Deformed N=2 theories, generalized recursion relations and S-duality
Marco Billo, Marialuisa Frau, Laurent Gallot, Alberto Lerda, Igor Pesando
aa r X i v : . [ h e p - t h ] F e b Preprint typeset in JHEP style. - PAPER VERSION
LAPTH 005/13
Deformed N = 2 theories, generalized recursionrelations and S-duality M. Bill´o , M. Frau , L. Gallot , A. Lerda , I. Pesando Universit`a di Torino, Dipartimento di Fisica and I.N.F.N. - sezione di TorinoVia P. Giuria 1, I-10125 Torino, Italy LAPTH, Universit´e de Savoie, CNRS9, Chemin de Bellevue, 74941 Annecy le Vieux Cedex, France Universit`a del Piemonte Orientale, Dipartimento di Scienze e Innovazione Tecnologica,and I.N.F.N. - Gruppo Collegato di Alessandria - sezione di TorinoViale T. Michel 11, I-15121 Alessandria, Italy billo,frau,lerda,[email protected]; [email protected]
Abstract:
We study the non-perturbative properties of N = 2 super conformal fieldtheories in four dimensions using localization techniques. In particular we consider SU(2)gauge theories, deformed by a generic ǫ -background, with four fundamental flavors or withone adjoint hypermultiplet. In both cases we explicitly compute the first few instantoncorrections to the partition function and the prepotential using Nekrasov’s approach. Theseresults allow to reconstruct exact expressions involving quasi-modular functions of the baregauge coupling constant and to show that the prepotential terms satisfy a modular anomalyequation that takes the form of a recursion relation with an explicitly ǫ -dependent term.We then investigate the implications of this recursion relation on the modular properties ofthe effective theory and find that with a suitable redefinition of the prepotential and of theeffective coupling it is possible, at least up to the third order in the deformation parameters,to cast the S-duality relations in the same form as they appear in the Seiberg-Wittensolution of the undeformed theory. Keywords: N = 2 SYM theories, instantons, recursion relations, S-duality. ontents
1. Introduction 12. The N = 2 ∗ SU(2) theory 4
3. The N = 2 SU(2) theory with N f = 4
4. Modular anomaly equations and S-duality 17 ǫ ǫ
5. Conclusions 23A. Modular functions 25B. The coefficient h of the SU(2) N f = 4 prepotential 26C. Reformulating the modular anomaly equations 27D. S-duality at order ( ǫ ǫ )
1. Introduction
Four-dimensional field theories with rigid N = 2 supersymmetry provide a remarkablearena where many exact results can be obtained; indeed N = 2 supersymmetry, not beingmaximal, allows for a great deal of flexibility but, at the same time, is large enough to guar-antee full control. This fact was exploited in the seminal papers [1, 2] where it was shownthat the effective dynamics of N = 2 super Yang-Mills (SYM) theories in the limit of lowenergy and momenta can be exactly encoded in the so-called Seiberg-Witten (SW) curvedescribing the geometry of the moduli space of the SYM vacua. When the gauge groupis SU(2), the SW curve defines a torus whose complex structure parameter is identified1ith the (complexified) gauge coupling constant τ of the SYM theory at low energy. Thiscoupling receives perturbative corrections at 1-loop and non-perturbative corrections dueto instantons, and the corresponding effective action follows from a prepotential F that isa holomorphic function of the vacuum expectation value a of the adjoint vector multiplet,of the flavor masses, if any, and of the dynamically generated scale in asymptotically freetheories or of the bare gauge coupling constant τ in conformal models (see for instance [3]for a review and extensions of this approach).Recently, N = 2 superconformal field theories (SCFT) have attracted a lot of atten-tion. Two canonical examples of SCFT’s are the N = 2 SU(2) SYM theory with N f = 4fundamental hypermultiplets and the N = 2 ∗ theory, namely a N = 2 SYM theory withan adjoint hypermultiplet. In both cases, the β -function vanishes but, when the hypermul-tiplets are massive, the bare coupling τ gets renormalized at 1-loop by terms proportionalto the mass parameters. Besides these, there are also non-perturbative corrections due toinstantons. As shown in [4] for the N = 2 ∗ theory and more recently in [5] for the N f = 4theory, by organizing the effective prepotential F as a series in inverse powers of a and byexploiting a recursion relation hidden in the SW curve, it is possible to write the variousterms of F as exact functions of the bare coupling. These functions are polynomials inEisenstein series and Jacobi θ -functions of τ and their modular properties allow to showthat the effective theory at low energy inherits the Sl(2 , Z ) symmetry of the microscopictheory at high energy. In particular one can show [4, 5] that the S-duality map on thebare coupling, i.e. τ → − /τ , implies the corresponding map on the effective coupling, i.e. τ → − /τ , and that the prepotential F and its S-dual are related to each other by aLegendre transformation.The non-perturbative corrections predicted by the SW solution can also be obtaineddirectly via multi-instanton calculus and the use of localization techniques [6, 7] . This ap-proach is based on the calculation of the instanton partition function after introducing twodeformation parameters, ǫ and ǫ , of mass dimension 1 which break the four-dimensionalLorentz invariance, regularize the space-time volume and fully localize the integrals overthe instanton moduli space on sets of isolated points, thus allowing their explicit evaluation.This method, which has been extensively applied to many models (see for instance [10] -[17]) can be interpreted as the effect of putting the gauge theory in a curved background,known as Ω-background [6, 7, 18], or in a supergravity background with a non-trivialgraviphoton field strength, which are equivalent on the instanton moduli space [19, 20].The resulting instanton partition function Z inst ( ǫ , ǫ ), also known as Nekrasov partitionfunction, allows to obtain the non-perturbative part of the SYM prepotential according to F inst = − lim ǫ ,ǫ → ǫ ǫ log Z inst ( ǫ , ǫ ) . (1.1)Actually, the Nekrasov partition function is useful not only when the ǫ parameters aresent to zero, but also when they are kept at finite values. In this case, in fact, the non-perturbative ǫ -deformed prepotential F inst ( ǫ , ǫ ) = − ǫ ǫ log Z inst ( ǫ , ǫ ) (1.2) See also [8, 9] for earlier applications of these techniques. ǫ -deformed) perturbative part F pert , one gets a generalized prepotential that canbe conveniently expanded as follows F pert + F inst = ∞ X n,g =0 F ( n,g ) ( ǫ + ǫ ) n ( ǫ ǫ ) g . (1.3)The amplitude F (0 , , which is the only one that remains when the ǫ -deformations areswitched off, coincides with the SYM prepotential F of the SW theory, up to the classicaltree-level term. The amplitudes F (0 ,g ) with g ≥ F (0 ,g ) W g , where W is the chiral Weyl superfield containing the graviphoton field strength as its lowest com-ponent. These terms were obtained long ago from the genus g partition function of the N = 2 topological string on an appropriate Calabi-Yau background [21] and were shownto satisfy a holomorphic anomaly equation [22, 23] which allows to recursively reconstructthe higher genus contributions from the lower genus ones (see for instance [24, 25]). Morerecently, also the amplitudes F ( n,g ) with n = 0 have been related to the N = 2 topolog-ical string and have been shown to correspond to higher dimensional F-terms of the type F ( n,g ) Υ n W g where Υ is a chiral projection of real functions of N = 2 vector superfields[26], which also satisfy an extended holomorphic anomaly equation [27, 28]. By takingthe limit ǫ → ǫ finite, one selects in (1.3) the amplitudes F ( n, . This limit, alsoknown as Nekrasov-Shatashvili limit [29], is particularly interesting since it is believed thatthe N = 2 effective theory can be described in this case by certain quantum integrablesystems. Furthermore, in this Nekrasov-Shatashvili limit, using saddle point methods it ispossible to derive a generalized SW curve [30, 31] and extend the above-mentioned resultsfor the SYM theories to the ǫ -deformed ones.By considering the Nekrasov partition function and the corresponding generalized pre-potential for rank one SCFT’s, in [32] a very remarkable relation has been uncovered withthe correlation functions of a two-dimensional Liouville theory with an ǫ -dependent centralcharge. In particular, for the SU(2) theory with N f = 4 the generalized prepotential turnsout to be related to the logarithm of the conformal blocks of four Liouville operators ona sphere and the bare gauge coupling constant to the cross-ratio of the punctures wherethe four operators are located; for the N = 2 ∗ SU(2) theory, instead, the correspondenceworks with the one-point conformal blocks on a torus whose complex structure parameterplays the role of the complexified bare gauge coupling. Since the conformal blocks of theLiouville theory have well-defined properties under modular transformations, it is naturalto explore modularity also on the four-dimensional gauge theory and try to connect it tothe strong/weak-coupling S-duality, generalizing in this way the SW results when ǫ and ǫ are non-zero. On the other hand, one expects that the deformed gauge theory shouldsomehow inherit the duality properties of the Type IIB string theory in which it can beembedded. Some important steps towards this goal have been made in [33] and also in[34] where it has been shown that the SW contour integral techniques remain valid alsowhen both ǫ and ǫ are non-vanishing. To make further progress and gain a more quanti-tative understanding, it would be useful to know the various amplitudes F ( n,g ) in (1.3) as3 xact functions of the gauge coupling constant and analyze their behavior under modulartransformations, similarly to what has been done for the SW prepotential F (0 , in [4, 5].Recently, by exploiting the generalized holomorphic anomaly equation, an exact expressionin terms of Eisenstein series has been given for the first few amplitudes F ( n,g ) of the SU(2)theory with N f = 4 and the N = 2 ∗ SU(2) theory in the limit of vanishing hypermul-tiplet masses [35]. This analysis has then been extended in [36] to the massive N = 2 ∗ model in the Nekrasov-Shatashvili limit, using again the extended holomorphic anomalyequation, and in [37] using the properties of the Liouville toroidal conformal blocks in thesemi-classical limit of infinite central charge. However, finding the modular properties ofthe deformed prepotential in full generality still remains an open issue.In this paper we address this problem and extend the previous results by adopting adifferent strategy. In Section 2, using localization techniques we explicitly compute thefirst few instanton corrections to the prepotential for the N = 2 ∗ massive theory withgauge group SU(2) in a generic ǫ -background. From these explicit results we then inferthe exact expressions of the various prepotential coefficients and write them in terms ofEisenstein series of the bare coupling. Our results reduce to those of [4, 5] when thedeformation parameters are switched off, and to those of [35] - [37] in the massless or inthe Nekrasov-Shatashvili limits. The properties of the Eisenstein series allow to analyzethe behavior of the various prepotential terms under modular transformations and alsoto write a recursion relation that is equivalent to the holomorphic anomaly equation ifone trades modularity for holomorphicity. The recursion relation we find contains a termproportional to ǫ ǫ , which is invisible in the SYM limit or in the Nekrasov-Shatashvililimit. In Section 3 we repeat the same steps for the N = 2 SU(2) theory with N f = 4and arbitrary mass parameters and also in this case derive the modular anomaly equationin the form of a recursion relation. In Section 4 we study in detail the properties of thegeneralized prepotential under S-duality, and show that when both ǫ and ǫ are differentfrom zero, due to the new term in the recursion relation, the prepotential and its S-dual arenot any more related by a Legendre transformation, an observation which has been recentlyput forward in [38] from a different perspective. We also propose how the relation betweenthe prepotential and its S-dual has to be modified, by computing the first corrections in ǫ ǫ . Finally, in Section 5 we conclude by showing that there exist suitable redefinitionsof the prepotential and of the effective coupling that allow to recover the simple Legendrerelation and write the S-duality relations in the same form as in the undeformed SW theory.The appendices contain some technical details and present several explicit formulas whichare useful for the computations described in the main text.
2. The N = 2 ∗ SU(2) theory
The N = 2 ∗ SYM theory describes the interactions of a N = 2 gauge vector multipletwith a massive N = 2 hypermultiplet in the adjoint representation of the gauge group.It can be regarded as a massive deformation of the N = 4 SYM theory in which the β -function remains vanishing but the gauge coupling constant receives both perturbativeand non-perturbative corrections proportional to the hypermultiplet mass. Using the lo-4alization techniques [6, 7] one can obtain a generalization of this theory by considering the ǫ -dependent terms in the Nekrasov partition function. In the following we only discuss thecase in which the gauge group is SU(2) (broken down to U(1) by the vacuum expectationvalue a of the adjoint scalar of the gauge vector multiplet). We begin by considering thenon-perturbative corrections. The partition function Z k at instanton number k is defined by the following integral overthe instanton moduli space M k : Z k = Z d M k e − S inst (2.1)where S inst is the instanton moduli action of the N = 2 ∗ theory. After introducing twodeformation parameters ǫ and ǫ , the partition function Z k can be explicitly computedusing the localization techniques. In the case at hand, each Z k can be expressed as a sumof terms in one-to-one correspondence with an ordered pair of Young tableaux of U( k )such that the total number of boxes in the two tableaux is k . For example, at k = 1we have the two possibilities: ( , • ) and ( • , ); at k = 2 we have instead the five cases:( , • ), ( • , ), ( , • ), ( • , ), ( , ); and so on and so forth. Referring for example tothe Appendix A of [39] for details, at k = 1 one finds Z ( , • ) = ( − ǫ + e m )( − ǫ + e m )( a + e m )( a + e m − ǫ − ǫ )( − ǫ )( − ǫ ) a ( a − ǫ − ǫ ) ,Z ( • , ) = ( − ǫ + e m )( − ǫ + e m )( a + e m )( a + e m − ǫ − ǫ )( − ǫ )( − ǫ ) a ( a − ǫ − ǫ ) , (2.2)where a uv = a u − a v with a = − a = a , and e m = m + ǫ + ǫ ǫ -background [40,35]. The 1-instanton partition function is therefore Z = Z ( , • ) + Z ( • , ) = (4 m − ( ǫ − ǫ ) )(16 a − m − ǫ + ǫ ) )8 ǫ ǫ (4 a − ( ǫ + ǫ ) ) . (2.4)At k = 2 the relevant partition functions are Z ( , • ) = ( − ǫ + e m )( − ǫ + e m )( − ǫ + ǫ + e m )( − ǫ + e m )( a + e m )( − ǫ )( − ǫ )( − ǫ + ǫ )( − ǫ ) a × ( a + e m + ǫ )( a + e m − ǫ − ǫ )( a + e m − ǫ − ǫ )( a + ǫ )( a − ǫ − ǫ )( a − ǫ − ǫ ) ,Z ( , ) = ( − ǫ + e m ) ( − ǫ + e m ) ( a + e m − ǫ )( a + e m − ǫ )( − ǫ ) ( − ǫ ) ( a − ǫ )( a − ǫ ) × ( a + e m − ǫ )( a + e m − ǫ )( a − ǫ )( a − ǫ ) . (2.5)5he contributions corresponding to the other Young tableaux at k = 2 can be obtainedfrom the previous expressions with suitable redefinitions. In particular, Z ( • , ) is obtainedfrom Z ( , • ) in (2.5) by exchanging a ↔ a , Z ( , • ) is obtained by exchanging ǫ ↔ ǫ ,and finally Z ( • , ) is obtained by simultaneously exchanging a ↔ a and ǫ ↔ ǫ . Thecomplete 2-instanton partition function is Z = Z ( , • ) + Z ( • , ) + Z ( , • ) + Z ( • , ) + Z ( , ) (2.6)but we refrain from writing its expression since it is not particularly inspiring. This pro-cedure can be systematically extended to higher instanton numbers leading to explicitformulas for the instanton partition functions.Following [7] we can cast these results in a nice and compact form. Indeed, defining q = e π i τ (2.7)where τ = θ π + i πg is the complexified gauge coupling constant of the N = 2 ∗ SYMtheory, the grand-canonical instanton partition function Z inst = ∞ X k =0 q k Z k (2.8)where Z = 1, can be rewritten as Z inst = X ( Y ,Y ) q | Y | ∞ Y i,j =1 2 Y u,v =1 " a uv + ǫ ( i − − ǫ ja uv + ǫ ( i − − e k vj ) − ǫ ( j − k ui ) × a uv + e m + ǫ ( i − − e k vj ) − ǫ ( j − k ui ) a uv + e m + ǫ ( i − − ǫ j (2.9)where the first line represents the contribution of the gauge vector multiplet and the secondline that of the adjoint hypermultiplet. Here k ui and e k ui denote, respectively, the numberof boxes in the i -th row and in the i -th column of a Young tableaux Y u and are related tothe Dynkin indices of the corresponding representation. These quantities can be extendedfor any integer i with the convention that k ui = 0 or e k ui = 0 if the i -th row or the i -thcolumn of Y u is empty. For example, for Y u = the k ui ’s are (2 , , , , . . . ) and the e k ui ’sare (1 , , , , . . . ). Moreover we have | Y | = X u,i k ui = X u,i e k ui = k . (2.10)It is quite straightforward to check that the expressions (2.2) and (2.5) are reproduced bythe compact formula (2.9) by selecting the appropriate Young tableaux.Following Nekrasov’s prescription, we can obtain the generalized non-perturbative pre-potential according to F inst = − ǫ ǫ log Z inst = ∞ X k =1 q k F k . (2.11)6n the limit ǫ ℓ →
0, the above expression computes the instanton contributions to theprepotential of the N = 2 ∗ SYM theory, while the finite ǫ -dependent terms represent furthernon-perturbative corrections. Notice that in the limit e m →
0, (2.9) correctly reduces tothe partition function of the N = 4 SYM theory, namely to the Euler characteristic of theinstanton moduli space (see for instance [41]). The compact expression (2.9) allows to “guess” the perturbative part of the partitionfunction by applying the same formal reasoning of Section 3.10 of [6]. Indeed, in (2.9) werecognize the following universal ( i.e. k -independent but a -dependent) factor ∞ Y i,j =1 2 Y u,v =1 u = v a uv + ǫ ( i − − ǫ ja uv + e m + ǫ ( i − − ǫ j (2.12)which, if suitably interpreted and regularized [6, 7], can be related to the perturbative partof the partition function of the N = 2 ∗ theory in the ǫ -background. According to this idea,we are then led to write F pert = ǫ ǫ X u,v =1 u = v ∞ X i,j =1 log a uv + ǫ ( i − − ǫ ja uv + e m + ǫ ( i − − ǫ j . (2.13)Using the following representation for the logarithmlog x Λ = − dds (cid:16) Λ s Γ( s ) Z ∞ dtt t s e − tx (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) s =0 (2.14)where Λ is an arbitrary scale and summing over i and j , we can rewrite (2.13) as F pert = ǫ ǫ X u,v =1 u = v h γ ǫ ,ǫ ( a uv ) − γ ǫ ,ǫ ( a uv + e m ) i (2.15)where (see also [18, 35]) γ ǫ ,ǫ ( x ) = dds (cid:16) Λ s Γ( s ) Z ∞ dtt t s e − tx (e − ǫ t − − ǫ t − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) s =0 . (2.16)This function, which is related to the logarithm of the Barnes double Γ-function, can beeasily computed by expanding for small values of ǫ and ǫ . As a result, F pert becomes aseries in inverse powers of a whose first few terms are F pert = 14 (cid:0) m − s (cid:1) log 2 a Λ − (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1) a − (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1)(cid:0) m + 6 p − s (cid:1) a − (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1)(cid:0) m + 176 m p + 160 p − m s − ps + 51 s (cid:1) a + O ( a − ) . (2.17)7ere we have used (2.3) and introduced the convenient notation s = ǫ + ǫ , p = ǫ ǫ . (2.18)One can easily check that in the limit ǫ ℓ → N = 2 ∗ SU(2) gauge theory (see for instance [4]).
The complete generalized prepotential is the sum of the classical, perturbative and non-perturbative parts. We now focus on the latter two terms which are directly related tothe Nekrasov partition function. Just like the 1-loop piece (2.17), also the instanton terms(2.11) can be organized as a series expansion for small values of ǫ and ǫ , or equivalentlyfor large values of a . Discarding a -independent terms, which are not relevant for the gaugetheory dynamics, we write F pert + F inst = h log 2 a Λ − ∞ X ℓ =1 h ℓ ℓ +1 ℓ a ℓ (2.19)where the coefficients h ℓ are polynomials in m , s and p which can be explicitly derivedfrom (2.17) as far as the perturbative part is concerned, and from the instanton partitionfunctions (2.9), after using (2.11), for the non-perturbative part. Here we list the first fewof these coefficients up to three instantons: h = 14 (cid:0) m − s (cid:1) , (2.20) h = (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1)(cid:16) − q − q − q + · · · (cid:17) , (2.21) h = (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1)(cid:16) m + 6 p − s − s q + 12 m + 18 p − s q + 8 m + 12 p − s q + · · · (cid:17) , (2.22) h = (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1)(cid:16) m + 176 m p + 160 p − m s − ps + 51 s − s q − m + 1392 m p + 1440 p − m s − ps + 3903 s q − m + 1008 m p + 960 p − m s − ps + 987 s q + · · · (cid:17) . (2.23)It is interesting to notice that all h ℓ ’s are proportional to (cid:0) m − s (cid:1) and, except for h ,also to (cid:0) m + 4 p − s (cid:1) . Explicit expressions for h ℓ with ℓ > ǫ ℓ → q . Moreprecisely, we expect that h ℓ are (quasi) modular functions of weight 2 ℓ that can be written8olely in terms of the Eisenstein series E , E and E (see Appendix A for our conventionsand definitions). This is indeed what happens. In fact we have h = 14 (cid:0) m − s (cid:1) , (2.24) h = 12 · (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1) E , (2.25) h = 12 · · (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1) × h (cid:0) m + 6 p − s (cid:1) E + (cid:0) m + 6 p − s (cid:1) E i , (2.26) h = 12 · · · (cid:0) m − s (cid:1)(cid:0) m + 4 p − s (cid:1) × h (cid:0) m + 272 m p + 240 p − m s − ps + 5 s ) E + 84 (cid:0) m + 64 m p + 60 p − m s − ps + 13 s (cid:1) E E + (cid:0) m + 944 m p + 960 p − m s − ps + 3323 s (cid:1) E i . (2.27)By using the small q expansion of the Eisenstein series one can check that the explicitinstanton contributions we have computed using localization techniques are correctly re-covered from the previous formulas. We stress that the fact that the various instantonterms nicely combine into expressions involving only the Eisenstein series is not obviousa priori and is a very strong a posteriori test on the numerical coefficients appearing in(2.20)-(2.23). It is quite remarkable that the explicit instanton results at low k can benicely extrapolated and allow to reconstruct modular forms from which, by expanding inpowers of q , one can obtain the contributions at any instanton number.We can also organize the generalized prepotential according to (1.3) and obtain theamplitudes F ( n,g ) as a series in inverse powers of a with coefficients that are polynomialsin E , E and E . The first few of such amplitudes are: F (0 , = m log 2 a Λ − m E a − m (5 E + E )5760 a − m (175 E + 84 E E + 11 E )2903040 a + · · · , (2.28) F (1 , = −
14 log 2 a Λ + m E a + m ( E + E )1536 a + m (175 E + 336 E E + 8 E )2903040 a + · · · (2.29) F (0 , = − m E a − m (5 E + E )2304 a − m (11 E + 6 E E + E )41472 a + · · · (2.30) F (2 , = − E a − m (5 E + 9 E )30720 a + m (175 E + 588 E E + 559 E )7741440 a + · · · (2.31) F (1 , = E a + m (25 E + 17 E )23040 a + m (385 E + 798 E E + 213 E )1935360 a + · · · (2.32) F (0 , = − m (5 E + E )3840 a − m (160 E + 93 E E + 17 E )414720 a + · · · (2.33)9erms with higher values of n and g can be systematically generated without any difficultyfrom the h ℓ ’s given in (2.24)-(2.27). The first term F (0 , represents the prepotential of the N = 2 ∗ SU(2) gauge theory and its expression (2.28) agrees with that found in [4] fromthe SW curve (see also [5]). The other terms are generalizations of those considered in [35]and more recently in [37] in particular limits ( m → ǫ →
0) where they drasticallysimplify.
The generalized prepotential (2.19) is clearly holomorphic by construction but does not havenice transformation properties under the modular group since the coefficients h ℓ explicitlydepend on the second Eisenstein series E which is not a good modular function. Indeed,under τ → aτ + bcτ + d with ad − bc = 1 , (2.34) E transforms inhomogeneously as follows E ( τ ) → ( cτ + d ) E ( τ ) + ( cτ + d ) 6 cπ i . (2.35)Therefore, in order to have good modular properties we should replace everywhere E with the shifted Eisenstein series b E = E + π i( τ − ¯ τ ) at the price, however, of loosingholomorphicity. This fact leads to the so-called holomorphic anomaly equation [22, 23](see also [28]). On the other hand, in the limit ¯ τ → ∞ , holding τ fixed so that b E → E ,we recover holomorphicity but loose good modular properties and obtain the so-calledmodular anomaly equation [42]. This equation can be rephrased in terms of a recursionrelation satisfied by the h ℓ coefficients which allows to completely fix their dependence on E . To this aim let us consider the explicit expressions (2.24)-(2.27) and compute thederivatives of h ℓ with respect to E . With simple algebra we find ∂h ℓ ∂E = ℓ ℓ − X i =0 h i h ℓ − i − + ℓ (2 ℓ − ǫ ǫ h ℓ − (2.36)with the initial condition ∂h ∂E = 0 . (2.37)We have explicitly checked this relation for several values of ℓ >
3; we can thus regardit as a distinctive property of the ǫ -deformed N = 2 ∗ low-energy theory. Notice that inthe limit ǫ ℓ →
0, (2.36) reduces to the recursion relation satisfied by the coefficients of theprepotential of the N = 2 ∗ SU(2) theory found in [4, 5] from the SW curve, and that thelinear term in the right hand side disappears in the so-called Nekrasov-Shatashvili limit[29] where one of the two deformation parameters vanishes and a generalized SW curvecan be introduced [30, 31].Eq. (2.36) can be formulated also as a recursion relation for the amplitudes F ( n,g ) defined in (1.3). To see this, let us first extract the a -dependence and, in analogy with102.19), write F ( n,g ) = − ∞ X ℓ =1 f ( n,g ) ℓ ℓ +1 ℓ a ℓ , (2.38)so that h ℓ = X n,g f ( n,g ) ℓ ( ǫ + ǫ ) n ( ǫ ǫ ) g . (2.39)Notice that the coefficients f ( n,g ) ℓ , which are polynomials in the hypermultiplet mass m ,have mass dimensions 2(1 + ℓ − n − g ); therefore f ( n,g ) ℓ = 0 if n + g > ℓ + 1 (this conditioncan be easily checked on the explicit expressions (2.28)-(2.31). These definitions must besupplemented by the “initial conditions” f (0 , = m , f (1 , = − , f (0 , = 0 (2.40)which are obtained from (2.24). Inserting (2.39) in the recursion relation (2.36), we obtain ∂f ( n,g ) ℓ ∂E = ℓ X n ,n g ,g ′ (cid:16) ℓ − X i =0 f ( n ,g ) i f ( n ,g ) ℓ − i − (cid:17) + ℓ (2 ℓ − f ( n,g − ℓ − (2.41)where the ′ means that the sum is performed over all n , n , g and g such that n + n = n and g + g = g . Eq. (2.41) shows that the coefficients f ( n,g ) ℓ and hence the amplitudes F ( n,g ) are recursively related to those with lower values of n and g , similarly to what canbe deduced from the holomorphic anomaly equation [35].
3. The N = 2 SU(2) theory with N f = 4 We now consider the N = 2 SU(2) theory with N f flavor hypermultiplets in the funda-mental representation of the gauge group. When N f = 4 the 1-loop β -function vanishesand the conformal invariance is broken only by the flavor masses m f ( f = 1 , ..., ǫ ℓ deformation parameters in the Nekrasov partition function, following the same pathdescribed in the previous section for the N = 2 ∗ theory. Using localization techniques, we can express the instanton partition functions Z k of the N f = 4 theory as sums of terms related to pairs of Young tableaux. Referring again theAppendix A of [39] for details, at k = 1 we have the following two contributions Z ( , • ) = 1( − ǫ )( − ǫ ) a ( a − ǫ − ǫ ) Y f =1 ( a + e m f ) ,Z ( • , ) = 1( − ǫ )( − ǫ ) a ( a − ǫ − ǫ ) Y f =1 ( a + e m f ) (3.1)11here e m f = m f + ǫ + ǫ ǫ -background . Summing the two terms (3.1)and setting a = − a = a , we get Z = Z ( , • ) + Z ( • , ) = − ǫ ǫ " a + X f 12 Pf m ,N = 316 X f 64 + · · · , F = 23 a 192 + · · · , F = 2701 a · · · (3.12)Combining these contributions with the perturbative one given by the first term in (3.8)and adding also the classical tree-level prepotential F cl = log x a , we obtain (cid:16) log x − log 16 + 12 x + 1364 x + 23192 x + 270132768 x + · · · (cid:17) a ≡ log q a (3.13)that is a redefinition of the instanton expansion parameter and hence of the gauge couplingconstant. By inverting this relation we get [43] x = 16 q (cid:0) − q + 44 q − q + · · · ) = θ ( q ) θ ( q ) (3.14)(see Appendix A for some properties of the Jacobi θ -functions). As pointed out for instancein [44], by setting q = e i πτ with τ = θπ + i 8 πg , (3.15)one can show that τ is precisely the modular parameter of the SW curve for the SU(2) N f =4 theory appearing in the original paper [2]. This gauge coupling constant receives onlycorrections proportional to the hypermultiplet masses and as such is the strict analogue ofthe gauge coupling constant τ of the N = 2 ∗ theory considered in Section 2. On the otherhand, using (3.14) one can show that x is a cross-ratio of the four roots of the original SWcurve [2]. From now on we always present the results for the N f = 4 theory in terms of q . Except for a customary overall factor of 2. F pert + F inst as in (2.19), the first coefficients h ℓ up to three instantons turn out to be h = 12 (cid:0) R − s + p (cid:1) , (3.16) h = 124 h R − T ) − R ( s − p ) + s − s p + 3 p i + 32( T + 2 T ) q − h R + 6 T ) − R ( s − p ) + s − s p + 3 p i q + 128 (cid:0) T + 2 T (cid:1) q + O ( q ) , (3.17) h = 1240 h R − RT + 768 N − R − T )( s − p )+ R (56 s − s p + 192 p ) − s + 36 s p − s p + 30 p i + 32 s ( T + 2 T ) q + h N + 384 RT − R + 6 T )( s − p ) + 8 R (4 s − s p + 5 p ) − s + 20 s p − s p + 6 p i q − T + 2 T ) (cid:0) R − s + 16 p (cid:1) q + O ( q ) . (3.18)We worked out also the expressions for a few other h ℓ ’s with ℓ > h ).As in the N = 2 ∗ model, also in the N f = 4 theory we can view the previous results asthe first instanton terms of the expansion of (quasi) modular forms in powers of q . In thiscase, however, besides the Eisenstein series E , E and E , also the Jacobi θ -functions θ and θ are needed. Matching the q -expansion of these modular functions with the explicitresults (3.16)-(3.18), guided by what we already obtained in the ǫ ℓ → E dependence, we find h = 12 (cid:0) R − s + p (cid:1) , (3.19) h = 124 (cid:0) R − s + p (cid:1)(cid:0) R − s + 3 p (cid:1) E − (cid:0) T θ − T θ (cid:1) , (3.20) h = 1144 (cid:0) R − s + p (cid:1)(cid:0) R − s + 3 p (cid:1)(cid:0) R − s + 4 p (cid:1) E − (cid:0) R − s + 4 p (cid:1)(cid:0) T θ − T θ (cid:1) E + 1720 h R − R (3 s − p ) + 4 R (27 s − s p + 49 p ) − s + 68 s p − s p + 30 p + 2304 N i E − (cid:0) R − s + p (cid:1)h T θ (cid:0) θ + θ (cid:1) + T θ (cid:0) θ + 2 θ (cid:1)i . (3.21) As compared to the results presented in [5, 44], here all masses have been rescaled by a factor of √ m there f = √ m here f for all f . 15e have checked that a similar pattern occurs also in other h ℓ ’s with ℓ > s and p as in (1.3),we can obtain the amplitudes F ( n,g ) , the first few of which are F (0 , = 2 R log a Λ − R E a + T θ − T θ a − R (5 E + E )180 a − N E a + RT θ (2 E + 2 θ + θ )6 a − RT θ (2 E + 2 θ + θ )6 a + · · · , (3.22) F (1 , = − 12 log a Λ + RE a + R ( E + E )48 a − T θ ( E + 4 θ + 2 θ )12 a + T θ ( E − θ − θ )12 a + · · · , (3.23) F (0 , = 12 log a Λ − RE a − R (2 E + E )36 a + T θ (2 E + 2 θ + θ )6 a − T θ (2 E − θ − θ )6 a + · · · , (3.24) F (2 , = − E a − R (5 E + 9 E )960 a + · · · , (3.25) F (1 , = E a + R (10 E + 11 E )360 a + · · · , (3.26) F (0 , = − E a − R (95 E + 49 E )2880 a + · · · , (3.27) F (3 , = 5 E + 13 E a · · · , F (2 , = − E + 17 E a · · · , (3.28) F (1 , = 95 E + 94 E a · · · , F (0 , = − E + E a · · · . (3.29)One can easily check that F (0 , in (3.22) completely agrees with the prepotential of the N = 2 SU(2) N f = 4 gauge theory as derived for example in [5]; notice also that in themassless limit, i.e. R, N, T , T → 0, these amplitude drastically simplify and preciselymatch the results presented in [35]. It is also interesting to observe that in the Nekrasov-Shatshvili limit where one of the ǫ ℓ ’s vanishes, the amplitudes F ( n, of the N f = 4 theoryreduce to those of the N = 2 ∗ theory upon setting T = T = N = 0 and R = m . Indeed,with these positions and rescaling a → a , the amplitudes F ( n, in (3.22)-(3.29) becometwice the corresponding amplitudes of the N = 2 ∗ theory given in (2.28)-(2.33). Thissimple relation does not hold away from the Nekrasov-Shatashvili limit: the amplitudes F ( n,g ) with g = 0 are in fact intrinsically different in the two theories, as a consequence oftheir different gravitational structure. Looking at the explicit expressions (3.19) - (3.21), it is not difficult to realize that the E dependence of the h ℓ ’s is quite simple and exhibits a recursive pattern. This points again tothe existence of a recursion relation among the h ℓ ’s involving their derivatives with respect16o E . Indeed, following the same steps described in Section 2.4, one can check that ∂h ℓ ∂E = ℓ ℓ − X i =0 h i h ℓ − i − + ℓ (2 ℓ − ǫ ǫ h ℓ − (3.30)with the initial condition ∂h ∂E = 0 . (3.31)This recursion relation has exactly the same structure of that of the N = 2 ∗ theory givenin (2.36), the only difference being in the numerical coefficients. Notice that the coefficientsof the quadratic terms can be matched by a rescaling of the h ℓ ’s, but those of the linearterms proportional to ǫ ǫ remain different for the two theories; this is another signal oftheir intrinsically different behavior when a generic ǫ -background is considered.Finally, if we expand the h ℓ ’s as in (2.39) we can reformulate the recursion relation(3.30) in terms of the partial amplitudes f ( n,g ) ℓ and get ∂f ( n,g ) ℓ ∂E = ℓ X n ,n ; g ,g ′ (cid:16) ℓ − X i =0 f ( n ,g ) i f ( n ,g ) ℓ − i − (cid:17) + ℓ (2 ℓ − f ( n,g − ℓ − (3.32)where the ′ means that the sum is performed over all n , n , g and g such that n + n = n and g + g = g . This relation has to be supplemented by the initial conditions f (0 , = 2 R , f (1 , = − , f (0 , = 12 (3.33)obtained from (3.19).In the next section we will analyze the implications of the recursion relations and inparticular their consequences on the modular transformation properties of the generalizedprepotential. 4. Modular anomaly equations and S-duality The fact that the generalized prepotential can be written in terms of (quasi) modularfunctions of the bare coupling constant allows to explore the modularity properties ofthe deformed theory and study how the Sl(2 , Z ) symmetry of the microscopic high-energytheory is realized in the effective low-energy theory. In the following we will concentrate onthe SU(2) N f = 4 theory, even if our analysis and our methods can be equally well appliedto the N = 2 ∗ SU(2) model. Furthermore, we will focus on the generator S of the modulargroup, corresponding to the following transformation of the bare coupling constant S : τ → − τ . (4.1)Notice that this action implies that the instanton counting parameter x in the Nekrasov’spartition of the N f = 4 theory, defined in (3.14), transforms as follows S : x → − x , (4.2)17s one can readily check from the properties of the Jacobi θ -functions. This type of trans-formation is consistent with the interpretation of x as a cross-ratio, a fact that is alsoexploited in the AGT correspondence with the Liouville conformal blocks [32]. As dis-cussed in [2], in the N f = 4 theory the modular group acts with triality transformationson the mass invariants (3.4); in particular one has S : R → R , T → T , T → T , N → N . (4.3)In the deformed theory, these rules have to be supplemented by those that specify howSl(2 , Z ) acts on the equivariant deformation parameters. Adapting to the present case theconsiderations made in [45] for the ǫ -deformed conformal Chern-Simons theory in threedimensions, we assume that S simply exchanges ǫ and ǫ with each other . In particularthis means that s = ǫ + ǫ and p = ǫ ǫ are left unchanged, i.e. S : s → s , p → p . (4.4)Using the rules (4.1) - (4.4) and the modular properties of the Eisenstein series and Jacobi θ -functions, it is easy to show that the coefficients h ℓ of the generalized prepotential (see(3.21) and Appendix B) transform as quasi-modular forms of weight 2 ℓ with anomalousterms due to the presence of the second Eisenstein series E , namely S : h ℓ ( E ) → τ ℓ h ℓ (cid:16) E + 6 π i τ (cid:17) = τ ℓ ℓ X k =0 k ! (cid:18) π i τ (cid:19) k D k h ℓ ( E ) (4.5)where we have introduced the convenient notation D ≡ ∂ E . These transformation rulesare formally identical to those of the underformed N f = 4 theory derived in [5] from theSW curve; however, since the h ℓ ’s satisfy a modified recursion relation with a new termproportional to ǫ ǫ , the practical effects of (4.5) in the deformed theory are different fromthose of the undeformed case, as we shall see momentarily.To proceed let us consider the pair made by a and its S-dual image S ( a ) ≡ a D , onwhich S acts as follows [1, 2]: S : aa D ! → − ! aa D ! = a D − a ! . (4.6)We therefore have S ( a ) = − a . (4.7)In the SW theory this relation is enforced by taking S ( a ) = 12 π i ∂ F ∂a (4.8) Note that this rule is consistent with the interpretation of ǫ and ǫ as fluxes of (complex) combinationsof NS-NS and R-R 3-form field strengths in a Type IIB string theory realization which rotate amongthemselves under S-duality. F is the undeformed effective prepotential which is related to its S-dual by a Legendretransform: F − S ( F ) = 2 π i a S ( a ) . (4.9)It seems natural to try the same thing also in the deformed theory. Recalling that theeffective generalized prepotential is F = π i τ a + h log a Λ − ∞ X ℓ =1 h ℓ ℓ +1 ℓ a ℓ (4.10)where the first term is the classical part, we therefore posit S ( a ) = 12 π i ∂F∂a = τ a + 12 π i ∞ X ℓ =0 h ℓ ℓ a ℓ +1 . (4.11)Applying the S-duality rules (4.1) and (4.5), we obtain S ( a ) = − S ( a ) τ + 12 π i ∞ X ℓ =0 τ l (cid:16) h ℓ + π i τ Dh ℓ + O ( τ − ) (cid:17) ℓ ( S ( a )) ℓ +1 (4.12)= − a + 1(2 π i τ ) " ∞ X ℓ =0 Dh ℓ ℓ a ℓ +1 − ∞ X ℓ,n =0 (2 ℓ + 1) h ℓ h n ℓ + n a ℓ +2 n +3 + O ( τ − ) . The expression in square brackets can be simplified using the recursion relation (3.30); likein the undeformed theory, the quadratic terms in the h ’s exactly cancel but, due to the new ǫ -dependent term in the deformed recursion relation, an uncanceled part proportional to ǫ ǫ remains. This simple calculation shows that in order to enforce the relation (4.7) when ǫ ǫ = 0, the standard definition (4.11) has to be modified by adding terms proportionalto ǫ ǫ in the right hand side. In the sequel we will work out explicitly the first correctionsand show how the relation (4.7) constrains the form of S ( a ). ǫ ǫ To organize the calculation, we introduce a set of generating functions ϕ ℓ for the coefficients h ℓ according to ϕ = − h log a Λ + ∞ X ℓ =1 h ℓ ℓ +1 ℓ a ℓ , ϕ ℓ +1 = − ∂ a ϕ ℓ for ℓ ≥ . (4.13)In particular we have the following relations with the generalized prepotential F : ϕ = π i τ a − F , (4.14a) ϕ = − π i τ a + ∂ a F , (4.14b) ϕ = 2 π i τ − ∂ a F ≡ π i τ − π i τ , (4.14c)where in the last line we have introduced the effective coupling τ .19s shown in Appendix C, the ϕ ℓ ’s form a ring under the action of D due to the gen-eralized modular anomaly equations (3.30). This ring structure will enable us to formallyexpress everything as functions of the ϕ ℓ ’s with coefficients that may depend on the product ǫ ǫ . Such a dependence is a consequence of the ǫ ǫ -term in the recursion relation (3.30)and is the only explicit dependence on the deformation parameters that will be relevantfor our purposes, all the rest being implicit inside the ϕ ℓ ’s and the h ℓ ’s therein.As argued above, the definition (4.11) for S ( a ), namely S ( a ) = τ a + ϕ / (2 π i), has tobe replaced by a new one containing terms proportional to ǫ ǫ , i.e. S ( a ) = τ a + ϕ π i + X π i (4.15)with X = ǫ ǫ X + ( ǫ ǫ ) X + · · · (4.16)where the X ℓ ’s have to be determined by imposing the constraint (4.7). Applying S-dualityto (4.15), it is straightforward to obtain S ( a ) = − a + 12 π i τ h(cid:0) τ S ( ϕ ) − ϕ (cid:1) + (cid:0) τ S ( X ) − X (cid:1)i ; (4.17)therefore, in order to satisfy the relation (4.7), the expression in the square brackets abovemust vanish. Expanding this condition in ǫ ǫ , we obtain the following constraints (cid:0) τ S ( ϕ ) − ϕ (cid:1)(cid:12)(cid:12)(cid:12) = 0 , (4.18a) (cid:0) τ S ( ϕ ) − ϕ (cid:1)(cid:12)(cid:12)(cid:12) n + n X k =1 (cid:0) τ S ( X k ) − X k (cid:1)(cid:12)(cid:12)(cid:12) n − k = 0 (4.18b)where the symbol (cid:12)(cid:12) n means taking the coefficient of ( ǫ ǫ ) n (notice that the ϕ ℓ ’s defined in(4.13) do not have any explicit dependence on ǫ ǫ and thus ϕ ℓ (cid:12)(cid:12) n = δ n ϕ ℓ ; for the samereason we also have X ℓ (cid:12)(cid:12) n = δ n X ℓ ).Let us now compute S ( ϕ ). Using the S-duality rules (4.1) and (4.5), and exhibitingtemporarily the dependencies on E and a which are the only relevant ones for our purposes,we obtain τ S (cid:0) ϕ ( E ; a ) (cid:1) = ∞ X ℓ =0 τ ℓ +10 h ℓ (cid:16) E + π i τ (cid:17) ℓ ( S ( a )) ℓ +1 = ∞ X ℓ =0 h ℓ (cid:16) E + π i τ (cid:17) ℓ (cid:16) a + ϕ π i τ + X π i (cid:17) ℓ +1 = ϕ (cid:16) E + 6 π i τ ; a + ϕ π i τ + X π i (cid:17) = e π i τ ( D + ζ ∂ a ) ϕ ( E ; a ) (cid:12)(cid:12)(cid:12) ζ = ϕ + X . (4.19)Actually, this is a particular case of the more general result τ ℓ S ( ϕ ℓ ) = e π i τ ( D + ζ ∂ a ) ϕ ℓ (cid:12)(cid:12)(cid:12) ζ = ϕ + X . (4.20)20xpanding the exponential and using the relation( D + ζ∂ a ) n ϕ ℓ (cid:12)(cid:12)(cid:12) ζ = ϕ + X = n X k =0 ( − k (cid:16) nk (cid:17) X k ( D + ζ∂ a ) n ϕ ℓ + k (cid:12)(cid:12)(cid:12) ζ = ϕ , (4.21)after some straightforward algebra to rearrange the various terms, we obtain τ ℓ S ( ϕ ℓ ) = ∞ X n =0 ( − n n ! (cid:16) X π i τ (cid:17) n Σ ( ℓ + n ) (4.22)where the functions Σ ( ℓ ) are defined byΣ ( ℓ ) = e π i τ ( D + ζ ∂ a ) ϕ ℓ (cid:12)(cid:12)(cid:12) ζ = ϕ . (4.23)As shown in Appendix C, these functions satisfy the remarkably simple relationΣ ( ℓ +1) = − τ τ ∂ a Σ ( ℓ ) (4.24)that is a consequence of the ring properties obeyed by the functions ϕ ℓ , which in turn aredue to the modular anomaly equations (3.30). In view of this we can therefore rewrite(4.22) as τ ℓ S (cid:0) ϕ ℓ (cid:1) = e ζ π i τ ∂ a Σ ( ℓ ) (cid:12)(cid:12)(cid:12) ζ = X . (4.25)Note that the bare coupling τ initially appearing in the right hand side of (4.20) has beendressed into the effective coupling τ .We now exploit this result and proceed perturbatively in ǫ ǫ to obtain explicit ex-pressions. At the zeroth order in ǫ ǫ , from (4.25) we have τ S ( ϕ ) (cid:12)(cid:12)(cid:12) = Σ (1) (cid:12)(cid:12)(cid:12) = ϕ (4.26)where the last equality follows from (C.13). The constraint (4.18a) is therefore identicallysatisfied. This is no surprise since we have already shown that the relation (4.7) must betrue up to terms proportional to ǫ ǫ . At the first order in the deformation parameters weget instead τ S ( ϕ ) (cid:12)(cid:12)(cid:12) = Σ (1) (cid:12)(cid:12)(cid:12) + X π i τ ∂ a Σ (1) (cid:12)(cid:12)(cid:12) = τ ϕ π i) τ − X ϕ π i τ (4.27)where we have used (C.13). Inserting this result into (4.18b) for n = 1, we obtain thefollowing equation for X S ( X ) (cid:12)(cid:12)(cid:12) = X τ − ϕ π i) τ . (4.28)To solve it we take advantage of the S-duality relations at the zeroth order in ǫ ǫ , whichare formally the same of the SW theory. Using (4.24) - (4.26) it is easy to show that S ( ϕ ) (cid:12)(cid:12)(cid:12) = ϕ τ τ , S ( ϕ ) (cid:12)(cid:12)(cid:12) = ϕ τ , (4.29)21nd also that S ( τ ) (cid:12)(cid:12)(cid:12) = − τ . (4.30)Equipped with these results, one can check that a solution to (4.28) is given by X = ϕ π i τ . (4.31)In conclusion we find that the S-dual image of a , which obeys the constraint (4.7) up toterms of order ( ǫ ǫ ) , is S ( a ) = τ a + ϕ π i + ǫ ǫ ϕ π i) τ + O (cid:0) ( ǫ ǫ ) (cid:1) . (4.32)Observing that ϕ = 2 π i ∂ a τ , we can rewrite the ǫ ǫ -term above also as ǫ ǫ π i) ∂ a log (cid:16) ττ (cid:17) , (4.33)so that (4.32) becomes2i π S ( a ) = ∂ a F + ǫ ǫ ∂ a log (cid:18) ττ (cid:19) + O (cid:0) ( ǫ ǫ ) (cid:1) . (4.34)In the following we will investigate the implications of this result, while we refer to Ap-pendix D for its extension to the second order in ǫ ǫ . An immediate consequence of the ǫ -correction in (4.34) is that S ( a ) is not simply pro-portional to the derivative of the prepotential F ; thus it is natural to expect that in thedeformed theory the S-dual of the prepotential is not simply given by a Legendre transfor-mation as it happens instead in the undeformed SW case (see (4.9)). In [38] the relationbetween F and S ( F ) has been conjectured to be a deformed Fourier transformation. Herewe reach the same conclusion, even though from a different perspective since for us alldeviations from the undeformed theory are parametrized by the product ǫ ǫ .We can compute the first ǫ -corrections to the relation between F and S ( F ) using thesame methods of the previous subsection and the results given in Appendix C. The startingpoint is the relation between ϕ and the generalized prepotential (see (4.14a)); from thiswe get the useful identity F − S ( F ) = 2 π i a S ( a ) + π i τ (cid:0) S ( a ) − aτ (cid:1) + S ( ϕ ) − ϕ . (4.35)Using (4.15) and (4.22) for ℓ = 0, we can easily rewrite the right hand side and obtain F − S ( F ) = 2 π i a S ( a ) + ϕ π i) τ + Σ (0) − ϕ − X π i τ (cid:0) Σ (1) − ϕ (cid:1) + 12 (cid:16) X π i τ (cid:17) (cid:0) Σ (2) + 2 π i τ (cid:1) + · · · (4.36) Note that actually the solution (4.31) is not unique, since in principle one could add to X a term Y such that S ( Y ) = Y /τ . A detailed analysis shows that a term of this type is of the form α ϕ / ( τ + 1)with α constant, which has different pole structure in τ with respect to X . We therefore do not considerthis possibility and take α = 0. X which are at least O (cid:0) ( ǫ ǫ ) (cid:1) . Notice thatthe difference Σ (1) − ϕ is of order ǫ ǫ , as one can see from (4.26); thus the linear termin X gives contributions O (cid:0) ( ǫ ǫ ) (cid:1) , like the X term. This means that the knowledgeof the first order correction X obtained in the previous subsection is enough to computethe correction to F − S ( F ) at order ( ǫ ǫ ) . This mechanism actually works at all orders,namely the k -th order coefficient of F − S ( F ) does not depend on X k but only on X j with j < k .Using the explicit value of X given in (4.31) and the expressions for the Σ ( ℓ ) ’s givenin Appendix C (see in particular (C.12) - (C.14)), after straightforward algebra we obtain F − S ( F ) = 2 π i a S ( a ) − ǫ ǫ (cid:16) ττ (cid:17) + ( ǫ ǫ ) (cid:16) ϕ (2i πτ ) + 1196 ϕ (2i πτ ) (cid:17) + O (cid:0) ( ǫ ǫ ) (cid:1) . (4.37)This shows that the simple Legendre transform relation (4.9), which holds in the under-formed theory, does not work any more when ǫ ǫ = 0, as argued, from a different point ofview, in [38]. In the next section, however, we will show that with a suitable redefinitionof the prepotential F and of the coupling constant τ it is possible to recover the standardLegendre transform relation also in the ǫ ǫ -deformed theory. 5. Conclusions The explicit first-order calculation of Section 4.1 shows that S ( a ) is not simply the deriva-tive of the prepotential F . However, it is still a total derivative, as is clear from (4.34).This feature is maintained also at the second order. Indeed, as shown in Appendix D, the( ǫ ǫ ) correction X can be chosen as X = 116 ϕ (2 π i τ ) + 2396 ϕ ϕ (2 π i τ ) + 1164 ϕ (2 π i τ ) = ∂ a (cid:16) − ϕ (2 π i τ ) − ϕ (2 π i τ ) (cid:17) , (5.1)so that we can rewrite (4.15) in the following way S ( a ) = 12 π i ∂ a b F (5.2)with b F = F + ǫ ǫ (cid:16) ττ (cid:17) − (cid:16) ǫ ǫ (cid:17) (cid:16) ϕ (2 π i τ ) + 1112 ϕ (2 π i τ ) (cid:17) + O (cid:0) ( ǫ ǫ ) (cid:1) . (5.3)In the deformed theory the S-duality transformation of the effective coupling τ does nothave a simple form; in fact, using (4.14c) and the transformation properties of ϕ , it iseasy to show that (see (D.7)) S ( τ ) = − τ − ǫ ǫ (cid:16) ϕ (2 π i) τ + 34 ϕ (2 π i) τ (cid:17) + O (cid:0) ( ǫ ǫ ) (cid:1) . (5.4)23owever, there is a modified effective coupling on which S-duality acts in a simple way.This is b τ ≡ ∂ a S ( a ) = 12 π i ∂ a b F = τ " − ǫ ǫ (cid:16) ϕ (2 π i τ ) + ϕ (2 π i τ ) (cid:17) + O (cid:0) ( ǫ ǫ ) (cid:1) . (5.5)One can easily check that S ( b τ ) = ∂ S ( a ) S ( a ) = − ∂ S ( a ) a = − b τ . (5.6)Thus, in the effective ǫ -deformed theory it is b τ , and not τ , that exhibits the same behaviorof the bare coupling τ under S-duality. We also observe that if one uses b τ , the expressionof the extended prepotential b F given in (5.3) simplifies and becomes b F = F + ǫ ǫ (cid:16) b ττ (cid:17) + (cid:16) ǫ ǫ (cid:17) ϕ π i b τ ) + O (cid:0) ( ǫ ǫ ) (cid:1) . (5.7)This result seems to suggest that it is possible to write the higher order corrections onlyin terms of ϕ which, being the triple derivative of the prepotential, is proportional to theYukawa coupling C aaa , the rank-three symmetric tensor playing a crucial rˆole in specialgeometry. Moreover, one can verify the simple Legendre transform relation b F − S ( b F ) = 2 π i a S ( a ) (5.8)up to terms of order ( ǫ ǫ ) . It is quite natural to expect that this pattern extends also tohigher orders.Our detailed analysis shows that when both deformation parameters ǫ and ǫ arenon-vanishing, besides the ǫ -dependent structures generated by the Nekrasov partitionfunctions, the effective theory seems to require a new series of explicit ǫ ǫ -corrections inorder to have S-duality acting in the proper way. These new corrections, being proportionalto inverse powers of the coupling constant, appear to correspond to perturbative terms athigher loops and are absent in the Nekrasov-Shatashvili limit. We find remarkable thatby using the modified prepotential b F and the modified coupling b τ , all S-duality relationsacquire the standard simple form as in the undeformed theory (see (5.2), (5.6) and (5.8)).It would be very nice to see whether these results admit an interpretation in the contextof special geometry or in a more general geometric set-up that allows to go beyond theperturbative approach in the deformation parameters we have used in this paper. It wouldbe interesting also to study the recursion relations obeyed by the functions ϕ ℓ ’s which canbe iteratively obtained from the viscous Burgers equation satisfied by ϕ (see (C.2)).We conclude by observing that all S-duality formulas we have derived for the SU(2) N f = 4 theory can be obtained for the N = 2 ∗ SU(2) theory as well. The only difference inthis case is that every explicit occurrence of ǫ ǫ has to be replaced by ( ǫ ǫ ) / cknowledgments We thank Francesco Fucito and Francisco Morales for very useful discussions. Thiswork was supported in part by the MIUR-PRIN contract 2009-KHZKRX. A. Modular functions We collect here some useful formulas involving the modular functions we used. θ -functions: The Jacobi θ -functions are defined as θ (cid:2) ab (cid:3) ( v | τ ) = X n ∈ Z q ( n − a ) e π i( n − a )( v − b ) , (A.1)for a, b = 0 , q = e π i τ . We simplify the notation by writing, as usual, θ ≡ θ (cid:2) (cid:3) , θ ≡ θ (cid:2) (cid:3) , θ ≡ θ (cid:2) (cid:3) , θ ≡ θ (cid:2) (cid:3) . The functions θ a , a = 2 , , 4, satisfy the “ aequatio identicasatis abstrusa ” θ − θ − θ = 0 , (A.2)and admit the following series expansions θ (0 | τ ) = 2 q (cid:0) q + q + q + · · · (cid:1) ,θ (0 | τ ) = 1 + 2 q + 2 q + 2 q + · · · ,θ (0 | τ ) = 1 − q + 2 q − q + · · · . (A.3) η -function: The Dedekind η -function is defined by η ( q ) = q ∞ Y n =1 (1 − q n ) . (A.4) Eisenstein series: The first Eisenstein series can be expressed as follows: E = 1 − ∞ X n =1 σ ( n ) q n = 1 − q − q − q + . . . ,E = 1 + 240 ∞ X n =1 σ ( n ) q n = 1 + 240 q + 2160 q + 6720 q + . . . ,E = 1 − ∞ X n =1 σ ( n ) q n = 1 − q − q − q + . . . , (A.5)where σ k ( n ) is the sum of the k -th power of the divisors of n , i.e., σ k ( n ) = P d | n d k . Theseries E and E are related to the θ -functions in the following way E = 12 (cid:0) θ + θ + θ (cid:1) , E = 12 (cid:0) θ + θ (cid:1)(cid:0) θ + θ (cid:1)(cid:0) θ − θ (cid:1) . (A.6)The series E , E and E are connected among themselves by logarithmic q -derivativesaccording to q∂ q E = 16 (cid:0) E − E (cid:1) , q∂ q E = 23 (cid:0) E E − E (cid:1) , q∂ q E = E E − E . (A.7)25lso the derivatives of the functions θ a have simple expressions: q∂ q θ = θ (cid:0) E + θ + θ (cid:1) , q∂ q θ = θ (cid:0) E + θ − θ (cid:1) , q∂ q θ = θ (cid:0) E − θ − θ (cid:1) . (A.8) Modular transformations: Under the Sl(2 , Z ) modular transformation τ → τ ′ = aτ + bcτ + d with a, b, c, d ∈ Z and ad − bc = 1 , (A.9)the Eisenstein series E and E behave as modular forms of weight 4 and 6, respectively: E ( τ ′ ) = ( cτ + d ) E ( τ ) , E ( τ ′ ) = ( cτ + d ) E ( τ ) . (A.10)The series E , instead, is a quasi modular form of degree 2: E ( τ ′ ) = ( cτ + d ) E ( τ ) + 6i π c ( cτ + d ) . (A.11)Under the generators T and S of the modular group the θ -functions and the Dedekind η function transform as follows T : θ ↔ θ , θ → θ , η → e i π η , S : θ → − τ θ , θ → − τ θ , θ → − τ θ , η → √− i τ η . (A.12) B. The coefficient h of the SU(2) N f = 4 prepotential Here we give the explicit expression of the coefficient h up to three instantons obtainedusing localization. It is given by h = 11344 h R − R T + 12288 RN + 26112 T − T T − T − (1792 R − RT + 10752 N )( s − p ) + R (1568 s − s p + 5376 p ) − T (9408 s − s p + 32256 p ) − R (496 s − s p + 5248 s p − p )+ 51 s − s p + 1102 s p − s p + 357 p i + 24 s ( T + 2 T ) q − h T + 2 T ) − RT + 2 N )(5 s − p )+ 48( R + 6 T )(16 s − s p + 7 p ) − R (2 s − s p + 7 s p − p )+ 3(16 s − s p + 168 s p − s p + 9 p ) i q + 32( T + 2 T ) h R + 6 T ) − R (280 s − p )+ 313 s − s p + 366 p i q + O ( q ) . (B.1)26ollowing the procedure described in the main text, we can rewrite h in terms of (quasi)modular functions according to h = 13456 (cid:0) R − s + p (cid:1)(cid:0) R − s + 3 p (cid:1)h R − R (5 s − p ) + 5 s − s p + 99 p i E − h R − R (5 s − p ) + 5 s − s p + 99 p i ( T θ − T θ ) E + 11440 (cid:0) R − s + 6 p (cid:1)h R − R (3 s − p ) + 4 R (27 s − s p + 49 p ) − s + 68 s p − s p + 30 p + 2304 N i E E − (cid:0) R − s + 6 p (cid:1)(cid:0) R − s + p (cid:1)h T θ (2 θ + θ ) + T θ ( θ + 2 θ ) i E + 8( T θ − T θ ) E + 16 h T ( θ + 2 θ ) θ − T ( θ + 2 θ ) θ − T T θ θ ( θ − θ ) i − h R − R (5 s − p ) + 21 s − s p + 31 p i ( T θ − T θ ) E + 1120960 h R − R s + 39424 R p + 67872 R s − R s p + 94304 R p − Rs + 134112 Rs p − Rs p + 47616 Rp − N ( s − p ) + 331776 N R − T + T T + T )+ 3323 s − s p + 46862 s p − s p + 6615 p i E . (B.2)By expanding the modular functions in powers of q as shown in Appendix A, one canrecover the instanton terms in (B.1). C. Reformulating the modular anomaly equations The modular anomaly equation (3.30) implies the following relation: Dϕ = 12 ϕ + ǫ ǫ ϕ (C.1)where the functions ϕ ℓ ’s have been defined in (4.13). Since the operators D and ( − ∂ a )commute, by applying the latter to (C.1) and remembering that ϕ ℓ +1 = − ∂ a ϕ ℓ , it isstraightforward to obtain the action of D on any ϕ ℓ and verify that these functions form aring under it. For example, at the next step we obtain Dϕ = ϕ ϕ + ǫ ǫ ϕ = ϕ (cid:0) − ∂ a ϕ (cid:1) + ǫ ǫ ∂ a ϕ . (C.2)which is, up to rescalings, the viscous Burgers equation ∂ t u = u∂ x u + ν∂ x u (C.3)with the viscosity ν proportional to ǫ ǫ .In Section 4 the S-duality requirements were formulated in terms of the quantities Σ ( ℓ ) defined in (4.23), which we rewrite here for convenience as follows:Σ ( ℓ ) = ∞ X n =0 n ! (2 π i τ ) n P ( ℓ ) n (C.4)27ith P ( ℓ ) n = ( D + ζ ∂ a ) n ϕ ℓ (cid:12)(cid:12)(cid:12) ζ = ϕ . (C.5)From this definition it is easy to show that the following relation holds: P ( ℓ +1) n = − ∂ a P ( ℓ ) n + n ϕ P ( ℓ +1) n − . (C.6)In turn, this implies that Σ ( ℓ +1) = − ∂ a Σ ( ℓ ) + ϕ π i τ Σ ( ℓ +1) . (C.7)Recalling that 2 π i τ − ϕ = 2 π i τ (see (4.14c)), this is tantamount to the recursion relation(4.24), namely Σ ( ℓ +1) = − τ τ ∂ a Σ ( ℓ ) , (C.8)which allows to easily obtain the expression of any Σ ( ℓ ) once Σ (0) is known.To determine Σ (0) we start considering the quantities P (0) n . Clearly, from (C.5) wehave P (0)0 = ϕ . The next case is P (0)1 = (cid:0) D + ζ ∂ a (cid:1) ϕ (cid:12)(cid:12)(cid:12) ζ = ϕ = Dϕ − ϕ = − ϕ + ǫ ǫ ϕ (C.9)where the last step follows from (C.1). By further applications of the operator (cid:0) D + ζ ∂ a (cid:1) ,we get P (0)2 = ǫ ǫ ϕ + (cid:16) ǫ ǫ (cid:17) ϕ ,P (0)3 = ǫ ǫ ϕ + (cid:16) ǫ ǫ (cid:17) (cid:0) ϕ ϕ + 5 ϕ (cid:1) + (cid:16) ǫ ǫ (cid:17) ϕ , (C.10)with similar expressions for higher values of n . In fact, it is possible to derive the generalexpression of the P (0) n ’s at the first orders in their explicit dependence on ǫ ǫ : P (0) n (cid:12)(cid:12)(cid:12) = δ n, ϕ − δ n, ϕ ,P (0) n (cid:12)(cid:12)(cid:12) = ( n − ϕ n ,P (0) n (cid:12)(cid:12)(cid:12) = ( n − (cid:16) ϕ (cid:16) ∂∂ϕ (cid:17) + 512 ϕ (cid:16) ∂∂ϕ (cid:17) (cid:17) ϕ n . (C.11)Inserting these results into (C.4), we obtainΣ (0) (cid:12)(cid:12)(cid:12) = ϕ − ϕ π i τ ) , Σ (0) (cid:12)(cid:12)(cid:12) = − 12 log (cid:16) − ϕ π i τ (cid:17) = − 12 log ττ , Σ (0) (cid:12)(cid:12)(cid:12) = 18 ϕ (2 π i τ ) + 524 ϕ (2 π i τ ) , (C.12)where from the second equality on we have used that 2 π i τ − ϕ = 2 π i τ .28pplying the recursion formula (C.8) we can easily get the explicit expressions for theΣ ( ℓ ) ’s that are needed in the calculations presented in Section 4 or in Appendix D. Theyare, for ℓ = 1: Σ (1) (cid:12)(cid:12)(cid:12) = ϕ , Σ (1) (cid:12)(cid:12)(cid:12) = 2 π i τ (cid:16) ϕ (2 π i τ ) (cid:17) , Σ (1) (cid:12)(cid:12)(cid:12) = 2 π i τ (cid:16) ϕ (2 π i τ ) + 23 ϕ ϕ (2 π i τ ) + 58 ϕ (2 π i τ ) (cid:17) ; (C.13)for ℓ = 2: Σ (2) (cid:12)(cid:12)(cid:12) = 2 π i τ (cid:16) ϕ π i τ (cid:17) , Σ (2) (cid:12)(cid:12)(cid:12) = (2 π i τ ) (cid:16) ϕ (2 π i τ ) + ϕ (2 π i τ ) (cid:17) ; (C.14)for ℓ = 3: Σ (3) (cid:12)(cid:12)(cid:12) = (2 π i τ ) (cid:16) ϕ (2 π i τ ) (cid:17) , Σ (3) (cid:12)(cid:12)(cid:12) = (2 π i τ ) (cid:16) ϕ (2 π i τ ) + 72 ϕ ϕ (2 π i τ ) + 4 ϕ (2 π i τ ) (cid:17) ; (C.15)for ℓ = 4 : Σ (4) (cid:12)(cid:12)(cid:12) = (2 π i τ ) (cid:16) ϕ (2 π i τ ) + 3 ϕ (2 π i τ ) (cid:17) ; (C.16)and finally for ℓ = 5 :Σ (5) (cid:12)(cid:12)(cid:12) = (2 π i τ ) (cid:16) ϕ (2 π i τ ) + 10 ϕ ϕ (2 π i τ ) + 15 ϕ (2 π i τ ) (cid:17) . (C.17) D. S-duality at order ( ǫ ǫ ) We start from the relation (4.18b) for n = 2, namely τ S ( X ) (cid:12)(cid:12)(cid:12) − X = − τ S ( ϕ ) (cid:12)(cid:12)(cid:12) − τ S ( X ) (cid:12)(cid:12)(cid:12) . (D.1)From (4.22) we read that τ S ( ϕ ) (cid:12)(cid:12)(cid:12) = Σ (1) (cid:12)(cid:12)(cid:12) − X π i τ Σ (2) (cid:12)(cid:12)(cid:12) − X π i τ Σ (2) (cid:12)(cid:12)(cid:12) + 12 X (2 π i τ ) Σ (3) (cid:12)(cid:12)(cid:12) . (D.2)Substituting the expressions of the various Σ ( ℓ ) (cid:12)(cid:12)(cid:12) k ’s given in Appendix C, and that of X given in (4.31), we get τ S ( ϕ ) (cid:12)(cid:12)(cid:12) = 2 π i τ (cid:16) ϕ (2 π i τ ) + 1324 ϕ ϕ (2 π i τ ) + 1332 ϕ (2 π i τ ) (cid:17) − ϕ π i τ X . (D.3)On the other hand, from (4.31) it follows that τ S ( X ) = 18 π i τ τ S ( ϕ ) S ( τ ) . (D.4)29he numerator of this expression can be computed from (4.22), yielding τ S ( ϕ ) = Σ (3) − X π i τ Σ (4) + · · · = Σ (3) (cid:12)(cid:12)(cid:12) + ǫ ǫ (cid:16) Σ (3) (cid:12)(cid:12)(cid:12) − X π i τ Σ (4) (cid:12)(cid:12)(cid:12) (cid:17) + · · · (D.5)= (2 π i τ ) ϕ (2 π i τ ) + ǫ ǫ (2 π i τ ) (cid:16) ϕ (2 π i τ ) + 134 ϕ ϕ (2 π i τ ) + 134 ϕ (2 π i τ ) (cid:17) + · · · where the last step follows from the results given in Appendix C. For the denominator wetake into account that S ( τ ) = S ( τ ) − S ( ϕ )2 π i = − τ − S ( ϕ )2 π i . (D.6)Resorting again to (4.22) to evaluate τ S ( ϕ ), we obtain S ( τ ) = − τ − π i τ " Σ (2) (cid:12)(cid:12)(cid:12) + ǫ ǫ (cid:16) Σ (2) (cid:12)(cid:12)(cid:12) − X π i τ Σ (3) (cid:12)(cid:12)(cid:12) (cid:17) + · · · = − τ − ǫ ǫ (cid:16) ϕ (2 π i) τ + 34 ϕ (2 π i) τ (cid:17) + · · · . (D.7)Inserting (D.5) and (D.7) into (D.4) and extracting the term of order ǫ ǫ , we find τ S ( X ) (cid:12)(cid:12)(cid:12) = − (2 π i τ ) (cid:16) ϕ (2 π i τ ) + 1116 ϕ ϕ (2 π i τ ) + 58 ϕ (2 π i τ ) (cid:17) . (D.8)Using this result and (D.3) into (D.1), we finally obtain the following constraint on X : τ S ( X ) (cid:12)(cid:12)(cid:12) − π i τ (cid:16) X π i τ (cid:17) = (2 π i τ ) (cid:16) ϕ ϕ (2 π i τ ) + 732 ϕ (2 π i τ ) (cid:17) . (D.9)Considering the structures involved in this relation, we can try an Ansatz such that X iswritten as a total derivative: X = ∂ a (cid:18) − λ ϕ (2 π i τ ) − λ ϕ (2 π i τ ) (cid:19) = λ ϕ (2 π i τ ) + 2( λ + λ ) ϕ ϕ (2 π i τ ) + 3 λ ϕ (2 π i τ ) . (D.10)With this position we find straightforwardly, using the formulæ of Appendix C, that τ S ( X ) (cid:12)(cid:12)(cid:12) − π i τ (cid:16) X π i τ (cid:17) = 2 π i τ (cid:16) (6 λ − λ ) ϕ ϕ (2 π i τ ) + (9 λ − λ ) ϕ (2 π i τ ) (cid:17) . 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