Blowup Equations for 6d SCFTs. I

We propose novel functional equations for the BPS partition functions of 6d (1,0) SCFTs, which can be regarded as an elliptic version of Gottsche-Nakajima-Yoshioka's K-theoretic blowup equations. From the viewpoint of geometric engineering, these are the generalized blowup equations for refined topological strings on certain local elliptic Calabi-Yau threefolds. We derive recursion formulas for elliptic genera of self-dual strings on the tensor branch from these functional equations and in this way obtain a universal approach for determining refined BPS invariants. As examples, we study in detail the minimal 6d SCFTs with SU(3) and SO(8) gauge symmetry. In companion papers, we will study the elliptic blowup equations for all other non-Higgsable clusters.


Introduction
Quantum field theories with the highest amount of symmetries, namely supersymmetry as well as conformal symmetry, in the highest possible dimension are 6d superconformal field theories.
The classification of such theories subdivides between two main classes of theories, namely the N = (2, 0) theories and the N = (1, 0) theories. The former have been studied intensively for a long time now and there are powerful techniques available for constructing their partition functions on various manifolds. The subject of N = (1, 0) theories, which preserve half of the supercharges of the (2, 0) theories, has recently enjoyed a resurgence due to a proposed classification of such theories in terms of F-theory compactifications on non-compact elliptic Calabi-Yau three-folds [1,2]. In this classification, the geometry of the base B of the Calabi-Yau manifold directly translates into the tensor multiplet sector of the 6d SCFTs where the number of tensor multiplets is given by the dimension of H 1,1 (B, Z) and the intersection form on B gives the couplings of these tensor multiplets to each other. Note that an action is not available as field strengths of tensor multiplets are constrained to be self-dual. Nevertheless, it is useful to write down a "formal" action on the tensor branch from which many properties of the theory and its compactifications can be deduced, see for example [3,4]. Furthermore, the base B of the Calabi-Yau is non-compact and all curve classes inside it are required to be simultaneously shrinkable to zero volume in order to restore conformal invariance of the resulting 6d theory at its tensionless limit. This gives strong constraints on the geometry and in particular forces all curves to be P 1 's which have negative self-intersection number. Moreover, for self-intersection numbers −n lower than −2 the elliptic fiber above the corresponding curve Σ becomes singular with a singularity type determined by Kodaira's classification of elliptic fibers [5,6]. The singularity becomes worse when n increases such that beyond n = 12 it becomes too bad for a smooth description of the Calabi-Yau threefold. The physical interpretation of these singularities is the emergence of a bulk gauge group (whose Lie Algebra g Σ is determined by the intersection form of the resolved singularity) in the 6d SCFT on its tensor branch. If we have two curves Σ 1 and Σ 2 with non-trivial intersection number and gauge groups, then the corresponding 6d theory will also have bi-fundamental matter in suitable representations of the arising gauge groups. In the current paper we want to focus on the cases where the base B contains only one curve with self-intersection −n, i.e. it is a certain decompactification limit of a Hirzebruch surface F n . Then the possible gauge groups which arise as a function of n are as follows 1 : These theories are known as the Non-Higgsable Clusters [6]. 2 Note that the E 7 Lie algebra appears twice, namely for the self-intersections −7 and −8. The difference is that in the −7 case there is also fundamental matter which is non-Higgsable. These theories are the subject of the current paper where we focus on the cases n = 3 and n = 4 and aim at testing a novel method for computing BPS partition functions of the corresponding minimal 6d SCFTs. Let us describe our procedure for computing such partition functions in the following. 1 The cases of n = 9, 10, 11 involve points of enhanced singularities on the base curve which must be resolved giving rise to additional curves in the base. 2 The Non-Higgsable Clusters also include three examples of intersecting chains of two or three curves, which we do not cover here.
In order to be able to compute a partition function for our 6d theories we first need to make a choice for a background geometry. As it turns out, an appropriate choice for 6d SCFTs is the Omega background R 4 × 1 , 2 T 2 [7]. This background not only regularizes the infinities arising from non-compactness of R 4 but also serves as a building block for computing partition functions for other backgrounds like superconformal indices and T 2 × S 4 partition functions [8,9]. As was observed in [7], instantons on this background arise from self-dual strings wrapping the T 2 and localized at a point on R 4 . From the F-theory point of view, such strings arise from D3 branes wrapping a curve Σ in the base which in our case is a (−n)-curve. The bulk gauge group will descend to a flavor symmetry on the worldvolume theory of these strings which is a 2d N = (0, 4) supersymmetric theory with R-symmetry SU (2) + × SU (2) R where SU (2) R is the R-symmetry of the 6d SCFT. The partition function on the tensor branch Z 6d , i.e. when the volume t b of the (−n)-curve in the base is non-zero, is the generating function of the elliptic genera E k of k strings up to a prefactor [7,10,11].
For 6d theories corresponding to the particular choices n = 3 and n = 4 the worldvolume theory of the strings is known in terms of a quiver gauge theory whose single gauge node is of rank k [11,12]. For the other cases, references [13,14] give some descriptions for the k = 1 subsector of a single string but a complete description including a computation scheme for all the E k is still lacking. In this paper we want to remedy this gap by providing a novel computation scheme for Z 6d which allows us to derive expressions for the E k recursively. This is done by using the so-called blow-up equations.
The blowup equations have their origin in the studies of Donaldson invariants [15][16][17][18]. But the version we are most interested in is the generalized version proposed and later proved by Göttsche, Nakajima, and Yoshioka. Nakajima and Yoshioka [19] first considered the 4d N = 2 SU (N ) supersymmetric Yang-Mills theory on the Omega background. The idea is to view R 4 ∼ = C 2 as the limit of its blow-up at the origin [19], denoted by C 2 , when one sends the size of the exceptional divisor P 1 to zero. Then U (1) 1 × U (1) 2 has a natural action on C 2 with two fixed points, one at the north pole and one at the south pole of the exceptional P 1 . Computing the partition function on the background C 2 × 1 , 2 T 2 through localization then contributes a product of two copies of the partition function on our original background while one has to sum now over non-trivial fluxes of the B-field through the exceptional divisor. This idea can be put into functional equations for the Nekrasov partition function [19] (see also [20]), and they were instrumental for Nakajima and Yoshioka to prove Nekrasov's conjecture [21]. Later together with Göttsche they generalized and then proved the blowup equations for 5d N = 1 SU (N ) super-Yang-Mills theories on the Omega background R 4 × 1 , 2 S 1 with a possible Chern-Simons term of level m [22][23][24]. On the other hand, the Nekrasov partition function of such a 5d theory can be computed by the refined topological string theory with target space the local toric Calabi-Yau threefold X N,m , which is the resolution of the cone over the Y N,m singularity [25,26]. This is an example of the geometric engineering [27]. Inspired by the consistency study of the exact quantization program of mirror curves of local Calabi-Yau threefolds [28][29][30][31][32], the Göttsche-Nakajima-Yoshioka blowup equations were reformulated completely in terms of the geometric data of the Calabi-Yau threefold X N,m [33]. The reformulation, however, was not complete, and the complete set of equations were provided in [34].
These geometrically reformulated or generalized blowup equations prove to be very powerful. First of all, just as in the case of the Göttsche-Nakajima-Yoshioka blowup equations [35], they can be used to compute the Nekrasov instanton partition functions [34]. Second, the generalized blowup equations open up possibilities for various directions of generalization. As we will see in the next subsection, the form of the generalized blowup equations is simple and universal, and it does not put any constraints on the target space of the topological string except that it has to be non-compact to allow for U (1) isometry crucial for the preservation of supersymmetry in the presence of the Omega background. This naturally poses the question of the validity of the generalized blowup equations beyond 5d SU (N ) SYM engineered by the X N,m geometries. Indeed it was checked in [34] that the generalized blowup equations are satisfied by some toric Calabi-Yau threefolds which engineer 5d SU (N ) gauge theories with matter. Moreover, what is fascinating is that the generalized blowup equations may even be valid for 6d SCFTs as the topological string theory on non-compact elliptic Calabi-Yau threefolds used in F-theory compactifications computes precisely Z 6d of these 6d SCFTs on the Omega background. As a first step it was checked in [33,34] that the simplest 6d SCFT, the E-string theory, respects half of the generalized blowup equations. The verification of the other half is a bit trickier, and it will be discussed in our upcoming work. In this paper, we demonstrate the validity of the generalized blowup equations through the already well-studied cases of n = 3 and n = 4 minimal SCFTs in the present paper, and illustrate their power by computing the elliptic genera as well as the BPS invariants with them. Furthermore, by reducing the n = 4 model down to the 5d SO(8) SYM, we verify the validity of the generalized blowup equations for the latter theory as well, which is also new. We will cover all the remaining minimal SCFTs in companion papers. In the next subsection we give a quick overview of the generalized blowup equations.

Overview of geometric blowup equations
Consider putting the refined topological string theory on a non-compact Calabi-Yau threefold X. Let H 2i (X, Z) be the homology groups of compact 2i-cycles. In particular H 2 (X, Z) includes compact curve classes {Σ i }, and H 4 (X, Z) compact divisor classes {D j }. We denote the complexified Kähler moduli of the compact curve classes by t i with Vol(Σ i ) = −Re(t i ), and the dimensions of the two homology groups by b := dim H 2 (X, Z) , g := dim H 4 (X, Z) . (1.1) Since X is not compact, these two numbers are not necessarily identical. We encode the intersection numbers of curve classes and divisor classes in a matrix It is always possible to find b − g independent linear combinations of the curve classes so that they have zero intersection number with any compact divisor. We call the corresponding Kähler moduli mass paramters and sometimes denote them by t m i , as they are interpreted as masses of hypermultiplets or instanton fugacity in 5d or 4d theories. It was first observed in [36] and later confirmed in many examples that the non-vanishing BPS invariants N d j L ,j R on a noncompact Calabi-Yau threefold respect a checkerboard pattern: there exists a b dimensional vector B with entries in Z 2 such that 3 The entries of the B field may be fractional if it is not expanded in integral curve classes.
holds for any non-vanishing N d j L ,j R . Then we could define the twisted refined free energy via where only in the worldsheet instanton contributions are the Kähler moduli t shifted by the vector B. We call B the B field as it combines into the Kalb-Ramond part of the complexified t.
The twisted free energy appears prominently in the geometric engineering of Nekrasov partition functions [25,26,32] as well as in the program of exact quantum mirror curves [28,29]. We also define the twisted partition function Note we do not put hat on Z pert since the Kähler moduli are not shifted there.
In terms of these quantities, the blowup equations can be reformulated in the following way Here r = (r i ), which we call a r field, is a b dimensional vector with entries in Z satisfying Two r fields r, r are equivalent if as the corresponding blowup equations can be identified by the shift n → n + n . The prefactor Λ is trivial in the sense that it only depends on the mass paramters t m but not on the true moduli [34]. It also depends on the choice of the r field, thus it gives rise to different blowup equations with different choices of the r field. For some choices of the r field, Λ vanishes all together, and we call the corresponding equations the vanishing blowup equations, while the other equations with non-vanishing Λ are called the unity blowup equations. Note that since the row vector of the C-matrix corresponding to a mass parameter is null, a multiplicative factor of Z which depends only on mass parameters but no other Kähler parameters decouples as its contributions to the blowup equations can be factored out of the summation in n and be absorbed in Λ. We will thus discard this type of components in Z. It was conjectured in [33,34] that for any non-compact Calabi-Yau threefold there is a finite but non-empty set of r fields so that the blowup equations (1.6) hold. It was further conjectured and verified for some toric Calabi-Yaus in [34] that the BPS invariants could be computed from the blowup equations using classical geometric data of X as input. Furthermore, Λ is modular invariant with respect to the monodromy group of the topological string moduli space. In this paper we demonstrate the validity of these statements for the partition functions of the minimal 6d n = 3, 4 SCFTs.
The partition function of a 6d SCFT on the Omega background R 4 × 1 , 2 T 2 can be split to three components (1.9) Here t b , τ and a are tensor modulus, complex structure of T 2 , and gauge fugacities (Wilson lines on T 2 ) respectively. Z pert contains perturbative contributions. Z 1-loop comes from Kaluza-Klein modes of 6d particle multiplets on T 2 . We denote it by the superscript 1-loop because it descends to 1-loop contributions in 4d when we shrink T 2 to a point. Finally Z ell (t b , τ, a, 1 , 2 ) splits by with Q ell = e t ell the counting parameter, and E k (τ, a, 1 , 2 ) the k string elliptic genus. By the F-/M-theory duality and the relation of the BPS sector of the M-theory with the refined topological string theory, the partition function of a 6d SCFT on the tensor branch is computed by the partition function of the refined topological string theory on the same Calabi-Yau threefold X encoding the BPS invariants on X, and the moduli t ell , τ, a are identified with linear combinations of the Kähler moduli of compact curve classes in X. In particular, Z ell includes the BPS states of M2 branes wrapping the base curve, while Z 1-loop the BPS states of M2 branes not wrapping the base curve at all. They combine into the component Z inst that encodes all the BPS invariants. Z pert basically encodes the intersection numbers of divisors in X. If we further decompactify X along the direction of the elliptic fiber in the M-theory picture, the 6d (1, 0) SCFT reduces to a 5d N = 1 SYM with the same gauge group on the Omega background, where the tensor modulus t b becomes the gauge coupling.
The organization of the rest of the paper is as follows. In Section 2 we compute the initial data for blowup equations. These include the curve-divisor intersection C-matrix, the B-field for the checkerboard pattern, Z pert and Z 1-loop . We give explicit expressions for these initial data for the cases of 6d SCFTs with SU (3) and SO(8) bulk gauge groups. In Section 3 we put everything together and first demonstrate the validity of the blowup equations order by order in terms of Q ell expansion and then proceed to recursively compute the elliptic genera of multiple strings as well as the corresponding BPS invariants. In Section 4 we study reductions of the blowup equations in the 5d limit, that is when one of the circles of the T 2 which is wrapped by the strings shrinks to zero radius. Finally, in Section 5 we present our conclusions and give an overview of applications and open problems.

Initial data for blowup equations
We explain here how to compute the initial data for the blowup equations: the curve-divisor intersection C-matrix, the B fields, as well as the perturbative and 1-loop partition functions Z pert , Z 1-loop , for 6d minimal SCFTs with no matter. As we note in subsequent subsections, these data are necessary if we wish to derive compact formulas of elliptic genera from the blowup equations, while the piece Z 1-loop is not needed if we wish to directly compute BPS invariants from the blowup equations.

Curve-divisor intersection matrix
The structures of the elliptic non-compact Calabi-Yau threefolds underlying the 6d minimal SCFTs are for instance discussed in [37]. Let the gauge group G be of rank r. There are g = r + 1 compact divisors. They result from the resolution of the singular elliptic fiber and they intersect with each other like the nodes of the Dynkin diagram of g. One of the divisors (c) n = 6 is special as it intersects with the base B and it corresponds to the affine node in the Dynkin diagram. We label the special divisor by D r+1 and the subsequent divisors D r , D r−1 , . . .. All these divisors are Hirzebruch surfaces F n i . The indices n i of these divisors in different 6d (1, 0) minimal SCFTs can be found in [37], and we give some examples in Figs. 2.1. The number of irreducible compact curves is b = r + 2. Of these r + 1 curves are the P 1 fibers of the Hirzebruch surfaces Σ i , i = 1, . . . , r + 1 and they stretch in the vertical direction. Their labelling follows the labelling of the underlying divisors. These curves satisfy where a i are marks of the affine Lie algebra g, and δ is the elliptic fiber. The last compact curve Σ b is in the horizontal direction, and it projects down to the compact −n curve Σ B in the base.
In accord with topological string calculations we choose it to be a Mori cone generator. It is always the P 1 base of the Hirzebruch surface in the center of the chain of F n i with the lowest index. It is therefore related to the base curve Σ B by In the case of n = 3, 4, we have We denote the volumes of these irreducible curves by t i (i = 1, . . . , r + 1) and t r+2 = t b . This geometric picture allows us to write down the intersection C-matrix where the (r + 1) × (r + 1) submatrix −A is minus the Cartan matrix of the affine Lie algebra g, and the last row depends on the indices of the Hirzebruch surfaces D i = F n i . It is then easy to see that the only mass parameter is which is the comlexified volume of the elliptic fiber δ. We will give the concrete expressions of the C-matrix of the SU (3) and the SO(8) theories in the example subsetions. Note that in most of the paper we expand Z ell in terms of instead of Q ell = e t ell as the curve associated to t ell may not be in an integral class. These parameters are related by 4

The B field
We would like to compute the b dimensional Z 2 B field which characterizes the checkerboard pattern of non-vanishing BPS invariants N d j L ,j R with identity Since the r.h.s. is linear in the curve class d, we only need to know the entries of the B field corresponding to each individual irreducible curves. Each irreducible curve can be embedded in an algebraic surface in the Calabi-Yau threefold X. Let us denote the curve and the surface where it is embedded by C and S respectively. The non-vanishing BPS invariants associated to this curve must have [33,39] 2j max where C 2 is the self-intersection number of the curve in the surface S, and K S the canonical class of S. We have then 2j L + 2j R + 1 = C 2 mod 2 . (2.10) Thus the entry of the B field corresponding to C is its self-intersection number in the surface S modulo two. On the other hand, the self-intersection number C 2 is identified with the degree of the normal bundle of C perpendicular to S. Recall that the normal bundle of a curve in a Calabi-Yau threefold has the form Since the two degrees n and −2 + n are equivalent modulo two, we can take either of them to be the entry of the B field corresponding to the curve C. Since we have a good understanding of irreducible curves and surfaces in the Calabi-Yau threefold underlying the 6d minimal SCFTs as we discussed in the beginning of the section, these numbers can be easily computed for each irreducible compact curve.

Perturbative partition function
The perturbative contribution Z pert (t, 1 , 2 ) = exp F pert (t, 1 , 2 ) has the following form The perturbative prepotential F pert,(0,0) is decided 5 by the intersection numbers of divisors Poincaré dual to the curve classes Σ i associated to t i . Since the Poincaré duality is only rigorously defined in a compact manifold, we should compute F pert,(0,0) cmp in a compact Calabi-Yau threefold where the non-compact Calabi-Yau threefold X is embedded and take an appropriate decompactification limit. Fortunately the compactification of the Calabi-Yau threefolds underlying the 6d SU (3) and SO(8) gauge theories have been constructed in [11], and we use the compact models there to compute F pert,(0,0) cmp which is subsequently reduced to F pert,(0,0) in the decompactification limit. On the other hand, once F pert,(0,0) is computed for the 6d gauge theory, we could obtain F pert,(0,0) 5d for the 5d gauge theory by further decompactifying the Calabi-Yau threefold X along the direction of the elliptic fiber while keeping the volumes of Σ i (i = 1, . . . , r) finite. 6 The latter is also given by the perturbative Nekrasov partition function [22] F Nek,pert (a, q, 1 , 2 ) = − 1 where a is the vector of Coulomb moduli, q is the instanton counting parameter, ∆ + is the set of positive roots of the Lie algebra g, and h ∨ G the dual Coxeter number of G. •, • is the invariant bilinear form 7 in the Lie algebra g. The dictionary between field theory parameters and geometric Kähler moduli is (see for instance [32]) where α i are simple roots, and t m the mass parameter whose associated curve, we recall, that does not intersect with compact divisors. Once we could identify the first line of (2.13) with F pert,(0,0) 5d , we could uplift the second line of (2.13) to obtain the perturbative genus one free energies F ,(1,0) , F ,(0,1) for the 6d theories, as we will do in example subsetions. In particular, we find in the examples of the n = 3, 4 theories The perturbative prepotential can also include terms linear in t. But they decouple from the blowup equations. 6 We send the volume of the curve class Σr+1 which intersects with the base to infinity. 7 Here we normalize it so that the longest root has norm square 2.

One-loop partition function
Z 1-loop has the contribution of the Kaluza-Klein modes on the 6d S 1 of the 6d particle multiplets. The contribution of a single supermultiplet of various types reads as follows [40] 8 where Q τ = e τ 9 , while Q G = e a , Q F = e m G are gauge and flavor symmetry fugacities, with powers * appropriate charges of the supermultiplets. The plethystic exponential is defined as In the case of 6d minimal SCFTs, there is no contributions from charged hypermultiplets, while the contributions of tensor multiplets only depend on the mass parameter τ and no other Kähler moduli and can thus be factored out of the blowup equations. Therefore in this paper we only consider contributions of 6d vector multiplets to Z 1-loop . The spectrum of vector multiplets and thus their total contribution to Z 1-loop can be computed by the refined topological string theory with (j L , j R ) = (0, 1/2) for vector multiplets. Here H vert 2 (X, Z) is the homology group of compact curves in the vertical direction, and it is generated by Σ i (i = 1, . . . , r + 1). Using N Σ 0,1/2 = 1, we obtain To be in line with the refined Gopakumar-Vafa formula of topological string theory [41,42], we suppress a term of 1/2 in [40]. 9 Note our convention here differs from the usual convention in the mathematics literature by a factor of 2πi. 10 Here M (j L ,j R ) Σ differs from that in [42] by 1 in order for the contributions of vector multiplets to be in the denominator, as they should. where ∆ + is the set of positive roots of the affine Lie algebra. By the identification of the imaginary root with the elliptic fiber and (2.5), the expression for Z 1-loop is equivalent to (2.21)

6d SU (3) theory
The non-compact Calabi-Yau threefold X underlying the 6d SU (3) model on the Omega background is the elliptic fibration over O(−3) → P 1 with the singular fiber resolved. As explained in [37], there are b = 4 compact irreducible curves and g = 3 compact irreducible divisors. The latter D 1 , D 2 , D 3 are three F 1 surfaces in the vertical direction intersecting at a common (−1)-curve Σ 4 = Σ b , which projects to the (−3)-curve in the base. The other three curves Σ i (i = 1, 2, 3) are the P 1 fibers of the Hirzebruch surfaces. This geometry is illustrated in Fig. 2.2. The intersection matrix of the curves Σ i (i = 1, . . . , 4) and the divisors D j (j = 1, . . . , 3) is in accord with the general structure (2.4). The understanding of the embedding of the curves Σ i in surfaces in X allows us to write down the B field B = (0, 0, 0, 1) , (2.23) following the discussion in section 2.2.
To compute the perturbative prepotential F pert,(0,0) , we follow [11] and take X as the decompactification limit of the compact Calabi-Yau threefold X, the elliptic fibration over F 3 , along the horizontal direction perpendicular to the (−3)-curve in the base. The compact model X can be realized as a hypersurface in a toric variety. Therefore its triple intersection numbers and thus the perturbative prepotential can be computed with the usual techniques in toric geometry (see for instance [43]). Then F pert,(0,0) of the non-compact model is obtained by integrating over the periods which remain finite in the decompactification limit [44]. In this way, we find We can further decompactify X along the direction of the elliptic fiber by sending the volume of one of the curves in the vertical direction to infinity. Let us take the limit 11 t 3 → ∞ and after following the same procedure of integrating over finite periods, we obtain This should coincide with the perturbative Nekrasov partition function for 5d N = 1 pure SYM with G = SU (3). Combining (2.13) and (2.14), the latter reads To identity the mass parameter t m in (2.25), we first write down the curve-divisor intersecton C-matrix of the 5d theory which can be obtained by removing in the 6d C-matrix (2.22) the third row corresponding to Σ 3 and the third column corresponding to the divisor D 3 containing Σ 3 . We find the curve Σ b − Σ 1 − Σ 2 does not intersect with any compact divisor. The corresponding mass paramter for the 5d theory should be With this identification, it is easy to see that F pert,(0,0),Nek 5d,SU (3) indeed coincides with F pert,(0,0) 5d,SU (3) from decompactification up to a pure mass parameter term We also notice that the 6d and the 5d perturbative prepotentials only differ by Since we wish to obtain the corresponding 5d gauge theory, we should decompactify the vertical curve which intersects with the base [37]. Nevertheless since the Kähler moduli of the three vertical curves appear to be on the equal footing in F pert,(0,0) 6d,SU (3) , we can choose any of them to decompactify, keeping the others intact. This implies that the decompactification limit is really obtained by This observation allow us to write down the perturbative contributions to genus one free energies of the 6d model. By writing down a generic linear ansatz for F pert,(1,0) , F pert,(0,1) , separating out t 3 + t b , and demanding the remaining piece coincides with the second line of (2.13), as well as imposing symmetry between t 1 , t 2 , t 3 , we fix F pert,(1,0) , F pert,(0,1) for the 6d SU (3) model uniquely to be The difference from the 5d free energy is By specializing (2.21), we get the one-loop partition function (2.34)

6d SO(8) theory
The non-compact Calabi-Yau threefold X underlying the 6d SO(8) model on the Omega background is the elliptic fibration over O(−4) → P 1 with the singular fiber resolved. As explained in [37] there are b = 6 compact irreducible curves and g = 5 compact irreducible divisors. The divisors D 1 , D 2 , D 3 , D 4 , D 5 = D c are Hirzebruch surfaces F 2 , F 2 , F 2 , F 2 , F 0 linking up with each other like the Dynkin diagram of so (8), where D c plays the role of the central node, while D 4 lays the role of the affine node and intersects with the base. D 1 , D 2 , D 3 , D 4 intersect with D c by the (−2) curves which are all homologously equivalent on D c . We take this curve to be Σ 6 = Σ b . The remaining irreducible curves Σ i (i = 1, . . . , 4) and Σ 5 = Σ c are the P 1 fibers of D i (i = 1, . . . , 4) and D c . This geometry is illustrated in Fig. 2.3. The intersection matrix of the curves Σ i (i = 1, . . . , 6) and the divisors D j (j = 1, . . . , 5) is in accord with the general structure (2.4). following the discussion in section 2.2. As in the case of SU (3) theory, we compute the perturbative prepotential F pert,(0,0) by following [11] and taking X as the decompactification limit of the compact Calabi-Yau threefold X, the elliptic fibration over F 4 , along the horizontal direction perpendicular to the (−4) curve in the base. The compact model X can also be realized as a hypersurface in a toric variety. Therefore its triple intersection numbers and thus the perturbative prepotential are readily computable using usual techniques in toric geometry. We then obtain F pert,(0,0) of the non-compact model by integrating over finite periods in the decompactification limit [44] and we get We further decompactify X along the direction of the elliptic fiber by sending t 4 → −∞. After following the same procedure of integrating over finite periods, we obtain This should coincide with the perturbative Nekrasov partition function for 5d N = 1 pure SYM with G = SO (8). Combining (2.13) and (2.14), the latter reads To identity the mass parameter t m in (2.38), we first obtain the curve-divisor intersecton Cmatrix of the 5d theory which is done by removing in the 6d C-matrix (2.35) the fourth row corresponding to Σ 4 and the fourth column corresponding to the divisor D 4 containing Σ 4 . We find the curve The corresponding mass paramter for the 5d theory should be With this identification, it is easy to see that F pert,(0,0),Nek 5d,SO (8) indeed coincides with F pert,(0,0) 5d,SO(8) from decompactification up to a pure mass parameter term . (2.42) We also notice that the 6d and the 5d perturbative prepotentials only differ by This implies that the decompactification limit is really obtained by This observation allow us to write down the perturbative contributions to genus one free energies of the 6d model. By writing down a generic linear ansatz for F pert,(1,0) , F pert,(0,1) , separating out t 4 + 1/2t b , and demanding the remaining piece coincides with the second line of (2.13), as well as imposing symmetry between t 1 , t 2 , t 3 , t 4 , we fix F pert,(1,0) , F pert,(0,1) for the 6d SO(8) model uniquely to be The difference from the 5d theory is Finally, by specializing (2.21), we get the 1-loop partition function (2.47)

Elliptic genera from blowup equations
In this section we put everything together, and demonstrate for the 6d SU (3) and SO (8) SCFTs the validity of blowup equations order by order in an expansion in Q b with the help of the wellknown results of the elliptic genera of these two theories [12,38,38]. Then we reverse the logic and show that the blowup equations can be used to solve the elliptic genera and BPS invariants, illustrating the power of blowup equations in the studies of 6d SCFTs

Constraint on r fields
We first find a mild condition on the r field and argue that the number of inequivalent and admissble r fields satisfying this condition is finite. We first rewrite the blowup equation (1.6) by moving the unshifted partition function Z(t, 1 , 2 ) to the other side of the equation where the dependence on r is always understood, and We have also used the fact that τ is the only mass parameter. When the l.h.s. of the blowup equations are expanded in terms of the Kähler moduli the perturbative partition function Z pert determines the leading order terms. The contributions of Z pert to the l.h.s of (3.1) reads where we have used (2.15). For the blowup equations to hold at the leading order of Q k , we must have Here I k is the set of integral vectors n that minimize f k (n) for true Kähler moduli (not mass parameters). The latter can be written as Note that the functions f k (n) for k = 1, . . . , r and f 0 (n) are dependent on the r field as well.
In order for the blowup equation to make sense, f k (n) as functions of the integral vector n must have a minimum for any k = 1, . . . , r + 2, which allows us to determine the valid r fields [34]. We find that in the case of the n = 3, 4 theories, the valid r fields satisfy in other words, the entry of the r field corresponding to τ is zero. With this feature we can write down the blowup equations as identities of Jacobi forms, as we will see in section 3.2, and proceed to prove the blowup equations order by order in terms of the Q b expansion using the modularity argument. We will call the r fields satisfying the constraint (3.6) admissible.
Interestingly, we notice that (3.6) is equivalent to a slightly stronger condition that f b (n) = f r+2 (n) in particular has a minimum for a real vector n. 12 To see this, let us recall that κ ijk are intersection numbers of divisors K i dual to the curve class Σ i , satisfying In the elliptic Calabi-Yau threefold underlying a 6d minimal SCFT with a bulk pure gauge theory, the curves Σ i (i = 1, . . . , r + 2) have mutual intersection numbers identical to minus the Cartan matrix of the affine lie algebra . . , r + 1) be the irreducible compact divisors coming from Σ i fibered over the P 1 in the base. We should thus have This condition is stronger because if a minimal real n exists, a minimal integral n must exist nearby; on the other hand, if a minimal integral n exists, there can be a non-integral flat direction of f b (n). 13 We understand that the Poincaré is only rigorously defined in a compact manifold, while the elliptic Calabi-Yau threefold here is non-compact. So we are presenting here an argument not a proof. We also checked the validity of (3.6) for the SU (3), SO(8) as well as some other 6d gauge theories.
We note that the two sets of divisors D i and K j should be related by the C-matrix so that(1.2) still holds. Let us come back to the discussion of the functions f k (n). Take the direction k = r + 2 = b. The coefficients can then be explicitly evaluated r+2 i,j=1 12) and the function f b (n) reads 14 which annihilates A when multiplied from the left. We thus get l · r = 0, which is the condition (3.6).
Let us now give a counting of inequivalent and admissible r fields. Given the condition (1.7), we can paramterize r fields by The domain of inequivalent v-vectors, defined to be the lattice Z b modulo the equivalence relation (3.17), has only a finite number of points along g directions, and extends freely along the remaining b − g directions. In practise, we can always make linear combinations of curve classes so that the last b − g rows of the intersection matrix C are empty, i.e. where v i , i = 1, . . . , g can only take a finite number of integral values, while the remaining entries denoted by * can take any value in Z. The equivalence condition for the truncated v-vectors defined byv Clearly the matrix C sub defines a lattice embedding Z g → Z g . Therefore the number of inequivalent truncated v-vectors is det C sub .
In the case of 6d minimal SCFT with a pure bulk gauge theory, there is only one mass parameter τ corresponding to the volume of elliptic fiber, and we have seen that the corresponding entry of r field must be zero. Therefore, the number of inequivalent and admissible r fields must be identical with that of inequivalent truncated v-vectors, which is det C sub . We mentioned in the previous sections that C barring the last row is identified with the opposite of the affine Cartan matrix. Hence in practice, we can construct the full rank square submatrix C sub of C by throwing away the row corresponding to the affine node in the Dynkin diagram. 15 Once all the inequivalent truncated v-vectors are found, we can convert them to r fields with the help of (3.16) and (3.6). In the 6d n = 3, 4 minimal SCFTs, we find that all admissible r fields give rise to valid blowup equations.

Recursion relations
Here we derive recursion relations of elliptic genera from the blowup equations. Later when we discuss individual models in sections 3.3 and 3.4, we will demonstrate the validity of these recursion relations and then inverse the logic solving elliptic genera from these relations.
The blowup equations for the partition function of a 6d SCFT can be written as follows (3.22) where it is understood that everything depends on the choice of r field. Here Z ell is the generating function of elliptic genera

23)
15 Note the mark associated to the affine node is 1. We can construct another full rank C sub by throwing away a different row corresponding to a different node. If the associated mark ai is greater than 1, the number of truncated v-vectors might be larger. But we cannot recover integral r fields from all integral truncated v-vectors because of (3.6). In the end the number of integral r fields is still the determinant of C sub constructed from throwing away the affine node.
where Z k is proportional to the k-string elliptic genus with a model-dependent prefactor, and it only depends on Kähler moduli of vertical curves t , = 1, . . . , r + 1. When Z ell is twisted with t shifted to t + πiB, Q b is multiplied with a phase (−1) B b , while Z k is unchanged, as t are volumes of (−2) curves and thus the corresponding entries of B vanish according to the discussion in Section 2.2. The function A is given by including the perturbative contribution 25) and the one-loop contribution with f 0 , f defined in (3.3). We don't have to put hat over Z 1-loop because entries of B associated to t ( = 1, . . . , r + 1) are all zero. By comparing the coefficients of Q k b on both sides of (3.22), we find the recursion relation (3.27) The expression above can be simplified due to the following observation. Given the expression (3.13) of f b (n) and the condition (3.6) on the r field, it is clear that where a = (a k ) the vector of marks. Similarly for k = 1, . . . , r + 1 is also invariant under the integral shift (3.28). Therefore, we could define representativesn of n by n =n + ma , m ∈ Z (3.30) so that no two representatives differ by aZ. Then the summation in (3.27) can be split into two steps where the summation of m gives a theta function. These are the equations whose validity we will demonstrate order by order through a modularity argument in the following example sections.
In the remainder of this section, we illustrate how to derive elliptic genera from the recursion relation (3.31). We start with k = 0. Given the expression (3.13) for f b (n), its value is already non-negative since the affine Cartan matrix A m is positive semi-definite. If the minimal value of f b (n) is greater than zero, the identity (3.31) could not hold at k = 0 unless Λ = 0. Thus we should have a vanishing blowup equation. If the minimal value of f b (n) is zero, there is a chance that (3.31) is satisfied at k = 0 and we get a non-vanishing Λ, which should result in a unity blow up equation. In the latter case, using Z 0 = 1, we find the following expression for Λ whereÎ b is the set of representativesn which minimize f b to zero. Let us now focus on unity blowup equations. As we will see in example sections, the associated r fields have zero entries except for r b : r = (0, . . . , 0, r b ); besides, one can always choose the representativen inÎ b to be the null vector. As a result, t are not shifted in Z k 1 , Z k 2 if either k 1 = k or k 2 = k. We can thus put the unity recursion relations for k ≥ 1 in a more explicit form The coefficients are .

(3.36)
I k is the summation on the r.h.s. of (3.31) with k 1 , k 2 < k, and thus are known data in a recursive calculation. The relations (3.33) can be solved to give compact expressions of Z k , following the procedure in [35] for a similar calculation for 5d gauge theories. The key observation is that do not depend on the choice of r fields. If there are at least three unity r fields, we can pick three of them, and write down three equations of the form (3.33), and combine them into the linear system (3.37) Here we only write down the most important parameters each function depends on. If the matrix M Λ k of coefficients Λ, Λ k on the l.h.s. of the linear system is of full rank, it can inverted unity (0, 0, 0, −1) (0, 0, 0, 1) (0, 0, 0, 3) vanishing (2, −2, 0, 1) (−2, 0, 2, 1) (0, 2, −2, 1) (0, −2, 2, 1) (2, 0, −2, 1) (−2, 2, 0, 1) to give us the answer of Z k in terms of I k and thus lower base degree partition functions. We will demonsrate that this method also works for 6d theories, except for the solution of Z 1 for the SU (3) model. Here and later in this paper we use the following notation of Jacobi theta functions with characteristics Indeed all the Λ only depend on τ and no other Kähler moduli. 16 16 We notice that Λ are all Jacobi forms of weight 1/2 with respect to the modular group acting on τ . This is a subgroup of the monodromy group Γ of the total modular space, and we thus seem to have a contradiction with the claim [34] that Λ is supposed to have weight zero with respect to Γ. To reconcile them, we recall that we have thrown away the contribution of the tensor multiplet to Z 1-loop . It is not difficult to verify with the help of the identities in Appendix A that if included it contributes to an additional factor η(τ ) −1 to Λ reducing the weight of the latter to zero.
As for the vanishing blowup equations, we only have to check one of them, as their r fields are related to each other by S 3 symmetry acting on the first three entries. Consider the r field (−2, 2, 0, 1). There are three sets of n which minimize f b (n) represented bŷ where Θs encapsulate contributions from perturbative partition functions and have the form , .

(3.46)
In the derivation of these expressions, the identities in Appendix A are very useful. With the reparametrisation subject to v 1 + v 2 + v 3 = 0, the identity (3.44) can be written as (3.48) This last identity can be proved by noticing that each term and therefore the total sum is a Jacobi form 17 for Γ(3) of weight 1 and index polynomial and by verifying that the first few terms in the Q τ expansion vanish, which we have checked up to very high orders.

Modularity at generic base degree
Here we given an argument for the validity of the recursion relation (3.31) by demonstrating that both sides of the equation (3.31) are multivariate meromorphic Jacobi forms of the same weight and index polynomial at any base degree k. Once this is achieved, after multiplying both side of (3.31) with the common denominator, we get an identity of multivariate weak Jacobi forms of the same weight and index, which can then be proved by plugging in the expression of Z k given in [12] and comparing the first few terms in the Q τ expansion. Consider a blowup equation with r = (r 1 , r 2 , r 3 , r b ) subject to the condition r 1 + r 2 + r 3 = 0. The perturbative contribution to the recursion (3.31) is For the contributions from vector multiplets, using the notation (A.7), we have D 1-loop =PE − (Bl (0,1/2,−2n 1 +n 2 + r 1 2 ) (q 1 , q 2 )Q 1 + Bl (0,1/2,2n 1 −n 2 + r 2 +r 3 2 ) (q 1 , q 2 )Q 2 Q 3 +Bl (0,1/2,n 1 −2n 2 + r 2 2 ) (q 1 , q 2 )Q 2 + Bl (0,1/2,−n 1 +2n 2 + r 1 +r 3 2 ) (q 1 , q 2 )Q 1 Q 3 +Bl (0,1/2,n 1 +n 2 + r 3 2 ) (q 1 , q 2 )Q 3 + Bl (0,1/2,−n 1 −n 2 + r 1 +r 2 where we used the property that D 1-loop (n) = D 1-loop (n) and set n 3 = 0. Then noticing Q τ = Q 1 Q 2 Q 3 and using the notation (A.14), we have (3.59) In Appenidx A, we showθ R (t) is a meromorphic Jacobi form of weight 0, and thus so is D 1-loop up to the prefactor. Finally, given the relation between Z k and the k-string elliptic genus E k for the SU (3) theory we also have (3.62) Using the fact that the elliptic genus E k (t , 1 , 2 ) is a meromorphic Jacobi form of weight zero and index polynomial [13,38] Ind It is not difficult to show that up to the common prefactor every term in (3.62) is a meromorphic Jacobi form 18 for Γ(3) of weight 1/2 and index polynomial which is indepdent of the summation indices n 1 , n 2 , k 1 , k 2 , and thus so is the total sum. On the other hand, if the blowup equation is of vanishing type, the l.h.s. of (3.31) vanish; if the blowup equation is of unity type, we have r = (0, 0, 0, r b ), and after plugging in the expression of Λ, we find the l.h.s. of (3.31) to be l.h.s. = e which is also a meromorphic Jacobi form of the same weight and the same index (3.65) up to the same prefactor. In both cases, after multiplied with a common denominator, the recursion relations (3.31) can be cast as identities of (weak) Weyl invariant Jacobi forms of identical weights and indices. As the ring of Jacobi forms is finitely generated, these identities can be proved by checking that when the correct r (Tab. 3.1) are plugged in the first few terms in Q τ expansion are correct. For instance when k = 0 we find (3.62) indeed reduces to the computation of Λ in the unity cases and the identity (3.48) in the vanishing cases. When k = 1 and with a unity r plugged in, the recursion relations reduce to We have used the variables v = (v 1 , v 2 , v 3 ) for Kähler moduli defined in (3.47) to make the Weyl symmetry of SU (3) more transparent, and we define v ij = v i − v j . Here and in the following we suppress the modular parameter of a theta function if it is simply τ . One can readily verify by using the expression (3.51) of E 1 in [12] and by the first few terms in the Q τ expansion that (3.67) is valid if r b is odd but not if r b is even. Similarly when the vanishing r = (−2, 2, 0, 1) is plugged in, we find the identity which can also be explicitly verified. The cases of the other vanishing r fields can be obtained by permutations of the components r 1 , r 2 , r 3 of the r field. We have also checked the cases of k = 2. Identities for k > 2 can be checked in a similar manner.

Recursion formula for elliptic genera
We would like to inverse the logic and illustrate here that it is possible to solve E k (k ≥ 2) for the SU (3) theory from the recursion relations (3.33) following the argument at the end of section 3.2.
In the case of 6d SU (3) theory, there are three unity r fields. The corresponding Λ k are • r 1 = (0, 0, 0, −1) Surprisingly, we find that at base degree one the matrix M Λ 1 is actually not of full-rank. Therefore one cannot invert M Λ 1 to solve Z 1 from the recursion relation.
That det M Λ 1 = 0 may have something to do with the curious coincidence that while the characteristics of the theta functions enjoy a Z 3 symmetry connected to the gauge group SU (3), the elliptic parameters of theta functions enjoy some S 3 symmetry. Note that the three types of theta functions . On the other hand, the three elliptic parameters for each theta function sum up to zero, and enjoy a S 3 symmetry. On the other hand, det M Λ k does not vanish at base degrees k > 1. For instance the leading order contribution in the Q τ expansion is We can thus obtain compact expressions of Z k from the linear equation (3.37) by inverting M Λ k . We do not give explicit formulas for Z k here as they are quite lengthy, and the results of Z k are already well known in the literature [12,33]. Instead we will compute and list the BPS invariants in the next subsection, which also serves as another check on the blowup equations.

Solving refined BPS invariants
Among the recursion relations those from unity blowup equations (3.33) are most useful as it is rather easy to solve them and obtain compact formulas of ellptic genera; recursion relations from vanishing blowup equations are rather complicated and it is difficult to get headway with them. Another way to solve the blowup equations (1.6) is to expand them in terms of all Kähler moduli Q i and extract equations of refined BPS invariants. There are two advantages to this method: one can equally easily extract equations from vanishing blowup equations and thus increase the number of available constraint equations; one can in fact start without the input of Z 1-loop but with only the truly perturbative data: the C-matrix, the B-field, and Z pert .

Base degree zero
Following the same analysis as in the SU (3) theory, we can find all the inequivalent and admissible r fields and divide the resulting blowup equations into unity and vanishing equations. The results are listed in Tab. 3.2.
For the unity blowup equations, at base degree k = 0 they reduce to the computation of Λ. We find the following results for the four r fields in the first row of Tab. 3.2 Summing over them, the lowest order blowup equation is , , , .

(3.86)
It is equivalent to the identity It can be proved by noticing that each summand and thus the total sum is a Jacobi form for Γ(4) of weight 7/2 and index polynomial and that the first few terms in Q τ expansion vanish, which we checked up to very high orders.
(3.92) It is equivalent to which can be similarly proved. Note that in order to write recursion relations in terms of appropriate Jacobi forms, we need to absorb one of the five Kähler moduli t 1 , t 2 , t 3 , t 4 , t c completely into τ . It is canonical to absorb t 4 associated to the affine node as one goes down from the affine Lie algebra to the simple Lie algebra. Here we choose to absorb t c so that the symmetry between t 1 , t 2 , t 3 , t 4 still survives.

Modularity at generic base degree
Following the example of the SU (3) theory, we show here that both sides of the recursion relation (3.31) for the SO(8) theory are meromorphic Jacobi forms of the same weight and index polynomial at any base degree k. When this is established, one can multiply both sides with the common denominator and obtain an identity of weak Jacobi forms and proceeds to prove it by comparing the first few terms in the Q τ expansin.
Consider a blowup equation with r = (r 1 , r 2 , r 3 , r 4 , r c , r b ) subject to the condition r 1 + r 2 + r 3 + r 4 + 2r c = 0. The perturbative contribution is with the notation r 5 = r c . In addition n .
We also have ×θ −n 1 +n 2 +n 3 −nc+ r 1 +rc ×θ −n 1 −n 2 +n 3 + r 1 +r 2 +rc 2 (t 1 + t 2 + t c )θ −n 1 +n 2 −n 3 + r 1 +r 3 +rc ×θ −n 1 −n 2 −n 3 +nc+ r 1 +r 2 +r 3 +rc 2 (t 1 + t 2 + t 3 + t c )θ −nc+ r 1 +r 2 +r 3 +2rc Up to the common prefactor, each summand happens to be a meromorphic Jacobi form 20 for Γ4 of the same weight 1/2 and index polynomial which is independent of the summation indices n i , k 1 , k 2 and thus so is the total sum. On the other hand, if the blowup equation is of vanishing type, the l.h.s. of (3.31) vanish; if the blowup equation is of unity type, we have r = (0, 0, 0, 0, 0, r b ), and after plugging in the expression of Λ, we find the l.h.s. of (3.31) to be l.h.s. = Q which is a meromorphic Jacobi form of the same weight and the same index (3.106). In both cases, after multiplied with a common denominator, the recursion relations (3.31) become identities of (weak) Weyl invariant Jacobi forms of identical weights and indices. As in the case of SU (3) theory, these identities can be proved by checking that when the correct r (Tab. 3.2) are plugged in the first few terms in Q τ expansion are correct. For instance, when k = 0 we find (3.105) indeed reduces to the computation of Λ in the unity cases, and the identities (3.87), (3.93) in the vanishing cases. When k = 1, let us first reparametrise the Kähler moduli by . (3.112) Here we define θ 1 (m i ± m k ) as θ 1 (m i + m k )θ 1 (m i − m k ). One can readily verify by the first terms in the Q τ expansion and using the expression of E 1 in (3.24) of [11] 21 that the identity holds only when r b is even. The vanishing equations at k = 1 and the cases of k ≥ 2 can be checked in a similar manner.

Recursion formula for elliptic genera
In the case of 6d SO(8) theory, there are four unity r fields. The corresponding Λ k are • r 2 = (0, 0, 0, 0, 0, 0) (3.118) • r 4 = (0, 0, 0, 0, 0, 4) (3.120) In this case, the matrix M Λ k constructed out of any three of the four unity r fields at any base degree have non-vanishing determinant and is thus of full rank. For instance when r 1 , r 2 , r 3 are used, the leading order contribution to det M Λ k is One can therefore invert M Λ k to solve for Z k or E k from the recursion relation. For instance, using the identity (3.4.2) with the unity r fields (0, 0, 0, 0, 0, 2), (0, 0, 0, 0, 0, 0), and (0, 0, 0, 0, 0, −2), we obtain the following expression for the one string elliptic genus: where ∆ is the determinant of the matrix We have seen in sections 2.5.1, 2.5.2 that the perturbative free energy of the 6d theory is reduced to that of the 5d theory through the limit lim : τ + c t m → −∞ , t m finite , (4.1) with a model-dependent constant c. When applied to Z inst this limit is equivalent to keeping finite terms in the limit Q τ = e τ → 0. The 6d one-loop partition function given by (2.21) becomes Using the formal identity 3) The last line of (4.2) could be written as which is precisely the 1-loop partition function of a 5d pure SYM theory [46]. Furthermore, the partition function component Z k of the 6d gauge theory is identified with the k-string elliptic genus by 22 (4.6) When reduced to 5d gauge theory by sending Q τ to 0, we recover 5d gauge instanton partiton functions Z 5d k by [12,38] 23 (4.8) 22 This relation coincides with that in [38] when n = 3, 4. The discrepany for n > 4 is due to that Q b defined in [38] is no longer the volume of a Mori cone generator. 23 The discussion in section 5.4 of [38] is slightly inaccurate.
Here q is the gauge instanton fugacity related to the 5d mass paramter t m by q = e tm . The first line in the dictionary (4.7) is then consistent with the observation (2.28), (2.41). These observations allow us to conclude that we can obtain the full partition function of the 5d pure SYM theory from the partition function of the 6d gauge theory throug the operation where Z dec (τ, t m 1 , 2 ) is the component that runs off in the limit (4.1), which is the exponential of the extra piece in the perturbative free energy given by (2.30),(2.33) combined for SU (3) and (2.43),(2.46) combined for SO(8) theories respectively. Here we use T for Kähler moduli in 6d instead of t to stress that there is one more Kähler modulus in 6d theories. We make it explicit that Z dec only depends on τ, t m and no other Kähler moduli. Besides, we are free to twist the partition functions in the sense of (1.4) and put hats over Z 5d , Z 6d . Then by multiplying both sides of the blowup equations for the 6d theory with an inverse power of Z dec and taking the limit lim , we get where we have defined As we will illustrate by the examples of SU (3) and SO(8) theories, if we expand Λ 6d and B dec in terms of Q τ , and keep only the coefficients of the lowest power on both sides of (4.10), we get the blowup equations for the 5d gauge theory Note that the dimension of n is reduced by 1 in the 5d blowup equations.

SU (3) model
In the case of 6d SU (3) model, we have concretely We take the Nekrasove partition function to be the partition function of the 5d gauge theory.
Z dec should include an extra piece from (2.29), and it reads Then B dec SU (3) = exp τ  where the remaining pieces in the ellipses are linear in t m and R and depend only on n 3 but not on n 1 , n 2 . Clearing, depending on the value of r 3 + r 4 , there are only one or two integral values of n 3 which minimize the power of Q τ . If we only keep the minimal power of Q τ in (4.10), although we still sum n 1 , n 2 over all integers, we only sum n 3 over one or two values. This is the reason the dimension of the summation index vector n is reduced by 1 in the 5d blowup equations. In the case where n 3 can take two values (this happens when r 3 + r 4 = 6k + 3, k ∈ Z), one 6d blowup equation splits to two 5d blowup equations. Note that given the definition of R in (3.2), when we sum n ∈ Z in the blowup equations, we are effectively summing 2R over all the r fields in the same equivalence class. Therefore when dimensionally reducing blowup equations, we can get the equivalence classes of r fields for the 5d theory simply by fixing the value of n 3 , as we prescribed above, and deleting the entry r 3 associated to t 3 .
This procedure also gives us immediately a way to compute Λ 5d from Λ 6d , which are given in section 3.3.1 where n min 3 is a value of n 3 that minimizes the power of Q τ . We list the 5d r fields reduced from 6d r fields as well as the corresponding Λ 5d in Tab. 4.1. They are consistent with a direct compute with 5d blowup equations. Note that Λ 5d for the 5d unity r fields (2, −2, −3), (2, −2, 3), (−2, 2, −3), (−2, 2, 3) cannot be derived by (4.16) though, and they are instead computed from 5d blowup equataions. They are also notably pairwise identical as they should since they descend from the same 6d vanishing blowup equations pairwise. Note that the last entry of 5d r field is r m related to r b by r m = r b − r 1 − r 2 .  The analysis is completely analogous as in the case of the SU (3) theory. The splitting of 6d r fields to pairs of 5d r fields happens when 2r 4 + r 6 = 8k + 4, k ∈ Z. We list the resulting 5d r fields and the corresponding Λ 5d in Tab. 4.2. Note that the last entry of 5d r field is r m related to r b by r m = r b − 2r 1 − 2r 2 − 2r 3 − 4r c . (4.21)

Conclusion and discussion
In this paper, we consider the n = 3, 4 minimal 6d SCFTs in the tensor branch. These theories are obtained by F-theory compactification on non-compact elliptic Calabi-Yau threefolds. We demonstrate that the elliptic genera of these theories, which encode the refined BPS invariants of the underlying Calabi-Yau threefolds, satisfy the generalized blowup equations. Furthermore, we illustrate that the generalized blowup equations can be used to solve the elliptic genera as well as the refined BPS invariants. We emphasize here that the generalized blowup equations is an extremely powerful tool for computing the BPS invariants of non-compact Calabi-Yau threefolds. All the currently existing techniques for computing BPS invariants in local geometry, being well established and very powerful, have their limitations in terms of accessible geometries. The topological vertex [42,47,48] is only applicable for toric geometries or generalizations thereof. The holomorphic anomaly equations [49,50] and the topological recursion [51][52][53] are useful only if the mirror geometry is known, and in particular if the number of compact divisors g 24 in the original geometry is low (g ≤ 2). The modular bootstrap [13,33,38,54,55] only works if the Calabi-Yau is elliptic and is most efficient if there is no singular elliptic fiber (see also [56]). On the other hand, the generalized blowup equations are more versatile than any of these individual methods. They have been applied in toric geometries [34], elliptic geometries (which are non-toric), and the cases where the number of compact divisors is greater than two (this paper). Up to this moment, there does not seem to be any restriction on the type of non-compact Calabi-Yau threefolds for which the generalized blowup equations are applicable.
Nevertheless, we have to point out that why the generalized blowup equations work still remains a mystery. The only case where the blowup equations have a rigorous mathematical proof is when applied on the X N,m geometries [24]. A better mathematical or physical understanding of the generalized blowup equations would be extremely desirable. For example, what is the relation between blowup equations and refined holomorphic anomaly equations? To anwser this question requires a non-holomorphic version of blowup equations. And what is the relation between blowup equations and refined topological vertex? For 6d SCFTs, this may involves the recently proposed elliptic topological vertex [57]. Furthermore the moduli space of the topological string theory usually contains both geometric and non-geometric phases. In this paper we only work deep in the geometric phase around the large volume limit. It is an interesting problem to study the blowup equations in the other phases of the moduli space as in [34]. Finally one could certainly push the computation of the BPS invariants for an almost infinite range of noncompact Calabi-Yau threefolds. The easiest targets and the most similar to what are studied here are those for the remaining cases of minimal 6d SCFTs, the results of which we will report in companion papers in the near future.
(A. 8) It has been argued from the 1 , 2 expansion of refined free energy and blowup equations [34] that under this condition the apparent denominator of Bl (j L ,j R ,R) (q 1 , q 2 ) can always be factored out so that Bl (j L ,j R ,R) (q 1 , q 2 ) = finite series in q 1 , q 2 . (A.9) We call (A.9) fundamental identities. Note that since Bl (j L ,j R ,−R) (q 1 , q 2 ) = Bl (j L ,j R ,R) (q −1 1 , q −1 2 ) , (A. 10) we only need to consider the cases with R ≥ 0.
In the following, we present some frequently used instances of the fundamental identities for small spins. As we have seen in the main text, the contribution of vector multiplets can always be factorized as products of T R (z) = PE − Bl (0,1/2,R) (q 1 , q 2 )Q z + Bl (0,1/2,−R) (q 1 , q 2 ) Q τ Q z .
(A. 16) In the case of R < 0 we can use the above expression for −R with 1,2 replaced by − 1,2 or equivalently q 1,2 replaced by 1/q 1,2 . In both cases,θ R (z) is a multivariate Jacobi form of weight zero and index quadratic form