Spheres, charges, instantons, and bootstrap: A five-dimensional odyssey

We combine supersymmetric localization and the conformal bootstrap to study five-dimensional superconformal field theories. To begin, we classify the admissible counter-terms and derive a general relation between the five-sphere partition function and the conformal and flavor central charges. Along the way, we discover a new superconformal anomaly in five dimensions. We then propose a precise triple factorization formula for the five-sphere partition function, that incorporates instantons and is consistent with flavor symmetry enhancement. We numerically evaluate the central charges for the rank-one Seiberg and Morrison-Seiberg theories, and find strong evidence for their saturation of bootstrap bounds, thereby determining the spectra of long multiplets in these theories. Lastly, our results provide new evidence for the F-theorem and possibly a C-theorem in five-dimensional superconformal theories.

A new angle to study (super)conformal field theories is the conformal bootstrap of local correlation functions. This method exploits unitarity, (super)conformal symmetry, and crossing symmetry to constrain the local operator spectrum and OPE coefficients, and has been successfully applied to various theories in dimensions from two to four , to non-supersymmetric five-dimensional theories such as the O(N ) models [115][116][117][118], and to six-dimensional superconformal field theories [119,120]. It is then natural to ask what can be learned about five-dimensional superconformal field theories, if we combine the power of supersymmetric localization with the bootstrap machinery.
Up until now, all known localization results in five dimensions have made no contact with correlation functions of local operators in the fixed-point theory. One aim of the current paper is to develop a systematic procedure for extracting such correlation functions from the supersymmetric partition function, which can be computed by localization. A reasonable starting point is to consider the two-point functions of the stress tensor and the conserved currents. Because conformal symmetry and the conservation laws fix their tensor structures uniquely, and supersymmetry generates the two-point functions of all other operators in the associated superconformal multiplets, the only independent physical quantities are the overall coefficients, namely, the conformal central charge C T and the flavor central charge C J . 3 We shall study conformal field theories on the five-sphere, which is conformally flat and naturally regulates infrared divergences. Since the stress tensor is coupled to the spacetime metric, it is no surprise that under small perturbations of the five-sphere metric g µν , the variations in the free energy 4 F S 5 ≡ − log Z S 5 (1.1)

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also carefully take into account the potential ambiguities in F S 5 due to local counter-terms in terms of the background fields. Although the connection between the five-sphere free energy and the central charges exists in bosonic conformal field theories as well, the relation is not practically useful unless the free energy at the fixed point can be reliably computed in a weakly-coupled (effective) description, such as in the infrared gauge theory. Supersymmetry thus comes as a crucial ingredient. Most of the known five-dimensional superconformal field theories admit gauge theory descriptions after mass deformations and renormalization group flows. The particular mass parameter m I that triggers the desired flow is identified (up to a numerical constant) with 1/g 2 YM [7]. This same mass parameter is also associated with the U(1) I instanton symmetry with the conserved current J U(1) I = i 8π 2 * Tr(F ∧ F ), (1.2) and sets the mass scale for the instanton particles in the gauge theory. It has been conjectured that the supersymmetric five-sphere partition function of the superconformal field theory deformed by the instanton particle mass m I is exactly captured by the infrared gauge theory [32,35,40]. 6 In particular, the undeformed free energy F S 5 for the fixed-point theory is obtained by sending m I → 0, which is the strong coupling limit in the gauge theory. Supersymmetry also puts strong constraints on the admissible counter-terms and their contributions to the deformed free energy. We now summarize the key results and discoveries of this paper.
Extracting C T and C J from the five-sphere free energy (section 3). We can consider supersymmetric deformations of the five-sphere partition function by turning on (real) squashing parameters ω i = 1 + a i for the metric (a i = 0 gives the round-sphere metric), and mass parameters M a for the flavor symmetries. These can be systematically studied by coupling the superconformal field theory to off-shell background supergravity [30,127]. It is straightforward to compute the dependence of the free energy F S 5 on these small supersymmetric deformations using conformal perturbation theory. We obtain the following expression for the quadratic terms in the squashing parameters a i , and similarly for the quadratic terms in the mass parameters M a ,

4)
6 A priori, one may worry about potential higher derivative deformations of the gauge theory Lagrangian that may modify the localization result. Because of the preserved supersymmetry, the partition function is protected from the Q-exact deformations of the localization Lagrangian. On general grounds, Q-closed deformations of the gauge theory Lagrangian should come from integrating out BPS particles (e.g. W-bosons and point-like instantons). Since such BPS states are all realized in the gauge theory and incorporated in the localization computation, it is plausible that the "bare" Lagrangian suffices for our purpose here. We emphasize that this is merely a heuristic argument.

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where r is the radius of the sphere. These formulae allow us to determine the central charges C T and C J from the deformed five-sphere partition function, which is computable by localization in the infrared gauge theory.
Supersymmetric counter-terms (section 2). A subtlety in the computation of the five-sphere free energy is that there may be potential ambiguities due to finite local counterterms, given by supersymmetric completions of various (mixed) Chern-Simons terms. For example, there are the counter-terms W ∧ dW ∧ dW, W ∧ Tr(R ∧ R), (1.5) for W the U(1) background gauge field and R the Riemannian curvature two-form. Such contributions capture our ignorance of contact-terms in the short distance limit of correlation functions. Their coefficients are quantized by demanding invariance under large background gauge transformations. Although the Chern-Simons terms are perfectly conformal, it turns out that in some cases their supersymmetric completions break conformal invariance. This means that if the relevant non-conformal contact-terms have fractional coefficients, we can choose to restore conformal invariance at coincident points at the expense of introducing a non-quantized supersymmetric Chern-Simons counter-term, thereby breaking large background gauge symmetry. In other words, we cannot simultaneously fulfill conformal invariance and background gauge invariance with supersymmetric regulators. This tension between conformal symmetry, background gauge symmetry, and supersymmetry is analogous to what happens in three dimensions [123,128], and signals the existence of a new superconformal anomaly in N = 1 theories in five dimensions. It turns out that these counter-terms do not affect the relations (1.3) and (1.4), but are crucial in understanding flavor symmetry enhancement on the five-sphere.
Instanton contributions to the five-sphere partition function (section 4). Having understood how to extract the central charges from localizing the gauge theory path integral, we apply this procedure to the "simplest" class of five-dimensional superconformal field theories, namely the rank-one Seiberg E N f +1 and Morrison-Seiberg E 1 theories. These theories are described in the infrared by a USp (2) super Yang-Mills theory coupled to N f fundamental hypermultiplets. The infrared flavor symmetry SO(2N f ) × U(1) I is enhanced to E N f +1 at the ultraviolet fixed point. 7 As usual in localization, the gauge theory path integral reduces to an integral over BPS configurations weighted by their classical action and one-loop determinants. The novelty in five dimensions is that the BPS locus also involves the so-called contact instantons for the gauge fields. 8 It was conjectured that for generic squashing, the contributions from 7 Such USp(2) gauge theories come with a discrete theta parameter θ ∈ {0, π}, prescribing how the two topological sectors characterized by π4(USp(2)) ∼ = Z2 are summed together in the path integral. For N f > 0, the massless infrared Lagrangian has symmetry O(2N f ) ∼ = Z2 SO(2N f ) acting on the hypermultiplets, and the Z2 normal subgroup which flips the sign of a single mass is equivalent to exchanging θ = 0 with θ = π [12,42,61]. Therefore, for N f > 0, the two theories at θ = 0 and θ = π are equivalent. This Z2 symmetry of the Lagrangian is not a symmetry of the gauge theory. 8 Upon circle compactification, the five-dimensional contact instantons reduce to the usual Yang-Mills instantons in four dimensions.

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contact instantons on the five-sphere are captured by gluing together three copies of the Nekrasov partition function Z 1,2,3 S 1 ×R 4 [38,40]. In other words, the conjecture states that 9 Z S 5 = [dλ]e −F ∨ eff (λ) Z 1 S 1 ×R 4 (λ)Z 2 S 1 ×R 4 (λ)Z 3 S 1 ×R 4 (λ), (1.6) where F ∨ eff (λ) is a cubic polynomial in λ, and is equal to the one-loop effective prepotential when every Coulomb branch parameter λ is larger than all the masses of the hypermultiplets. We shall address some key issues with (1.6) and write down a precise formula, where the chemical potentials in the Nekrasov partition function Z S 1 ×R 4 are substituted by the deformation parameters (squashing and masses). Although it is well understood that the flavor chemical potentials for the hypermultiplets involve imaginary shifts, the shift for the U(1) I instanton number chemical potential has not been determined in the literature. We fix this ambiguity by demanding that the five-sphere partition function exhibits enhanced flavor symmetry E N f +1 in the ultraviolet. Our formula also renders key physical quantities real at arbitrary instanton order.
A generic squashed five-sphere can be viewed as a circle-fibration over a compact base manifold. The three S 1 ×R 4 patches are located at the three fixed points of the U(1) ×U(1) isometry of the base. In the weak coupling limit m I → ∞, each Z S 1 ×R 4 is a series in e −m I , whose coefficients are computed by the Witten index of a certain "generalized" ADHM quantum mechanics. However, we are interested in the strong coupling limit m I → 0 in order to probe the superconformal fixed point. Ideally, one would hope to re-sum the entire series in e −m I and then re-expand around m I = 0, akin to what has been done for the fivedimensional N = 2 super Yang-Mills theory [32,38,40]. However, the re-summation for N = 2 super Yang-Mills relies on the known modularity properties of the instanton series, that follow from a relation to the six-dimensional N = (2, 0) "parent" theories. Since such properties are not known for the N = 1 gauge theories, we proceed by numerically evaluating the instanton contributions up to fourth order in the e −m I expansion, and setting m I → 0. Miraculously, we shall observe that the free energy F S 5 appears to converge (with squashing and mass deformations turned on), at least up to cubic order in the deformation parameters. We thus reliably compute the undeformed free energy and the central charges C T and C J .
Symmetry enhancement in the five-sphere partition function (section 4). Based on the fiber-base duality of the M-theory or type-IIB five-brane setup, it was observed in [49] that the Nekrasov partition function Z R 4 ×S 1 for the Seiberg theories exhibit not just the manifest SO(2N f ) × U(1) I flavor symmetry but the full enhanced E N f +1 . This directly implies that the superconformal index is also invariant under the Weyl group action of the enhanced flavor symmetry on the flavor fugacities [33,48]. Here φ collectively denotes the gauge field holonomies. The 9 Notice that each Z S 1 ×R 4 contains perturbative and instanton contributions.

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situation is less clear for the five-sphere free energy F S 5 , due to the appearance of F ∨ eff (λ) in (1.6). However, we shall see that after incorporating appropriate local counter-terms to the gauge theory action, F S 5 is indeed invariant under the enhanced Weyl group action on the mass parameters. Our arguments for flavor symmetry enhancement in F S 5 formally requires the inclusion of instanton contributions to all orders in e −m I . Nevertheless, at low orders in e −m I , we already find solid numerical evidence for the enhancement. In particular, we compare the values of C J for the E N f +1 flavor symmetry computed on one hand by SO(2N f ) mass deformations, and on the other by a U(1) I mass deformation, and find beautiful agreement. These checks provide extra confidence for the convergence of the instanton expansion at strong coupling.
Superconformal bootstrap (section 5). After reliably computing the values of the central charges C T and C J in the Seiberg theories, we make a connection to the conformal bootstrap of the four-point functions of moment map operators, the superconformal primaries that generate the flavor current multiplets. The semi-definite programming technique generates constraints on the intermediate spectrum and the OPE coefficients, some of which involving the stress tensor and the flavor current multiplets depend on the central charges C T and C J . More specifically, demanding that the flavor symmetry be SU(2), E 6 , E 7 , and E 8 , we obtain lower bounds on the central charges C T and C J . The bootstrap method is most powerful when a conformal field theory saturates the bound, in which case, one can use the extremal functional method to determine the non-BPS spectrum and their OPE coefficients [85,88]. We find strong evidence for the saturation by the rank-one Seiberg theories, and thereby make predictions for the dimension of the lowest non-BPS operator appearing in the intermediate channel. One can also in principle solve for the OPE data in these theories, going much beyond the data that supersymmetric localization alone can provide. We demonstrate the power of this method in this paper, and invite for further explorations.

Supersymmetric five-sphere partition function
Any five-dimensional conformal field theory can be put on a five-sphere in a canonical manner by the stereographic mapping from R 5 . The five-sphere free energy F S 5 should then capture physical information about the conformal field theory. However, such quantities generally suffer from divergences and require regularization. The sphere being compact eliminates potential infrared divergences. However, at short distances the theory is indistinguishable from that on flat space, which leads to ultraviolet divergences of the form [129] F S 5 ∼ (Λr) 5 + (Λr) 3 + (Λr) + finite. (2.1) The explicit r-dependence breaks conformal invariance, but these divergent pieces can always be removed by introducing local diffeomorphism-invariant counter-terms of the schematic form where R denotes the background curvature. Such counter-terms are obviously nonconformal. However, they compensate the non-conformal pieces in (2.1), giving rise to a regularization scheme, which actually preserves conformal invariance on the sphere. This is to be contrasted with the even-dimensional case, in which the five-sphere free energies of conformal field theories are of the form F S 2n ∼ (Λr) 2n + (Λr) 2n−2 + · · · + a log(Λr) + finite, (2.3) where the logarithmic divergence cannot be cancelled by a local diffeomorphism-invariant counter-term. In that case, the coefficient a becomes a physical observable of the conformal field theory, the well-known trace anomaly. The presence of the log-term also means that the finite term in F S 2n is ambiguous. In contrast, the finite piece of F S 5 is unambiguous and an intrinsic observable of the conformal field theory, since there is no diffeomorphisminvariant counter-term dependent on the background curvature that could shift its value.
In other words, with any ultraviolet diffeomorphism-invariant regularization scheme, we obtain the same answer for the finite part of F S 5 . From now on, we write F S 5 to mean the finite part after subtraction by the counter-terms in (2.2). In unitarity theories, F S 5 is a real number. We can deform from the conformal five-sphere background either by (a) putting the conformal field theory on a general Riemannian manifold (M, g µν ), or (b) if the conformal field theory has global symmetry, by coupling the global symmetry currents J µ to background gauge fields W µ . Upon such deformations, the free energy F M (g, W ) typically suffers from ambiguities due to contact-terms among the conserved currents and the stress tensor of the form written in momentum space for convenience. The coefficients κ JJJ and κ T T J are real due to unitarity. 10 Such ambiguities can be classified by the possible local counter-terms, which involve the background metric and gauge fields, subject to diffeomorphism and gauge invariance. 11 For example, at cubic order in the background fields, we can have (mixed) five-dimensional Chern-Simons terms 12 which are conformal. On generic backgrounds, these counter-terms give nonzero contributions to F M (g, W ), which is generally complex. In supersymmetric theories, the contact-term ambiguities are further constrained by insisting on a supersymmetric regularization scheme. Such a scheme can be systematically JHEP03(2018)123 formulated by coupling the superconformal field theory to off-shell background N = 1 supergravity, and its flavor symmetries to background N = 1 vector multiplets. We should only consider counter-terms in the supergravity that are manifestly supersymmetric. Due to the presence of additional fields in the background multiplets, there are a number of non-conformal supersymmetric counter-terms. 13 Depending on which symmetries of the theory we want to preserve at short distances (on nontrivial backgrounds), we have different choices for the coefficients of the counter-terms. As we shall see, the tension between supersymmetry, conformal invariance and invariance under large background gauge transformations leads to a new superconformal anomaly for five-dimensional superconformal field theories. 14 2.1 Off-shell supergravity and conformal field theories on five-spheres Before going into the details of the supersymmetric free energy of five-dimensional superconformal field theories, let us first clarify how supersymmetry is preserved on a round sphere, and discuss the relevant off-shell supergravity theories.
When we place a five-dimensional superconformal field theory on the sphere via the stereographic map, the full superconformal symmetry F (4) is formally preserved. Hence, the background coupling is naturally described by conformal supergravity. In particular, the superconformal stress tensor multiplet couples to the standard Weyl multiplet which contains the metric g µν , the dilaton D, an su(2) R gauge field V ij µ , a two-form field v µν , the gravitino ψ i µ , and the dilatino χ i . Similarly, the superconformal current multiplets couple to five-dimensional vector multiplets where W µ is the background gauge field for the flavor symmetry, M denotes the scalar mass parameter, Ω i α is the gaugino, and Y ij is a triplet of auxiliary scalars. However, due to ultraviolet divergences, the partition function and the correlation functions need to be regulated. On a round sphere, the maximal subalgebra that can be preserved by the regulators is su(4|1). The eight supercharges are parametrized by symplectic-Majorana Killing spinors ε i satisfying where t = i 2r σ 3 , and r is the radius of the sphere. Therefore, to describe the coupling between the (regulated) superconformal field theory and the (deformed) five-sphere background, we should consider Poincaré supergravity, whose gauge algebra is su(4|1), instead of conformal supergravity which gauges the entire F (4). 13 Stated differently, the correlation functions of currents themselves have conformal contact-terms as in (2.4), but supersymmetry implies that the correlation functions of the other operators in the current multiplets must have non-conformal contact-terms, if the former are present.
14 This is analogous to the superconformal anomaly of three-dimensional N = 2 superconformal field theories studied in [123,128]. In that case, the anomalies originate from contact-terms in the two-point functions of current multiplets, whereas here they appear in the three-point functions.

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Off-shell Poincaré supergravity can be obtained by introducing two gauge-fixing compensators to the standard Weyl multiplet in conformal supergravity [130,131]. One of them needs to be a vector multipletŴ 9) and the other can either be a hypermultiplet, a linear multiplet, or a non-linear multiplet. We choose the other compensator to be a linear multiplet whereN andL ij are scalars,Ê µ is a divergenceless vector, andφ i α is their fermionic partner. As we shall see, this ensures that upon gauge-fixing, the resulting Poincaré supergravity admits the desired supersymmetric five-sphere background.
We gauge-fix the dilatation and superconformal transformations by setting 15 and break the SU(2) R symmetry down to U(1) R by imposinĝ Here µ and L are arbitrary constants of mass dimension one and two, respectively. To summarize, the off-shell Poincaré supergravity multiplet contains the component fields [131] A supersymmetric five-sphere background can be obtained by setting all fermionic fields in the Weyl multiplet and the compensator multiplets to zero, as well as by requiring the vanishing of their supersymmetric variations (2.14) Here D µ denotes the covariant derivative D µ ≡ ∇ µ − V µ . Furthermore, F (Ŵ ) and F (V ) are the field strengths ofŴ µ and V ij µ respectively. In particular, given the Killing spinor equation (2.8), all variations in (2.14) vanish if we set and all other background fields to zero. 15 The standard Weyl multiplet also contains a gauge field bµ for the dilatation symmetry. Here we set bµ = 0 in order to fix the special conformal gauge symmetry.

Counter-terms and a new superconformal anomaly
The counter-term ambiguities in the supersymmetric five-sphere free energy are characterized by local diffeormorphism-and supersymmetry-invariant functions of the Poincaré supergravity multiplet (2.13), and also the vector multiplets (2.7) if the superconformal field theory has global symmetries. Equivalently, we can write these counter-terms in terms of the Weyl multiplet (2.6) and the two compensator multiplets, while keeping in mind the gauge-fixing conditions (2.11) and (2.12). For ease of notation, we take the latter approach. Let us first examine the dimensionful bosonic counter-terms in (2.2). There is no supersymmetric completion of the leading Λ 5 divergence, which simply reflects the universal feature of vanishing cosmological constant for supersymmetric field theories. The Λ 3 divergent term can be completed by the supersymmetric Einstein-Hilbert action [130,[132][133][134]. Finally, the Λ divergent term is completed by the supersymmetrized R 2 interactions, which come with three parameters associated to the three independent structures: Ricci squared R µν R µν , Ricci-scalar squared R 2 , and Weyl tensor squared C µνρσ C µνρσ [131,[135][136][137]. 16 These divergent counter-terms allow us to regularize the five-sphere free energy while preserving the massive subalgebra su(4|1). Due to the absence of marginal counter-terms purely in terms of the Poincaré supergravity multiplet (2.13), the regularized supersymmetric five-sphere free energy is real and free of (finite) ambiguities. 17 This freedom from ambiguities is no longer the case once we couple the conserved currents in the superconformal field theory to background vector multiplets W f . The regularized free energy F S 5 can be shifted by marginal counter-terms that are all supersymmetric completions of various (mixed) Chern-Simons terms. Below we shall only keep track of the couplings between the scalars M f in the vector multiplet and the Riemannian curvature R. 18 By unitarity, the overall coefficients κ 1,2,3 T T J and κ JJJ are real numbers.

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This contact-term breaks conformal invariance at short distances, which leads to a nonvanishing one-point function of the scalar N on the five-sphere. 22 If κ 1,2 T T J is non-integral, we have the option of removing this non-conformal contact-term by adding a counter-term of the form (2.17) with the non-integral coefficient thereby breaking the background large gauge invariance explicitly. 23 Either way, the fractional part of κ T T J is a measure of the superconformal anomaly and a physical quantity of the theory. The same can be said about the contact term in the four point function (2.23) As is usual with anomalies, the change of these contact-term coefficients under renormalization group flows will be protected and their fractional parts should also be preserved under dualities. We shall discuss these anomalies in general five-dimensional N = 1 field theories in more details in a separate paper [141].
The fact that the above marginal counter-terms always involve either one or three (non-compensator) vector multiplets implies that F S 5 is unambiguous when there is no coupling between the current multiplets and background gauge fields. With background vector multiplets, the ambiguities of F S 5 (W f ) are constrained to be O(W f ) and O(W 3 f ). Most importantly, the O(W 2 f ) piece, which -as we shall observe in section 3.2 -carries information about the flavor central charge C J , is free from ambiguities. The linear and cubic ambiguities in F S 5 (W f ) will also play an important role in section 4.2.3 in identifying the symmetry enhancement in five-sphere localization.

From the partition function to central charges
In a conformal field theory in d spacetime dimensions, the stress tensor two-point function is constrained by conformal symmetry to take the form Similarly, if we consider a conformal field theory that has a flavor symmetry G generated by the flavor currents J a µ . Then the two-point functions of the flavor currents are constrained by conformal symmetry 22 The operator N has scaling dimension four, thus on the five-sphere it can mix with the identity opera- This is another way to interpret the fact that the one-point function N S 5 is non-vanishing, which is similar to what happens to exactly marginal operators in four-dimensional N = 2 superconformal field theories [140]. 23 The more precise definition of the five-dimensional Chern-Simons terms involve the extension to an auxiliary six-manifold. The fact that we are using a fractional Chern-Simons counter-term means that the partition function now depends on the data of the bulk six-manifold.

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and conservation laws to take the form The normalization of the stress tensor T µν and the flavor currents J a µ are specified in appendix A, and the explicit forms of the conformally covariant structures I µν,σρ (x) and I µν (x) are given in (A.1). In the following, we denote by C G J the flavor central charge of the flavor group G.
This section introduces precise relations between the central charges C T and C J and supersymmetric partition functions on five-spheres for five-dimensional superconformal field theories. In particular, C T can be probed by adding a nontrivial squashing to the five-sphere metric, and C J can be probed by turning on a mass-deformation for flavor symmetries. We first present the explicit formulae relating C T and C J to the respective deformations of the five-sphere partition functions, and test them for the (simple) case of a free hypermultiplet. In sections 3.1, 3.2 and 3.3, we prove these relations by explicitly coupling the theory to background supergravity. An important ingredient in the proof is the supersymmetric background for a five-sphere with generic squashing, which is discussed in appendix B.
Supersymmetric field theories can be put on nontrivial geometries by coupling to offshell background supergravity [127]. When appropriate background superfields are turned on and enough supersymmetry is preserved, the partition function on such backgrounds is generally well-behaved along renormalization group flows. Preserving supersymmetry is crucial because generic interacting superconformal field theories in five dimensions are strongly coupled, and computations are only viably performed in the infrared Lagrangian theory. Supersymmetry allows one to extrapolate these infrared computations to the ultraviolet fixed points (see section 4 for explicit examples). In such curved backgrounds, the stress tensor is coupled to the background metric. Thus, the dependence of the partition function under infinitesimal deformations of the geometry away from the round sphere is captured by correlation functions of the stress tensor. Conversely, the deformed partition function encodes the stress tensor correlators, in particular the stress tensor two-point function, and therefore determines the conformal central charge C T at the superconformal fixed points. The coupling of the stress tensor multiplet to the background supergravity multiplet will be introduced in section 3.1. It depends on the squashing parameters ω i = 1 + a i , i = 1, 2, 3, which appear in the squashed sphere metric (B.5). According to the above discussion, we expect C T to be proportional to the quadratic order coefficients in the free energy F = − log Z under infinitesimal deformations from the round sphere. The precise relation we derive is From here on out, we shall set the five-sphere radius r = 1.
For theories with flavor symmetry G, one can perform a different type of deformation, by coupling the flavor current multiplet to a background vector multiplet. The detailed coupling depends crucially on the mass parameters M a (a = 1, . . . , dim(G)) and will be -13 -JHEP03(2018)123 introduced in sections 3.2 and 3.3. The precise relation between the flavor central charge C G J for G and the mass-deformed round-sphere partition function is given by Suppose the superconformal field theory flows to a gauge theory in the infrared with an (infrared) flavor group K ⊆ G acting on the hypermultiplets. Then the mass term of the hypermultiplet in the action on the round S 5 (ω i = 1) is given by where M ∈ k ≡ Lie(K) is the mass matrix. The relation between the flavor central charge C K J of the flavor group K and the mass-deformed free energy is where Tr(·) is the Killing form defined in (A.8). If the embedding ι : k → g ≡ Lie(G) is known, the flavor central charge C G J of the ultraviolet flavor group G is related by the embedding index (reviewed in appendix C) Free hypermultiplet: a check. In d dimensions, the values of C T for a scalar φ and a Dirac spinor ψ are [142]  For the flavor central charge C J for the SU(2) flavor symmetry of a free hypermultiplet, a calculation analogous to [142] gives (3.10) The partition function of a mass-deformed free hypermultiplet on a squashed sphere is given by [36,39] where ω 1 , ω 2 , ω 3 are the squashing parameters of the squashed five-sphere metric (B.5), and the mass matrix in (3.5) is chosen to be M = imσ 3 . The definition and relevant properties of the triple sine function S 3 are reviewed in appendix F. Using the integral representation (F.5) for S 3 , we find that the free energy F ≡ − log Z is given by the expression .

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In the second term, the series expansions of the integrand in small a i ≡ ω i − 1 and in m commute with the integral. To quadratic order in these parameters we thus find (3.13) which indeed gives the correct C T and C J applying the formulas (3.3) and (3.6).
3.1 Linearized coupling for the stress tensor multiplet and C T The primary operator content of the five-dimensional N = 1 stress tensor multiplet B[3; 0, 0; 0] (appendix D explains the notation) can be summarized as where each entry [d 1 , d 2 ] (2J R ) ∆ labels the representation of the corresponding bosonic conformal primary under the bosonic subgroup of F (4) [143,144]. The d 1 , d 2 and 2J R are the Dynkin labels of the so(5) × su(2) R . The Poincaré supercharges acting on the superconformal primary Φ generate the entire stress tensor multiplet.
The normalization of the individual operators are fixed by requiring (3.16) where Q satisfies the five-dimensional supersymmetry algebra (see appendix A.3), and T µν is canonically normalized as in appendix A.1.
Any five-dimensional superconformal field theory can be put on a curved manifold by coupling the stress tensor multiplet to the background N = 1 standard Weyl multiplet, which contains the dilaton D, the metric g µν , an su(2) R gauge field V ij µ , a 2-form field v µν , the gravitino ψ i µ , and the dilatino χ i . To preserve supersymmetry in the "rigid limit" [127], we require the background fields to have trivial variations under a subset of the N = 1 supercharges. This requirement boils down to the conditions ψ i µ = χ i = 0 as well as The preserved supercharges are parametrized by ξ i and η i subject to the above conditions. The round five-sphere background is given by (2.15), but we do not need to worry about the compensator multiplets here. 24 In appendix B, we derive the general squashed 24 Note that the compensator multiplets (2.9) and (2.10) do not couple to physical operators in the superconformal field theory, and therefore do not enter the linearized coupling. The only way they could show up is in the potential counter-term ambiguities, which is absent here as explained in section 2.1.

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supersymmetric five-sphere background, which is parametrized by three real parameters ω 1,2,3 . Here we shall show that the leading order effect of infinitesimal squashing on F S 5 is precisely captured by the conformal central charge C T . To linear order in perturbation theory around the round S 5 background, we have the coupling 25 We expand the squashed five-sphere partition function for the conformal field theory in the squashing parameters a i = ω i − 1 around a i = 0. The first order terms in this expansion vanish as a consequence of the vanishing one-point functions of the stress tensor multiplet operators. At second order, the contributions are captured by the integrated two-point functions of Φ, B µν and J ij µ . Note that a general feature of the supersymmetric squashed five-sphere background is that Hence, we do not need to consider the two-point functions of T µν to second order in a i . Thus, from (3.18), it follows immediately that to second order in the squashing parameters. Now the squashed five-sphere background in appendix B expanded to linear order in a i gives To evaluate the integrated two-point functions (contracted with spacetime tensors) (3.20), it is convenient to transform to the stereographic coordinates (see appendix B.4), in which case we have (3.22) 25 The coefficients in the linearized coupling are fixed by (3.16) and (3.17). One can also extract the coefficients from the coupling between standard Weyl multiplet and hypermultiplets in [52,138,139].

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The flat-space two-point functions of Φ, J ij µ , and B µν are given by which can be derived from (3.16) and the superconformal Ward identities. Using stereographic projection, we obtain the two-point functions on the five-sphere, where we introduced a frame 25) and the SO(6) invariant distance, To simplify the integrated two-point functions, we first take advantage of the SO(6) invariance of the measure, to rotate y to 0. This allows us to extract a factor of the unit five-sphere volume The remaining integral over d 5 x √ g has power law divergences at small x and needs to be regularized. Here we use the dimensional regularization (analytic continuation using the Gamma function) .
The choice of regularization does not affect the result, as follows from the counter-term analysis in section 2.2. Let us start with the first term in (3.20), which has contributions from JJ . It can be evaluated to Next we consider the contribution from BB , which leads to

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Finally, the remaining ΦΦ term integrates to Putting everything together, we end up with the previously advertised equation 3.2 Linearized coupling for the current multiplet and C J Now, let us consider a five-dimensional N = 1 superconformal field theory with flavor symmetry G. The primary operator content of a flavor current multiplet D[3; 0, 0; 2] (appendix D explains the notation) is where a is the adjoint label. More explicitly, the primary operators N a and J a µ are related to the superconformal primary L a ij by We can introduce mass deformations for the superconformal field theory whilst still preserving supersymmetry, by coupling to background off-shell supergravity and vector multiplets. The values of the background fields will be constrained by requiring that their off-shell supersymmetry variations vanish. Here we are interested in mass-deformed fivedimensional superconformal field theory on the five-sphere, in which case the background supergravity fields take values as in (2.15). The coupling to a background vector field with components W a µ , M a , Ω ai , Y aij , (3.35) takes the form [139] where the terms involving fermions are suppressed. The supersymmetry variation of the background gaugino is where ε j is a symplectic-Majorana spinor satisfying the Killing spinor equation (2.8) on the five-sphere. It is easy to see that with M a constant element of g = Lie(G), gives the desired supersymmetry-preserving mass deformation

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As explained in section 2.2, while the O(M ) and O(M 3 ) terms in the mass-deformed five-sphere free energy are ambiguous, the O(M 2 ) dependence is not, and is in particular determined by The two-point functions of N a and L a ij on flat space are related to the two-point function of J µ by (3.34) and the superconformal Ward identities. The two-point functions on the fivesphere are then given by stereographically projecting the flat space two-point functions, which results in where s(x, y) is the geodesic distance (3.26) on the sphere. Performing the double integral by dimensional regularization using (3.28), we end up with

Relation to the gauge theory partition function
As explained in the introduction, part of the ultraviolet global symmetries are preserved in the infrared gauge theory phase. They are realized either by the flavor symmetries of the hypermultiplets or by the topological U(1) I instanton symmetry. To compute C J from (3.42) by localization for either of these subgroups (see section 4), we need to correctly normalize the mass matrix, or equivalently normalize the conserved current multiplet (now no longer superconformal) in the infrared gauge theory. We fix the normalization by demanding that the current J a µ is canonically normalized as in (A.4) and (A.5). This prescription is unambiguous (irrespective of the gauge coupling) thanks to the Ward identity of the flavor symmetry. The normalization of the other bosonic conformal primaries in the current multiplet are fixed by supersymmetry (3.34). In particular, the OPE between the moment map operators, at the ultraviolet fixed point, takes the form with the prescribed normalization. Below we shall identify the normalized infrared current multiplets associated to both the hypermultiplet flavor symmetry and the U(1) I instanton symmetry. Suppose we gauge a subgroup H ⊂ USp(2N ). The commutant (centralizer) then gives the flavor symmetry of the hypermultiplets. For simplicity, let us focus on one simple factor K of the commutant subgroup, and consider the charged hypermultiplets that transform in the representation (R H , R K ) of H ×K. The action for N massive (gauged) hypermultiplets where we labeled fields in the dynamical vector multiplets as We have suppressed the R H and R K indicesȦ = 1, · · · , dim(R H ) and A = 1, · · · , dim(R K ), and definedψḂ with the invariant tensors IȦḂ and I AB given by restrictions of the invariant tensor of USp(2N ). To be explicit, the IȦḂ and I AB are normalized such that The mass matrix M is Hermitian, takes values in k = Lie(K), and couples in (3.44) to the moment map operators as where T a denote the Hermitian generators of k with normalization given in (A.10). The normalization in (3.48) needs to be specified before we use (3.42) to compute C J from localization. Since the normalization is independent of the gauge coupling, we can work in the weak coupling limit, and use the two-point functions for free hypermultiplets, 26 Therefore, the normalized current is given by and the normalized moment map operators are fixed by supersymmetry to be Tr(M 2 ), (3.54) or equivalently, where M is the mass matrix in (3.44), and C 2 (R K ) is the Dynkin index associated with R K . At the ultraviolet fixed points of five-dimensional theories, the global symmetry is typically enhanced to a larger group G ⊃ K. If G is simple (or a simple factor), then we can obtain C G J from C K J through the embedding index and similarly when K is replaced by the U(1) I instanton symmetry. In appendix C, we provide some details about the embedding indices appearing in the Seiberg exceptional superconformal field theories.

Seiberg and Morrison-Seiberg exceptional theories
A special class of five-dimensional superconformal field theories are the theories with exceptional E n (n = 1, . . . , 8) flavor symmetry proposed by Seiberg [7], and E 1 = U(1) by Morrison and Seiberg [8]. For n < 6, the flavor groups are E 5 = SO(10), E 4 = SU (5), (1), and E 1 = SU (2). For each flavor group, there is a family of theories of labeled by their ranks, and the rank-N theory has an N -dimensional is the Weyl group of USp(2N ). The family of Seiberg E 8 theories arise in the low energy limit of N D4 branes probing a nine-dimensional E 8 singularity, which is constructed out of seven D8-branes on top of an O8-orientifold plane at infinite string coupling (table 1). The Seiberg E 8 theories can be deformed by the dimension-four R-symmetry-singlet scalar primary N a in the E 8 flavor current multiplet (see (3.33)), and flow to low energy theories with smaller flavor symmetry. The flows fall into two categories, as depicted in figure 1: • Flow 1. If the deformation by N a breaks the E 8 flavor group to E n (n < 8), then the infrared fixed point is the Seiberg E n theory of the same rank. There exist similar flows from E n to E m (and E 1 ) for n > m. • Flow 2. If the deformation by N a breaks the , then the low-energy theory is N = 1 USp(2N ) Yang-Mills coupled to N f fundamental hypermultiplets together with a single antisymmetric hypermultiplet (the latter is decoupled for rank-one). 27 Along Flow 2, the operator N a flows to the Yang-Mills Lagrangian density in the infrared. In the language of the infrared, the inverse Yang-Mills coupling squared 1/g 2 YM is the mass parameter that parameterizes the renormalization group flow. The infrared gauge theories contain instanton particles of mass m I = 4π 2 /g 2 YM , which are charged under the instanton symmetry U(1) I . Furthermore, along Flow 2, it was assumed in [32,35,40] that with respect to a particular supercharge Q, the Yang-Mills term is Q-closed, while all other irrelevant operators are Q-exact. 28 Another possibility is as we explained in footnote 6. Consequently, the partition function is insensitive to those irrelevant couplings. The supersymmetric partition function can be computed by standard localization techniques, in which a Q-exact term is added to the action that formally does not change the partition function [25][26][27][28][29][30]. When the coefficient of the Q-exact term is tuned to be large, the path integral localizes to the fixed-point loci of the action of the supercharge Q. The result from 27 The antisymmetric hypermultiplet transforms under an additional mesonic SU(2) m flavor symmetry.
However, since our focus in this paper is on the rank-one theories, we shall omit this additional SU(2) m from most of our discussions (in particular, we omit its corresponding chemical potential from the partition function), and refer the reader to [145]. 28 Similar arguments were made for five-dimensional maximal N = 2 super Yang-Mills [32,40]. In fact, the deformations of maximally supersymmetric gauge theories that preserve 16 supersymmetries have been classified in [146][147][148][149][150]. Based on this classification, there is only one irrelevant deformation that could potentially not be Q-exact -the supersymmetric completion of Tr(Fµν FνρFρσFσµ). Thus the claim of [32,40] is essentially the statement that this particular term does not arise in the compactification of the six dimensional N = (2, 0) theory to five dimensions.
where Z pert is the one-loop partition function at the perturbative fixed point, n labels the nontrivial fixed points, S n ∝ m I is the classical action and Z n the one-loop determinant of the Q-exact localizing term, both at the nth fixed point. The goal of this section is to explicitly compute the supersymmetric five-sphere partition function for the Seiberg exceptional theories. We borrow the matrix model expression for the localized path integral [31,32,39] and a recipe for computing the instanton contributions [38,40,48] from the literature. The resulting free energy is a function of the squashing parameters ω i = 1 + a i (i = 1, 2, 3), the instanton particle mass m I , and the masses m f (f = 1, . . . , N f ) of the fundamental hypermultiplets. The m f appear in the mass matrix in the hypermultiplet action (3.44) as 29 which is a Cartan generator of SO(2N f ) in the vector representation. In section 4.1, we first discuss the dependence of the partition function on the central charges, and certain properties that follow from the Weyl group of the exceptional flavor symmetry. Since only the SO(2N f ) × U(1) I subgroup is manifest in the gauge theory, these properties provide highly nontrivial checks on the precise triple factorization formula, which is presented in section 4.2, and explicitly (numerically) evaluated in section 4.3.

Central charges and constraints from Weyl group
We apply the relations derived in section 3 between the central charges C T , C J and the five-sphere free energy to the specific case of the Seiberg E N f +1 theories, with the mass matrix given in (4.3).  29 Although the physical mass matrix M in (3.5) is Hermitian, when performing localization, M has be to analytically continued to an anti-Hermitian matrix for the convergence of the localized path integral [30,31]. 30 In appendix E, we fix the normalization of the U(1)I instanton current multiplet by requiring that the minimal charge of the U(1)I symmetry is one. This gives M = imI ≡ 4π 2 i/g 2 YM in (3.36).
where the embedding indices are computed in appendix C to be By the embedding map given in [33,48], the SU(2) mass matrix is ( 1 4 m 1 − 1 2 m I )iσ 3 , and the U(1) mass parameter is M = 7 4 m 1 + 1 2 m I . From (3.4) and (3.6), the free energy at quadratic order in the mass parameters is (4.7) (2) part is enhanced to SU(3) at the ultraviolet fixed point, with embedding index I su(2) →su(3) = 1. By (3.4) and (3.6), the free energy at quadratic order in the mass parameters is The Weyl group of the enhanced flavor group E N f +1 acts on the mass parameters as linear transformations on m f and m I . Up to possible counter-terms at linear and cubic orders in the mass parameters (see section 2.2), the free energy F (ω i , m f , m I ) should be invariant under the enhanced Weyl group. For example, there is a second degree Weyl group invariant polynomial which is unique for N f = 0, 3, 4, . . . , 7, and the second equality in (4.5) is a consequence of the enhanced Weyl group. The Weyl group actions on the mass parameters m f and m I are the same as the Weyl group actions on the chemical potentials m f and m I , which are specified in (4.26).
The above properties are only expected to be true for the exact five-sphere partition function at the ultraviolet fixed point. When computed using the gauge theory description, all instanton saddles must be taken into account, and the instanton particle mass m I -24 -

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should be taken to zero in order to hit the ultraviolet fixed point. A finitely truncated free energy would not exhibit enhanced Weyl group invariance; in particular, the flavor central charges C SO(2N f ) J and C U(1) I J would not satisfy the relations (4.5). Since the instanton expansion (4.1) becomes uncontrolled at m I = 0, a priori, one could not hope that the partition function computed by a finite-order truncation of the series gives a good approximation to the exact partition function. However, by explicit numerical computation up to four instantons in section 4.3, we see miraculously that the expansion (4.1) seems to converge even at m I = 0, and the coefficients of the terms violating enhanced Weyl group diminish at higher and higher orders! In particular, the violation of the relation (4.5) between C

Heuristic argument for triple factorization
It was conjectured in [32,38,40] (also see [151,152] for reviews) that the five-sphere partition function of five-dimensional N = 1 gauge theories can be computed by a Coulomb branch integral that "glues" three copies of the Nekrasov partition function. The heuristic argument of this triple factorization goes as follows. In the localization computation of the partition function on the squashed five-sphere background (B.5), one adds a large Q-exact term to the action. At the fixed-point loci of Q, the hypermultiplets have vanishing vacuum expectation values, while the gauge fields in the vector multiplets satisfy the "contact instanton equation" [29,30], and the scalar fields in the vector multiplets satisfy Here, F ≡ F − iκ −1 φ dY; the Reeb vector ξ µ , scalarsκ andβ, and one-form Y are defined in appendix B.2. A class of solutions to the scalar field equations (4.11) is φ =κλ, (4.12) with λ being a constant that takes values in the Cartan subalgebra of the gauge group, and [ F µν , λ] = 0, by the first equation in (4.11). The five-sphere partition function is computed by an integral over the Coulomb branch parameters λ, of the schematic form where S cl,n is the classical action of the n-th solution to the contact instanton equation, and Z n is the one-loop determinant in that background. The most general solution to the contact instanton equation is unknown, but there is a class of solutions given by embedding the flat space instanton solutions as follows. We can -25 -

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view the squashed five-sphere metric as a (singular) T 3 fibration over the base S 2 , which we parameterize with constrained coordinates y i satisfying y 2 1 + y 2 2 + y 2 3 = 1. The T 3 fiber degenerates to S 1 at three points on the S 2 , (y 1 , y 2 , y 3 ) = (1, 0, 0), (0, 1, 0), and (0, 0, 1). (4.14) At these three points, the Reeb vector ξ µ generates closed orbits along the S 1 , and the contact instanton equation (4.10) reduces to the instanton equation on R 4 . One can embed the R 4 small instanton solution into S 5 , localized at one of the three points and constant along the direction of the Reeb vector. When the Coulomb branch parameters λ take generic values in the Cartan subalgebra, the field strength of the small instanton must also be in the Cartan subalgebra, such that it commutes with λ. The instanton solutions at different degenerate points can be superposed since the field strengths are all in the Cartan subalgebra. Consider a solution with instanton numbers k 1 , k 2 , and k 3 at the three degenerate points. The classical action was computed in [40] to be where m I = 4π 2 /g 2 YM is the mass of the instanton particle. At each degenerate point, the geometry is locally S 1 × R 4 , and it was conjectured that the one-loop determinants factorize into three copies of the twisted partition function on S 1 ×R 4 associated to the three degenerate points. The S 1 × R 4 partition function suffers from infrared divergences, which can be regularized by the twisted boundary condition (z 1 , z 2 , t) ∼ (z 1 e − 1 , z 2 e − 2 , t + 1), where t is the coordinate of the S 1 with unit radius, z 1 and z 2 are the complex coordinates on the two two-planes R 12 , R 34 ∈ R 4 . The twisted S 1 × R 4 partition function is nothing but the Nekrasov partition function Z S 1 ×R 4 [25,26].

Triple factorization formula for rank-one
We focus on the rank-one Seiberg exceptional theories. The five-sphere partition function is computed by the Nekrasov partition function via the formula where the exponent S 0 is The contour C in the undeformed case lies slightly above the real axis, −∞ + i to ∞ + i for small > 0, and in the general case is such that when continuously deforming from the undeformed case, no pole crosses the contour. 31 In the region λ ≥ m f for all f , what 31 We do not have a first-principle derivation of this contour prescription, but it seems to be the only prescription that gives sensible results. For instance, naively choosing the contour to be on the real axis gives a free energy that diverges in the undeformed limit (when the instantons are included).

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we call F ∨ eff has the interpretation as the one-loop effective prepotential on the squashed five-sphere. 32 In F ∨ eff , the quadratic term in λ comes from the on-shell action (4.15), and the linear and cubic terms in λ are from the one-loop determinants of the vector and hypermultiplets. 33 The arguments of the Nekrasov partition function Z S 1 ×R 4 ( 1 , 2 , α, m f , m I ) are the following: α is the Coulomb branch parameter, m f with f = 1, . . . , N f are the chemical potentials for the SO(2N f ) flavor symmetry, and m I is the chemical potential for the U(1) I instanton symmetry.
The Nekrasov partition function Z S 1 ×R 4 ( 1 , 2 , α, m f , m I ) receives perturbative and instanton contributions, The perturbative part was computed in [33] using the Atiyah-Singer equivariant index theorem, and the result is summarized in appendix G.1. The instanton partition function organizes into a sum of contributions from different instanton numbers, 20) which was obtained in [33,48] by computing the Witten indices of the ADHM quantum mechanics arising in the D-brane configuration of table 1, with the D0-brane worldline along the zeroth direction. In appendix G.2, we review the ADHM quantum mechanics and the computation of the Witten indices.
The substitution rules for the chemical potentials 1 , 2 , α, and m f in terms of the Coulomb branch parameter λ, the squashing parameters ω i , and the mass parameters m f and m I can be determined by studying the m I → ∞ limit of this formula, and demanding 32 We thank Hee-Cheol Kim for correspondence on this point. 33 Without loss of generality, let us assume mN f ≤ mN f −1 ≤ · · · ≤ m1. We conjecture that the one-loop effective prepotential on the squashed five-sphere is for mn+1 ≤ λ ≤ mn. This expression reduces to (4.17) when λ ≥ m f for all f . The functions B3,3(z | ω) come from the triple sine functions in (4.21) by applying (F.10) for Im(z) ≥ 0 and (F.11) for Im(z) < 0. In the flat space limit ωi → 0, the five-sphere one-loop effective prepotential behaves as F eff ∝ F R 4 eff /(ω1ω2ω3) (with the identification of mI with 1 2 m0 in [9]). Thus, it is plausible that F eff is the generalization of F R 4 eff to the squashed sphere. In particular, the pieces of (4.18) inhomogeneous in (λ, mI, m f ) should be produced by the one-loop effective coupling between the background supergravity multiplet and the U(1) vector multiplet (on the Coulomb branch).
which was obtained by a direct one-loop computation in [31,32,39]. The triple sine function S 3 (z | ω) is defined and discussed in some detail in appendix F, and S 3 (z | ω) is its z-derivative. The comparison between the m I → ∞ limit of (4.16) and (4.21) is reviewed in appendix G.1. Given the on-shell action (4.15), we expect that m I should be substituted by 2πm I ω i . However, we conjecture that there is an additional imaginary shift − N f 2 πi in the substitution rule for the U(1) I chemical potential m I .

Instanton particle mass shift and flavor symmetry enhancement
We presently give arguments for the − N f 2 πi shift in the substitution rule for m I in the triple factorization formula (4.16) for the five-sphere partition function of rank-one Seiberg exceptional theories. As shown in [49], the Nekrasov partition function exhibits manifest enhanced flavor symmetry when expanded in the shifted Coulomb branch parameter We define root ). A Weyl reflection w α associated to a root vector α acts on a 34 The factor of S 3 (0 | ω) coming from the contributions of the zero modes in the vector multiplet is of critical importance, but is often ignored in the literature. In particular, only when this factor is included will the five-sphere free energy take the form (3.3). Furthermore, the S 3 (0 | ω) factor is essential for the gauge theory partition function (4.21) to be invariant under a simultaneous rescaling ωi → c ωi with c ∈ R when the mass parameters mI and m f are turned off; this property is consistent with (but not implied by) (3.3) and continues to hold when instanton contributions are included.
The character is invariant under the Weyl reflections We define the shifted character weight , which is also invariant under the Weyl reflections on m, (4.28) Rewriting the triple factorization formula (4.16) in terms of Z S 1 ×R 4 , we obtain × (2 cyclic perms on ω i ) .
) is a weight vector. By (4.28), we find that the expression in the square bracket in (4.29) is invariant under the enhanced Weyl group actions (4.26). Below we find that F ∨ eff is invariant under the enhanced Weyl group after taking into appropriate counter-terms, thus the full partition function respects the enhanced symmetry. This gives strong evidence for the − N f 2 πi chemical potential shift in Z S 1 ×R 4 .
We now show that F ∨ eff is indeed invariant under the enhanced Weyl group, after an appropriate subtraction of counter-terms. In terms of the shifted Coulomb branch parameter λ, we have

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The cubic term in λ is obviously Weyl-invariant. After recognizing the Weyl-invariant polynomial (4.9), we see that the linear term in λ is also Weyl-invariant. The λ-independent terms are not Weyl-invariant, but they can be canceled by the admissible counter-terms classified in section 2.2. 36 We end this section by pointing out an interesting observation on the Nekrasov partition function Z S 1 ×R 4 . We conjecture that the states contributing to the Nekrasov partition function for the Seiberg exceptional theories satisfy 37 where q is the charge under the USp(2) gauge group, R is the representation of the E N f +1 flavor group and q R is the charge under its center.
For E 2 , the quotient (4.33) is identified with the Z 7 subgroup of the center U(1) ⊂ E 2 .
The generators of Z 8−N f are represented by the following weight vectors ρ e , (4.34) Any weight vector ρ ∈ Λ e N f +1 weight can be written as ρ = n ρ e + α, for some n ∈ Z 8−N f and α ∈ Λ e N f +1 root . If R is an irreducible representation, then the characters obey where ξ R is an (8 − N f )-th root of unity. The Z 8−N f center charge q R of a representation R is defined by Finally, the property (4.32) follows from a conjectural identity for the Z S 1 ×R 4 defined in (4.23), 37) 36 We emphasize here that these Weyl non-invariant terms should not be confused with the potential superconformal anomalies of the fixed point superconformal field theory with EN f +1 flavor symmetry. Instead, they imply that the regularization scheme used in the gauge theory localization computation only preserves the SO(2N f ) × U(1)I subgroup. In other words, the EN f +1 preserving scheme can be implemented in the localization computation by including the corresponding counter-terms from the beginning. Note that this is not an issue for the superconformal index because these counter-terms all vanish on the supersymmetric S 1 × S 4 background. 37 This relation between the gauge and the Z8−N f center charges was also observed in the ray operator indices [56].

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which in terms of the ordinary Nekrasov partition function Z S 1 ×R 4 is equivalent to 38 (4.38)

Numerical results
We proceed with the direct numerical evaluation of the triple factorization formula (4.16), and compute the undeformed free energies F 0 as well as the central charges C T and flavor central charges C J using (4.4), (4.7), and (4.8) for the rank-one Seiberg exceptional and Morrison-Seiberg E 1 theories. The results up to four instantons are summarized in tables 2, 3, and 4. We close this section with the following remarks.
• The instanton contributions are small compared to the perturbative result, especially for larger N f .
• For E n , n = 3, 4, . . . , 8, the flavor central charge C En J can be obtained by the dependence of the free energy on either the masses of the fundamental hypermultiplets, or on the mass of the instanton particle. The two methods may give differing results at finite instanton number, but should ultimately agree to be consistent with the enhanced flavor group. Table 5 tracks the differences of the two up to four instantons. We see that the difference indeed diminishes with higher instanton numbers, especially in the E 5 case, providing a strong check of the triple factorization formula (4.16). This also suggests that the instanton expansion (4.1) may be convergent even at the ultraviolet fixed point, where m I = 0.
• In general, the coefficients of the terms which are not invariant under the Weyl group can be nonzero at finite instanton number. We computed the expansion of the free energy in both m I and m f up to cubic order, and after adding counter-terms to cancel the last line of (4.31), observed that such coefficients all appear to diminish at higher and higher instanton numbers. This check is another piece of evidence for the convergence of the instanton series at m I = 0.
• The values of the undeformed free energy given in table 2 show that the Seiberg exceptional theories connected by renormalization group flows (Flow 1 in figure 1) have larger −F 0 and C T in the ultraviolet than in the infrared. This observation hints towards a five-dimensional version of the F -or C-theorem [66-75, 77, 78]. 39 38 We checked this identity up to instanton number five. 39 In three-dimensional N = 2 superconformal field theories, there are counter-examples to a C-theorem for CT [153]. In five-dimensional non-supersymmetric conformal field theories, there is also a counterexample described in [4]. However, it is plausible that with eight supercharges, a C-theorem could hold for five-dimensional N = 1 superconformal field theories. We hope to investigate this possibility in the near future.   Table 5. The differences between the values of the flavor central charge C G J computed using the fundamental hypermultiplet masses and using the instanton particle mass, at each instanton number in the rank-one Seiberg exceptional theories.

Superconformal bootstrap
In previous sections, the values of the conformal central charge C T and the flavor central charge C J have been computed for the Seiberg exceptional theories, based on computations of the squashed sphere partition functions. In this section, we study these theories by the superconformal bootstrap, exploiting the superconformal and flavor symmetries in these theories. The values of C T and C J are related to certain OPE coefficients involving the BPS scalars residing in the flavor current multiplets, and allow us to pinpoint the Seiberg exceptional theories in the space of unitary solutions to bootstrap. We then present the results of the numerical bootstrap. As a check of our numerics, we first consider theories with SU(2) flavor symmetry that may have higher spin conserved currents, and compare the bootstrap bounds with a single free hypermultiplet. Then we go on to consider interacting theories with flavor groups E 1 ∼ = SU(2), E 6 , E 7 , and E 8 , and compare with the Seiberg exceptional theories.
We stress here that our goal is not to compute C T or C J in certain theories by bootstrap, but to demonstrate that the bootstrap (the extremal functional method) in principle systematically solves certain strongly interacting theories such as the Seiberg exceptional theories, by providing evidence that their central charges saturate bootstrap bounds.

Superconformal bootstrap with flavor symmetry
Flavor symmetries of a five-dimensional superconformal field theory are realized by the conserved currents in the D [2] superconformal multiplets [143,144]. The superconformal primary of the D[2] multiplet is the moment map operator L a ij , which transforms in the adjoint representation of the flavor group. Its top component L a 11 is 1 2 -BPS. These moment map operators furnish the so-called Higgs branch chiral ring, and their expectation values, subject to the ring relations, parametrize the Higgs branch vacuum moduli space M H of the superconformal field theory.

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The superconformal bootstrap of the four-point functions of the moment map operators can be treated in a uniform fashion across spacetime dimensions three, five, and six [120]. 40 This section reviews the setup. For simplicity, we assume for now that the moment map operators reside in a single flavor current multiplet. After contracting the SU(2) R indices on each operator L ij (x) with auxiliary variables Y i to form L(x, Y ) ≡ L ij (x)Y i Y j , the four-point function takes the form where is related to the number of spacetime dimensions by = d−2 2 , and the cross ratios u, v, w are defined as 41 By superconformal symmetry and the fact that L 11 is 1 2 -BPS, the OPE coefficients in L × L for all operators residing in each superconformal multiplet are linearly related to a single structure constant; otherwise, without the 1 2 -BPS condition, there would be multiple superconformal blocks and multiple independent structure constants. This implies that the four-point function G can be expanded in superconformal blocks, where X labels superconformal multiplets allowed by the selection rules for L × L. We refer the reader to [120,154] for the expressions for the blocks. Like the four-point function, the blocks are second order polynomials in w −1 , In unitary theories, the expansion coefficients λ 2 X are non-negative. In a consistent conformal field theory, the operator product expansions must be associative, which entails the crossing symmetry constraints, The first equality is solved by imposing certain selection rules on the intermediate primary operators, so we only need to consider the second equality. By the use of superconformal Ward identities, this second equation can be reduced to

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Expanding this equation in superconformal blocks, we have where I is the putative spectrum of non-identity superconformal multiplets. Let us relax the assumption that the scalars reside in a single flavor current multiplet, and label them by an adjoint index a. The four-point function now has extra indices and its superconformal block decomposition takes the form where P abcd I is the projection matrix that projects onto the contributions from intermediate operators transforming in the representation R I . Under crossing, the contributions from different R I mix together, and thus the crossing equation with flavors becomes where the crossing matrix F J I is defined as The crossing matrices can be computed by the methods of [155], and the results for the flavors groups of interest are listed in table 6.
Putting everything together, the full system of bootstrap equations are subject to the following selection rules: 1. Symmetric representations in adj × adj appear with + J R even, and antisymmetric ones with + J R odd.   The bootstrap makes contact with the previous sections by a relation between the OPE coefficients for the stress tensor and flavor current multiplets and the central charges C T and C J . The formulae are [120]

Solving theories by the extremal functional method
The linear functional method exploits the non-negativity of the coefficients in the (super)conformal block expansion in unitary theories, and puts nontrivial bounds on the operator dimensions and OPE coefficients. This section reviews this method. We then explain how the extremal functional method solves the theories that saturate bootstrap bounds.

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The idea is to consider a vector valued linear functional α I on functions of u, v, and act on the bootstrap equations (5.12)  In the following, we keep the sum over the I index implicit, and write (K X ) J as K X ,J . Each linear functional that satisfies implies a bound on the OPE coefficients .  [85,88]. In practice, the above extremization procedure can only be performed within a finitedimensional subspace of linear functionals. The following is a convenient basis. Define z andz by (5.19) so that crossing u ↔ v is equivalent to (z,z) ↔ (1 − z, 1 −z), and consider the space of linear functionals at derivative order Λ: The optimal bounds at higher and higher derivative orders become tighter and tighter, and the most stringent bound is obtained by extrapolating Λ to infinity. Other practicalities with the bootstrap numerics have been discussed in [120], to which the reader is referred. The most interesting bounds to consider are perhaps on the OPE coefficients for the stress tensor and flavor current multiplets, since they are related to the central charges C T and C J , which have been computed in previous sections.

Numerical bounds
This section presents the results of the numerical application of the linear functional method to bootstrap superconformal field theories with flavor groups SU(2), E 6 , E 7 , and E 8 . In all cases considered, the bounds on central charges are extrapolated to infinite derivative order Λ → ∞ using two ansatzes, quadratic The variation among extrapolations with different ansatzes serves as an estimate for the extrapolation error.

Free hypermultiplet
As a first step, we make assumptions that should single out the free hypermultiplet as a solution to the crossing equations -SU(2) flavor symmetry, the existence of higher spin To further check that the extremal theory minimizing C J is indeed a free hypermultiplet, we employ the extremal functional method to determine the spectrum appearing in the D[2]×D [2] OPE in the theory with minimal C J , and compare with the known spectrum of a free hypermultiplet. Figure 3 shows the gaps (lowest scaling dimension) at various derivative orders, and extrapolations to infinite derivative order. We find the gaps in the

SU(2) flavor symmetry
Next, we bootstrap interacting unitary superconformal field theories with SU(2) flavor symmetry. Interacting means that the B[0] >0 multiplets containing higher spin conserved currents are absent. Figure 4 maps out the allowed region in the C T − C J plane, where there appears to be a sharp corner with the minimal allowed C J . Also shown are the fourinstanton values of (C J , C T ) in the rank-one Seiberg E 1 theory, and the perturbative values in the higher-rank theories borrowed from upcoming work [145]. Figure 9 shows the lower bounds on C T and C J at various derivative orders, and extrapolations to infinite derivative order. Assuming that the four-instanton values of C T and C J are good approximations to the exact values, the extrapolations suggest that neither the bound on C T nor that on C J is saturated by the rank-one Seiberg In particular, they suggest that the sharp corner does not approach the rank-one Seiberg E 1 theory at infinite derivative order. We are not aware of a candidate theory that sits at the sharp corner. Nonetheless, the rank-one Seiberg E 1 theory does appear to sit close to the boundary of the allowed region. To examine this further, figure 6 shows the lower bounds on C J when C T is set to the four-instanton value of C T in the rank-one Seiberg E 1 theory, and extrapolated to infinite derivative order, giving lim Λ→∞ min C J = 17.9(1). (5.28) Assuming that the four-instanton values of C T and C J are good approximations, the extrapolation supports the hypothesis that the rank-one Seiberg E 1 theory sits at the boundary of the allowed region. If this hypothesis is true, then we can employ the extremal functional method to determine the spectrum of long multiplets appearing the D[2] × D [2] OPE in the rank-one Seiberg E 1 theory. Figure 7 shows the gaps (the lowest scaling dimension) at various derivative orders, and extrapolations to Λ → ∞ using the quadratic ansatz (5.23) and linear ansatz (5.24). We find the gaps in the 1 and 5 channels to be 0.02 0.04 0.06 0.08 0.10 1/Λ 0 5 10 15 min C J SU(2) with C T set to rank-one Seiberg E 1 Figure 6. Lower bounds on C J with C T set to the value in the rank-one Seiberg E 1 theory, for interacting theories with SU(2) flavor group, at various derivative orders Λ and extrapolated to infinite order using the quadratic ansatz (5.21) (solid) and the linear ansatz (5.22) (dashed). Also shown is the value of C J in the rank-one Seiberg E 1 theory (dotdashed). Finally, we bootstrap interacting unitary superconformal field theories with E 6 , E 7 , and E 8 flavor symmetries. Figure 8 maps out the allowed regions in the C T − C J plane. For each flavor group, the allowed region has two corners, corresponding to the minimal C J and the minimal C T , and the four-instanton values of (C J , C T ) in the rank-one Seiberg exceptional theory sit near the corner with the minimal C J . To examine the corners further, figure 9 shows the lower bounds on C T and on C J at various derivative orders, and extrapolations to infinite derivative order. And figure 10 shows the values of C J and C T that minimize C T and C J , respectively. The results as summarized in tables 7 and 8 provide strong evidence for the rank-one Seiberg E 6 , E 7 , and E 8 theories having the minimal C J among all interacting theories with E 6 , E 7 , and E 8 flavor symmetry, respectively. We are not aware of candidate theories that saturate the extrapolated bound on C T .
We suspect that the slight discrepancies between the extrapolated bootstrap bounds on C J and the four-instanton values in the rank-one Seiberg exceptional theories, as well as the discrepancies in the values of C T at min C J , disappear when even higher derivative orders are included. 42 In the zoomed-in plots of figure 10, the values of C T at min C J exhibit upward trends at high derivative orders that potentially diminish the discrepancies. 43 Assuming that the rank-one Seiberg E 6 , E 7 , and E 8 theories saturate the lower bounds on C J , we employ the extremal functional method to determine for each theory the spectrum appearing in the D[2] × D [2] OPE. We find that the long multiplets that achieve the lowest scaling dimension have zero spin and appear in the representations 650 of E 6 , 1539 of E 7 , and 3875 of E 8 . For these channels, figure 11 shows the gaps (the lowest scaling dimension) at various derivative orders, and table 9 summarizes the extrapolated gaps at infinite derivative order.
In the rank-one Seiberg E N f +1 theory, the structure of the Higgs branch moduli space M H is particularly simple: it is given by the one-instanton moduli space of E N f +1 [7], which can be described by a complex algebraic variety with holomorphic coordinates L a ≡ L a 11 , subject to the Joseph ideal relations [64,156] L ⊗ L| J = 0, Sym 2 (adj) = (2 adj) ⊕ J. (5.30) Consequently, in the D[2] × D [2] OPE, the D [4] can only appear in the 5 of SU(2), the 2430 of E 6 , the 7371 of E 7 , and the 27000 of E 8 . We confirmed this expectation by numerically observing that α D [2],adj [K D [4],r ] are parametrically much smaller for r in the above representations than for r in the other representations. 42 Our estimates for the errors in the bootstrap data due to spin truncation are around 1% [120]. Therefore, the only meaningful discrepancies occur in the E6 case. 43 In light of the results for E1 in section 5.3.2, there is the possibility that the rank-one Seiberg theories generally do not saturate the absolute lower bound on CJ (it is conceivable that only the E8 case does), but lie on the lower boundaries of the allowed regions close to the kinks. In this scenario, the extremal functional method still solves these theories. Determining which scenario is correct requires considerably more computational power, and is left for future work.  Table 7. The extrapolated lower bounds on C T and C J for interacting theories with flavor groups G = E 6 , E 7 , E 8 at infinite derivative order, and compared to the four-instanton values in the rank-one Seiberg exceptional theories.
G lim Λ→∞ C T at min C J C 4−inst   Table 9. Predicted gaps (the lowest scaling dimension) in the long multiplets in the Seiberg exceptional theories, and the flavor group representations in which they transform.

Discussion and outlook
In this paper, we carefully analyzed the coupling of five-dimensional N = 1 superconformal field theories with mass deformations to squashed-sphere backgrounds, and presented a precise triple factorization formula for computing the free energy in rank-one theories that incorporates instanton contributions. Along the way, we classified the admissible supersymmetric counter-terms, discovered a new superconformal anomaly, and derived relations to the conformal and flavor central charges. The knowledge of the counter-terms allowed us to elucidate the invariance of the five-sphere free energy under the enhanced Weyl group actions, both formally and supplemented with strong numerical evidence. Using the triple factorization formula, we numerically computed the central charges in the rankone Seiberg exceptional and Morrison-Seiberg E 1 theories. Finally, we made connections between the central charges and the OPE data, and studied the numerical bootstrap of the four-point function of moment map operators. We found strong evidence for the saturation of the bootstrap bounds by the rank-one Seiberg theories, and extracted the spectra of long multiplets in these theories.     Figure 9. Lower bounds on C T (left) and C J (right) for interacting theories with E 6 , E 7 , and E 8 flavor groups, at various derivative orders Λ and extrapolated to infinite order using the quadratic ansatz (5.21) (solid) and the linear ansatz (5.22) (dashed). Also shown are the four-instanton values of C T and C J in the rank-one Seiberg exceptional theories (dotdashed).    The Seiberg theories are holographically dual to type-I' string theory on AdS 6 × HS 4 , which in a certain low energy limit is captured by Romans F (4) supergravity. In upcoming work [145], we examine the higher-rank Seiberg theories and their holographic duals. In particular, we compute the central charges C T and C J at large-rank using matrix model techniques, and compare with certain couplings in Romans F (4) supergravity. In another paper [141], we investigate the new five-dimensional superconformal anomaly explained in section 2.2, as well as its implications for dualities. Other arenas for further exploration include a proof of the F -or a C-theorem for N = 1 superconformal field theories in five dimensions, of which the numerical results of section 4.3 are suggestive, and a better (physical) understanding of our contour prescription for the gauge theory sphere partition function. A final observation is that the undeformed five-sphere free energy F 0 sits at a local maximum with respect to infinitesimal supersymmetric squashing and mass deformations, as the signs in (3.3)- (3.6) show. This property is reminiscent of the F -maximization in three-dimensional N = 2 superconformal field theories [71], except that here the maximization is automatic since there is no mixing between the non-Abelian SU(2) R R-symmetry and flavor symmetries.

A Conventions and normalization
This appendix summarizes the various conventions and normalization adopted in this paper.

A.1 Normalization of the central charges C T and C J
The conformally covariant structures I µν,σρ (x) and I µν (x) that appear in the stress tensor two-point function (3.1) and the flavor current two-point function (3.2), are defined by The stress tensor has a canonical normalization coming from the normalization of the dilatation operator D, which in radial quantization is defined by the following integral of the stress tensor, The normalization of D (and hence T µν ) is such that the state |O corresponding to an operator O with scaling dimension ∆ has eigenvalue ∆ under D, With this normalization, the coefficient C T in (3.1) is physical and called the conformal central charge.
Similarly, the flavor currents have a canonical normalization coming from the normalization of the flavor charges Q a , which in radial quantization are defined by the following integral of the flavor currents, The canonical normalization requires that the flavor charges acting on the states J a µ corresponding to the flavor currents under the state-operator correspondence give 44 where f abc is the structure constant of the flavor group G normalized by where h ∨ is the dual Coxeter number. With this normalization, the coefficient C J in (3.2) is physical and called the flavor central charge. When the flavor symmetry is U(1), we normalize the current by where |φ is a state carrying the elementary U(1) charge in the theory. 44 In our convention, Q a is an anti-hermitian operator. In particular, f abc is purely imaginary for compact flavor groups.

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Let g be the Lie algebra of the flavor group G. For x, y ∈ g, we define the Killing form (x, y) = Tr(xy) ≡ 1 2h ∨ tr adj (xy), (A.8) and tr adj (·) is the trace in the adjoint representation. We can pick a basis {T a } for the Lie algebra g, such that Then the normalization (A.6) implies The traces in other representations are linearly related to the Killing form. For example, we have tr fund (·) = Tr(·) for the fundamental representation of SU(N ) or USp(2N ), and tr vec (·) = 2Tr(·) for the vector representation of SO(N ).

A.3 Five-dimensional supersymmetry conventions
In flat space, the five-dimensional supersymmetry algebra is where U i j are the SU(2) R R-symmetry generators.

B Supersymmetric backgrounds for squashed five-spheres
In this appendix, we compute the five-dimensional backgrounds preserving rigid (global) supersymmetry for a generic squashed five-sphere with the minimal U(1) × U(1) × U(1) isometry. The goal is to write down the background fields in the standard (irreducible) Weyl multiplet (see [139] and also [138]). As a matter of fact, one can explicitly show that the standard Weyl multiplet is precisely equivalent to the conformal boundary of Romans F (4) gauged supergravity. We start by providing the appropriate field redefinitions which relate the study of supersymmetric backgrounds performed in reference [52], arising at the conformal boundary of Romans F (4) supergravity, to the standard Weyl multiplet. 45

B.1 Standard Weyl multiplet and its relation to Romans F (4) supergravity
The irreducible standard Weyl multiplet consists of 32 + 32 bosonic and fermionic degrees of freedom given in terms of the vielbein e µ a , two su(2) Majorana fermions ψ i µ and χ i , two real bosons b µ and D, an su(2) gauge field V ij µ , and a real antisymmetric tensor v ab . The supersymmetry conditions for the standard Weyl multiplet read where ε i and η i are the Killing spinors and conformal Killing spinors, respectively, K a the generators of special conformal transformation, and with Q i and S i the supercharges and their conformal cousins. The covariant derivatives are defined as follows, where ω µ ab is the spin connection. Finally, V µν ij is the field strength of V µ ij and for our purposes, 45 See also [157].

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Standard Weyl multiplet Conformal boundary of Romans F (4) Table 10. Dictionary between the fields in the standard Weyl multiplet in the notation of [139], and the fields arising from expanding Romans F (4) gauged supergravity at the conformal boundary [52].
We may translate the conditions (B.1) into equivalent supersymmetry conditions arising in the analysis of [52]. We explicitly write down the translation for the fields between [139] and [52] in table 10. Notice that in the last line, we denote by C µ the generator of Weyl transformations; a field of given Weyl weight w is acted on by the covariant derivative as D µ ≡ ∂ µ + w C µ . In the following, we shall set C = b = 0. To avoid confusion, we also denote byε andη the Killing spinors in the analysis of [52].
We pick the following five-dimensional metric, where φ i are periodic coordinates, φ i ∼ φ i + 2π, y i are constrained coordinates such that y 2 1 + y 2 2 + y 2 3 = 1, and ω i = 1 + a i , for i = 1, 2, 3, (B.6) are the squashing parameters, that govern the deformation away from the round sphere. Now, let us introduce the following frame, -54 -

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where for ease of notation we introduced the definitions We now present the U(2) structure, defined by a function S, a one-form K 1 and real and complex two-forms J and Ω, respectively, which are explicitly given by which can be explicitly written as Using the U(2) structure equations introduced in [52], 46 we can solve for the background fields -the function α, the one-form K 2 , the U(1) ⊂ SU(2) gauge field a and the two-form b. We find the general solution 47 (B.13) 46 Notice that the field D -or equivalently X2 -does not appear in these equations. However, it is completely fixed in terms of the other fields and the U(2) structure by (for instance) solving the second equation in (B.1). Given this and by taking a U(1) ⊂ SU(2) truncation of the background gauge field, equation (B.1) is equivalent to (B.12) as proved in [52]. 47 Here we pick a particular gauge for the U(1) ⊂ SU(2) gauge field a, such that i ξ a = 0.

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which is parameterized by the arbitrary (real) function α. Furthermore, we used the notation (recall that ω i = 1 + a i ) We can compute X 2 or (equivalently) the scalar field D in the standard Weyl multiplet The background fields then read (B.17) We can now translate those background fields (α, a µ , b µν , X 2 ) into the ones appearing in the standard Weyl multiplet, (V µ ij , v µν , D), by table 10. In particular, one can show that the Killing spinor and the conformal Killing spinor (as well as their complex conjugates) solve the Killing spinor equations for the standard Weyl multiplet given in (B.1), with the explicit five-dimensional gamma matrices We find the following explicit expansions for the background fields in the standard Weyl multiplet 48

B.4 Stereographic coordinates
For the evaluation of integrated two point functions on the round five-sphere in section 3.1, it is more convenient to use the stereographic coordinates x 1,2,3,4,5 . The relation to the {y i , φ i } coordinates is x 4 = y 2 sin φ 2 1 + y 3 sin φ 3 , x 5 = y 3 cos φ 3 1 + y 3 sin φ 3 .

C Embedding index
Let g be a Lie algebra. The Killing form (x, y) for x, y ∈ g is defined in (A.8). We denote the Cartan subalgebra of g by h, and the dual vector space by h * (the space of linear functions from h to R). The Cartan element H α ∈ h associated to a vector α ∈ h * is defined by (H α , H) = α(H) for any H ∈ h. The Killing form on the vector space h * is defined by α, β = (H α , H β ) for α, β ∈ h * . 48 The SU(2) gauge field Vµ ij is pure gauge at leading order. We shall pick a gauge here in which it is vanishing (it will leave the remaining background fields invariant).

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Consider a subgroup G of the flavor group G, and let g be the Lie algebra of G , with ι : g → g the embedding map. The Killing forms of g and g are linearly related by I g →g × (x, y) = (ιx, ιy) for any x, y ∈ g , (C.1) where I g →g is called the embedding index. Since the flavor current two-point function (3.2) is proportional to the Killing form, the flavor central charge C G J associated to the subgroup G is related to C G J by In this appendix, we compute the embedding indices I so(2N f ) →e N f +1 and I u(1) I →e N f +1 . Consider a sublattice inside the root lattice of e N f +1 generated by the simple roots {α 2 , α 3 , . . . , α N f +1 }, the labeling of which is specified in figure 12. The roots {α 2 , α 3 , . . . , α N f +1 } form a subdiagram of Dynkin type D N f , and the roots of e N f +1 that are inside this sublattice generate an so(2N f ) subalgebra. The embedding so(2N f ) → e N f +1 of the Cartan elements is given by where α so i , i = 1, . . . N f denote the simple roots of so(2N f ). And the embedding index (C.1) of so(2N f ) → e N f +1 can be computed by where A ij = α i , α j and A so ij = α so i , α so j are the Cartan matrices of e N f +1 and so(2N f ), respectively.
The instanton number U(1) I is defined as the commutant of this subgroup. Note that with this definition, the simple root of u(1) I is not α 1 , which has nontrivial intersection with α 3 . Instead, we identify where ρ i for i = 1, . . . , N f +1 are the fundamental weights that satisfy ρ i , α j = δ ij . Under this identification, α I is orthogonal to the roots α 2 , α 3 , . . . , α N f +1 , and with a normalization fixed by the condition (A.7). The embedding u(1) I → e N f +1 of the Cartan elements is given by The embedding index of u(1) I → e N f +1 can be computed by where the inner product of the u(1) I Lie algebra is normalized as (H u(1) I , H u(1) I ) = 1.

D Review of superconformal representation theory
The five-dimensional superconformal algebra is F (4), which contains the bosonic subalgebra so(2, 5) × su(2) R . There are sixteen fermonic generators: eight supercharges Q A α and eight superconformal supercharges S α A , where α = 1, . . . , 4 and A = 1, 2 are the so(5) and su(2) R spinor indices, respectively. Superconformal primaries are operators that are annihilated by all the superconformal charges S α A . A highest weight state of F (4) is a superconformal primary that is also a highest weight state of the maximal compact subalgebra so(2) × so(5)×su(2) R . Representations of the superconformal algebra are generated by successively acting with the supercharges Q A α and the lowering generators of so(5) × su(2) R on the highest weight states. While some descendants of a highest weight state can appear to have zero norm, in unitary theories, they must be decoupled, and the shortened multiplets are referred to as short multiplets.
Each superconformal multiplet can be labeled by the charges ∆, J ± , J R of its highest weight state under the Cartan of so(2)×so(5)×su(2) R , where J ± are the Cartan generators of the su(2) + × su(2) − ⊂ so(4) ⊂ so (5). All the charges are real for unitary representations of the Lorentzian conformal algebra so(2, 5) × su(2) R . The short representations are classified into A, B, D types, satisfying the following conditions [143,144,158,159] A : ∆ = 2J + + 3J R + 4, for J + , J − , J R ≥ 0, where d 1 = 2J − and d 2 = 2J + − 2J − are the so(5) Dynkin labels. Due to OPE selection rules, in the bootstrap analysis, we only have to consider multiplets whose superconformal primaries are in the symmetric rank-representation of so (5). We denote such representations by The ∆, subscripts for D-type multiplets and the ∆ subscript for B-type will be omitted since their values are fixed by (D.1) and (D.4).

E Instanton particle mass term
As discussed in section 4, the Seiberg exceptional theories can be mass-deformed by a weight-four scalar in the flavor current multiplet. Consequently, they flow to USp(2N ) gauge theories in the infrared. The general form of the mass deformation is given in (3.39).
In this appendix, we derive the relation between the mass parameter and the Yang-Mills coupling of the infrared gauge theory. Let us consider a USp(2N ) vector multiplet, which contains a gauge field A a µ , a symplectic-Majorana spinor λ ai α , a scalar φ a , and scalars D a ij all in the adjoint representation of SU(2) R . The index a ∈ {1, . . . , (2N + 1)N } labels the adjoint representation of the gauge group USp(2N ). The leading terms in the supersymmetry variation of the component fields are (E.1) The U(1) I instanton current multiplet can be constructed from bilinears of the vector multiplet together with higher order corrections. The superconformal primary L ij of the instanton current multiplet takes the form (E.2) 49 We use 2JR since it is the Dynkin label of su(2)R.

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The other two primaries N and J λ are related to L ij via (3.34), and therefore given by Note that the operators in (E.2) and (E.3) are all properly normalized, in the sense that the instanton number current J λ has the canonical normalization (A.7). 50 Substituting the above into the action (3.39), and comparing with the standard Yang-Mills kinetic term, we find (E.6)

F Triple sine function
In this appendix, we define the "Barnes triple sine" function, and list a plethora of properties that are useful in the main text. Let us start by defining the Barnes multiple zeta function, where Re w > 0, Re s > N and ω 1 , . . . , ω N > 0. This function is meromorphic in s, with simple poles at s = 1, . . . , N . One can then define the Barnes multiple gamma function Finally, the multiple sine function is defined in terms of the Barnes gamma function as  where ω tot = N i=1 ω i . For the purpose of this paper, there is a more convenient (equivalent) way of writing the triple sine function. One can prove that the above definition is equivalent to [160][161][162] S N (w | ω) = exp (−1) N πi N ! B N,N (z | ω) + (−1) N I N (z | ω) , (F.5) where we have introduced the integral .

(F.6)
The integral is over the real axis with the exclusion of the (essential) singularity at x = 0 by a small half-circle reaching into the positive half-plane. Furthermore, we denote by B N,N (z | ω) the generalized/multiple Bernoulli polynomials, which can be explicitly computed by expanding and solving t N e zt N j=1 (e ω j t − 1) = ∞ n=0 t n n! B N,n (z | ω) (F.7) order-by-order. In our case of interest, N = 3, we have There is yet another definition of the triple sine function in terms of the generalized q-Pochhammer symbols, which are defined as 51 (1 − pq j 1 q k 2 ) for |q 1 |, |q 2 | < 1, which is a meromorphic function of z (p = e 2πiz ) [161]. The triple sine function can be written as [161] S 3 (z | ω) = e − πi 6 B 3,3 (z| ω) (e Notice as compared with [161], we write (p; q1, q2) ≡ (p; q1, q2) ∞ .

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Similarly, we can rewrite it as × (2 cyclic perms on ω i ). (F.12) Without loss of generality, let us assume Im( ω 1 ω 2 ), Im( ω 2 ω 3 ), Im( ω 1 ω 3 ) > 0. Then we have the following formula for the derivative S 3 (0 | ω), The perturbative part of the Nekrasov partition function on S 1 × R 4 was computed in [33] using the Atiyah-Singer equivariant index theorem. It can be expressed in terms of the plethystic exponential where f vec and f hyper are the "single particle" indices, cosh(m f ).

(G.2)
Here, Z vec and Z hyper are the one-loop determinants of the vector and hypermultiplets on the zero-instanton background. Similarly, they can also be expressed in terms of the q-shifted factorials as Following [163,164], we show that the full partition function (4.16) in the weak coupling m I → ∞ limit agrees with the perturbative partition function (4.21) expressed in terms -63 -

G.2 ADHM quantum mechanics
In the string theory realization of five-dimensional Seiberg exceptional theories, the instantons are described by D0-branes moving in a D4-D8/O8 background [48]. The directions of the various branes are summarized in table 1. The low energy theory on k D0-branes is an N = 4 O(k) gauged quantum mechanics with SU(2) R + × SU(2) R − R-symmetry and SU(2) + × SU(2) − flavor symmetry corresponding to the rotations on the four-planes R 1234 and R 5678 , respectively. The field content of the supersymmetric quantum mechanics is summarized in table 11. The vector and Fermi multiplets, which arise from the D0-D0 strings are in the adjoint representation of the gauge group O(k), and the hyper-and twisted hypermultiplets are in the symmetric representation. The D0-D4 and D0-D8 strings are in the bi-fundamental representation of the gauge group O(k) and their flavor groups, USp(2N ) and SO(2N f ), respectively.
Consider an N = 2 subalgebra inside the N = 4 supersymmetry algebra. The Witten index is defined as where ± = 1 ± 2 2 , Q and Q † are the supercharges in the N = 2 subalgebra, J R + , J R − are the Cartan generators of the SU(2) R + and SU(2) R − R-symmetry groups, and J ± , F f and H i of the SU(2) ± , SO(2N f ), and USp(2N ) flavor groups respectively. The eigenstates with the same nonzero eigenvalues of the Hamiltonian H = {Q † , Q} are paired up and exchanged by the action of the supercharges Q and Q † . Their contributions to the trace (G.10) cancel, and hence the Witten index is independent of the inverse temperature β.
The Witten index was computed using supersymmetric localization in [48,165]. The result can be expressed as a multi-dimensional contour integral over a variable φ valued in   the maximal torus of the complexified gauge group. The real part of φ is the scalar zero mode inside the vector multiplet, and the imaginary part is the holonomy of the gauge field along the time circle. The integrand is then given by the one-loop determinants of the field content listed in table 11. It was shown in [48,165,166] that the precise contour prescription is given by the Jeffrey-Kirwan residue. The Witten indices for the quantum mechanics of different k can be combined into a generating function where we introduced the chemical potential m I for the U(1) I instanton symmetry. In the following, we shall restrict to rank-one. The instantons in the five-dimensional gauge theory have the interpretation as D0branes bound with D4-branes. However, the Witten index receives contributions from both the bound and unbound D0-branes. The instanton part of the Nekrasov partition function is a ratio of two generating functions of Witten indices [48],