure 1.We plot the cone for S 3 α which is simply the lattice spanned by integers n 1 , n 2 ≥ 0. For various values of α 1 , α 2 we plot the regions B t,α and Bt,α 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 top depending on the values of α 1 , α 2 .From left to right: (i) where x y is the remainder of x/y.Similarly, the bounds for negative t ′ are the following: For α 1 , α 2 = 1 these reduce to bounds appearing in [20] for the reduction from S 3 to S 2 .In order to see how cancellations work in the general case, note that and similarly for a − t α 2 ̸ = 0. Therefore, depending on whether a + t α 1 = 0 and/or a − t α 2 = 0, only the term for ñ1 = 0 and/or ñ1 = t/α 1 α 2 survives.Thus, employing the condition in (4.5), we find We can now substitute in (4.2): For α 1 , α 2 = 1 this expression reproduces the perturbative partition function for S 3 .For generic values of α 1 , α 2 the slicing along t top only allows functions whose charge t is a multiple of α 1 or α 2 .
Exotic Theory.We follow a similar procedure here, where Plots of the slices B t,α , Bt,α of Z 2 ≥0 are shown in Figure 2. Again, only modes on the n 1 , n 2 -axis survive after cancellations.Explicitly, the bounds on ñ1 = n 1 α 1 ∈ Z ≥0 are the following: We plot the cone for S 3 α which is the lattice spanned by integers n 1 , n 2 ≥ 0. For various values of α 1 , α 2 we plot the regions B t,α and Bt,α 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 ex depending on the values of α 1 , α 2 .From left to right: (i) (4.12) 9 We henceforth combine t ≥ 0 and t ′ ≤ 0 into a single t ∈ Z.
Similarly to the previous case, due to (4.8), a mode survives cancellations if either a + t α 1 = 0 or a − t α 2 = 0.The same bounds hold for t ′ = −t.Combining the contributions from both t and t ′ in (4.3) and employing (4.5), we find Again, for α 1 , α 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 (3.3).

Finite Quotient and Reduction
In section 3 we explained that the partition function on the branched lens space L 3 α (h, ±1) can be obtained simply by imposing the condition (3.4) on t.Doing so for Z top  α is obtained by taking the limit h → ∞ as explained in section 3.2.This gives ) It is important to note that the first and second factor only contribute at α(m) ∈ α 1 Z, resp.α(m) ∈ α 2 Z.For α 1 , α 2 = 1 and ϵ 1 = 1 one recovers 11 the result for a smooth S 2 [14,51] while for α 1 = α 2 we reproduce [5].Note that ϵ 1 is different in (4.16) and (4.17) according to (3.3).
10 See [12] for related work. 11To match with the expression on S 2 we need to perform a constant shift of the Coulomb branch parameter a → a − im.Such shift has also been discussed in [20,21].It will also appear in Appendix B in order to match with the result for the spindle in [11].
At the end of this section we want to point out an interesting observation.In [11] the partition function on CP 1 (N 1 ,N 2 ) × S 1 has been computed for two different backgrounds, called twist and anti-twist (here 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 CP 1 (N 1 ,N 2 ) is in perfect agreement with (4.16) and (4.17) for twist, resp.anti-twist under the identification α i = N i , i = 1, 2. We will have more to say about this matching in section 7.
5 Partition Function on S 5 α /S 1 In this section we compute the partition function for an N = 2 vector multiplet, at each flux sector but at the trivial instanton sector, on spaces B 2 α 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 N = 1 vector multiplet on the branched cover of S 5 .Unlike r = 2, the perturbative partition function on S 5 α 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 α i .

Finite Quotient and Reduction
Contrary to the case of r = 2, here both products in the triple sine function (3.2) 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 (xn 1 + yn 2 + z) in terms of which we can write the triple sine function as Here, ω 3 is obtained from (2.10) as top: (5.4) In section 3 we explained that the partition function on L 5 α (h, ±1) can be obtained from (5.1) simply by imposing the condition (3.4) on t.For fixed m this gives where ∆ denotes the root set of G.The partition function on the quotient space B 1 α is obtained by taking the limit h → ∞ as explained in section 3.2.This gives (5.8) Note that the slices B t,α in the exotic case are non-compact and the products infinite.Thus, in order to make sense of (5.8) they need to be suitably regulated, e.g. using zeta-function regularisation.
The classical action S cl 4 on B 2 α is obtained from the one on L 5 α (h, ±1) by taking the limit h → ∞ such that the product g 2 YM,4d := g 2 YM • h is kept fixed: ρ 5 tr a 2 , (5.9) where we have introduced the shorthand . Note that, contrary to the r = 2 case, the classical action has no flux-dependence12 .

Examples
The essential information needed to compute (5.7), resp.(5.8) for explicit examples are the slices B m,α .In this subsection we consider two different types of slicings corresponding to choices where and illustrate them for specific examples 13 .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 α
≥0 are depicted in Figure 3 for topological and exotic theories.For h → ∞, only a single slice survives and contributes at the flux sector t = α(m).We show the (n 1 , n 2 )projection of such slices for the topological and exotic theories for two different choices of m in Figure 4 and 5. Looking at these slices we notice that n 2 only comes in multiples   of α 2 = 2.For generic values of α 1 , α 2 , α 3 one can check that n i comes in multiples of α i .
Let us introduce ñi := n i α i ∈ Z and rewrite the partition functions as follows: where, the new slices are defined for α(m) (5.12) The slices B m,α are shown in Figure 6 for α(m) = 12.Note that whenever no light blue points appear in the plots, then all points contribute to the slice.For generic values of α(m) the slices do not necessarily start from the origin and then certain shifts depending on α(m) α 1 and α(m) α 2 need to be included in the definition of B m,α .
In the case α 1 , α 2 , α 3 = 1 the result for both topological and exotic theories agree with those on CP 2 in [20].More interestingly, Z top B 2 α is in agreement with the equivariant index on weighted projective space CP 2 (N 1 ,N 2 ,N 3 ) in [23] under the identification (5.13) Specifically, for any value of α(m) we can match 15 both, the size and the shape of B m,α with the slices of [23].However, in [23] 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 ̸ = 1 momentarily.Slices B t,α ⊂ Z 3 ≥0 for k 2 = 3, k 3 = 5 and different values for t are shown in Figure 7.For h → ∞, only a single slice survives and contributes at the flux sector t = α(m).We show the (n 1 , n 2 )-projection of these slices in 8.We can now understand the dependence on k 1 .Let us recall that the slices B m,α are defined to be the pairs (n 1 , n 2 ) ∈ Z 2 ≥0 satisfying: (5.14) If k 1 = 1 this condition translates into: (5.15) Instead, if k 1 ̸ = 1, only pairs (n 1 , n 2 ), for which there exists n 3 ∈ Z ≥0 such that (5.14) holds, contribute.We present slices for B 2 2,2,1 in Figure 9. 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   .16)and satisfying (5.15) contribute to the slice B m,α .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 α and Z B 2 α agree with the respective oneloop determinant on CP 1 (N 1 ,N 2 ) and 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 α , resp.B 2 α .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 branch indices of B 2 α .Rather than weighted projective space, for choice (ii) the corresponding orbifold is an orbifold projective space [53].

Factorised Expressions
In the previous section we have computed the one-loop determinant for fixed m and the classical part of the partition function on B 2 α .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.[21] 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 CP 2 [20] with the only difference being an overall rescaling by 1/(α 1 α 2 α 3 ) in front of the shift proportional to α(m) and some shifts depending on α(m) α 1 and α(m) α 2 .Thus, we focus on the factorisation of choice (ii) here.Let us start by expressing the five-dimensional perturbative partition function (3.2) as a product of contributions from neighbourhoods Ĉ2 ϵ 1 ,ϵ 2 × S 1 around the fixed fibres (i.e. the vertices of the toric diagram; Ĉ2 denotes the respective branched cover of C 2 ).As for an ordinary S 5 , the factoised expression for the perturbative partition function on S 5 α is determined by a choice of imaginary part of the equivariance parameters 16 ϵ 1 , ϵ 2 , and thus of the vector x.Then, we have the following factorisation property: where i labels the three fixed fibres and the local equivariance parameters for the T 3 -action are ϵ i 1 , ϵ i 2 , β −1 i .These, together with the parameter s i can be read off from Table 1.The Table 1.Local equivariance parameters for the topological (left) and exotic (right) theories.For ease of notation we relabeled β−1,top Υ i -functions are defined as follows: The lattices D i , D ′ i ⊂ Z 2 depend on the imaginary part of ϵ 1 , ϵ 2 and are closely related to those appearing in [21] 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 β −1 2 , β −1 3 do not lead to points on the lattice.This happens because the distance from the shift by β −1 1 is not integer.
• When k 1 ̸ = 1 there is an extra complications as not all points in the regions D i , D ′ i have to be considered, as it is obvious from (5.16).
Thus, we define the regions D i , D ′ i as in eq. ( 6.3) [21] but consider all values of 17 (i, j) ∈ R 2

≥0
such that when they are shifted by β −1 2 , β −1 3 they belong to the sublattice defined in (5.16).Once the factorisation of the perturbative partition function on S 5 α is determined, the one-loop contributions at each topological sector for L 5 α (h, ±1) and B 2 α follow simply by restricting t: 16 Each individual contribution is affected by the choice of imaginary part.However, the partition function is independent of this choice. 17As in [21], depending on the regularisation, i, j can be greater than 0 or 1.
Note that, as for quasi-toric four-manifolds [21], 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 α and weighted projective space CP 1,2  N , the factorisation also agrees.

Discussion
So far we have computed the one-loop determinants at all flux sectors and classical action for the 2d N = (2, 2) vector multiplet on a branched cover B 1 α of S 2 , (4.16)-(4.18)and the 4d N = 2 vector multiplet on a branched cover B 2 α of CP 2 , (5.7)-(5.9).We obtain these results from dimensional reduction of the 3d N = 2 and 5d N = 1 vector multiplet on branched spheres S 3 , resp.S 5 .These are principal S 1 -fibrations over B 1,2  α and the reduction is implemented, according to [21], 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.

N
Let us now draw the connection to weighted projective spaces.First, [24] showed that for the case of only a single non-trivial branch index α 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 2πin/q (n = 0, . . ., q − 1) of the fields associated to the covering.On the other hand, in 2d [22] showed that the theory on S 2 with a 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 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 /Z q .
In section 4 we have observed that both, the one-loop determinants for topological and exotic theories on B 1 (α 1 ,α 2 ) agree with those on weighted projective space CP 1 (α 1 ,α 2 ) for every flux sector.This proves (7.1) also for the case α 1 ̸ = α 2 .Note that the factorisation can be traced back to the following resummation property [11] of the index for a corresponding twisted Dolbeault complex on CP 1 α (see Appendix B for notation): The one-loop determinant on CP 1 α is extracted from this index.For α 1 = q, α 2 = 1 the left-hand side precisely yields the product in (7.1).A similar resummation occurs for the complex of the topological twist on CP 2 N [23] and, indeed, we observed in section 5 that for choice (i) the one-loop determinant for the topological twist on B 2 (α 1 ,α 2 ,α 3 ) agrees with the one on CP 2 (α 2 α 3 ,α 1 α 3 ,α 1 α 2 ) .We therefore make the following Conjecture 7.1.For a fixed flux sector m, the one-loop determinant for the 4d N = 2 vector multiplet for exotic theories agrees on the two spaces B 2 α and CP 2 N , for gcd(α i , α j ) = 1, i = 1, 2, 3 and the weight vector N as in (5.13).
Note that we require N in terms of the branch indices α to be such that the singular locus on CP 2 N matches precisely the branch locus on B 2 α (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 α for choice (ii) and orbifold projective spaces.

Full Partition Function on CP 2
N .Let us now comment on instantons in the theory on CP 2 N .By a standard argument [26,28], the instanton part of the partition function is given by a product of Nekrasov partition functions on C 2 [30][31][32][33][34][35] 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 The shifts in a and the local equivariance parameters are discussed in section 6.For the topological twist we have ℓ = 3 (i.e.instantons at all three fixed points) while for the exotic theory we have ℓ = 2 (i.e.anti-instantons at one fixed point).Consequently, the full partition function on weighted projective space CP 2 N reads with S cl 4 in (5.9) and Z B 2 α in (5.7), resp.(5.8).We believe a similar expression holds for orbifold projective spaces employing the shifts in a and the equivariance parameters in Table 1.

Future Directions
Branched Coverings of Toric Sasakian Manifolds.A natural extension of this work is to consider the N = 1 vector multiplet on branched covers M α of other five-dimensional toric Sasakian manifolds M .We obtain these in a similar fashion by extending the periodicities of the three angles.We again find free S 1 -actions for these spaces (from analysing corresponding locally free actions on the unbranched five-manifold; these do not necessarily coincide with the Reeb anymore) along which we can reduce.The B t,α involved in quotienting will now be slices not of Z r ≥0 but of the dual moment map cone of M .With these modifications we would obtain the partition function on four-dimensional branched covers of quasi-toric manifolds [21,29].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 [54] which are S 1principal bundles over branched covers B 1 α × B 1 α ′ .According to our argument, it should be possible to reproduce the one-loop determinant including flux contributions on CP 1 α ×CP 1 α ′ .
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 Z α i -subgroup of U (1) acting trivially.Consequently, the singularities are of the form C r−1 /Z α 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 HP 1 α , which is an S 4 with orbifold singularities at the two poles of the form C 2 /Γ for a finite subgroup Γ ⊂ SU (2).One could envision to obtain instantons in 4d from flat SU (2)-connections on finite quotients of S 7 .which p takes the form D 2 × I d−2 → D 2 × I d−2 , (z, x) → (z α , x).The positive integer α is called the branch index of ỹ 18 .
For our case, we take M = S 2r−1 and L to be a codimension 2 subspace From the toric viewpoint which we take frequently, S 2r−1 can be seen as a Lagrangian T rfibration 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 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 with angles ϕ ∈ [0, π/2] and ∆θ i = 2π.In order to move to a branched covering S 3 α , we can simply extend the periodicity of the azimuthal angles to ∆ θi = 2πα i , which we now write as θi , to distinguish from the canonical periodicity of θ i .The branch locus is comprised of the two S 1 -fibres where ϕ = 0 and ϕ = π/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 α is an α 2 -fold (resp.α 1 -fold) covering of S 3 .Then C 1 ⊂ L has a closed neighbourhood homeomorphic to D 2 × S 1 on which the covering takes the form (z, e iθ 2 ) → (z α 1 , e iα 2 θ 2 ), i.e. points on p −1 (C 1 ) have branch index α 1 ; similarly, points on p −1 (C 2 ) have branch index α 2 (which we record in the subscript α = (α 1 , α 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 θ i .
For the round metric on S 3 , the corresponding metric on S 3 α is given (in terms of the 2π-periodic angles) by Also from this metric we see that close to the fibres at ϕ = 0 (resp.ϕ = π/2) there is a conical singularity with surplus angle 2πα 1 (resp.2πα 2 ).Clearly, the T 2 -action on S 3 lifts to S 3 α 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, Z)-transformations) by the vector field 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 α 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.over a weighted projective space19 CP r−1 N .Note that CP r−1 (1,...,1) = CP r−1 and (A.9) is simply the Hopf fibration.This case has been studied in [20].
The orbifold structure of CP r−1 N can be inferred from (A.8) as follows: denote by [z] a point in CP r−1 N with z j ̸ = 0 for some j ∈ {1, . . ., r} and let m be the greatest common divisor of the corresponding set of weights N j .Then the isotropy group Γ [z] of [z] is isomorphic to Z m .In particular, we have Γ [z] ≃ Z N i for [z] = [0 : . . .: 0 : z i : 0 : . . .: 0] with z i ̸ = 0 and, due to gcd N = 1, at points where z i ̸ = 0 for all i ∈ {1, . . ., r}, Γ [z] is trivial.
Three Dimensions.In this case, (A.9) simply corresponds to a Seifert fibration of S 3 with two exceptional fibres (N 1 , β 1 ), (N 2 , β 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 impose20 N 1 β 2 + N 2 β 1 = 1.The base space, which topologically is S 2 , has conical singularities at the two poles with deficit angle 2π/N 1 and 2π/N 2 , respectively.This space is commonly referred to as a spindle.
The orbifold fundamental group of CP 1 N is trivial, its second integral cohomology is Five Dimensions.In this case, the singularity structure of (A.9) can be more interesting.
Viewing CP 2 (N 1 ,N 2 ,N 3 ) as a toric orbifold for the moment, the ith facet F i of its polytope is located at z i = 0. Hence, points in the interior Fi have isotropy group isomorphic to Z gcd(N j ,N k ) , where j, k ̸ = i.Moreover, the facet F i itself corresponds to a spindle 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 Z N k , where i, j ̸ = k.The singularity structure is displayed in Figure 10.As an example, convenience, let us start by introducing some notation from [11]: where m ± ∈ Z, n ∈ Z and α denote the roots of the gauge algebra.The notation ⌊x⌋ denotes the floor function and σ = ±1 for the topologically twisted and exotic theory, respectively (or the twist and anti-twist in the language of [11]).
The one-loop contribution to the partition function for the 3d vector multiplet on CP 1 α × S 1 can be obtained from an index computation of a twisted Dolbeault operator and the index reads Here, q ± = q 1/α 1,2 and ω ± = e 2πi/α 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: where (z; q) n is the n-th q-Pochhammer symbol

B.2 Dimensional Reduction
In order to match with our results in 2d we compute the small radius-limit where the S 1factor shrinks.Our computation follows the dimensional reduction for the superconformal index in [51].We set where β 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: as r ∈ 2Z and n ∈ Z.In the limit β → 0 the following relation holds: where This result is the one-loop at each flux sector on CP 1 α .The extra power of β, which is needed to recover the result on S 2 starting from the superconformal index with σ = −1 for α 1 = α 2 = 1, is discussed in section 8.1 of [51].

B.3 Spindle
First we rewrite we find that (B.16) The two expressions above simplify noting that (B.17 We are now ready to substitute into the one-loop determinant (B.12).
Topological Twist.This is the case of σ = 1 and we find This constant shift is such that also the classical contribution to the partition function matches [20].
Exotic.This is the case of σ = −1 and we find where the first term contributes only if α(m) ∈ α 1 Z and the second only if α(m) ∈ α 2 Z.This agrees with (4.17) for ϵ 1 = 1 and replacing a ↔ η.Also in this case the classical contributions can be shown to match what we found reducing from the branched cover.

(4. 15 )
The partition function on the quotient space B 1

2 Figure 9 .
Figure 9. Slices for B 2 2,2,1 .Left side: B m,α of the topologically twisted theory for α(m) = 3.Right side: B m,α of the exotic theory for α(m) = 3.Only points in light blue contribute to the slices.The shape of the slice is as for CP 2 [20] but, here, only certain pairs (n 1 , n 2 ) contribute.

Figure 10 .
Figure 10.Left-hand side: singularity structure of CP 2N 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 α 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.