Superconformal Partition Functions and Non-perturbative Topological Strings

We propose a non-perturbative definition for refined topological strings. This can be used to compute the partition function of superconformal theories in 5 dimensions on squashed S^5 and the superconformal index of a large number of 6 dimensional (2,0) and (1,0) theories, including that of N coincident M5 branes. The result can be expressed as an integral over the product of three combinations of topological string amplitudes. SL(3,Z) modular transformations acting by inverting the coupling constants of the refined topological string play a key role.


Introduction
Topological strings have been defined perturbatively, but it is certainly interesting to ask whether one can find a non-perturbative definition for them. In a strong sense topological strings, which capture the BPS content of the deformations of the superconformal theories, compute relevant amplitudes for supersymmetric partition functions of superconformal theories. Thus one idea is to reverse the statement and define non-perturbative topological strings using supersymmetric partition functions.
The relation between topological string partition functions and superconformal index for N = 1 5d theories has been explored in [1,2]. The aim of this paper is to extend this relation in two directions: Given the relation between superconformal partition functions and topological strings we come up with both a definition of nonperturbative topological strings on the one hand, and also a proposal for how to use topological strings to compute certain supersymmetric partition functions. In particular, we focus on the partition function of N = 1 superconformal theories in 5d on S 5 and superconformal N = (2, 0) and N = (1, 0) theories in 6d on S 5 × S 1 . The perturbative parts of the superconformal partition functions were computed for certain gauge theories on S 5 [3][4][5][6][7], and using this ingredient and the condition that the BPS content captured by topological strings behaves as the fundamental degrees of freedom of the theory, an idea advanced in [2], we propose not only a way to compute the full answer for superconformal partition functions on S 5 , but also a non-perturbative definition for topological strings. Moreover by viewing 6d (2, 0) and (1, 0) superconformal theories compactified on S 1 as a supersymmetric system in 5d, we are able to also compute the superconformal index for a large class of (2, 0) (and in particular N coincident M5 branes) and (1, 0) theories in 6 dimensions.
The highly non-trivial aspect of this proposal is that the full non-perturbative aspect of the topological partition function enters because we have coupling constants of topological strings inverted. In particular, roughly speaking the proposal for the non-perturbative topological string partition function Z np takes the form (which will be made more precise later in the paper) 1 Z np (t i , m j , τ 1 , τ 2 ) = Z top (t i , m j ; τ 1 , τ 2 ) Z top (t i /τ 1 , m j /τ 1 ; −1/τ 1 , τ 2 /τ 1 ) · Z top (t i /τ 2 , m j /τ 2 ; τ 1 /τ 2 , −1/τ 2 ) where t i , m j are normalizable and non-normalizable Kahler classes, and τ 1 , τ 2 are the two couplings of the refined topological strings. Of course to define exactly what this means we have to be more precise and we use the BPS degeneracies captured by topological strings to give a precise meaning to Z np . Furthermore, the superconformal partition function on S 5 is written in terms of this composite non-perturbative Z by where m j are interpreted as mass parameters and τ 1 , τ 2 can be viewed as squashing parameters for S 5 . The relevant 5d theories we consider can be viewed as compactification of M-theory on singular loci of Calabi-Yau manifolds where some 4-cycles have shrunk [8][9][10][11][12]. For a subset of these, which geometrically engineer a gauge theory [13], Z top can be identified with the 5d gauge theory partition function [14], with τ i = i . We can also consider 6d superconformal theories: There are two classes of them, with (2,0) or (1, 0) supersymmetry. A large class of these theories can be obtained as F-theory on elliptic 3-folds (in the case of (2, 0) it corresponds to a constant elliptic fiber). Compactifying these theories on a circle down to 5 dimensions leads to dual descriptions involving M-theory on elliptic Calabi-Yau threefolds. Upon further compactification on S 5 , we can use the resulting non-perturbative topological string on elliptic Calabi-Yau threefold to compute the partition function on S 5 . This leads to the partition function of the 6d theory on S 1 × S 5 , i.e. it leads to the computation of the 6d superconformal index, where m j correspond to fugacities for flavor symmetry and τ 1,2 correspond to parameters of supersymmetric rotations on S 5 . Moreover one of the fugacities m i corresponds to the Kahler class τ of the elliptic fiber. This will correspond to the extra parameter in the superconformal (1, 0) theory. For the (2, 0) theory the superconformal index depends on 4 parameters. In this case the corresponding topological theory computes the partition function of N = 2 * gauge theory in 5 dimensions and the mass and coupling constant of the gauge theory correspond to the two additional parameters 1 By analytic continuation this can also be written in the form Z np = Z top · Z top · Z top . of the (2, 0) 6d index (see [5] for a related discussion). Thus, we are able to compute the superconformal index for N = (2, 0) systems in 6 dimensions.
We can also consider Lagrangian defects of topological strings. These lead to 3d theories living on the non-compact part of the M5 brane wrapping the Lagrangians.
Upon compactification on S 3 these can also be viewed as a non-perturbative completion of the open topological string, which has already been considered in [15][16][17][18] The organization of this paper is as follows: In section 2 we review the relation between open topological strings and the S 3 partition function of M5 branes wrapping Lagrangians in CY. In section 3 we study the partition function of the N = 1 superconformal theories in 5 dimensions. In section 4 we propose a non-perturbative definition of topological strings which can be used for the computation of these amplitudes. In section 5, we offer a possible explanation of our results from M-theory. In section 6 we discuss the connection with 6d superconformal indices and in particular compute the superconformal index for coincident M5 branes. In section 7 we present our conclusions. Some more technical aspects of the paper are presented in Appendices A,B and C.

SCFT on squashed S 3 and open topological strings
One of the common themes that have emerged in the study of superconformal theories in various dimensions is the important role played by the BPS states that arise when one moves away from the superconformal fixed point (see [2] and references therein).
In particular it was shown in [2] that the superconformal index in diverse dimensions is deeply related to BPS spectrum and this data can be used to fully compute the index in N = 2 theories in d = 3 and N = 1 theories in d = 5. These correspond to partition functions on S 2 × S 1 and S 4 × S 1 respectively. Here we are interested in computing the partition functions of these theories on S 3 and S 5 , respectively. To this end, it is instructive to review the case of N = 2 superconformal theories on the squashed three-sphere S 3 b . This class of theories is particularly simple, since away from the superconformal point only a finite number of BPS particles appear, which are in one-to-one correspondence with the electrically charged fields of the SCFT. The full partition function for these theories has been computed exactly [19,20] and indeed we will see that it can be reinterpreted in terms of contributions coming from the BPS particles (as occurs in a similar context in [15][16][17]).
For ω 1 = ω 2 , the SO(4) isometry group of S 3 gets broken to U (1) × U (1). The ratio of the equivariant parameters for the two rotations is We now recall the partition function for superconformal gauge theories on the squashed three-sphere, whose gauge and matter content are provided respectively by vector and chiral multiplets. Away from the superconformal point, many of these theories can be constructed from M-theory as the worldvolume theories of M5-branes wrapping S 3 b times a Lagrangian submanifold of an appropriately chosen Calabi-Yau threefold X. The geometry of X determines the BPS content of the theory, and the superconformal theory is recovered in the IR (shrinking the size of the Lagrangian to zero).
Let g be the Lie algebra of the gauge group G, and h its Cartan subalgebra.
Let h i , i = 1, . . . , rank(G) be a basis for h. We denote a generic element of h by φ = φ i h i , and for an arbitrary weight ν of g we write φ ν = ν, φ . By localization, the computation of the partition function of the SCFT reduces to an integral over h, with contributions from one-loop determinants for the chiral and vector multiplets: where ∆ + is the set of positive roots of G. The classical action can contain Chern-Simons and FI terms, and produces a factor of where k i is the CS level and ξ i ∈ R is the FI-term. For abelian factors we can also have additional off-diagonal CS interactions as well as mixed CS terms with flavors symmetries.
If we include matter fields in a (not necessarily irreducible) representation R of the gauge group G, for each weight in R we obtain a chiral multiplet. The one-loop contribution to the partition function is where the double sine function S 2 (z|ω 1 , ω 2 ) is defined in Appendix A.1. The vector multiplet, on the other hand, contributes a factor of (taking into account the shift in spin s = 1/2) where by ∆ we mean the set of roots of G. Note that for a spin s field we get a shift of ( 1 2 + s, Putting all the pieces together, the partition function is Thus to each multiplet α corresponds a factor of S 2 (z α |ω 1 , ω 2 ) ±1 , where the argument of the double sine function depends on the data attached to the multiplet. Note that for the vector multiplet the β∈∆ S 2 (iφ β + ω 1 + ω 2 |ω 1 , ω 2 ) is equal to a q-deformed Vandermonde. The double sine has simple modular transformation under the S transformation of SL(2, Z). Indeed, when τ = ω 1 /ω 2 ∈ H the double sine function can be written in the following suggestive form (A.11): where we have defined ζ α = 2πiz α /ω 2 ,ζ α = ζ α /τ ,τ = −1/τ , and q = exp(2πiτ ) and q = exp(−2πi/τ ). The exponential prefactors come from the (2, 2) multiple Bernoulli polynomial (A.8), Under an S modular transformation that takes τ →τ and ζ α →ζ α , On the other hand, the double sine function does not transform into itself under the T transformation τ → τ + 1, so we cannot complete this to a full SL(2, Z) action.
We would now like to clarify the relation with BPS states and open topological string theory. For this purpose, it is convenient to strip away the prefactors from the double sine function and define Using the building block of the double sine function we can write down the contribution of particles of charges n i , n j under U (1) gauge factors and flavor factors respectively with central terms (x i , m j ) (before gauging) and spins s: Thus we would get for many particles a partition function of the form: where we have included the prefactor (involving the exponential of the quadratic form) which is added at the end depending on the FI terms and the CS levels (see [21] for a thorough discussion of these terms). To obtain the final partition function we have to integrate over the scalars in the U (1) vector multiplets leading to In the next section we discuss how this can be presented in the context of 3d theories living on M5 branes wrapped on special Lagrangian 3-cycles, using open topological string amplitudes.

Topological String Reformulation
We now use topological strings to reformulate this partition function (see also [17]). It is known that open topological strings captures the BPS content of M5 branes wrapped on special Lagrangian cycles of Calabi-Yau threefold [22]. For simplicity we will focus on the unrefined case here (but will extend the discussion to the refined case when considering the closed string sector). Consider M-theory compactification on a Calabi-Yau threefold, and consider a number of M5 branes wrapping some special Lagrangian cycles. Then M2 branes ending on M5 branes constitute the BPS states of the theory.
The partition function of topological strings captures this. In particular we have (up to quadratic exponential prefactor) 2 : For our purposes it is more convenient to define a slightly shifted version of the topological string amplitude given bỹ where q = exp(2πiτ ) and N n i ,n j ,sa denote the number of BPS states with the corresponding charges as spin. We will drop the tilde in the rest of the paper as we will be mainly discussing this shifted version. The unshifted version can be recovered by shifting the τ back.
We now simply ask what would the partition function of this theory be if we were to put it on the squashed S 3 ? Even though we have no a priori Lagrangian description of this theory we will assume, as in [2], that the BPS states can be treated as elementary degrees of freedom. Using the fact that double sine computes the corresponding term we would thus naturally get where by definition what we mean by Z top at −1/τ is the product expression we have given. Notice that the factor of (−1) s in the expansion, which for even s does not seem to affect the perturbative Z top , will be relevant under the τ → −1/τ , which we include in the definition of Z top at −1/τ .  Such a computation has been carried out in [3,4] for certain BPS particles which appear as the perturbative part of the partition function of N = 1 superconformal field theory on S 5 with non-abelian gauge group and matter in an arbitrary representation R. We review this result and propose a generalization of it to particles of arbitrary spin. This is also important to us for another reason: As in the 3d case, even if the gauge theory is non-abelian, the computations can be entirely recast in terms of an integral over the abelian Coulomb branch parameters, where the non-abelian aspects are reflected by the existence of additional BPS states in the computation. This allows us to formulate our final result in term of an integral over the Coulomb branch.
In the perturbative computation, the path integral localizes on the Cartan subgroup of the gauge group, and the hyper and vector multiplets, which correspond respectively to the matter and gauge content of the theory, contribute the following one-loop determinants evaluated on the localization locus: where µ are the weights in the representation and φ is an element of the Cartan, and where ∆ + denotes the positive roots of the gauge group.
In appendix B we show that these expressions can be recast in terms of triple sine functions [23][24][25][26] as and up to a prefactor which can be reabsorbed into the cubic prepotential. The triple sine function is defined as a regularized infinite product over three indices: In the theory on S 3 an interesting deformation was obtained by introducing squashing parameters ω 1,2 , and the one-loop determinants were found to be built out of factors of S 2 (z|ω 1 , ω 2 ). In our current setup, it is also very natural to move away from this limit and consider an analogous deformation by three parameters ω 1,2,3 . That is, we conjecture that one can formulate a deformation of the theory on squashed S 5 , which can be embedded in C 3 as and that each occurrence of S 3 (z|1, 1, 1) gets replaced by S 3 (z|ω 1 , ω 2 , ω 3 ). The SO (6) isometry of S 5 gets broken to is given by ω i /ω j .
The hyper and vector multiplet one-loop determinants become and Putting all these contributions together, the perturbative contribution to the partition function (choosing units where the radius of S 5 = 1) is which comes from the tree level Lagrangian (where we have included the effect of ω i being turned on). Notice that this term is the exponential of a cubic polynomial where C captures the cubic content of the prepotential term, where we view 1/g 2 Y M as a scalar in an ungauged vector multiplet. Just as in the 3d case the non-abelian measure factors have disappeared and we can interpret the integrand as the contribution of the electric BPS states in an abelian theory, as we go away from the conformal fixed point on the Coulomb branch. However, unlike in the 3d case, here there are more BPS states than those captured by the perturbative content of the theory. In fact, the five-dimensional theory will have an infinite number of BPS states, including ones which carry instanton charge. Our proposal is that the full partition function on squashed S 5 is simply given by the contribution over all BPS states and not just the electric ones. In other words, we propose: where each Z α is a contribution from a BPS particle written in terms of triple sine function (and its generalization), and Z 0 (φ) = e C(φ,m) is the effective semi-classical contribution and is a polynomial of degree 3 in φ and m. By Z α we mean the determinant contributions coming from the individual BPS states with the exponential prefactor stripped off (see the next section for more details). This proposal fits naturally with the computation in [3][4][5] where the main missing ingredient was the contribution of instantons to the partition function. Here we are proposing that the BPS content of the theory, which includes instanton charged states, completes the computation.
In the case where the superconformal theory comes from a Calabi-Yau threefold, C can be related to the classical properties of the CY and captures the classical prepotential term, as well as genus 1 corrections which are linear in φ and m. In the unrefined case C is simply given by where J(φ, m) denotes the Kahler form on the CY which is parameterized by φ, m and c 2 is the second Chern class of the CY where the genus 0 piece can be read off from [27,28] and the genus 1 piece from [29]. In the refined case where τ 1 + τ 2 = 0 this We will choose normalizations where the Kahler class is given by 2πiT . In this normalization we can write this as and C ijk denotes the triple intersection and c i 2 the second Chern class in this basis. In the next section we show how topological strings capture this partition function elegantly, leading on the one hand to the full partition function for N = (1, 0) theories obtained by compactification of M-theory on toric CY threefold, and on the other hand to a non-perturbative definition of topological string.

Non-perturbative topological strings and the partition function on S 5
Consider M-theory on Calabi-Yau threefolds. It is known that topological strings capture the BPS content of M2 branes wrapped over 2-cycles of the Calabi-Yau [31,32].
Furthermore, in the case of toric threefolds (which lead to N = (1, 0) theories of interest to us here) we can consider a refinement of the BPS counting [33]. The relation between the topological string partition function and BPS state counting is given by Note that we have stripped off the classical terms, and below when we restore the classical pieces we will make it clear.
and the s i just capture the spin content (not including the I L factor): It will be useful for us to slightly change the definition of topological strings (as in the open sector discussed in the 3d context) by shifting 4 one of the couplings by 1: Since we will be mainly dealing with this object we will be calling it Z top and drop the tilde. Of course one can recover the usual definition of topological string by shifting back the coupling by 1.
In order to connect this to the partition function on S 5 we need to know how each field contributes to the partition function. Consider a field with spins (s 1 , s 2 ) (coming as part of a BPS multiplet). Then we already know that when (s 1 , s 2 ) = 0 the contribution is given by a shifted triple sine function: Moreover for a vector multiplet (0, 1/2) which has (s 1 , s 2 ) = (± 1 2 , ± 1 2 ) we get Now comes the main point. The connection to non-perturbative topological strings come to life thanks to a remarkable formula (equation (A.12)) for the triple sine function: where we have shifted the argument of the triple sine by the universal term ∆ = (ω 1 + ω 2 + ω 3 )/2, and we set T = z/ω 3 , Furthermore, q = exp(2πiτ 1 ) and t = exp(−2πiτ 2 ) and similarly for the other variables.
Each infinite product in this expression is convergent when Im τ 1 > 0 > Im τ 2 , but similar convergent expressions can be obtained in other regions (see Appendix A.1).
The expression for the triple sine function also includes an exponential prefactor which comes from the (3,3) multiple Bernoulli polynomial (A.9) with shifted argument, Taking z = z 0 = k i t i + l j m j for the hypermultiplets and and z = z 0 ± (ω 1 + ω 2 + ω 3 )/2 for the vector multiplets and choosing the gauge ω 3 = 1, one finds that the numerator in (4.1) gives precisely the contributions of the hyper and vector multiplets to the topological string partition function! Similarly when s 1 = s 2 = s and z = z 0 + s(ω 1 + The numerator in this expression also captures the contribution to the topological string partition function of a BPS states with spin (s, s). It is thus natural to propose that the triple sine function also gives the determinant for spin (s, s) states.
This triple product structure involving topological string contributions has a simple generalization for arbitrary spin (s 1 , s 2 ): . which we propose to be giving the determinant contribution for spin (s 1 , s 2 ) states.
Note that for s 1 = s 2 this differs from the triple sine function. Taking the product over all the BPS states, which we need to do according to our proposal for the computation of the partition function over S 5 , we obtain where in the above, in addition to the product over the BPS states, we have included the cubic prefactor Z 0 = exp(C(t i , m j ; τ 1 , τ 2 )). We can rewrite this expression as follows: .
The numerator is precisely the topological string partition function, and we can also relate the two factors in the denominator to the topological string partition function: and The prime signifies that these two factors of the topological string have SU (2) L and SU (2) R exchanged, which is equivalent to replacing (s 1 , s 2 ) with (−s 1 , s 2 ) (or equivalently (s 1 , −s 2 )) for each BPS state. In fact, not worrying about regions of convergence, we can use the identity to rewrite the product of BPS contributions simply as the product of three factors of the topological string partition function: Equation (4.2) can be viewed as defining a non-perturbative completion of topological string, in the sense that the two additional factors are non-perturbative, as they involve at least one τ i → −1/τ i . At the end of this section we will explain the analytic properties of Z as a function of τ i . Just to complete our discussion, in order to compute the S 5 partition function we simply have to integrate this over the directions in t i :

Contribution of the massless vector multiplet
The massless vector multiplets also make a contribution to the partition function. These contributions do not depend on the moduli but depend on the squashing parameters.
Therefore they can be brought out of the integrals over the Coulomb branch. These terms are given in the topological string context by powers of the MacMahon function.
If we have U (1) r gauge theory this leads, as discussed in [2], to If we use our prescription to compute the contribution of this factor to the partition function we get a factor of This has a zero for each U (1) reflecting the fact that we have to delete the zero mode associated to the Coulomb branch parameters and instead integrate over it, which is part of the prescription. This is equivalent to replacing S 3 with its derivative S 3 evaluated at 0. In other words, the contributions for the massless vector multiplet to the partition function is

An Example: SU(2) gauge theory
Here we present one example of how the computation is done. The case we focus on is a toric 3-fold that engineers SU (2) gauge theory coming from the O(−2, −2) → P 1 × P 1 geometry. We consider the partition function of this theory on the squashed five-sphere.
As discussed, we predict the full partition function to be , is the refined topological string partition function for the P 1 × P 1 geometry of Figure 1, which was obtained in [34] (which is the same as Nekrasov's partition function for the 5d SU (2) theory [14] with i = τ i ): The classical piece C(a, 1 ) is given by The partition function involves sums over Young diagrams. We use the following notation: ν t is the transpose of ν; |ν| denotes the number of boxes in ν; ν i is the number of boxes in the i-th column of ν; ||ν|| 2 = i ν 2 i ; for a box s = (i, j) in the i-th column and l-th row of ν, a(s) = ν t j − i and (s) = ν i − j; and, lastly, κ(ν) = 2 s∈ν (j − i). Recall that we need to shift τ 1 → τ 1 + 1 in these formulas to obtain the Z SU (2) appearing in the integrand.

Analytic properties of Z
The triple sine function (as discussed in Appendix A.1) is defined only when all three ω i are in the same half plane. If this is satisfied, the triple sine function is well defined and is an entire function which has zeroes at a lattice of points corresponding to n i t i + k j m j = (n 1 + 1 2 )τ 1 + (n 2 + 1 2 )τ 2 + (n 3 + 1 2 ) (see Appendix A.1). Similarly the function C s 1 ,s 2 (n i t i + k j m j |τ 1 , τ 2 ) has zeros and poles at values of n i t i + k j m j which can be read off from equation (C.6). It is natural to also expect that C s 1 ,s 2 is well-defined only when all three ω i are in the same half plane. The non-perturbative topological string partition function is made up of an infinite product of such functions which we conjecture to exist.

A possible derivation from M-theory
In this section we propose an explanation for the triple product structure that arises when one introduces squashing parameters for S 5 . We start by recalling in more detail the M-theory setup that computes the topological string partition function. We pick a non-compact toric Calabi-Yau threefold X, and take the remaining five-dimensional space to be the Taub-NUT space T N times the M-theory circle S 1 . We express Taub-NUT space in terms of complex variables (z 1 , z 2 ) and introduce a twist: as we go around S 1 , we rotate (z 1 , z 2 ) → (e 2πiτ 1 z 1 , e 2πiτ 2 z 2 ) (and do a compensating twist on X to keep it supersymmetric). We denote this twisted space by (T N × S 1 ) τ 1 ,τ 2 . Then it is known that [30] Z top (X, τ 1 , τ 2 ) = Z M −theory (X × T N × S 1 ) τ 1 ,τ 2 .
The M-theory partition function counts the number of M2-branes wrapping cycles in X, which project to points in Taub-NUT space. When the equivariant parameters are turned on, the particles are concentrated around the origin z 1 = z 2 = 0.
We can also consider the open string sector of topological strings, which corresponds to adding M 5 branes wrapping a Lagrangian submanifold L ⊂ X and the Melvin cigar (M C) subspace of (T N × S 1 ) τ 1 ,τ 2 , which has the geometry of S 1 × C τ 1 . Here S 1 is the M-theory circle, and C τ 1 is the plane in T N with rotation parameter τ 1 (but we could as well have chosen our M5-branes to fill C τ 2 ). In topological string theory, wrapping an M5 brane on K = M C × L translates to placing a τ 1 -brane on L [22] (see [15,35] for a discussion of the refined case). The problem of counting worldsheet instantons ending on L translates to counting the states of a gas of M2-branes which wrap two-cycles of X with boundary on L; the M2 branes project to points on the Melvin cigar. Turning on equivariant parameters again forces these particles to be concentrated at the tip of the cigar, which is located at z 1 = z 2 = 0. Then the M5 brane partition function in this setup is the same as the open topological string partition function: To obtain the partition function of the resulting theory on squashed S 3 we take a second copy of the Melvin cigar, which we denote by M C, and glue it to the first one along the common boundary (as was suggested in the topological string context in [15,16] and discussed in detail in [17]). This operation can be visualized most clearly by regarding the squashed S 3 as a torus fibration over the interval, as in Figure 2, and the T 2 is the one we have discussed away from the tips of M C and M C. Each Melvin cigar fills out a solid torus, and we glue the two after performing an S modular transformation which interchanges the two circles in M C. The only subtlety is that we need to ensure that the two cigars are twisted in a compatible way. In particular the complex structure parameter as seen from the viewpoint of one tip is different from that of the other end. This forces us to rescale the rotation parameter for M C tô Moreover the topological string has opposite orientation on the M C suggesting complex conjugation of the topological string amplitude, which is equivalent to inversion of Z.
The partition on S 3 b then is just the product of the topological string factors on the two hemispheres 5 , .
The main lesson we extract from the open string case is that for generic choices of the rotation parameters the topological string (or, equivalently, M-theory) computation localizes at the fixed points of the equivariant action on C 2 . In discussing aspects of closed strings we will have to recall that when we have a more complicated geometry made of patches which look like C 2 τa,τ b × S 1 , we would expect by localization to With this picture in mind, we wish to study the partition function on S 5 . We view S 5 as a circle fibration over CP 2 . Moreover CP 2 itself can be viewed as consisting of a T 2 over a triangle, as is familiar in the context of toric geometries (see e.g. [36]).
Thus we can think about the squashed five-sphere as a S 1 × S 1 × S 1 = T 3 fibration over a triangle, where each circle in the fiber gets rotated by a different parameter τ i (see Figure 3). In the interior of the triangle all three circles have finite size, but along the edges one of them shrinks to zero size, and the vertices are the points where two of the circles degenerate. We find it convenient to label by v i the vertex where the i-th circle of the fiber does not degenerate. We also denote by e ij the edge that connects v i and v j . It is easy to convince oneself that the neighborhood of v i looks like S 1 i × C j × C k , where i = j = k and each circle in the fiber corresponds to a different factor in the geometry. So from each vertex we expect a contribution of Z closed top 6 . To figure out the 6 Up to the factor of (−1) F because the corresponding S 1 in this case is shrinkable inside S 5 and appropriate parametrization at each vertex, one can start by setting the equivariant parameters to be (τ 1 , τ 2 , 1) at v 3 , so that we get a factor of Z closed top ( t, τ 1 , τ 2 ). We can reach the two other vertices by moving along the edges e 31 and e 32 . At v 1 the role of the M-theory circle is played by the first circle, so for the gluing along the edge to be consistent we are required to rescale the equivariant parameters by 1/τ 1 . This gives us a factor of Z closed top ( t/τ 1 , 1/τ 1 , τ 2 /τ 1 ). Similarly we learn that v 2 contributes a factor of Z closed top ( t/τ 2 , τ 1 /τ 2 , 1/τ 2 ). Collecting the contributions from the three vertices, we find that M-theory on squashed S 5 computes As explained in section 4, we can rewrite this expression in convergent form as where the factors in the denominator are to be computed with the SU (2) L and SU (2) R spins exchanged.
The non-perturbative open topological string fits very nicely in this picture: the fiber over an edge e ij consists of two non-degenerate circles S 1 i and S 1 j , which play inverted roles at the two vertices. This means that over each edge we have a squashed If we were to choose the e 23 edge, we would obtain To make this into a rigorous derivation for arbitrary toric Calabi-Yau, we would need to have a way to compactify the full M-theory on S 5 , which will necessarily involve some unconventional fields being turned on (similar to what was found in the 4d case [37]). It is natural to conjecture, given what we are finding, that such a setup gives a different spin structure compared to the usual case where S 1 is not contractible. This explains the origin of the shift τ 1 → τ 1 + 1 in the previous sections.
should be consistent, at least in the case of non-compact Calabi-Yau's. In the subset of cases where the CY engineers a gauge theory, where Z top is identified with the Nekrasov partition function, it should be possible to rigorously derive this result from the localization arguments in the path-integral.

Superconformal Indices in Dimensions
It is natural to ask whether the techniques we have introduced can be used to compute superconformal indices in 6 dimensions. This is natural because this involves computations of the amplitudes on S 5 × S 1 . Moreover, compactification on S 1 leads to a 5 dimensional theory, of the type we have studied. Also, as in the lower dimensional case studied (such as S 1 × S 4 ) turning on the fugacities and supersymmetric rotations of the S 5 should correspond to introducing squashing parameters for S 5 .
In this section we show how this can be done. The generic case of interest is superconformal theories with N = (1, 0) supersymmetry. A special case of these are the (2, 0) theories. We will discuss each one in turn.
The R-symmetry for this case is Sp(2). Let R denote its Cartan. The superconformal index in this case can be defined as follows [41]: where J ij denote the rotation generators of SO(6) acting on S 5 , and F i are charges associated to flavor symmetries (where we have only kept the terms which appear nontrivially in the partition function). The choice of the parameters q 1 , q 2 is motivated from connection with the rotations in 4d, already discussed in the context of 5d theories.
The basic idea, similar to relating the 4d index to 3d partition functions [42][43][44], is to connect the 6d index to our 5d setup by compactifying this theory on S 1 . The only subtlety is to identify the charges as well as the relation of the parameters in the lower dimensional theory with the higher dimensional theory. In the context of compactification of the 6d theory on a circle, we would need to enumerate the resulting 5d BPS states (including winding of 6d BPS strings around the S 1 ) and simply apply the formalism we have developed to this 5d theory. Here the 5d theory will have a tower of BPS states with a specific structure due to the fact that it is coming as a KK reduction from a one higher dimensional theory. If this theory is dual to M-theory on a CY then from the perspective of this 5d theory we can enumerate all BPS states using topological strings. Then using the three combinations of them and integrating over the scalars in the gauge multiplets yields the partition function on S 5 , thus effectively computing the index of the 6d theory.
Note that from the perspective of the 5d BPS counting, the KK momentum should appear as a special flavor symmetry. In the context of F-theory on elliptic CY and its duality with M-theory upon compactification on S 1 (as we will review below), this will turn out to be the winding number over an elliptic fiber. We will denote the Kahler class of the elliptic fiber by τ and define q = exp(2πiτ ), where τ is the Kahler modulus of the elliptic fiber (the reason for this terminology will become clear later).
Let M i = exp(2πim i ), where m i denote the non-dynamical fields (coming from nonnormalizable Kahler moduli). The question is what is the relation between the 5d parameters q, q 1 , q 2 , m i with the parameters appearing in the 6d index q, q 1 , q 2 , m i ? A similar situation was studied in the relation between superconformal index in 4d and the partition function in 3d [42][43][44]. In that case the squashing parameter are rescaled by a factor of R, the radius of the circle. We propose a similar relation in this case.
Using the fact that the Kahler class of the elliptic fiber in F-theory is related to R by 2πiτ = 1 R we are led to In computing the partition function on squashed S 5 we need to integrate over the dynamical fields. Let t i denote the scalars associated to the resulting gauge fields in 5d coming from 6d tensor multiplets, which are normalizable (corresponding to normalizable Kahler moduli of the CY). Then we obtain the formula . This naturally follows from our formalism. It is a general proposal regardless of whether or not we have a topological string realization of the theory: The Z top factor simply denotes the BPS partition function. However the question is how to compute the BPS partition function. If we can relate it to an actual topological string then we have techniques for its computation; the most convenient one for this purpose is the Ftheory construction, because of the duality between F-theory compactified on S 1 and M-theory on the same space [45]. Thus in 5 dimensions we obtain the theory involving M-theory on an elliptic 3-fold. Luckily topological strings on elliptic 3-folds have very nice properties and have been studied extensively [46][47][48][49][50][51]. The relation between 6d and 5d theories via F-theory/M-theory duality has also been studied in [52].
As an example, consider the superconformal theory associated with a small E 8 instanton. In the F-theory setup, this corresponds to F-theory with vanishing P 1 in the base of F-theory [8,40]. After compactification on S 1 , this gives an elliptic 3-fold containing 1 2 K3 (obtained by the elliptic fibration over the P 1 ). This theory has 10 Kahler classes: One elliptic fiber class τ , the base t b and eight mass parameters m i (to be identified with the Cartan of E 8 ). τ corresponds to momentum and t b corresponds to the winding of the 6d tensionless string along the circle [50]. The unrefined topological string for this theory was studied in [46][47][48]50]. To obtain the index for this theory we have to integrate over the t b . Similarly a large class of (1, 0) theories can be obtained by considering F-theory where the base contains more blow ups on C 2 (see [53] for a recent discussion related to this). This would entail blowing up a multiple of times, each corresponding to a Kahler parameter t i , which we will have to integrate over in computing the index (the corresponding U (1) vector multiplet in 5d arises from the 6d tensor multiplet in the same multiplet as the blow up parameter t i ). A subset of such blowups are the toric ones. These are in one-to-one correspondence with 2d Young diagrams [54]. Elliptic threefolds over these spaces, in the limit of blowing down all the 2-cycles, should correspond to a (1, 0) conformal theory. The case of a Young diagram with a single row with k entries corresponds to k small E 8 instantons. It would be interesting to study this large class of (1, 0) theories given by a Young diagram. In particular it should be interesting to compute the corresponding refined topological strings for this background. The topological string partition functions for this class of theories seem to enjoy the following perturbative modular property under the inversion of the Kahler class of the elliptic fiber [47][48][49]51]: Note the asymmetric role in the modular transformation for the dynamical fields t i versus the non-dynamical fields m j which correspond to flavor symmetries 7 . In the context of our non-perturbative completion, as we will see later in the context of the theory of M5 branes, this relation receives additional non-perturbative factors. This turns out to be rather important for simplifying the computation of the 6d index as we will discuss in section 6.5.
More generally we can consider instead of C 2 the A n−1 orbifold as the base of Ftheory. If we do not add any further blow ups, this gives the A n−1 , (2, 0) theory, which we discuss in the next section (the above modular property turns out to be important later when we compute, in our formalism, the index of an M5 brane). If in addition we also blow up the points in the base we get among the various possibilities the small E 8 instantons in the A n−1 geometry, as (1, 0) superconformal theories of the type studied in [39].
The same reasoning as in the case of (1, 0) superconformal theories leads to the following picture. The 5d theory we obtain by compactifying the (2, 0) theory is an ADE Yang-Mills theory with 16 supercharges. Turning on the fugacity Q m corresponds to turning on a mass m for the adjoint field, where Q m = e 2πim (for the identification of this with R 2 − R 1 generator of R-symmetry see [55]). In other words we can view the resulting theory as N = 2 * theory in 5d. Let Z top (t i , ; τ, τ 1 , τ 2 , m) capture the BPS partition function for this 5d theory where t i denotes the Cartan of ADE. This partition function can be explicitly evaluated for the A n−1 case using the instanton calculus [14,56] or the refined topological string [34] on the periodic toric geometry [33]. The D and E should be in principle possible, either using geometric engineering or instanton calculus for N = 2 * .
Then to compute the index we have where we have taken into account the relation between the 5d parameters and 6d parameters. In order to gain insight into the mechanics of this computation we show how it works for the simplest case, namely a single M5 brane, which corresponds to A 0 theory and recover the result of [41]. This lends support to our general proposal and more specifically to the identification of the squashing parameters and Kahler classes with the parameters appearing in the 6d superconformal index. The case of A 0 theory is particularly simple because we have no integrals to perform. In that case the nonperturbative Z we obtain is exactly the same as the perturbative one! This ends up being related to the modularity of the topological string partition function on elliptic threefolds. Moreover we discuss the possibility that this may be the general story for all (1, 0) and (2, 0) theories in section 6.5. We also show the setup for the computation for the higher A n−1 theories in the refined topological vertex formalism. We also give the expression for the index for the A 1 case in the unrefined setup as an integral over three factors of topological string amplitudes.

Index for a single M5 brane
As discussed above the case for single M5 brane corresponds to studying topological strings for N = 2 * U (1) theory in 5 dimensions. This corresponds to a periodic toric edge. The topological string partition function for this theory was worked out in [58] (see also [59]) and the result is given by 8 (after shifting τ i → τ i + 1): where Q m = e 2πim , q = e 2πiτ , q 1 = e 2πiτ 1 , and q 2 = e 2πiτ 2 , and we have included one factor of MacMahon function which is somewhat ambiguous in the computation of the refined topological string. The refined topological string captures the Kahler moduli dependence of the amplitudes and does not fix the terms purely depending only on q 1 , q 2 . In fact we will need to multiply the above expression by 1/η(q 1 ) for reasons that we will explain below, where η(q 1 ) is the Dedekind eta-function.
The spectrum of this theory consists of a tower of hyper multiplets of mass 2πi(m+ kτ ) (one for each integer k) and a tower of tensor multiplets with mass 2πikτ . This is as expected, because the reduction of a single M5 brane on a circle leads exactly to such a multiplet, where 2πiτ is identified with 1/R, with R the radius of the circle taking us from 6 to 5 dimensions. It is important to rewrite the above partition function in a more symmetric way: Let us redefine Q m by Then the partition function is totally symmetric in (q, q 1 , q 2 ), if we in addition include a factor of 1/η(q 1 ) which is ambiguous for the refined topological vertex. To see this, we have to rewrite everything in terms of positive powers of q 2 : where we delete the i = j = k = 0 terms for the second term in the numerator. The manifest permutation symmetry between q, q 1 , q 2 is expected from the fact that in the 6d they become the parameters associated to the three rotation planes. Note also that 8 We thank A. Iqbal for a very helpful explanation of this result and its modular properties.
the way we have rewritten the numerator corresponds to the fact that a tensor multiplet It is remarkable that taking the three copies of the five-dimensional partition function led to an answer which is perturbative in τ 1 , τ 2 , and we offer an explanation of it below.
Likewise, the contributions from the η factors simplify. From η(τ 1 )η(τ 2 ) we get, up to prefactor: We thus end up with A glance at equations (6.1) and (6.3) reveals that the only difference between the perturbative answer and the full non-perturbative result is a rescaling of (m, τ, τ 1 , τ 2 ) → (m/τ, −1/τ, τ 1 /τ, τ 2 /τ ), which is the correct map between the 5d and 6d parameters, as discussed above. We now offer an explanation of the fact that the non-perturbative completion of the Z top resulted in the same function in modular transformed variables. As discussed before (and which can be verified explicitly for this example), we expect a pertubative modularity of the topological string partition functions of elliptic Calabi-Yau threefold of the form: Instead what we have found in this example is that Note that the additional terms in the denominator are non-perturbative in the topological string coupling constants and thus can be viewed as a non-perturbative completion of the modularity of topological strings. We will comment on the implication of this for possible simplification for the general computation of the index of all 6d theories in section 6.5.
The same result could also have been derived from the relation between the triple sine and elliptic gamma functions (equation (A.14)), which we also report here: .
Then, using equation (A.17), we can write and . Likewise, and similarly for η(τ 1 ) and η(τ 2 ). Writing Z np U (1) = exp n I(q n m , q n , q n 1 , q n 2 ) n , we get Deleting the zero mode of G 2 (0) correspond to deleting the −1 in the above expression.
provided that we identify which is in accord with the transformation of the basis used in that paper compared to ours in writing the index.

Multiple M5 branes
Similarly we can consider multiple M5 branes. This was studied in [33] in the unrefined topological string formalism (where q 1 q 2 = 1) which can easily be generalized to the refined one (which was not developed at the time). For N M5-branes the toric geometry will involve N parallel lines wrapping the periodic direction of the toric base. See Figure   5 for the case with N = 2. The topological string will depend on one mass parameter m, on the periodic size τ , and on N − 1 moduli t i which correspond to relative separation of the horizontal lines. These are the parameters that we need to integrate over in evaluating the 6d index. It would be interesting to perform this computation in detail [? ]. This involves gluing 2N vertices of the refined topological vertex, and a sum over 3N Young diagrams attached to the internal edges, just as in the unrefined case (where τ 1 + τ 2 = 0) studied in detail in [33]. In that case, the answer for topological string partition function is given by , where M (q) is the MacMahon function, q = e 2πiτ 1 , Q = e 2πiτ , Q F = e 2πia , and Q m = e 2πim , where we have denoted e 2πiτ by Q instead of q since q parametrizes the unrefined topological string coupling constant. Also, h(i, j) = ν i − i + ν t j − j + 1 is the hook length for a box (i, j) ∈ ν, and C k (ν 1 , ν 2 ) can be computed from where f ν (q) = (i,j)∈ν q j−i . This can be extended to the refined computation which we denote by Z top (τ, a, m; τ 1 , τ 2 ), from which we would compute the full index by doing the integral over the a variable for the Z np .
6.5 Z top = Z np in 6d?
As we have seen in the context of computation of the superconformal index for a single M5 brane, the non-perturbative completion of Z top yields again Z top with modular transformed variables. This raises the question whether this is always true, namely 10 : However, as already discussed, we expect from the perturbative modularity of Z top a relation of almost this form, namely This is almost of the naive form we expected, except that t i , the dynamical variables which we need to integrate over, are not transformed under τ → −1/τ . This strongly suggests that the non-perturbative completion of the above equation is simply This would be consistent with the fact that the BPS states of the elliptic 3-fold should organize according to a tower of KK modes and for each such tower the identity 6.2 would transform the answer back to the original form except in the modular transformed variables. This would give a dramatic simplification for the computation for the 6d case.
Namely we would get (taking into account the change of parameters from 5d to 6d): where Z top is the same as the 5d gauge theory partition function (including the cubic prefactor). We are currently investigating this theory [61]. Similarly if the elliptic fibration of F-theory is constant the same construction will lead to an N = 2 theory. Here we will have one extra flavor symmetry (the analog of the mass in the N = 2 * theory discussed before) which will play the role of the additional parameter t that one can add to the index in the context of N = 2 theories in d = 4 [62]: It would be interesting to study these and explore connections with the computations already done in the literature (see [63] and references therein for examples of such computations).

Conclusion
We have provided evidence that the partition function of superconformal theories on These results complement that in [2]  The ideas in these papers suggest that the BPS states in a supersymmetric theory (with enough supersymmetry) go a long way in defining the superconformal fixed points they come from. It would be very interesting to see whether this can be made into a systematic method for defining the full superconformal theory.

Note added
After the completion of this paper a number of other papers appeared [65][66][67]  In this appendix we provide the definition and relevant properties of the multiple sine and multiple elliptic gamma functions [23][24][25][26]. We begin by defining the multiple zeta functions ζ r (z, s|ω) = ∞ n 1 ,...,nr=0 for z ∈ C and Re s > r. We adopt the notation ω = (ω 1 , . . . , ω r ) and n · ω = n 1 ω 1 + · · · + n r ω r . We require that all ω i ∈ C lie within the same half of the complex plane. By analytic continuation the domain of definition of multiple zeta functions can be extended to s ∈ C.
Similar expressions can be obtained for other regions by using the invariance of the triple sine function under exchange of ω 1 , ω 2 , ω 3 .

B Triple sine formulas for hyper and vector multiplets
In this appendix we recast the one-loop hyper and vector multiplet contributions to the 5d partition function on unsquashed S 5 as computed in [4] in terms of triple sine functions.

B.1 Hypermultiplets
We wish to show that the one-loop partition function for a hypermultiplet in the representation R of the gauge group, whose weights we denote by µ, is equal to µ S 3 (iφ µ + 3/2|1, 1, 1) −1 .

(C.1)
We would like to express this in a form analogous to the definition of the triple sine function, equation (A.2). Assuming that this function has similar analytic properties to the triple sine function, we can read off the zeros α i and poles β j of this function from its definition and express it as a regularized infinite product, , which is valid up to an exponential prefactor. In particular, from the denominator of (C.1) we get where ξ = z + τ 1 (s 1 + 1/2) + τ 2 (s 2 + 1/2) + (s 1 + 1/2)).