The exact 8d chiral ring from 4d recursion relations

We consider the local F-theory set-up corresponding to four D7 branes in type I' theory, in which the exact axio-dilaton background tau(z) is identified with the low-energy effective coupling of the four-dimensional N=2 super Yang-Mills theory with gauge group SU(2) and Nf=4 flavours living on a probe D3 brane placed at position z. Recently, an intriguing relation has been found between the correlators forming the chiral ring of the eight-dimensional theory on the D7 branes and the large-z expansion of the tau profile. Here we apply to the SU(2), Nf=4 theory some recursion techniques that allow to derive the coefficients of the large-z expansion of tau in terms of modular functions of the UV coupling. In this way we obtain exact expressions for the elements of the eight-dimensional chiral ring that resum their instanton expansions, previously known only up to the first few orders by means of localization techniques.


Introduction
F-theory is a very interesting framework for building string models that may be potentially relevant for phenomenology (for reviews see, for instance, Ref.s [1,2]). It also represents an intriguing arena from the more formal point of view, as it is supposed to incorporate the non-perturbative corrections of type IIB string compactifications by geometrizing them in a very non-trivial way.
By considering local type I ′ models containing D7 branes and an O7 plane, a remarkable relation has been recently pointed out [3,4,5] between the profile of the axio-dilaton field τ in F-theory and certain correlators in the eight-dimensional gauge theory living on the D7 branes that provide its microscopic description. This relation reads Tr m 2ℓ a 2ℓ . The parameters m i = z i /(2πα ′ ) correspond to the locations z i of the D7 branes when they are displaced from the orientifold plane. In this paper we will focus on the case in which there are N f = 4 D7 branes, supporting an SO (8) gauge theory in eight dimensions. This is the local limit of F-theory considered long ago by Sen in Ref. [6], where he proposed that the exact profile of τ is given by the effective low-energy coupling of the N = 2 super Yang-Mills theory in four dimensions with gauge group SU (2), N f = 4 fundamental flavours and τ 0 as UV coupling. This non-trivial relation can be understood [7] by considering a D3 brane probe of the geometry created by the D7 branes and the O7 plane: indeed, the D3 branes supports an Sp(1) ∼ SU (2) gauge theory with four flavours (plus a decoupled hypermultiplet in the antisymmetric representation), and its gauge kinetic term couples to the axio-dilaton field. From this perspective, the parameters m i are the flavour masses. When the complex scalar field φ that belongs to the gauge multiplet on the D3 takes a vacuum expectation value φ = (a, −a) , (1.3) i.e. when the D3 brane and its orientifold image are placed at z = ±2πα ′ a, the exact axiodilaton τ (a) represents the effective coupling of this gauge theory. As such, it is encoded in the corresponding Seiberg-Witten (SW) curve [8,9] of which it describes the complex structure parameter. Eq. (1.1) was put forward in Ref. [3] based on the computation of the first few Dinstanton corrections to the D3 coupling in the D3/D7 system as a series in the nonperturbative parameter q = e iπτ 0 . These corrections were found to match the first few terms in the q-expansion of the 8d chiral ring elements obtained in Ref. [10] via localization techniques. In [4] the same relation has been understood entirely from the D7 brane point of view by showing how the D-instantons that correct the chiral ring correlators also modify the source terms for τ , hence its profile. Adopting this point of view, in Ref. [5] the relation has been proven at all instanton orders and extended to any number of D7 branes in presence of an O7 plane, both in a flat background and in a R 4 /Z 2 orbifold.
The exact expression of τ encoded in the SW curve implicitly contains, via Eq. (1.1), all information about the 8d chiral ring correlators. However, to extract the exact expression of a given correlator, we must be able to single out a specific term in the 1/a expansion of τ . This is not trivial, but it can be done systematically by using recursive techniques, akin to the Matone relation [11]. Here we will rediscuss this type of recursions, gathering an understanding that allows us to apply them also to the N f = 4 case, where the structure of the SW curve is complicated by the presence of several different invariants constructed with the flavour masses. In this way we are able to obtain exact expressions for the 8d chiral ring elements that resum their instanton expansions, previously known only up to the first few orders. We find this a remarkable by-product of the deep relation between the 4d effective physics on the D3 brane and the 8d theory on the "flavour" D7 branes encoded in Eq. (1.1). Let us note that for conformal theories there exists also another recursive approach, based on the modular anomaly equation [12,13], which allows to partially fix the coefficients of the large-a expansion of the effective coupling τ ; this approach, which has been used in Ref. [14] for the so-called N = 2 * theory (also known as mass deformed N = 4 theory) where there is a single mass invariant, could also be applied to the N f = 4 model, but it is less efficient than the one we are going to discuss.
The structure of this paper is as follows: in Section 2 we discuss in general the recursion relations for rank one N = 2 theories, and describe the procedure to follow when the SW curve is not in a factorized form. In Section 3 we discuss the SU(2) N f = 4 theory and show how to obtain from its SW curve a recursion relation that yields the various correlators of the "flavour" theory, whose properties are presented in Section 4 together with comments and concluding remarks. Finally, in the Appendices we give some more technical details and discuss the recursion relation in the N = 2 * model seen as a particular case of the N f = 4 theory.

Recursion relations for rank one N = 2 theories
The SW curve for N = 2 super Yang-Mills theories with gauge group SU(2) is a torus and can be thus described as an algebraic surface via a cubic equation of the form where the precise expression for the roots E ℓ depends on the gauge-invariant coordinate on the Coulomb moduli space on the masses (if there is matter) and on the dynamically generated scale Λ (or the bare UV coupling τ 0 in the conformal cases). All these dependencies are here summarized by the variable z. The complex structure parameter τ of the torus describes the complexified IR coupling of the gauge theory according to where θ eff and g eff are the θ-angle and the Yang-Mills coupling constant at low-energy, and is related to the anharmonic ratio κ of the roots as follows where the θ's are the Jacobi θ-functions (see Appendix A for our conventions). In the semiclassical regime, i.e. when u is large, we have where the second equality follows from Eq. (1.3). At a generic point u on the moduli space we have a symplectic section (a(u), a D (u)) given by the periods of the SW differential, such that where ω 1 and ω 2 are the periods of the torus with The low-energy physics can be described by an effective theory for an abelian multiplet with lowest component a and a prepotential F(a) such that The SW curve (2.1) encodes the exact expressions for the physical quantities in the corresponding gauge theory, including its effective coupling τ . These quantities admit a large-a expansion which exhibits, beside the tree-level and perturbative terms, also nonperturbative contributions from all instanton sectors which can be computed directly using localization techniques in the multi-instanton calculus [15]. To extract the instanton expansion from the SW curve one has to determine τ (a). This can be done by first obtaining the expression of τ (u) by inverting Eq. (2.4), and then by determining u as a function of a by inverting the first relation in Eq. (2.6). This procedure is straightforward but may become rather cumbersome in practice.
A more efficient way to proceed is to write the prepotential F(a), and hence τ (a), as an expansion for large a with unknown coefficients and to obtain a recursion relation for the latter. This can be done by expanding the right hand side of Eq. (2.4) around a specific value of τ corresponding to the semiclassical limit, and the left hand side of Eq. (2.4) around particular values of the roots E ℓ of the SW curve that correspond to this limit. This is basically the idea behind the recursion relation originally devised in Ref.s [11,16] for the pure SU(2) theory. In this case the SW curve is [8] so that the anharmonic ratio of the roots defined in Eq. (2.4) iŝ In these expressions we have introduced a "hat" sign to take into account the fact that the complex structure of the SW curve (2.10) turns out to be related to the gauge theory parameters by [8] as opposed to Eq. (2.3); moreover, the parameterû is related to the vacuum expectation value a in the semi-classical regime byû ∼ a 2 /2, to be contrasted with Eq. (2.5) 1 . In Ref. [11] it was shown that that the quantityû/Λ 2 satisfies a differential equation whose solution isû ; (2.13) from this result the identificationκ = θ 4 2 (τ )/θ 4 3 (τ ) immediately follows. One extra ingredient that is needed to proceed is the relation between u (orû) and a, which generalizes the classical one given in Eq. (2.5). Such a relation is provided [11] through the prepotential F(a) by means of ∂F(a) ∂Λ . (2.14) Inserting this in the left hand side of Eq. (2.13) and using Eq. (2.9) in the right hand side, one obtains a non-trivial equation for the prepotential from which, by expanding in inverse powers of a, one can derive a recursion relation for the coefficients of this expansion. For dimensional reasons these coefficients correspond to different powers of Λ 4 , i.e. to different instantonic sectors 2 , and hence this recursion relation allows to reconstruct the higher instanton contributions starting from the lower ones.
Things are a bit different in conformal theories. In this case the relation (2.14) is replaced by with τ 0 being the bare UV coupling. Inserting Eq. (2.15) in the left hand side of Eq. (2.4) and replacing in the right hand side τ via Eq. (2.9), one generates a recursion relation for the coefficients of the large-a expansion of the prepotential in which the q dependence is exact. However, not always the SW curve is given or known in the factorized form (2.1) considered so far. For example, for the SU(2) theories with N f ≤ 4 massive flavours the SW curves are written as cubic polynomials in a non-factorized form [9] for which it is not easy or practical 3 to find the three roots E ℓ . Thus, in these cases the recursion relation 1 To avoid this change of conventions and normalizations, in place of the curve (2.10) one could use for the pure SU(2) theory the isogenic SW curve with complex structure τ = 2τ in agreement with Eq. (2.3). The anharmonic ratio of the roots would then read cannot be directly obtained by applying the above procedure. Nevertheless, a simple generalization exists and a recursion relation can be implemented also in these cases 4 . In fact, by shifting the variable x if needed, it is always possible to put a cubic polynomial in a Weierstraß form: In this description the complex structure τ can be directly related to the coefficients G 2 and G 3 by forming the combination , (2.18) and identifying it with the "absolute modular invariant" by writing where E 4 and E 6 are the Eisenstein series of weight 4 and 6, respectively. By equating the right hand sides of Eq.s (2.18) and (2.19) we obtain the relation between τ and u, and then we can proceed as described above and establish a recursion relation by exploiting either Eq. (2.15) or Eq. (2.14) depending on whether or not the theory is conformal.
In the next section we will apply this method to the SU(2) theory with N f = 4 massive flavours whose SW curve is known in a non-factorized form [9], and explicitly derive a largea expansion of its effective coupling τ in which each coefficient is determined exactly as a function of q by means of a recursion relation. As explained in the Introduction, via Eq. (1.1) this is tantamount to finding the exact expression of the elements of the 8d chiral ring on the "flavour" D7 branes.

The SU(2) N f = 4 theory
The SU(2) N f = 4 theory for vanishing masses is conformal and the corresponding SW curve is just a torus of complex structure τ 0 , representing the UV coupling [9]. Such a torus can be given a simple description as the locus of a factorized cubic equation in Weierstraß form where the three roots e ℓ , satisfying e 1 + e 2 + e 3 = 0, are the following functions of τ 0 : with θ a being the Jacobi θ-functions 5 . The coefficients g 2 and g 3 can then be expressed in terms of the Eisenstein series E i as so that the absolute modular invariant J 0 ≡ J(τ 0 ), in accordance with Eq.s (2.18) and (2.19), becomes Notice that all the functions of τ 0 involved in the above definitions are expressible as power series in q defined in Eq. (2.16). When masses are turned on, the equation of the curve is modified as described in Ref. [9]. The result is still a cubic polynomial which can be written as with η being the Dedekind η-function and, for ℓ = 1, 2, 3, Here R, T ℓ and N are invariants of the flavour group SO(8) that are, respectively, quadratic, quartic and sextic in the masses m i : When the masses are set to zero, it is straightforward to see that Eq. (3.5) reduces to the equation 7 of a torus of complex parameter τ 0 . For non-zero masses, by a suitable shift 8 6 Note that A 2 is proportional to the discriminant of the cubic equation and can be written as in x, the curve can still be cast in the Weierstraß form (2.17) with coefficients G 2 and G 3 depending on u, on the flavour invariants, and on q. Their explicit expressions are It is easy to check that in the massless limit we have with g 2 and g 3 given in Eq. (3.3). The modular invariant J can then be explicitly determined in terms of u, of the flavour invariants and of q by Eq. (2.18), which we rewrite as (3.12) The complex structure τ , namely the exact IR complexified gauge coupling, is in turn implicitly determined by the modular invariant, to which it is related by Eq. (2.19), that we rewrite as . (3.13) Comparing these two expressions for J we can establish a relation between u and τ . However, in order to make contact with the standard field theory results, we have to write everything in terms of a. The ingredients that are needed for this purpose, namely the functions u(a) and τ (a), are provided by the prepotential F(a) through Eq.s (2.15) and (2.9), respectively. At this point, when both Eq.s (3.12) and (3.13) give J as a function of a, we can proceed in two distinct ways. On the one hand, we can expand Eq. (3.13) in powers of Q = e πiτ , (3.14) and Eq. (3.12) in powers of q, and then compare the two expansions, thus finding order by order a relation between Q and q in which the dependence on a (and the mass invariants) is exact [3]. On the other hand, we can expand both Eq.s (3.12) and (3.13) in (inverse) powers of a and, by comparing the two expansions, obtain a recursion relation for their coefficients which allows to determine exactly the full dependence on q.
We now show that using the first approach we can easily reconstruct the 1-loop corrections to the gauge coupling constant and the prepotential of the N f = 4 theory. Later, we will exploit the second approach and study the recursion relation.

Tree-level and 1-loop terms
Expanding Eq. (3.13) for small Q, we obtain likewise, using Eq. (3.10) and then expanding Eq. (3.12) for small q, we get Equating these two expressions and using the classical approximation (2.5) to replace u with 2a 2 , we deduce that where the second line follows from the definitions (3.9) of the mass invariants. The square root in Eq. (3.17) represents the complete 1-loop correction to the UV coupling as a function of the mass deformations and of the classical vacuum expectation value a. By taking the logarithm of both sides and using Eq. (1.2), after some simple algebra, we can rewrite Eq. (3.17) as follows which indeed is the correct expression for the gauge coupling constant of the massive SU(2) N f = 4 theory in the 1-loop approximation. We can also write this result in terms of the prepotential F(a) in accordance with Eq. (2.9). Introducing for later convenience the quantities for ℓ ≥ 0, the prepotential that follows from Eq. (3.18) is then up to possible a-independent terms. One can easily check that this agrees with the perturbative expression obtained with standard field theory methods (see, for example, Ref. [17]) up to constant terms which can always be absorbed into a redefinition of the (arbitrary) scale Λ.
To obtain the q-dependent terms in the prepotential, one can go to higher order in the q-expansion, as discussed in Ref. [3]. Alternatively, one can expand the modular invariant J given by Eq.s (3.12) and (3.13) in inverse powers of a and by comparing the two expansions establish a recursion relation for their coefficients which fixes the complete q-dependence. This is the approach we are going to discuss in the following.

Initial condition
In order to successfully implement a recursion relation, we need to know, as an initial condition, the exact expression in q of the first sub-leading term of u(a) for large a. In other words, writing we need to determine the function λ(q). To do this, we start by inverting the above relation, obtaining From this it readily follows that the first period ω 1 of the torus has the expansion On the other hand, given the Weierstraß form (2.17), the period ω 1 can be expressed as [18] where F (a, b, c; z) is the hypergeometric function and the normalization has been chosen so that the leading behaviour 1/(2 √ 2 u 1/2 ) is correctly reproduced. From this expression we can obtain the term δω 1 linear in R as a perturbation around the massless case; then, by writing it as according to Eq. (3.23), we can read off λ(q). Let us now give some details. From Eq. (3.10) we easily find where the dots stand for terms of higher order in the mass deformations which are not relevant for our present purposes. Moreover, from Eq. (3.12) we get On the other hand, by varying Eq. (3.24) we obtain Taking into account that F 1 12 , 5 12 , 1; 1 J 0 = E 1/4 4 , and using Eq.s (3.26) and (3.27), after some algebra involving the properties of the Eisenstein series and the θ-functions collected in Appendix A, we get Upon comparison with Eq. (3.25), we thus obtain Inserting this into Eq. (3.21) and using the resulting expression for u(a) in Eq. (2.15), we can easily derive the leading terms of the semiclassical expansion of the prepotential F(a), namely It is interesting to remark that this structure has also been obtained in Ref. [19] using the AGT conjecture [20] and the Zamolodchikov formula for the 4-point conformal blocks in the two-dimensional Liouville theory 9 .

Recursion relation
By comparing the two expressions (3.20) and (3.31), we are immediately led to write the following expansion for the complete prepotential (3.32) The coefficients h ℓ , to be determined, are the generalizations of those defined in Eq. (3.19) when non-perturbative instanton corrections are taken into account. Via Eq. (2.9), the expansion (3.32) corresponds to writing the IR coupling τ as The absence in the effective prepotential of any dependence on the scale Λ other than that arising at 1-loop implies that or, equivalently, that Tr m 2 = Tr m 2 . This relation, which is explicitly confirmed by instanton calculations [10], can be understood also as a simple consequence of the scaling dimensions of the instanton moduli space which do not allow to generate any nonperturbative contribution to Tr m 2 .
The next ingredient is obtained from Eq. (2.15) which, together with the expansion (3.32) and Eq. (3.35), implies that with λ(q) given in Eq. (3.30). Now, we are ready to establish the desired recursion relation. On the one hand, we expand 1/J around τ 0 , getting The difference (τ − τ 0 ) is parametrized as in Eq. (3.33), while the derivatives of 1/J can be straightforwardly computed from Eq. (3.13) in terms of the derivatives of the Eisenstein series given in Appendix A, taking into account that Eq. (2.16) implies that ∂ τ τ 0 = iπ q∂ q . For instance, the first two derivatives are On the other hand, we can compute the difference δ(1/J) via Eq. (3.12), by expanding the coefficients G 2 and G 3 given in Eq. (3.10) with respect to the values they assume in the massless case, writing It is not particularly useful here to spell η 2 and η 3 in detail, but we just remark that they depend on u and on the flavour invariants. With easy manipulations we then find We can now equate the two different expressions of δ 1 J , given in Eq.s (3.37) and (3.40), order by order in the large-a expansion that is obtained substituting Eq.s (3.33) and (3.36) in the first and in the second one, respectively.
At the first non trivial order, namely 1/a 2 , Eq. (3.37) gives, through Eq. (3.38), On the other hand, Eq. (3.40) reduces to With a bit of algebra, from Eq. (3.40) we get instead Comparing these two expressions, and taking into account Eq. (3.30), we find At the next-to-next order, we are able to determine the coefficient h 2 , which appears in the 1/a 6 term of Eq. (3.37), since the corresponding term in the expansion of Eq. (3.40) contains only quantities already determined, namely h 0 , λ and q∂ q h 1 . This pattern, which is easily implemented on a symbolic computation program like Mathematica, continues at all orders, and allows us to determine recursively the coefficients h ℓ . The explicit results up to h 4 are From the modular transformation properties of the Eisenstein series and the θ-functions, we see that the coefficients h ℓ are almost modular forms of degree 2ℓ; the failure to be exact modular forms is due to the appearance of the Eisenstein series E 2 whose modular transformations are anomalous. Note that the recursive method we have described fixes completely the coefficients h ℓ , differently from the recursion relation based on the modular anomaly equation [14] which only determines the E 2 dependence. In models with many different structures like the N f = 4 theory, this is a big computational advantage. In the next section we are going to analyze these results and comment on their properties.

Discussion of the results and comments
The explicit formulas for the coefficients h ℓ obtained in the previous section allow us to read, via Eq. (3.34), the exact expression for the elements of the SO(8) chiral ring of the eight-dimensional theory on the D7 branes. For example, from Eq. (3.45) we have Expanding the modular functions in powers of q, we can obtain the various instanton contributions. Explicitly, we have where in the final step we used the definitions (3.9) of the mass invariants. One can check that this result completely agrees with the one obtained via localization techniques in Ref.s [10,21] from direct multi-instanton calculations performed in the D7/D(-1) brane system of type I ′ . Furthermore, the non-perturbative part of the quartic correlator (4.1) matches precisely against the exact results for the BPS-saturated quartic coupling in the dual Heterotic string (see, for instance, the discussion in Ref. [21]). This analysis can be extended to the higher elements of the SO(8) chiral ring, and again we find perfect agreement with all results existing in the literature on this matter. The details for the chiral correlators up to Tr m 10 are given in Appendix B, together with their expansions up to the first few instantons.
We observe that by setting T 1 = T 2 = N = 0 and retaining only the dependence on the quadratic invariant R, our results reduce to those found in Ref. [14] with a different method; indeed with these positions the SU(2) N f = 4 theory reduces to the so-called N = 2 * model (also known as mass deformed N = 4 SU(2) theory) that was studied in that reference. In Appendix C we present an alternative derivation of the recursion relation for the N = 2 * model and also discuss the decoupling limits to the pure SU(2) theory.
Another interesting remark is that the coefficients h ℓ given in Eq.s (3.45) -(3.48) satisfy with the initial condition ∂h 0 /∂E 2 = 0. This recursion relation fixes the E 2 dependence of all coefficients h ℓ and could be used to reconstruct them in analogy to what has been done for the N = 2 * SU(2) model in Ref. [14]. Eq. (4.3) can be given a nice interpretation in terms of the modular anomaly equation [12,13]. Let us consider the combination (a D − τ 0 a), which, using Eq.s (2.8) and (3.32), can be written as and study its modular transformation properties. We recall that under an S modular transformation we have (4.5) moreover we assume that the coefficients h ℓ are almost modular forms of weight 2ℓ transforming under S as follows and that the "anomalous" term δh ℓ arises only through the fact that h ℓ depends on E 2 , i.e. it is of the form with α ℓ modular forms of weight 2ℓ. Taking into account the modular properties of E 2 (see Eq. (A.10)), this implies that Applying these rules to Eq. (4.4), on the one hand we have while on the other hand we have (4.10) Comparing the right hand sides of these equations, and using the form given in Eq.
h m h n a 2(m+n+1) (4.11) from which, after a suitable relabeling of the indices, the recursion relation (4.3) and its initial condition easily follow. It is interesting to notice that Eq. (4.11) is nothing but the mode expansion of the following partial differential equation which is a type of inviscid Burgers' equation.
We conclude by observing that the recursive methods we have described in this paper could be generalized in several ways. In particular it would be very intersting to apply them to models with gauge groups of higher rank corresponding to SW curves of higher genus and use them to find further connections with their F-theory interpretation. Another interesting possibility would be to apply these techniques to the gravitational corrections of the 4d Yang-Mills theories and establish a connection with the topological amplitudes at higher genus. Finally, it would be nice to study the relation between the elements of the SO(8) chiral ring we have found and the corresponding amplitudes in the dual Heterotic string, generalizing the connection already established for the first element Tr m 4 in Ref.s [21,10] using the D-instanton interpretation. In particular, it would be interesting to see how the modular anomaly we have found in our expressions for the chiral ring elements might be related to the holomorphic anomaly for the dual Heterotic amplitudes. We hope to return to some of these issues in the near future.
do not indicate this dependence explicitly, except when some confusion is possible, and we write θ a for θ a (0|τ ), E 2 for E 2 (τ ) and so on.
The Jacobi θ-functions are defined as for a, b = 0, 1. We simplify the notation by writing, as usual, The functions θ a , a = 2, 3, 4, satisfy the "aequatio identica satis abstrusa" The Dedekind η-function is defined by 3) The first Eisenstein series can be expressed as follows: σ 3 (n) q 2n = 1 + 240q 2 + 2160q 4 + 6720q 6 + . . . , where σ k (n) is the sum of the k-th power of the divisors of n, i.e., σ k (n) = d|n d k . The series E 4 and E 6 are expressible as polynomials in the θ-functions according to (A.5) The series E 2 , E 4 and E 6 are connected among themselves by logarithmic q-derivatives and form a sort of a "ring": Tr m 10 = 1 4032 the simple factorized form so that one can establish a recursion relation based on Eq. (2.4). Indeed the anharmonic ratio of the roots is found to be is the corresponding ratio of the roots in the massless case.
On the one hand, we can use Eq. (C.2) to express the difference (κ − κ 0 ) in terms of u, and then plug in the large-a expansion of the latter in the form of Eq. (3.36). On the other hand, we can Taylor expand κ, seen as the function of τ given by Eq. (2.4), around κ 0 : and then insert the large-a expansion of (τ − τ 0 ) given in Eq. (3.33). Comparing order by order the two different expansions of (κ − κ 0 ) we can recursively determine the unknown coefficients h ℓ of the expansion exactly in q and in R, finding in the end that These expressions correspond to setting T i = N = 0 in the N f = 4 results given in Eq.s (3.45) -(3.48), and coincide with what was found in Ref. [14].
Decoupling limits to the pure SU(2) theory: As already discussed in Ref. [9], it is possible to recover the pure SU(2) theory from the N = 2 * model by sending the mass invariant R to infinity and at the same time q to zero, so as to keep the combination Λ 2 = 2Rq (C. 6) finite. Indeed, expanding the θ-functions, in this limit from Eq. (C.2) we find which coincides with the result (2.11) derived from the SW curve (2.10) of the pure SU(2) theory. Notice that above we have taken into account the fact that u = 4û according to what we have explained after Eq. (2.12).
Let us now consider the complex structure τ given in Eq. (3.33); using the h ℓ 's of Eq. (C.5), it is not difficult to check that in this limit one gets τ →τ = i π log 4a 2 Λ 2 + 1 2πi 3 2Λ (C.10) In the decoupling limit one gets indeed