F-theoretic vs microscopic description of a conformal N=2 SYM theory

The F-theory background of four D7 branes in a type I' orientifold was conjectured to be described by the Seiberg-Witten curve for the superconformal SU(2) gauge theory with four flavors. This relation was explained by considering in this background a probe D3 brane, which supports this theory with SU(2) realized as Sp(1). Here we explicitly compute the non-perturbative corrections to the D7/D3 system in type I' due to D-instantons. This computation provides both the quartic effective action on the D7 branes and the quadratic effective action on the D3 brane; the latter agrees with the F-theoretic prediction. The action obtained in this way is related to the one derived from the usual instanton calculus \`a la Nekrasov (or from its AGT realization in terms of Liouville conformal blocks) by means of a non-perturbative redefinition of the coupling constant. We also point out an intriguing relation between the four-dimensional theory on the probe D3 brane, which has a SO(8) flavor symmetry, and the eight-dimensional dynamics on the D7 branes. On the latter, SO(8) represents a gauge group and the flavor masses correspond to the vacuum expectation values of an adjoint scalar field m: what we find is that the exact effective coupling in four dimensions is obtained from its perturbative part by taking into account in its mass dependence the full quantum dynamics of the field m in eight dimensions.


Introduction and motivations
Phenomenologically viable string models based on consistent D-brane configurations have attracted a lot of attention in the last years [1]- [3]. In such constructions it is necessary to take into account possible non-perturbative corrections due D-instantons and (wrapped) euclidean branes; for a review see, for example, [4]. Some of these instantonic branes reproduce gauge instantons [5]- [8], other provide inherently stringy (or "exotic") instanton effects; these latter can be responsible of important terms in the effective action which would be perturbatively forbidden [9]- [11]. Recently there has been much progress in the explicit computation of such contributions, both ordinary and exotic, at least in supersymmetric cases (again, see [4] and references therein).
Another framework where phenomenological models with highly desirable features can be set up, and where in particular Grand Unified Theories occur naturally and consistently [12]- [14], is represented by F-theory compactifications [15] (for reviews see, for instance, [16,17]). F-theory gives a non-perturbative geometric description of type IIB backgrounds containing D7 branes and orientifold planes; it somehow resums the non-perturbative corrections arising from certain instantonic branes. Understanding in detail how this resummation takes place would improve our knowledge of the relation between the two descriptions. This could be useful for a better comprehension of further non-perturbative effects in Ftheory through a lift of their type IIB counterparts, a subject that is recently 1 receiving quite some attention [19]- [24].
In this paper we work out a simple, yet non-trivial, example where we are able to compute the D-instanton effects in the IIB description and show that they reconstruct the F-theory curve. This example was considered by A. Sen in [25], and is given by the compactification of F-theory on the orbifold limit of an elliptically fibered K3 surface, for which the complex structure modulus τ of the fiber is constant. This background was shown to correspond to the so-called type I theory, which is T-dual to type I theory compactified on a 2-torus T 2 , and thus possesses one O7 plane at each of the four fixed points of T 2 with four D7 branes on top of it. Focusing on the vicinity of one orientifold fixed-plane, and allowing the four D7 branes to move out of it, Sen conjectured that the corresponding F-theory background should be described by the Seiberg-Witten (SW) curve [26] for the N = 2 superconformal Yang-Mills theory with gauge group SU (2) and N f = 4 flavors [27]. This relation was later explained by T. Banks, M. Douglas and N. Seiberg [28] by considering a D3 brane in this background, which indeed supports such a fourdimensional gauge theory on its world-volume, with SU(2) realized as Sp (1). The SO (8) flavor symmetry of the N f = 4 theory is nothing else but the gauge group on the D7 branes.
Here we explicitly compute the non-perturbative corrections to the D7/D3 system in type I due to D-instantons. This requires to identify the spectrum of moduli, i.e. of excitations of the strings with at least one end-point on the D-instantons, and the moduli action that arises from disks with at least part of their boundary on a D-instanton, which was already discussed in [29]. To obtain the non-perturbative effects it is necessary to integrate over the moduli; we explicitly perform this integration by applying the by-now standard techniques based on the BRST structure of the moduli spectrum and action and its deformation by means of suitable RR backgrounds [30]- [34]. This induces a complete localization of the integral, similarly to what happens in supersymmetric instanton calculus in field theory [35]- [40]. These techniques have been recently applied to similar brane setups, such as the D7 system in type I [31,33], and the D7/D3 system on T 4 /Z 2 [34].
D-instanton effects induce corrections to both the quartic effective action on the D7 branes and the quadratic effective action on the D3 brane. We use the prescription proposed in [34] to disentangle the two contributions. The non-perturbative action on the D7's turns out to be exactly the same of the D7 system in type I theory considered in [31]. The nonperturbative effective coupling on the D3 brane agrees with the one extracted from the SW curve, that is with the F-theoretic prediction.
One interesting question is the precise relation between the eight-dimensional quartic effective action on the D7 branes and the F-theory curve. This problem was already addressed in the past using the duality of certain F-theory compactifications, including the one corresponding to type I theory, to heterotic models. Despite some interesting results [41,42], this relation is not yet totally clear. Since in our case the F-theory curve is nothing else but the SW curve encoding the effective theory on the D3 probe, another way to state the above question is: what is the relation between the non-perturbative effective actions on the D7 branes and on the D3 brane? More generally, how are the quantum dynamics on the D7's and that on the D3 related to each other?
We do not have a full answer to this question, but we uncover an intriguing relation that goes as follows. On the D7's, the SO(8) flavor symmetry represents a gauge group and the flavor masses m i correspond to the vacuum expectation values of an adjoint scalar field m(X): we find that the exact effective four-dimensional coupling is obtained from its perturbative part by taking into account in its mass dependence the eight-dimensional quantum dynamics of the field m, and in particular the so-called "chiral ring" formed by the correlators Tr m 2l . In this way the F-theory geometry is related explicitly to these eight-dimensional quantities. It would be very interesting to investigate whether such sort of perturbative propagation of the full quantum dynamics on a brane stack (in our case, the D7's) to another one (in our case the D3 probe) takes place also in other systems 2 .
The theory that, in our example, lives on the D3 brane is a four-dimensional conformal N = 2 theory. This class of theories have recently attracted much attention in relation to the so-called AGT conjecture put forward by L. F. Alday, D. Gaiotto and Y. Tachikawa [44]. This conjecture relates the effective actions obtained from usual instanton calculusà la Nekrasov [35] to suitable correlators of the Liouville theory in two dimensions [45] (see also [46]- [48]). In particular, the non-perturbative action for the SU (2) theory with N f = 4 can be extracted from the 4-point functions on the sphere. It is interesting to compare these results to what we get in the D7/D3 system in type I , where the conformal theory is realized as an Sp(1) gauge theory. It turns out that the effective action derived from the Nekrasov's prescription for the SU (2), N f = 4 case or, more efficiently, from its AGT realization in terms of Liouville conformal blocks does not coincide, at first sight, with our results, nor with the SW curve proposed in [27]. There is a discrepancy already at the massless level: in this case from the SW curve (and from our microscopic computation) we see that the treelevel coupling receives no corrections. When computed following Nekrasov's prescription for SU (2), or by the AGT method, instead, the coupling gets non-perturbatively modified 3 . This suggests that a redefinition of the coupling constant is needed in order to compare the two approaches; after such a redefinition is performed, remarkably the two methods are reconciled and the two results agree completely also in the massive case.
The paper is subdivided into several sections as we now describe. In Section 2 we introduce the model and its F-theory description through the SW curve; from this curve we extract the instanton expansion of the effective coupling. In Section 3 we describe the microscopic computation of D-instanton contributions in our model. The resulting effective action on the D7 branes is discussed in Section 4, while in Section 5 we write the D-instanton induced effective action on the D3 brane, which is in full agreement with the Ftheoretic description. In Section 6 we put forward our conjecture about the exact effective coupling on the D3 being determined by its perturbative part plus the eight-dimensional dynamics of the mass parameters. Finally, in Section 7 we discuss how our results compare to Nekrasov instanton calculus (or its AGT realization) for the SU (2) theory with N f = 4, and in Section 8 we present our conclusions. In the appendices some technical details and the extension of some results to asymptotically free cases with N f < 4 are given.

F-theory and the D7/D3 system in type I
In F-theory compactifications over an elliptically fibered manifold [15], the complex structure modulus τ of the fiber corresponds to the varying axio-dilaton profile of a suitable type IIB compactification on the base manifold. In [25] A. Sen studied F-theory on an elliptically fibered K3 surface, which is conjectured to be dual to heterotic string compactified on a two-dimensional torus. He considered the particular case in which the K3 is at the orbifold limit in moduli space where it is described by the following curve in Weierstrass form Here z is the coordinate on the base of the fibration and with f I constants. The absolute modular invariant of this curve 3 A discrepancy in this sense was already noticed at the two-instanton level in [49,50], where the direct integration over the moduli was performed without resorting to localization techniques.
is z-independent, and so is its complex structure modulus τ which can be determined from J by inverting the relation where the ϑ a 's are the Jacobi theta-functions and η is the Dedekind function. By studying the metric on the base space, it can be seen that the latter has the geometry of a torus orbifold of the type T 2 /Z 2 , with Z 2 acting as parity reflection along T 2 , in which the f I 's appearing in (2.2) correspond to the points of T 2 that are fixed under the Z 2 action.
This specific F-theory background can be identified with the so-called type I theory, namely type IIB compactified on a torus T 2 and modded out by where ω is the world-sheet parity reversal, F L is the left-moving space-time fermion number and I 2 the inversion on T 2 . This is the T-dual version of type I theory compactified on T 2 , and possesses four O7 orientifold planes located at the points of T 2 that are fixed under I 2 (see Fig. 1a). Each orientifold plane carries (−4) units of 7-brane charge, which need to be neutralized by putting 16 D7 branes transverse to T 2 . If we place them in groups of 4 over each orientifold plane, the tadpole cancellation becomes local and the axio-dilaton is constant over T 2 . From now on we take a local perspective and focus on one of the orientifold fixed planes (say, the one at z = f 1 ) and its associated stack of 4 D7 branes.

The local case
The action of the orientifold projection Ω is such that each group of 4 D7 branes supports an eight-dimensional theory with gauge group SO (8). Indeed, the massless degrees of freedom of the 7/7 open strings build up an eight-dimensional chiral superfield in the adjoint representation of SO (8), whose first few terms are where f M N is the field-strength, σ is the gaugino, m is a complex scalar, and (X, Θ) are the eight-dimensional super-coordinates. If the D7 branes are moved away from the orientifold plane, i.e. if we give a diagonal vacuum expectation value to the scalar field m, the charges no longer cancel locally, and correspondingly the solution of the equation of motion for τ displays logarithmic singularities at the orientifold and D7 brane locations. Let us parametrize the region near the orientifold fixed point with a coordinate 4 w and let the D7 branes and their images be located at w = ±m i / √ 2 with i = 1, . . . , 4 (see Fig. 1b). This corresponds to choose the following vacuum expectation values 5 4 With respect to the global coordinate z used above, w ∝ (z − f1). 5 Notice that m is a complex field in the adjoint of SO(8), i.e. it is a complex antisymmetric matrix. If m were real, its eigenvalues mi would be imaginary.
w-patch Figure 1: a) The initial set-up: 4 D7 branes (denoted by the red circles) are placed in the transverse torus T 2 at each of the O7 fixed planes (denoted by the green crosses). b) In the local limit near one of the fixed planes, the D7 brane displacements with respect to the original position are given by m i / √ 2 (the image branes, denoted by dashed circles, sit at −m i / √ 2). A D3 brane probe (depicted as a blue dot) is placed in a (its image being in −a).
With this choice the axio-dilaton becomes where τ 0 is the "bare" coupling. This solution does not make sense everywhere, since Im τ (proportional to the inverse string coupling) becomes negative close to the orientifold location (w = 0). As is well known, this problem is cured by non-perturbative corrections.

F-theoretic description and D3 brane probes
In [25] A. Sen proposed an F-theoretic interpretation of the local set-up introduced above in which the exact axio-dilaton profile is given by the complex structure modulus τ of the auxiliary Seiberg-Witten (SW) curve that describes the quantum moduli space of the N = 2 SYM theory in four dimensions with gauge group SU (2) and N f = 4 fundamental flavors [27]. The appearance of this seemingly unrelated four-dimensional gauge theory was explained in [28] by considering D3 brane probes in the orientifold background described in the previous section. As is well known, the orientifold projection that leads to an orthogonal gauge group on the D7 branes gives rise to a symplectic gauge group on the D3's. A single D3 and its image at an orientifold fixed point support a Sp(1) ∼ SU(2) gauge group. The degrees of freedom of the 3/3 strings fill up a four-dimensional N = 2 chiral multiplet in the adjoint representation of Sp(1), namely where we denoted the four-dimensional super-coordinates as (x, θ). Notice that in this orientifold background the 3/3 spectrum contains also a neutral hypermultiplet 6 .
The open strings stretching between the D3's and the D7's contain precisely the degrees of freedom of four fundamental hypermultiplets. From the D3 point of view, the SO(8) Chan-Paton group on the D7 branes represents the global flavor symmetry group, while the D7 brane positions m i appear as masses for the hypers, since the 3/7 strings become stretched. It turns out that the neutral hypermultiplet from the 3/3 sector is also completely decoupled from the flavored matter, and thus it will be ignored in our subsequent analysis.
Moving the probe D3 brane away from the orientifold fixed point corresponds to explore the Coulomb moduli space of its world-volume theory. In the D3 brane action the effective coupling in front of the quadratic gauge Lagrangian for the chiral multiplet in the Cartan direction is exactly given by the axio-dilaton field, that is, we have the identification where θ YM is the theta-angle and g YM the gauge coupling. This coupling receives perturbative corrections at the one-loop level only, and for N f flavors it takes the form 7 where a parametrizes the vacuum expectation value of the adjoint Sp(1) scalar field, namely For N f = 4 this expression is in agreement (a part from the finite renormalization term 2 ln 16 on which we will come back later) with Eq. (2.8), obtained by solving the field equations for the axio-dilaton sourced by the D7 branes and O7 planes. In this comparison we see that a represents the position of the probe D3 brane and hence corresponds to the coordinate w. For large a, the following expansion holds Tr m 2l a 2l , (2.13) where in the second line we used Eq. (2.7). This expansion will turn out to be extremely useful in the following. As is well known, from this field theory point of view it is possible to go beyond the perturbative results and derive the exact effective coupling on the moduli space of the D3 gauge theory from the appropriate SW curve. This coupling is then the exact axio-dilaton configuration for our set-up, i.e. it represents the F-theoretic solution. 7 In the N f = 4 case, the scale Λ cancels out.

The SW curve for the D3 gauge theory
The SW curve for the SU(2) SYM theory with N f = 4 massive fundamental flavors was proposed in [27] to be given by a torus described by the equation where the cubic polynomial P 3 (x) is Here, for = 1, 2, 3, we have introduced while R, T and N are invariants of the flavor group SO(8) that are, respectively, quadratic, quartic and sextic in the masses m i (see Appendix B for details). The three quartic invariants T satisfy the relation T 1 + T 2 + T 3 = 0, and hence we can use as independent quantities T 1 and the Pfaffian Pfm ≡ m 1 m 2 m 3 m 4 , (2.17) since The quantities e and A are functions of the "bare" coupling τ 0 , namely Notice that all these functions can be expanded in power series with respect to q = e πiτ 0 , (2.21) which represents the weight of a gauge instanton configuration. Finally, the parameterũ appearing in (2.16) is given byũ where the second equality follows from (2.12). Notice that in the massless case (m i = 0) the SW curve (2.14) reduces to the equation of a torus with τ = τ 0 .
(2. 25) In the massive case, instead, the complex structure τ associated to the curve (2.14) differs from the "bare" coupling τ 0 . It represents the exact coupling in the low-energy effective theory on the D3 probe and hence the exact axio-dilaton background associated to the D7 brane configuration. We shall now extract from the SW curve the instanton expansion of τ in the region of large a. By a suitable shift in x, the SW curve (2.14) can be put in Weierstrass form where the coefficients g 2 and g 3 can be determined explicitly in terms of u, of the mass invariants and of q. The absolute modular invariant of the curve has then an explicit expression in terms of the same parameters. This expression is such that j diverges as q → 0; indeed The modular invariant j is related to the complex structure modulus of the torus by the expression given in (2.4), which we rewrite here for convenience: (2.29) Inverting this relation yields the effective coupling as a function of j, namely Then, using the expansion of j given in (2.28), we can organize the effective coupling as a series in q 2πi τ = 2πi τ 0 + k f k (m, u) q k (2.31) in which each coefficient f k can in turn be expanded for large u.
The instanton expansion is then obtained by rewriting, in the semi-classical regime, the modulus u in terms of the vacuum expectation value a. To this end, let us recall that the periods of the holomorphic form dx/y on the Weierstrass torus have an explicit expression in terms of j. In the region where j is large, one period has the following closed-form expression −R 2 + 6T 1 4 + 12 Pfm q + 6(R 2 + 6T 1 ) q 2 + 48 Pfm q 3 + . . .   The effective coupling τ encoded in the SW curve contains a perturbative part in perfect agreement 9 with Eq. (2.13) for N f = 4 plus a series of non-perturbative corrections δ l , and thus takes the form Tr m 2l + δ l a 2l , (2.38) 8 The expansions are rather cumbersome, so we limit ourselves to order q 3 and to order 1/a 8 . This already provides a very strong check against the result of the explicit instanton computations presented in the following sections. 9 A part from the finite renormalization constant 2 ln 16.
where the explicit expressions for δ l can be deduced from Eq. (2.37). In particular, the first few corrections are In the following we will provide a microscopic derivation of this result by introducing stacks of D-instantons in our orientifold model.

D-instanton corrections
We now discuss the non-perturbative effects obtained by adding k D-instantons to the D3/D7 brane system described so far 10 . The D(-1)'s describe ordinary gauge instantons in the four-dimensional N = 2 Sp(1) gauge theory with N f = 4 realized on the worldvolume of the D3 branes, while they represent exotic instantons with respect to the eightdimensional N = 2 SO(8) gauge theory realized on the D7 branes, as discussed in [51,31,33]. In the following we describe the moduli spectrum and its BRST structure and compute the D-instanton partition function using localization methods.

Instanton moduli spectrum
The open strings with at least one end-point on the D(-1)'s account for the instanton collective coordinates (or moduli) and can be distinguished into three classes corresponding to open strings with both end-points on the D-instantons (the (-1)/(-1) sector), and to open strings with one end-point on the D-instantons and the other either on the D3 branes or on the D7 branes (the (-1)/3 sector and the (-1)/7 sector respectively). This system has already been considered and described in [29], but for completeness we now briefly recall its main features, taking advantage of the analysis presented in [34] for a very similar set-up.
(-1)/(-1) sector The string excitations in this sector carry Chan-Paton (CP) factors that are (k × k) matrices in a definite representation of SO(k) selected by the orientifold projection Ω. They also organize in representations of the Lorentz group which in our case is as dictated by the presence of the D7 and D3 branes.
In the NS sector the two bosonic moduli along the D7 transverse directions, with ADHM dimension of (length) −1 , are odd under Ω and thus transform in the anti-symmetric representation of SO(k). They can be organized in complex conjugate moduli χ andχ that are singlets of the four SU(2)'s in (3.1) and carry charge +1 and −1 under the U(1) factor. The eight bosonic moduli along the D7 longitudinal directions, with ADHM dimension of (length) +1 , are instead even under Ω and hence transform in the symmetric representation of SO(k). They form two vectors for each of the two SO(4) factors in (3.1) that we denote, respectively, a µ (µ = 0, . . . , 3) and a m (m = 4, . . . , 7). The diagonal components Tr a µ define the (center of mass) position of the D-instantons in the four-dimensional space that is longitudinal to both the D7's and the D3's, and thus they can be identified with the four-dimensional space-time coordinates x µ . The other four diagonal components Tr a m represent the position in the remaining four longitudinal directions of the D7 branes, and together with the x's they can be identified with the eight-dimensional coordinates X used in the previous section.
A similar analysis can be performed in the Ramond sector of the (-1)/(-1) strings. Here we have sixteen fermionic moduli which we can group into four sets Mα a , M αȧ , N αa and Nαȧ with α,α, a andȧ labelling the spinor representations of the four SU(2)'s in (3.1). We have denoted by M 's and N 's the components with positive and negative SO(2) chiralities, respectively, that correspond to eigenvalues +1 and −1 under Ω so that the M 's, with ADHM dimension of (length) +1/2 , are in the symmetric representation of SO(k), while the N 's, with dimension of (length) −3/2 , are in the anti-symmetric representation of SO(k).
(-1)/3 sector The CP factors of the open strings connecting the D-instantons with the D3 brane and its orientifold image are (k × 2) matrices transforming in the fundamental representation of both SO(k) and Sp (1). The open strings with opposite orientation, stretching between the D3's and the D(-1)'s, carry CP factors that are (2 × k) matrices but they should not be counted as independent because of the identification enforced by the orientifold projection.
The first four directions of the (-1)/3 strings have mixed Dirichlet-Neumann boundary conditions. In the NS sector we therefore find two bosonic moduli, with ADHM dimension of (length) 1 , in the spinor representation of SU(2) + which we denote by w α . However, if we want to exhibit also the SO(k) and Sp(1) indices, we should write w αRU (with R = 1, . . . , k and U = 1, 2). For the opposite string orientation the bosonic excitations arew αU R , but the following orientifold identification holds [29]: In the R sector we find four fermionic moduli, with dimension of (length) 1/2 , which we denote as µ a and µȧ since they are space-time scalars but spinors of the internal Lorentz group. The fermionic moduliμ a andμȧ arising from open strings with opposite orientation are subject to orientifold identifications similar to that displayed in (3.2).
(-1)/7 sector In this sector the open string excitations carry CP factors that are (k × 8) matrices transforming in the fundamental representation of both SO(k) and SO (8). Again the orientifold projection enforces identifications with the excitations of open strings with opposite orientations, which therefore should not be counted as independent. Since the (-1)/7 strings have eight directions with mixed Dirichlet-Neumann boundary conditions, the physical states are only in the R sector where we find one fermionic modulus µ with ADHM dimension of (length) +1/2 . To exhibit also the SO(k) and SO (8) indices we should write µ RI (with R = 1, . . . , k and I = 1, . . . , 8). The physical excitations of the 7/(-1) strings, whose CP factors are (8 × k) matrices, correspond to the modulusμ which is subject to the following orientifold identification The absence of physical bosonic moduli in this sector is typical of exotic instanton configurations.

BRST structure of moduli space
The moduli space we have described above admits a BRST structure that will play a crucial rôle for the localization of the integral over the D-instantons. The BRST charge Q can be obtained by choosing any component of the supersymmetry charges preserved on the brane system. The supersymmetry charges are invariant under SO(k), Sp(1) and SO (8), but transform as a spinor of the SO(4) × SO(4) subgroup of the Lorentz group (3.1), so that the choice of Q breaks it to the SU(2) 3 subgroup which preserves this spinor. In our case we take which corresponds to identify the spinor indices α and a of the first and third SU(2)'s in (3.1). After this identification is made, the fermionic moduli Mα a and M αȧ can be renamed as M µ=αα and M m=aȧ , and paired with a µ and a m into BRST doublets such that Qa µ = M µ and Qa m = M m . Similarly, in the fermionic sector of the (-1)/(-1) strings the singlet component N = αa N αa has the right transformation properties to qualify as BRST partner ofχ, i.e. Qχ = N . Likewise, in the (-1)/3 sector the fermionic moduli µ α=a can be paired with the bosons w α and form another BRST doublet such that Qw α = µ α . The remaining fields N c = (τ c ) αa N αa (τ c being the three Pauli matrices), Nαȧ, µȧ and µ are unpaired, and should be supplemented with auxiliary fields having identical transformation properties. Denoting such fields D c , Dαȧ, hȧ and h , respectively, we therefore have QN c = D c , QNαȧ = Dαȧ and so on.
In the usual instanton theory the auxiliary fields collect the D-and F-terms of the gauge theory on the D(-1)'s, and the corresponding D-and F-flatness conditions are the ADHM constraints on the instanton moduli space (see for example [8,37,40] for details). In our case we have an extension of this construction to a sort of generalized "exotic" instanton moduli space. More precisely, the seven auxiliary moduli D c and Dαȧ, of scaling dimension (length) −2 linearize the quartic interactions among the a µ 's and the a m 's, and in the explicit string realization correspond to vertex operators that are bi-linear in the fermionic string coordinates [8,31]. In particular, the triplet D c disentangles the quartic interactions of the a µ 's and the a m 's among themselves, while the quartet Dαȧ decouples the quartic interactions between the a µ 's and the a m 's. Finally, the dimensionless auxiliary moduli hȧ disentangle the quartic interactions between a m and w α , while h completes the BRST multiplet in the (-1)/7 sector. In the end only one modulus, namely χ, remains unpaired and therefore Qχ = 0. All this is summarized in Tab. 1. Since the scaling dimension of the BRST charge is (length) −1/2 , the dimensions of the components (Ψ 0 , Ψ 1 ) of any BRST doublet are of the form (length) ∆ and (length) ∆−1/2 . Thus, recalling that a fermionic variable and its differential have opposite dimensions, the measure on the instanton moduli space Here, the first term in the exponent accounts for the unpaired modulus χ in the antisymmetric representation of SO(k), while n b (n f ) denotes the number of BRST multiplets whose lowest components Ψ 0 are bosonic (fermionic). From Tab. 1 it is not difficult to verify that Hence, Eq. (3.6) implies that the full measure dM k is dimensionless, in agreement with the conformality of the theories on both the D3's and the D7's. For all the moduli we have listed above, including the auxiliary ones, it is possible to write vertex operators of conformal dimension one and use them to obtain the moduli action S(M k ) by computing disk amplitudes along the lines discussed in [7,8,30,31]. The result is the moduli action derived in [29] (see in particular Eq. (3.3) of that paper). A first generalization that will be necessary in the following is the introduction of the interaction terms between the moduli and the Sp(1) adjoint scalar φ of the 3/3 sector or the SO(8) adjoint scalar m of the 7/7 sector. As explained for example in [30], such interaction terms can be derived from mixed disk amplitudes with a portion of the boundary on the D3's or the D7's where a scalar field φ or m can be emitted. The result of these computations is the action S(M k ; φ, m). It turns out that this can be obtained from S(M k ) with a simple effective rule corresponding to a shift of the χ moduli. More precisely, we have to make the following replacements 11 Furthermore, one can prove that this action is BRST exact, namely for a suitably defined fermion Ξ and that Q is nilpotent up to infinitesimal transformations of SO(k), Sp(1) and SO(8) parameterized respectively by χ, φ and i m. This means that on any modulus we have where the T 's act in the specific SO(k), Sp(1) and SO(8) representations given in Tab. 1. For our later purposes it is enough to consider the Cartan directions of the various groups. We label the Cartan components of the SO(k) parameters of Q by χ = {χ r } with r = 1, . . . , rank SO(k), those of Sp(1) by φ = {a}, and those of SO(8) by m = {m i / √ 2} with i = 1, . . . , 4. The latter two choices are in agreement with the choices and normalizations used in the previous section, see in particular Eq.s (2.12) and (2.7). Using these ingredients, one can easily see that Q 2 in (3.10) corresponds to infinitesimal Cartan actions which can be diagonalized in any representation by going to the basis provided by the weights. Indeed, denoting respectively by π, σ and ρ the weights of the SO(k), Sp(1) and SO (8) representations under which a given modulus transforms, we can rewrite (3.10) as To fully localize the integral over moduli space it is necessary to use a BRST charge that is equivariant with respect to all symmetries of the model. In our case these include also the residual Lorentz symmetry SU(2) 3 defined in (3.4), besides the SO(k), Sp(1) and SO (8) symmetries considered so far.

Moduli integration via localization and instanton partition function
As explained for example in [35]- [40], the fully equivariant cohomology can be obtained by deforming the moduli action with parameters associated to "rotations" along the Cartan directions of the residual Lorentz group. For our model, these rotations can be parameterized by = { A } with A = 1, . . . , 4, subject to the constraint Although only three out of the 's are independent variables, in all calculations it is convenient to use all four of them and impose the relation (3.12) only at the very end. From the string point of view the -deformation can be obtained by switching on a RR background on the D7 brane world-volume corresponding to graviphoton field strengths of the type This RR background induces new -dependent terms in the moduli action (3.9) which then gets replaced by a deformed action S(M k ; φ, m, ). The -dependent terms can be explicitly derived by computing mixed open/closed string amplitudes on disks with insertions of the moduli vertex operators on the boundary and of the vertex operators representing the graviphoton field strengths F in the interior, as shown in [30,31,32] for similar systems. The net result of these computations amounts to replace the old BRST charge Q with a deformed charge Q satisfying the following relation which is a direct generalization of (3.11). Here we have taken the deformation parameters A to be of dimensions (length) −1 and have denoted by γ the weights of the representation of the residual Lorentz group (3.4) under which a given modulus transforms. Using the rules explained in [34], one can show that where γ are the weights of the three SU(2)'s, namely γ = 0 for a modulus in the 1, γ = ± 1 2 for a modulus in the 2 and so on. The explicit values of · γ for all BRST doublets of moduli are collected in Tab. 2.
The -deformed moduli action is still BRST exact (with respect to the new BRST charge Q), namely for a suitable fermion Ξ. This deformed BRST structure allows to perform rescalings of various moduli and show that the k-instanton partition function (with N k a normalization factor) can be computed exactly in the semiclassical approximation. Indeed, the complete localization of the integral over moduli space around isolated fixed points implies that Z k is given by the (super)-determinant of Q 2 evaluated at the fixed points of Q [35]- [40]. As we already mentioned, the moduli χ andχ appear very asymmetrically in the BRST formalism: χ parametrizes the SO(k) rotations, whileχ falls into one of the BRST doublets. Moreover, the contribution of the (χ, η) multiplet to the super-determinant cancels against an identical contribution coming from the neutral component in (N c , D c ) with identical transformation properties and opposite statistics. Taking this into account, the super-determinant of Q 2 takes a simple product form in terms of the Q 2 -eigenvalues given in (3.14) and the k-instanton partition function (3.17) can be expressed by the localization formula Here ∆(χ) is the Vandermonde determinant representing the Jacobian factor for the diagonalization of the integration variables χ. The factor P (χ) arises from the integration over the BRST doublets containing N c and Nαȧ which transform in the representation of SO(k). It is given by where we have introduced the following combinations 20) and denoted by r the number of null weights in the anti-symmetric representation of SO(k) (namely, the rank of the group) and by + the set of its positive weights.
The factor P (χ) in the denominator of (3.18) arises from the integration over the BRST doublets containing a µ and a m , and is given by where again r is the number of null weights in the symmetric representation of SO(k) and + is the set of its positive weights.
Finally, the last factor P (χ) in (3.18) collects the contributions from the integration over the BRST doublets containing the moduli with only one end-point on the D-instantons, namely the µ 's from the (-1)/7 sector, and the w α 's and µȧ's from the (-1)/3 sector. It reads where r is the number of null weights in the fundamental representation of SO(k) and + the set of its positive weights.
More explicit expressions for ∆(χ), P (χ), P (χ) and P (χ) are given in Appendix C. Here we just remark that the integrand in (3.18) is a rational function and that the integrals over χ r have to be computed according to the prescription of [52] as contour integrals after giving the deformation parameters A 's a positive imaginary part such that Im 1 > Im 2 > · · · > Im 4 > Im In this way the calculation of the D-instanton partition function Z k is reduced to the sum of the residues of a rational function with simple poles.

Non-perturbative prepotentials
In order to derive the non-perturbative contributions to the world-volume effective actions of the space-filling branes we have to sum over all instanton numbers and consider the "grand-canonical" instanton partition function where we have conventionally set Z 0 = 1. Then, we have to switch off the -deformations. However, in doing this we have to pay attention to a couple of points. In fact, as is clear from (3.17), the integrals appearing in Z k run over all moduli, including also the "center of mass" super-coordinates X and Θ. In presence of the -deformations it is rather easy to see that the integration over this eight-dimensional super-space yields a volume factor growing as 1/( 1 2 3 4 ) in the limit of small A 's. Therefore, to obtain the integral over the centered moduli and hence the contributions to the brane effective actions, this volume factor has to be removed before turning off the -deformations. In addition, we have to take into account the fact that the k-th order in the q-expansion receives contributions not only from genuine k-instanton configurations but also from disconnected ones, corresponding to copies of instantons of lower numbers k such that k = k. To isolate the connected components we have to take the logarithm of Z.
The singularity structure of log Z with respect to the -deformations turns out to be the following: Here, F n.p. and F n.p. are expressions which are regular when the deformations are switched off; the subscript refers to the non-perturbative nature of these quantities, while the superscripts signal their eight-dimensional, respectively four-dimensional, nature. Since the A 's have dimension of (length) −1 , F n.p. must have mass-dimension four, while F (4) n.p. must have mass-dimension two. Indeed, F n.p. is a quartic expression in m i and A , without any dependence on a. Therefore we can conclude that F (8) n.p. represents the non-perturbative quartic prepotential for the D7 brane world-volume theory. On the other hand, F n.p. is quadratic in A , a and m i ; since 1/( 1 2 ) represents the regulated (super)volume in the first four directions, we can conclude that F (4) n.p. is the non-perturbative quadratic prepotential for the D3 brane world-volume theory 12 .
The prepotentials F (8) n.p. and F (4) n.p. contain "gravitational" corrections encoded in the deformation parameters A which, as we have described in Section 3.3, arise from the RR closed string sector. In this paper we are not interested in this dependence, and we limit ourselves to the prepotentials  It should not be a surprise that the D-instanton partition function Z in presence of D7 and D3 branes contains both an eight-dimensional and a four-dimensional information. Indeed, depending on the point of view, the D(-1)'s can be considered alternatively either as (exotic) instantons for the eight-dimensional theory on the D7's or as (ordinary) instantons for the four-dimensional theory on the D3's. In the following sections we will consider in detail the non-perturbative prepotentials (3.26), give their explicit expressions for the first instanton numbers (up to k = 5), analyze their properties and discuss their relations with the F-theory perspective and the SW curve we discussed in the previous section. 12 Note that terms in F  proportional to f . We fix this kind of ambiguity by requiring that all terms assigned to F (8) n.p. have the structure and symmetry properties that are appropriate for a correct eight-dimensional interpretation as a prepotential.

Eight-dimensional effective prepotential and chiral ring
The non-perturbative prepotential for the eight-dimensional gauge theory on the D7 branes can be computed using formulae (3.25) and (3.26). Up to instanton order k = 5, we find This prepotential is independent on the vacuum expectation value a of the D3 adjoint multiplet, and coincides exactly 13 with the result found in [31] for the D7/D(-1) system in type I . Using the quadratic and quartic SO(8) invariants defined in Appendix B, we can rewrite the prepotential (4.1) as follows

(4.2)
This non-perturbative part has to be added to the tree-level prepotential and to the perturbative 1-loop contribution which, up to terms that depend on the complex structure U of the torus transverse to the D7 branes and that we do not write for brevity, is given by [51]  As shown in [31], the above results up to order k = 5 are consistent with heterotic/type I duality [41,29,53,54] and may be extended to all orders in q. Indeed, the complete prepotential may be written as where .

(4.6)
13 Notice that our conventions are changed with respect to [31] in that we now have q = exp(πiτ0) to agree with the conventions of [26], while there we wrote q = exp(2πiτ ). Moreover, we choose the overall normalization of the instanton partition function in a way which corresponds to sending q → −q, i.e. to changing the sign in front of the Pfaffian terms. Finally, the field m we use here corresponds to the field φ of [31]; however, because of our conventions in Eq. (2.7), the eigenvalues φi of the latter correspond to −imi/ √ 2.
Expanding the ϑ-functions and the Dedekind function in powers of q, one can easily check the agreement with the perturbative and non-perturbative expressions reported above. Up to τ 0 -independent terms that can be presumably absorbed by a (finite) renormalization of the 1-loop contribution (4.4) which was not considered in [31], we can further rewrite the eight-dimensional prepotential as follows Notice that the single-trace quartic structure present in T 1 , whose tree-level coupling is πiτ 0 (see Eq. (4.3)), has an effective coupling given by ln κ(q) = ln q + ln 16 − 8q + 12q 2 − 32 3 This single-trace structure is conformal at the perturbative level but from (4.8) we see that it receives both a finite 1-loop renormalization (the ln 16 term above) and a series of instanton contributions. In Section 7 we will find again this same non-perturbative redefinition (or more precisely its inverse) in the four-dimensional SU(2) SYM theory living on the D3 branes. From the prepotential F (8) we obtain the eight-dimensional SO(8) effective action on the D7 branes by promoting the diagonal vacuum expectation values m i to the full chiral superfield M (X, Θ) given in (2.6) and integrating over the eight-dimensional chiral superspace, namely Among various other terms, this effective action contains quartic couplings of the type Tr t 8 f 4 with t 8 being the totally anti-symmetric eight-index tensor appearing in various string amplitudes [55] and f being the field strength, that under the heterotic/type I duality exactly match the heterotic results [31]. As we discussed in Section 2.2, the mathematical description of the non-perturbative effects in the eight-dimensional type I theory should be identical to the non-perturbative description of the four-dimensional N = 2 SYM theory with gauge group SU (2) and N f = 4 (massive) flavors. For such a theory, which possesses a global SO(8) flavor symmetry, it was argued in [26] that the full symmetry of the effective action is SL(2, Z) whose generators act as standard modular transformations on the tree-level gauge coupling accompanied by triality transformations on the flavor SO(8) representations. Therefore, following the arguments of [25], we expect that such SL(2, Z) be a symmetry of the eight-dimensional effective action (4.9), in which now SO(8) is the gauge group. We now show that this is indeed the case. For simplicity we assume that the background axionic value be vanishing, i.e. we take Re τ 0 = 0.
Let us first consider the transformation S defined as This is the combination of the S modular transformation on the coupling constant τ 0 with the SO(8) triality transformation that exchanges the vector representation (associated to T 1 ) with one of the spinorial representations (associated to T 2 ) leaving fixed the other (associated to T 3 = −T 1 − T 2 ) as well as the quadratic invariant R. Using the modular properties of the Jacobi ϑ-functions (see Eq. (A.5)), it is easy to see that 11) and hence that the first two terms of the prepotential (4.7) are invariant under S. Furthermore, for the case at hand we have so that also the R 2 term in (4.7) is invariant under S. Now let us consider the transformation T defined as This is the combination of the T transformation acting on τ 0 with the triality transformation that exchanges the two spinor representations of SO (8), leaving fixed the vector representation (and the quadratic invariant R). From the modular properties under T given in (A.6), it is easy to check that (4.14) and hence that the prepotential (4.7) transforms under T as follows Even if the prepotential is not invariant, the change produced in the effective action is unobservable because where in the last step c 4 is the topological index of order four which is an integer number.
Since the effective action shifts by an integer multiple of 2πi, we can conclude 14 that the theory is invariant also under T . Thus, the symmetry group of the quantum SO(8) gauge theory living on the D7 branes is the one generated by S and T , which, as shown in Appendix B.1, is isomorphic to SL(2, Z) in agreement with our expectations. Besides the non-perturbative effective action and its symmetry properties, another useful set of information on the eight-dimensional theory is provided by the quantum vacuum expectation values of the composite operators Tr m 2l , forming the so-called "chiral ring" of the model. These have been computed in [33] via localization techniques with suitable insertions in the instanton partition function for a generic number of D7 branes. When this number is four, i.e. for our SO (8)   It is interesting to observe that the non-perturbative part of these expressions exactly coincides with the non-perturbative corrections for the coupling constant of the fourdimensional SU (2) gauge theory with N f = 4 as reported in Eq. (2.39). We will elaborate more on this relation in Section 6.

Four-dimensional effective prepotential
The non-perturbative prepotential for the four-dimensional Sp(1) gauge theory on the D3 branes can be computed by extracting the sub-leading 1/( 1 2 ) divergence in the deformed D-instanton partition function using formulae (3.25) and (3.26). The result of these calculations, which are technically similar to those reported in [34], can be written as follows where the first five instanton contributions are explicitly given by   To the above non-perturbative prepotential we must add the tree-level term where in the last step we replaced the adjoint field φ with its vacuum expectation value according to (2.12), and the perturbative 1-loop contribution given by (see for example [56]) (5.8) Note that for N f = 4 the only dependence on the UV cut-off Λ is in terms proportional to i m 2 i log(a/Λ) which, however, do not contribute to the effective action. The total prepotential n.p. This completely agrees with the effective coupling reported in Eq. (2.37) and derived from the SW curve for the SU(2) N f = 4 theory [26]. Thus, we can conclude that the nonperturbative features predicted by the SW curve for this theory can be interpreted as D-instanton effects in the D7/D3 system of type I . Furthermore, this agreement puts on a very solid ground the prescription described in Eq. (3.25), which, as a matter of fact, was successfully tested already in [34] in a different context by exploiting the heterotic/type I duality.

An intriguing relation
By comparing the non-perturbative terms of the effective coupling (5.11) with the vacuum expectation values (4.17) of the operators Tr m 2l forming the chiral ring of the eightdimensional theory living on the D7 branes, we find that (up to the available orders in q and in 1/a) our results take the following very intriguing and suggestive form Tr m 2l a 2l . (6.1) In other words, the non-perturbative corrections to the effective coupling on the D3's are obtained from its perturbative expression (2.13) after promoting the mass parameters m i , which classically correspond to the vacuum expectation values of the SO(8) adjoint field m on the D7 branes according to Eq. (2.7), to a bona-fide quantum field of which the appropriate expectation values have to be taken. We conjecture 16 that this relation actually extends to all orders in q and in 1/a 2 , and in Fig. 2 we give a graphical representation of it. The effective gauge coupling τ on the probe D3 brane (the blue dot) receives perturbative contributions. They arise from loops of open string states suspended between the D3 and the D7 branes (red circles), their images (dotted red circles) and the orientifold (green cross). These diagrams have a dual interpretation as tree-level closed string exchanges which are depicted as straight lines (the lines to the image D7 branes are not shown to avoid clutter). If we take into account the full quantum dynamics, including the nonperturbative effects, of the D7 fields whose eigenvalues correspond to the D7 positions (this procedure is graphically indicated by the circular red-shaded area including the D7 branes) and we still communicate this information to the probe D3 brane in a perturbative fashion, we get the exact effective coupling, i.e.
With simple formal manipulations we can rewrite the relation (6.1) as follows: from which, after using (5.10) in the left-hand side, we obtain Note that a ∂ a F (4) − 2F (4) is the same combination that appears in the Matone relation [58] for the pure SU(2) SYM theory, and hence we can regard Eq. (6.4) as a sort of generalization of this relation to the SU(2) theory with four massive flavors.
In conclusion, it can be said that knowing the chiral ring in the eight-dimensional theory is equivalent to knowing the four-dimensional prepotential. Conversely the eightdimensional prepotential can be read off directly from the four-dimensional one by means of This equation is a consequence of the particular relation 1 4 ! Tr m 4 = q ∂ ∂q F n.p. explained in [33], and of Eq.s (6.1) and (5.10).

Comparison with the SU(2) instanton calculus
In this last section we pursue the comparison of our results with the outcome of the standard supersymmetric instanton calculus for the SU(2) theory with N f flavors.

Instanton calculus
The computation of instanton effects in field theories with N = 2 supersymmetry, SU(N ) gauge group and N f fundamental hypermultiplets was performed by N. Nekrasov exploiting localization techniques [35,38]. In fact, it is very easy to extract from his work explicit expressions for the prepotential, at least up to three instantons; for instance, the result for the SU(2) gauge theory with N f = 3 is reported here in Appendix E (see in particular Eq. (E.3)). This result fully agrees, up to irrelevant constant rescalings of the mass parameters and of the dynamical scale, with the limit in which we decouple one flavor from the four-dimensional prepotential (5.9) obtained for the D3/D7 system in type I . This agreement represents another very reassuring check of our procedure.
Although originally proposed for asymptotically free theories, and thus limited to N f < 4 in the case of the SU(2) gauge theory, Nekrasov's prescription can formally be extended without any problem also to the conformal theory with N f = 4. We will describe the outcome of Nekrasov's prescription and elaborate on its relation with our results (and the SW approach) in the next subsection by pushing our analysis up to five instantons. To do so we resort, as a very efficient computational tool, to the recently discovered AGT connection that allows to obtain the instanton partition functionà la Nekrasov for the SU(2) N f = 4 gauge theory from the conformal blocks of a two-dimensional Liouville conformal field theory [44].
Before giving some details on this relation, let us anticipate the key points of our findings. Denoting by τ uv the tree-level coupling in this approach, and by One can check that this is the inverse of the expansion given in (4.8), so that we are led to identify x with κ(q) and write which coincides with the closed-form expression proposed in [44] (see also [46]- [48], and [59] where that expression was first proposed). As a pure matter-of-fact remark, it is interesting to notice that the tree-level coupling ln x utilized in Nekrasov's approach corresponds to the effective coupling ln κ(q) that appears in front of the SO(8) invariant structure T 1 in the eight-dimensional action on the D7 branes (see Eq.s (4.6) and (4.7)). This is another signal of a non-trivial interplay between quantum effects in four dimensions on the D3's and in eight dimensions on the D7's, in addition to the one described in Section 6.
Turning to the massive case, one finds that the Nekrasov/AGT prepotential for the SU(2) N f = 4 theory contains, at each order in its instanton expansion, several terms which do not appear in our findings. However, if we start from our complete prepotential (5.9) and substitute our expansion parameter q in terms of x according to Eq. (7.4), namely we obtain an exact matching with the Nekrasov/AGT prepotential (up to constant terms which do not contribute to the effective action). Let us now show this in some more detail.

The Nekrasov prepotential from the AGT realization
In [44] a remarkable relation between the deformed instanton partition function for N = 2 SU(2) gauge theories in four dimensions and the correlation functions of the Liouville theory in two dimensions has been discovered. In this correspondence the central charge c of the Liouville theory and the deformation parameters 1 and 2 of the instanton partition functions are related as follows c = 1 + 6 Q 2 = 1 + 6 ( 1 + 2 ) 2 1 2 , (7.7) while the logarithm of the conformal block of four Liouville operators [45] is related to the non-perturbative prepotential F n.p. of the N = 2 SU(2) theory with four massive hypermultiplets as computed in Nekrasov's approach [35]- [39]. More precisely we have is the conformal block of four Liouville primary fields with conformal dimensions ∆ i , located at ∞, 1, x and 0, and factorized in the channel (∆ 1 − ∆ 2 ) ∼ (∆ 3 − ∆ 4 ) with an intermediate state of conformal dimension ∆. These dimensions can be parameterized as follows where a is the vacuum expectation value of the SU(2) adjoint scalar field, and the µ i 's are related to the masses m i of the fundamental hypermultiplets according to (7.10) The dressing factor (1 − x) 2µ 1 µ 3 has been inserted to decouple a U(1) factor [44] and get the instantonic partition function for SU (2) and not for U (2). Note that in the literature there are different expressions for such a factor, containing also terms that depend on the background charge Q. However, these terms are not relevant in the limit 1 , 2 → 0 we are considering here 17 . Using the expression of the conformal blocks [45] (see also [46]- [48]) and expanding in powers of x, we have where the first few coefficients are (7.14) In Appendix D we report also the explicit expressions for F 4 and F 5 (see Eq.s (D.5) and (D.6)), as well as some other technical details on this approach. To this non-perturbative prepotential we must add the tree-level contribution F (4) tree = a 2 ln x , (7.15) and the x-independent 1-loop part 16) which coincides with (5.8) except for a small difference in the last term where the effect of a finite renormalization pointed out in [50] has been taken into account.
The total prepotential n.p. does not agree, at first sight, with the prepotential F (4) we have found in Section 5 (see Eq.s (5.1)-(5.9)). Indeed, one can easily see that in the massless case (m i = 0) the nonperturbative part F (4) n.p. is vanishing while F (4) n.p. is not, and that the numerical coefficients as well as the various structures are different in the two expressions. However, one can explicitly check that all a-dependent structures of F (4) and F (4) are exactly mapped into each other if q and x are related as in (7.6) (or equivalently as in (7.4)). Let us show this in some cases. Consider for example the terms in F (4) that are proportional to a 2 , namely a 2 ln x − ln 16 + 1 2 x + 13 64 Upon using (7.4), they simply become a 2 ln q, that is exactly the a 2 term of F (4) given in (5.7). Now consider the terms in F (4) that are proportional to 1/a 2 , namely x 3 +. . . (7.19) where the first x-independent contribution arises from F 1−loop . If we use the relation (7.6), we can rewrite (7.19) as 20) which exactly coincides with the part of F (4) proportional to 1/a 2 obtained in Section 5.
Likewise one can check that the agreement persists in all other a-dependent terms, and up to five instantons using the expressions reported in Appendix D.
The only mismatch is in the constant a-independent part, which however does not contribute to the four-dimensional effective action obtained from the prepotential by promoting a to the full-fledged chiral superfield Φ and integrating over the chiral four-dimensional superspace. Thus, this mismatch is irrelevant. We also note that while the non-perturbative prepotential F n.p. obtained from the D7/D3/D(-1) system is fully invariant under the SO(8) flavor symmetry acting on the massive multiplets, the prepotential F n.p. derived above contains the structure i<j m i m j which is not an SO(8) invariant. Since this structure appears only in the a-independent part, it does not contribute to the effective action, which indeed is SO(8) invariant. It is interesting also to notice that all terms proportional to i<j m i m j could be removed from the prepotential by changing the dressing factor in the AGT relation (7.8), and using (1 − x) − 1 Finally, the a-independent terms proportional to i m 2 i that were absent in F n.p. could be reabsorbed with a suitable redefinition of the UV cut-off appearing in the 1-loop part of the prepotential. As we have already remarked, all these ambiguities in the constant piece are not significant and do not influence the effective action of the four-dimensional gauge fields.

Summary and conclusions
We have considered the local system composed of four D7 branes and a D3 brane near one O7 fixed plane in type I theory on a T 2 torus. We have computed by means of localization techniques the non-perturbative corrections due to D-instantons to the effective actions on the D7 branes (for which they correspond to exotic instantons) and on the D3 brane. On the D3, which supports a Sp (1), N f = 4 theory, they represent gauge instantons. The effective coupling on the D3 coincides with the exact axio-dilaton field, and indeed we find that it agrees with the F-theoretic prediction put forward long ago by Sen [25]. We thus show by an explicit microscopic computation how F-theory resums D-instanton effects. Our results are consistent with the outcome of usual instanton calculusà la Nekrasov [35] for the SU (2), N f = 4 conformal theory, or with its AGT reformulation in terms of 2d Liouville blocks [44], upon a non-perturbative redefinition of the tree-level coupling.
We point out a very interesting relation between the effective coupling τ on the D3 (that is, the modular parameter of the F-theory curve) and the eight-dimensional dynamics on the D7-branes, and specifically the "chiral ring" correlators Tr m 2l , where m(X) is the eight-dimensional field in the adjoint of SO(8) whose eigenvalues m i appear as mass parameters on the D3. Indeed, we find that the exact expression for τ is obtained from its perturbative part replacing the occurrences of the masses with the corresponding chiral ring correlators in eight dimensions.

Acknowledgments
We thank R. Argurio, L. Ferro and especially M. Frau, F. Fucito, J.F. Morales and R. Poghossian for several useful discussions. M. B. thanks the K.I.T.P, Santa Barbara, and the organizers of the workshop "Strings at the LHC and in the Early Universe" for hospitality during the initial stage of this work. This research was supported in part by the National Science Foundation under Grant No. NSF PHY05-51164.

A. Notations and theta-function conventions
Couplings: We use the following names for the various quadratic gauge couplings: • τ 0 is the tree-level coupling in the D3/D7 system in type I , i.e. the asymptotic value of the axio-dilaton field. In this set-up the gauge theory on the D3 is realized as a Sp(1) theory with four flavors.
• τ uv is the tree-level coupling in the usual field-theoretical description of the SU (2) theory with N f = 4 flavors.
• τ is the exact effective coupling, which depends on the tree-level coupling, the vacuum expectation value a of the adjoint scalar parametrizing the Coulomb moduli space and on the mass parameters. This coupling must agree in the two descriptions.
Theta-functions: We adopt the following definition for the Jacobi ϑ-functions: The following properties are relevant for us: We will also use the transformation properties under the modular group generators, which are S :

B. Flavor invariants
The parameters m i correspond to the components along the Cartan directions of Spin (8) of the vacuum expectation values of the D7/D7 scalar field m(x) introduced in Eq. (2.7). In a given representation R, the element m i H i of the algebra is diagonal in the basis of weights w A (A = 1, . . . dim R), with eigenvalues m · w A . Spin(8) has an S 3 group of external automorphisms ("triality") that permutes its three 8-dimensional representations: the vector (v), chiral spinor (s) and antichiral spinor (c), remapping the weights of each of these representations in those of another one. This S 3 can be generated by any two exchanges, say Given the form of the weights of these representations, we can view these triality operations as acting on the Cartan parameters m i , as follows: The SW curve for the SU(2) theory with N f = 4 flavors, described in Section 2.2.1, is written in terms of SO (8) The triality generators act on the invariants T as follows: In other words, the triality group S 3 permutes the T invariants. which is invariant also under triality.
The following combinations are relevant for us: (B.7) B.1 Triality extension of SL(2, Z) Consider an SL(2, Z) group generated by S and T subject to the relations In our case this group is the modular group acting on the bare coupling τ 0 . Now consider also the permutation group S 3 generated by g 1 and g 2 , subject to the relations In our case this group is the triality group of SO(8), we have described above. Let us then construct the semi-direct product of SL(2, Z) and S 3 , which is generated by the following combined actions: T = (T , g 1 ) , S = (S, g 2 ) . (B.10) These two generators obey S 2 = 1 , ( S T ) 3 = 1 ; (B.11) therefore this group is again isomorphic to SL(2, Z).

C. Details on the D-instanton computation
The expressions of the factors P (χ), P (χ) and P (χ) appearing in the integrand of the formula (3.18) for the instanton partition function are given respectively in Eq.s (3.19), (3.21) and (3.22) in terms of the weights of the relevant representations of the SO(k) instantonic symmetry group. Let us recall the form of these weight vectors.
Weight sets of SO(2n + 1) This group has rank n. If we denote by e i the versors in the R n weight space, then • the set of the 2n + 1 weights π of the vector representation is given by ± e i , 0 with multiplicity 1 ; (C.1) • the set of n(2n + 1) weights ρ of the adjoint representation (corresponding to the two-index antisymmetric tensor) is the following: ± e i ± e j (i < j) , ± e i , 0 with multiplicity n ; (C.2) • the (n + 1)(2n + 1) weights of the two-index symmetric tensor 18 are ± e i ± e j (i < j) , ± e i , ± 2 e i , 0 with multiplicity n + 1 .
Weight sets of SO(2n) This group has rank n. If we denote by e i the versors in the R n weight space, then • the set of the 2n weights π of the vector representation is given by • the set of n(2n − 1) weights ρ of the adjoint representation (corresponding to the two-index antisymmetric tensor) is the following: ± e i ± e j (i < j) , 0 with multiplicity n ; (C.5) • the n(2n + 1) weights of the two-index symmetric tensor 19 are ± e i ± e j (i < j) , ± 2 e i , 0 with multiplicity n . (C.6) Inserting these expressions into Eq. (3.19) we have From Eq. (3.21) we get instead (C.8) 18 In fact, this is not an irreducible representation: it decomposes into the (n + 1)(2n + 1) − 1 traceless symmetric tensor plus a singlet. One of the 0 weights corresponds to the singlet. 19 Again, this is not an irreducible representation, since it contains a singlet.

D. Details on the AGT correspondence
The four-point conformal block B ∆ ∆ 1 ∆ 2 ;∆ 3 ∆ 4 (x) appearing in the AGT relation (7.8) can be written [45] as a formal series according to Here Y = {n 1 ≥ n 2 ≥ · · · ≥ n l > 0} is a partition, |Y | = n 1 + n 2 + · · · + n l is its order, Q ∆ (Y, Y ) is the correlator of two descendants of a primary state of dimension ∆, namely where L −Y |∆ = L −n 1 L −n 2 · · · L −n l |∆ (D. 3) with L n being the Virasoro generators, and finally γ ∆ 1 ∆ 2 ∆ (Y ) is given by For further details, see for example [46]. The explicit evaluation of Q ∆ (Y, Y ) and its inverse is a straightforward but tedious task. To perform the algebra we used the open-source program REDUCE and, exploiting the AGT relation (7.8), were able to compute 20 the non-perturbative prepotential F to the fifth order in x. The first three coefficients F 1 , F 2 and F 3 were reported in the main text in Eq.s (7.12), (7.13) and (7.14) respectively. Here we give the expressions for the fourth and fifth coefficients. We have E. Decoupling limits to N f = 3, 2, 1, 0.
From the prepotential we derived in Eq.s (5.2)-(5.6) for the N f = 4 case, it is possible to deduce the prepotential for the asymptotically free theories with N f = 3, 2, 1, 0 by decoupling in turn the massive fundamental hypermultiplets. We will now compare the results obtained in this way with those following from Nekrasov's prescriptions for the instanton calculus, which were originally given just for these asymptotically free cases. We will find full agreement in all terms contributing to the effective action. This enhances our confidence in our results, and shows that the difference with Nekrasov's formulas (or, equivalently, with the AGT methods) discussed in Section 7 only occurs in the conformal case N f = 4. Let us start decoupling one flavor and get the N f = 3 theory by taking the limit On the other hand, from Nekrasov's paper [38], using Eq. (3.16) and the relevant definitions given in the previous pages therein, we can extract the following expression of the SU (2) prepotential for N f = 3 flavors with masses M i : The two expressions (E.2) and (E.3) match after taking into account a different normalization of the masses and of the dynamical scale, namely The only difference is the constant term proportional to (M 1 + M 2 + M 3 ) in the first line of (E.3), which however does not contribute to the effective action upon integration over the chiral superspace, and thus is irrelevant.
The agreement persists for lower values of N f , upon taking further decoupling limits. For instance, the pure SU (2) case (N f = 0) can be reached from our N f = 4 results in Eq.s (5.2)-(5.4) by sending m 1,2,3,4 → ∞ , with Λ 4 ≡ q Pfm fixed . (E.5) The resulting prepotential reads and coincides with Nekrasov's result if we set Λ 4 =Λ 4 /4. The above results are perfectly consistent also with the instanton expansion of the effective coupling derived from the SW curves [26,27] for SU (2) gauge theories with N f = 0, 1, 2, 3.
Also the relation (6.1) between the exact gauge coupling τ on the D3 branes and the eight-dimensional chiral ring for the conformal N f = 4 case can be extended to the nonconformal cases with lower N f . We can check this by exploiting the results of [33], where the first chiral ring expectation values Tr m 2l have been given in the generic case of N f D7-branes supporting an SO(2N f ) group. Therefore we can compare these expressions to the effective coupling derived from the quadratic prepotentials we just described. Let us consider as an example the N f = 3 case. Deriving the prepotential (E.2) with respect to a or, equivalently, taking the limit (E.1) on the non-perturbative effective coupling (5.11), we find   On the other hand, from [33] we can deduce, in our conventions, the following results for from which Eq. (6.1) easily follows. The full effective coupling τ can thus be obtained from its perturbative expression (2.13) upon taking into account the eight-dimensional quantum dynamics of the mass parameters m i . The chiral ring is mathematically defined for any number N f of D7 branes but, as emphasized in [33], only for N f > 4 (resp. N f = 4 ) it can be associated to an eightdimensional SO(2N f ) gauge theory with negative (resp. vanishing) β function for which a weak coupling analysis is reliable. Our analysis in the cases N f = 4 and N f < 4 shows that the chiral ring for those values of N f can be given a sensible meaning in the four-dimensional theory.