Deformed theories, generalized recursion relations and S-duality
We study the non-perturbative properties of super conformal field theories in four dimensions using localization techniques. In particular we consider SU(2) gauge theories, deformed by a generic -background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton corrections to the partition function and the prepotential using Nekrasov’s approach. These results allow to reconstruct exact expressions involving quasi-modular functions of the bare gauge coupling constant and to show that the prepotential terms satisfy a modular anomaly equation that takes the form of a recursion relation with an explicitly -dependent term. We then investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective 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-Witten solution of the undeformed theory.
Four-dimensional field theories with rigid supersymmetry provide a remarkable arena where many exact results can be obtained; indeed supersymmetry, not being maximal, allows for a great deal of flexibility but, at the same time, is large enough to guarantee full control. This fact was exploited in the seminal papers [1, 2] where it was shown that the effective dynamics of super Yang-Mills (SYM) theories in the limit of low energy and momenta can be exactly encoded in the so-called Seiberg-Witten (SW) curve describing the geometry of the moduli space of the SYM vacua. When the gauge group is SU(2), the SW curve defines a torus whose complex structure parameter is identified with the (complexified) gauge coupling constant of the SYM theory at low energy. This coupling receives perturbative corrections at 1-loop and non-perturbative corrections due to instantons, and the corresponding effective action follows from a prepotential that is a holomorphic function of the vacuum expectation value of the adjoint vector multiplet, of the flavor masses, if any, and of the dynamically generated scale in asymptotically free theories or of the bare gauge coupling constant in conformal models (see for instance  for a review and extensions of this approach).
Recently, superconformal field theories (SCFT) have attracted a lot of attention. Two canonical examples of SCFT’s are the SU(2) SYM theory with fundamental hypermultiplets and the theory, namely a SYM theory with an adjoint hypermultiplet. In both cases, the -function vanishes but, when the hypermultiplets are massive, the bare coupling gets renormalized at 1-loop by terms proportional to the mass parameters. Besides these, there are also non-perturbative corrections due to instantons. As shown in  for the theory and more recently in  for the theory, by organizing the effective prepotential as a series in inverse powers of and by exploiting a recursion relation hidden in the SW curve, it is possible to write the various terms of as exact functions of the bare coupling. These functions are polynomials in Eisenstein series and Jacobi -functions of and their modular properties allow to show that the effective theory at low energy inherits the symmetry of the microscopic theory at high energy. In particular one can show [4, 5] that the S-duality map on the bare coupling, i.e. , implies the corresponding map on the effective coupling, i.e. , and that the prepotential and its S-dual are related to each other by a Legendre transformation.
The non-perturbative corrections predicted by the SW solution can also be obtained directly via multi-instanton calculus and the use of localization techniques [6, 7]111See also [8, 9] for earlier applications of these techniques.. This approach is based on the calculation of the instanton partition function after introducing two deformation parameters, and , of mass dimension 1 which break the four-dimensional Lorentz invariance, regularize the space-time volume and fully localize the integrals over the 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  - ) can be interpreted as the effect of putting the gauge theory in a curved background, known as -background [6, 7, 18], or in a supergravity background with a non-trivial graviphoton field strength, which are equivalent on the instanton moduli space [19, 20]. The resulting instanton partition function , also known as Nekrasov partition function, allows to obtain the non-perturbative part of the SYM prepotential according to
Actually, the Nekrasov partition function is useful not only when the parameters are sent to zero, but also when they are kept at finite values. In this case, in fact, the non-perturbative -deformed prepotential
represents a very interesting generalization of the SYM one. By adding to it the corresponding (-deformed) perturbative part , one gets a generalized prepotential that can be conveniently expanded as follows
The amplitude , which is the only one that remains when the -deformations are switched off, coincides with the SYM prepotential of the SW theory, up to the classical tree-level term. The amplitudes with account instead for gravitational couplings and correspond to F-terms in the effective action of the form , where is the chiral Weyl superfield containing the graviphoton field strength as its lowest component. These terms were obtained long ago from the genus partition function of the topological string on an appropriate Calabi-Yau background  and were shown to satisfy a holomorphic anomaly equation [22, 23] which allows to recursively reconstruct the higher genus contributions from the lower genus ones (see for instance [24, 25]). More recently, also the amplitudes with have been related to the topological string and have been shown to correspond to higher dimensional F-terms of the type where is a chiral projection of real functions of vector superfields , which also satisfy an extended holomorphic anomaly equation [27, 28]. By taking the limit with finite, one selects in (3) the amplitudes . This limit, also known as Nekrasov-Shatashvili limit , is particularly interesting since it is believed that the effective theory can be described in this case by certain quantum integrable systems. Furthermore, in this Nekrasov-Shatashvili limit, using saddle point methods it is possible to derive a generalized SW curve [30, 31] and extend the above-mentioned results for the SYM theories to the -deformed ones.
By considering the Nekrasov partition function and the corresponding generalized prepotential for rank one SCFT’s, in  a very remarkable relation has been uncovered with the correlation functions of a two-dimensional Liouville theory with an -dependent central charge. In particular, for the SU(2) theory with the generalized prepotential turns out to be related to the logarithm of the conformal blocks of four Liouville operators on a sphere and the bare gauge coupling constant to the cross-ratio of the punctures where the four operators are located; for the SU(2) theory, instead, the correspondence works with the one-point conformal blocks on a torus whose complex structure parameter plays the role of the complexified bare gauge coupling. Since the conformal blocks of the Liouville theory have well-defined properties under modular transformations, it is natural to explore modularity also on the four-dimensional gauge theory and try to connect it to the strong/weak-coupling S-duality, generalizing in this way the SW results when and are non-zero. On the other hand, one expects that the deformed gauge theory should somehow inherit the duality properties of the Type IIB string theory in which it can be embedded. Some important steps towards this goal have been made in  and also in  where it has been shown that the SW contour integral techniques remain valid also when both and are non-vanishing. To make further progress and gain a more quantitative understanding, it would be useful to know the various amplitudes in (3) as exact functions of the gauge coupling constant and analyze their behavior under modular transformations, similarly to what has been done for the SW prepotential in [4, 5]. Recently, by exploiting the generalized holomorphic anomaly equation, an exact expression in terms of Eisenstein series has been given for the first few amplitudes of the SU(2) theory with and the SU(2) theory in the limit of vanishing hypermultiplet masses . This analysis has then been extended in  to the massive model in the Nekrasov-Shatashvili limit, using again the extended holomorphic anomaly equation, and in  using the properties of the Liouville toroidal conformal blocks in the semi-classical limit of infinite central charge. However, finding the modular properties of the deformed prepotential in full generality still remains an open issue.
In this paper we address this problem and extend the previous results by adopting a different strategy. In Section 2, using localization techniques we explicitly compute the first few instanton corrections to the prepotential for the massive theory with gauge group SU(2) in a generic -background. From these explicit results we then infer the exact expressions of the various prepotential coefficients and write them in terms of Eisenstein series of the bare coupling. Our results reduce to those of [4, 5] when the deformation parameters are switched off, and to those of  -  in the massless or in the Nekrasov-Shatashvili limits. The properties of the Eisenstein series allow to analyze the behavior of the various prepotential terms under modular transformations and also to write a recursion relation that is equivalent to the holomorphic anomaly equation if one trades modularity for holomorphicity. The recursion relation we find contains a term proportional to , which is invisible in the SYM limit or in the Nekrasov-Shatashvili limit. In Section 3 we repeat the same steps for the SU(2) theory with and arbitrary mass parameters and also in this case derive the modular anomaly equation in the form of a recursion relation. In Section 4 we study in detail the properties of the generalized prepotential under S-duality, and show that when both and are different from zero, due to the new term in the recursion relation, the prepotential and its S-dual are not any more related by a Legendre transformation, an observation which has been recently put forward in  from a different perspective. We also propose how the relation between the prepotential and its S-dual has to be modified, by computing the first corrections in . Finally, in Section 5 we conclude by showing that there exist suitable redefinitions of the prepotential and of the effective coupling that allow to recover the simple Legendre relation 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 which are useful for the computations described in the main text.
2 The SU(2) theory
The SYM theory describes the interactions of a gauge vector multiplet with a massive hypermultiplet in the adjoint representation of the gauge group. It can be regarded as a massive deformation of the SYM theory in which the -function remains vanishing but the gauge coupling constant receives both perturbative and non-perturbative corrections proportional to the hypermultiplet mass. Using the localization techniques [6, 7] one can obtain a generalization of this theory by considering the -dependent terms in the Nekrasov partition function. In the following we only discuss the case in which the gauge group is SU(2) (broken down to U(1) by the vacuum expectation value of the adjoint scalar of the gauge vector multiplet). We begin by considering the non-perturbative corrections.
2.1 Instanton partition functions
The partition function at instanton number is defined by the following integral over the instanton moduli space :
where is the instanton moduli action of the theory. After introducing two deformation parameters and , the partition function can be explicitly computed using the localization techniques. In the case at hand, each can be expressed as a sum of terms in one-to-one correspondence with an ordered pair of Young tableaux of U() such that the total number of boxes in the two tableaux is . For example, at we have the two possibilities: and ; at we have instead the five cases: , , , , ; and so on and so forth. Referring for example to the Appendix A of  for details, at one finds
where with , and
At the relevant partition functions are
The contributions corresponding to the other Young tableaux at can be obtained from the previous expressions with suitable redefinitions. In particular, is obtained from in (8) by exchanging , is obtained by exchanging , and finally is obtained by simultaneously exchanging and . The complete 2-instanton partition function is
but we refrain from writing its expression since it is not particularly inspiring. This procedure can be systematically extended to higher instanton numbers leading to explicit formulas for the instanton partition functions.
Following  we can cast these results in a nice and compact form. Indeed, defining
where is the complexified gauge coupling constant of the SYM theory, the grand-canonical instanton partition function
where , can be rewritten as
where the first line represents the contribution of the gauge vector multiplet and the second line that of the adjoint hypermultiplet. Here and denote, respectively, the number of boxes in the -th row and in the -th column of a Young tableaux and are related to the Dynkin indices of the corresponding representation. These quantities can be extended for any integer with the convention that or if the -th row or the -th column of is empty. For example, for the ’s are and the ’s are . Moreover we have
Following Nekrasov’s prescription, we can obtain the generalized non-perturbative prepotential according to
In the limit , the above expression computes the instanton contributions to the prepotential of the SYM theory, while the finite -dependent terms represent further non-perturbative corrections. Notice that in the limit , (12) correctly reduces to the partition function of the SYM theory, namely to the Euler characteristic of the instanton moduli space (see for instance ).
2.2 Perturbative part
The compact expression (12) allows to “guess” the perturbative part of the partition function by applying the same formal reasoning of Section 3.10 of . Indeed, in (12) we recognize the following universal (i.e. -independent but -dependent) factor
which, if suitably interpreted and regularized [6, 7], can be related to the perturbative part of the partition function of the theory in the -background. According to this idea, we are then led to write
Using the following representation for the logarithm
where is an arbitrary scale and summing over and , we can rewrite (16) as
This function, which is related to the logarithm of the Barnes double -function, can be easily computed by expanding for small values of and . As a result, becomes a series in inverse powers of whose first few terms are
Here we have used (6) and introduced the convenient notation
One can easily check that in the limit this expression correctly reproduces the 1-loop prepotential of the SU(2) gauge theory (see for instance ).
2.3 Generalized prepotential
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 to the Nekrasov partition function. Just like the 1-loop piece (20), also the instanton terms (14) can be organized as a series expansion for small values of and , or equivalently for large values of . Discarding -independent terms, which are not relevant for the gauge theory dynamics, we write
where the coefficients are polynomials in , and which can be explicitly derived from (20) as far as the perturbative part is concerned, and from the instanton partition functions (12), after using (14), for the non-perturbative part. Here we list the first few of these coefficients up to three instantons:
It is interesting to notice that all ’s are proportional to and, except for , also to . Explicit expressions for with can be systematically derived from the generalized prepotential but they are not needed for our considerations.
Building on previous results obtained in limit from the SW curve [4, 5] and on the analysis of  for the massless case, we expect that the expressions (23)-(26) are just the first terms in the instanton expansion of (quasi) modular functions of . More precisely, we expect that are (quasi) modular functions of weight that can be written solely in terms of the Eisenstein series , and (see Appendix A for our conventions and definitions). This is indeed what happens. In fact we have
By using the small expansion of the Eisenstein series one can check that the explicit instanton contributions we have computed using localization techniques are correctly recovered from the previous formulas. We stress that the fact that the various instanton terms nicely combine into expressions involving only the Eisenstein series is not obvious a priori and is a very strong a posteriori test on the numerical coefficients appearing in (23)-(26). It is quite remarkable that the explicit instanton results at low can be nicely extrapolated and allow to reconstruct modular forms from which, by expanding in powers of , one can obtain the contributions at any instanton number.
We can also organize the generalized prepotential according to (3) and obtain the amplitudes as a series in inverse powers of with coefficients that are polynomials in , and . The first few of such amplitudes are:
Terms with higher values of and can be systematically generated without any difficulty from the ’s given in (27)-(30). The first term represents the prepotential of the SU(2) gauge theory and its expression (31) agrees with that found in  from the SW curve (see also ). The other terms are generalizations of those considered in  and more recently in  in particular limits ( or ) where they drastically simplify.
2.4 Recursion relations
The generalized prepotential (22) is clearly holomorphic by construction but does not have nice transformation properties under the modular group since the coefficients explicitly depend on the second Eisenstein series which is not a good modular function. Indeed, under
transforms inhomogeneously as follows
Therefore, in order to have good modular properties we should replace everywhere with the shifted Eisenstein series at the price, however, of loosing holomorphicity. This fact leads to the so-called holomorphic anomaly equation [22, 23] (see also ). On the other hand, in the limit , holding fixed so that , we recover holomorphicity but loose good modular properties and obtain the so-called modular anomaly equation . This equation can be rephrased in terms of a recursion relation satisfied by the coefficients which allows to completely fix their dependence on .
with the initial condition
We have explicitly checked this relation for several values of ; we can thus regard it as a distinctive property of the -deformed low-energy theory. Notice that in the limit , (39) reduces to the recursion relation satisfied by the coefficients of the prepotential of the SU(2) theory found in [4, 5] from the SW curve, and that the linear term in the right hand side disappears in the so-called Nekrasov-Shatashvili limit  where one of the two deformation parameters vanishes and a generalized SW curve can be introduced [30, 31].
Notice that the coefficients , which are polynomials in the hypermultiplet mass , have mass dimensions ; therefore if (this condition can be easily checked on the explicit expressions (31)-(34). These definitions must be supplemented by the “initial conditions”
where the means that the sum is performed over all , , and such that and . Eq. (44) shows that the coefficients and hence the amplitudes are recursively related to those with lower values of and , similarly to what can be deduced from the holomorphic anomaly equation .
3 The SU(2) theory with
We now consider the SU(2) theory with flavor hypermultiplets in the fundamental representation of the gauge group. When the 1-loop -function vanishes and the conformal invariance is broken only by the flavor masses (. Furthermore there are non-perturbative effects due to instantons which are nicely encoded in the exact SW solution . We now discuss the generalizations of these effects induced by the deformation parameters in the Nekrasov partition function, following the same path described in the previous section for the theory.
3.1 Instanton partition functions
Using localization techniques, we can express the instanton partition functions of the theory as sums of terms related to pairs of Young tableaux. Referring again the Appendix A of  for details, at we have the following two contributions
where . The partition function contains a part proportional to , a part which does not depend on and a part containing in the denominator. In all -dependent terms the flavor masses always occur in SO(8)-invariant combinations and thus in all such terms we can always express the mass dependence using the following quadratic, quartic and sextic SO(8) invariants:
The -independent terms in (47), instead, are not invariant under the SO(8) flavor group. Indeed, neither nor respect this invariance. Notice, however, that -independent terms in the partition functions, and hence in the prepotential, are irrelevant for the gauge theory dynamics, and thus can be neglected. We will always do so in presenting our results.
The explicit expressions of the partition functions for are rather cumbersome; nevertheless it is possible to write the (grand-canonical) instanton partition function in a quite compact way using the connection with the Young tableaux. Indeed, denoting as the instanton counting parameter and using the same notations introduced in Section 2.1, we have
Here the second line represents the contribution of the gauge vector multiplet and the last line that of the fundamental hypermultiplets. From this expression, by selecting the appropriate Young tableaux, one can obtain the various terms of the instanton partition function and their dependence on the parameters.
3.2 Perturbative part
Also in the theory one can deduce the perturbative contribution to the partition function from the “universal” factor of the grand-canonical instanton partition function (49), namely
which, after using (17) and summing over and , becomes