Partition functions of non-Lagrangian theories from the holomorphic anomaly

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the $\Omega$ background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators.


Introduction
In [1,2] it was shown that in a suitable limit, the Argyres-Douglas (AD) limit, the moduli space of a massive N = 2 supersymmetric gauge theory of the Yang-Mills type leads to isolated superconformal field theories (SCFT).A first attempt to classify such theories appeared in [3,4] while more recent results were obtained in [5][6][7][8][9][10][11][12][13][14].In this paper we will be concerned with the rank-one version of the AD SCFT's as well as of those of the Minahan-Nemeschansky (MN) type [15].Being such SCFT's isolated and strongly coupled, their analytic treatment is troublesome given that a Lagrangian description is not available.To circumvent these difficulties, at least five different strategies have appeared in the literature: The conformal bootstrap, the AGT duality, the matrix-model methodology, the largecharge expansion, and the geometric approach based on the Ω-background technology.The numerical conformal bootstrap is a method that exploits the constraints coming from the symmetries of the theory to give numerical estimates of the parameters of interest [16][17][18][19][20].The AGT duality, in its original formulation, relates the partition function of a four-dimensional SQCD with four massive flavors with a four-point correlator of a twodimensional conformal field theory [21,22].Making some (or all) of these points collide leads to rank-one SCFT's [23][24][25][26][27][28].Similar ideas have also been used to provide matrixmodel representations for the partition function in a class of AD theories [29][30][31][32][33][34][35][36].Both the AGT and the matrix-model technology have been useful to study AD theories with large deformation parameters (see also [37]).Another original perspective has been explored in the context of the large-charge expansion [38][39][40][41][42][43][44][45][46], where it was suggested that one can have an approximate description of such strongly coupled SCFT's in terms of an universal effective field theory.Finally, using the genus expansion of the Ω background, as well as ideas coming from localization [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61], it has been possible to study chiral/anti-chiral correlators of non-Lagrangian theories [45,62].Such analytic results, even if based on the first two leading terms in the expansion of the prepotential for small curvatures, show surprisingly good agreement with the numerical bootstrap method [18] as well as with the large-charge expansion [43,44].To improve the analytic estimate of [45,62] and get an exact result, one should incorporate higher curvature terms in the prepotential.This is the main motivation of this paper.
To accomplish this task we use the recursion equations following from the refined holomorphic anomaly.The latter was originally investigated in topological field theories [63], and then revisited in [64][65][66][67][68][69][70][71][72][73][74][75] after the introduction of the Ω background [47][48][49][50][51][52].We want to emphasize that, when approaching the AD point, it is essential to employ the holomorphic anomaly equation.Indeed this technique, while providing expressions which are perturbative in the Ω-background parameters, is exact in all the other parameters of the theory.This is an important difference with respect to localization techniques à la Nekrasov, which instead cannot be used in the context of strongly coupled field theories.
All rank-one SCFT's of the AD and MN type are characterized by the dimension of their Coulomb-branch parameter and they can be treated in a uniform way.First, we specialize the holomorphic anomaly equation to a specific one-parameter family of deformations of these SCFTs.This allows us to compute the free energy exactly in the deformation parameters and order by order in the Ω background parameters 1,2 .When we turn off the deformation and go to the conformal point, we discover significant simplifications.More precisely, we find that their partition function can be expressed as an infinite sum of confluent U-hypergeometric functions where a is the local coordinate on the Coulomb branch, δ = 2 , 3 depending on the SCFT, E I are the Eisenstein functions evaluated at the fixed value of the modular parameter τ * = i , e πi 3 , and γ a constant determined by the conformal dimension of the Coulombbranch operator (see Sec. 3.1 for more details).Finally, the coefficients c n are pure rational numbers depending only on the phase of the Ω background.They are determined by the gap conditions [66,73], that ensure consistency of the expansion near singular monopole points.In order to compute them, deforming away from the conformal point is essential.Nevertheless, we check that their value is independent of the particular deformation we choose.We will also show that in the so-called Nekrasov-Shatashvili limit (NS) [76], i.e. 1 → 0, the summation in (1.1) undergoes a non-trivial re-organization in terms of a different set of functions.This limit is relevant for the study of quantum-mechanical anharmonic oscillators.
This paper is organized as follows.In Section 2 we review the holomorphic anomaly equation and explain how to solve it recursively.In Section 3 we specialize this algorithm to the isolated rank-one conformal field theories and show that important simplifications occur leading to (1.1).In Section 4 we focus on the example of the sphere ( 1 = 2 ) which is relevant for the computation of the extremal correlators of these SCFTs.In Section 5 we discuss the NS limit.We conclude in Section 6 with a few hints for further investigations.Several technical details as well as conventions are relegated to five appendices. 2 The Ω-background prepotential

Holomorphic anomaly equation
We consider rank-one N = 2 supersymmetric (in general non-Lagrangian) theories living on an Ω-background specified by the parameters 1 , 2 and by a Seiberg-Witten (SW) geometry.We denote by (a, a D ) the SW periods, by u the Coulomb-branch parameter and omit the dependence on all remaining parameters: couplings and masses.The partition function on the Ω-background can be written as with F the prepotential.The theory prepotential is regular in the limit 1 , 2 → 0 so it can be expanded as 3) The F 0 (a) term represents the theory prepotential in flat space which can be determined out of the SW geometry.Higher derivative terms are given by the reduced partition function that unlike Z has a regular limit when the Ω-background is turned off.This function will be the main object of our study.We introduce the IR coupling . (2.5) The partition function Z can be alternatively viewed as a function of q or as a function of a.In particular, one can express Z(q) in terms of the Eisenstein's series E 2 (q), E 4 (q), E 6 (q) that form a basis of quasi-modular functions, see App.B for all the relevant definitions. 1  All F g (q)'s have weight zero and a(q) has weight one.S-duality covariance constrains the dependence of the partition function on E 2 .Indeed the full dependence on this form is determined by the anomaly equation [66,69,73,77] 2 In (2.6) the derivatives on the l.h.s. is carried out keeping E 4 , E 6 constant.In the r.h.s. of (2.6) the partition functions is meant as a function of q and a. Writing one finds the recursive equation (2.8) For the "reduced" prepotential F(q, β) = F(q, β) − F 0 (q) one finds or equivalently, using (2.2), (2.10) 1 As we will see later, in specific cases it is convenient to replace E4(q), E6(q) with different modular functions.
2 Throughout this paper we consider the holomorphic version of the anomaly equation, obtained by Equations (2.10) allows to compute F g recursively starting from F 1 (q, β) up to E 2 -independent terms.On the other hand F 1 (q, β) is determined in terms of ∂a ∂u (q) and the discriminant ∆(q) characterising the dynamics in flat space via the formula (2.11)

Seiberg-Witten elliptic curve
The functions ∂a ∂u (q) and ∆(q) entering F 1 are described by the SW elliptic geometry.For a recent discussion see also [78].We write the Seiberg-Witten curve in the Weierstrass form 3y ) The SW periods are given by and the complex coupling parameter q introduced in (2.5) can also be given in terms of The functional dependence u(q) and ω 1 (q) is determined by solving the elliptic geometry formulae for u(q) and ω 1 (q) in terms of E 4 (q) and E 6 (q).Once this is done, all functions of u can be viewed as functions of q.For example, the discriminant is given by (2.17) and the first gravitational correction becomes To compute higher derivative terms, we need derivatives with respect to a, that can be translated into derivatives with respect to q using the chain rule where ξ is a modular form of weight −3 and Here we have used (2.16) and computed the derivatives with respect to τ using where c g, (β, E 4 , E 6 ) is a modular form of weight 6g − 6 − 2 and h g (β, E 4 , E 6 ) is a modular function of weight 6g − 6, known as holomorphic ambiguity, which cannot be determined using (2.10).
The holomorphic ambiguities are fixed by imposing the so-called "gap conditions" [65,66,73,79], that determine the behavior of the prepotential near the points where the elliptic curve degenerates.Rank-one SCFT's can be deformed in such a way that the discriminant of their SW curve takes the particularly simple form leading to n equivalent singularities in the u-plane.
According to (2.17), the zeroes of the discriminant u ∼ u 0 correspond to the point where q = 0.This limit can be studied, expanding a(q) = q dq q ξ(q ) (2.26) for small q, and inverting the series to get q(a) for small a. Plugging this into the F g , the holomorphic ambiguities h g are determined by requiring the gap conditions [73,75] where Table 1: SW data for isolated rank-1 N = 2 SCFTs and B k are the Bernoulli numbers.We will work out two different choices of Ω background, i.e. β = 1 and β = 0.In these cases (2.27) becomes: It is important to stress that in more complicated setups,4 equations (2.16) or (2.22) are hard to solve or admit several inequivalent solutions, often related to each other by modular transformations.In such cases (see App. C for details), the F g 's transform non-trivially under modular transformations and a different set of gap conditions on their modular transformed F D g is required at q D → 0. The basis of modular functions to use is also adapted according to such situations.

The partition function
Rank-one conformal field theories can be realized in F-theory as a single D3-brane, probing a singularity built out of a certain number N 7 of coinciding mutually non-perturbative 7branes [80].The low-energy dynamics on the D3-brane is described by a SW elliptic curve specified by a single Coulomb-branch parameter u, a u-independent modular parameter τ and a discriminant ∆(u) ∼ g 3 2 − 27g The conformal dimension d of the Coulomb-branch parameter is given by that follows from the requirement that the SW period ∂a ∂u be of dimension 1−d and therefore the conformal dimension of the holomorphic differential be [dx/y] = 1 − d.From (3.1) and (3.3) it follows that N 7 is an integer, multiple of 2 or 3, and smaller than 12.In Table 1 we collect the SCFT data for all possible choices of N 7 . 5In the theories of type A the modular form E 4 vanishes, whereas E 2 , E 6 are constants.Similarly in the theories of type B the modular form E 6 vanishes, whereas E 2 , E 4 are constants.Therefore, in all these cases, the free energy is a function of β and of the following dimensionless quantities where For these SCFTs one finds where d is the conformal dimension of the Coulomb-branch operator (see Table 1).The first correction to the SW prepotential takes the general form6 By dimensional analysis, the higher corrections take the form where f g (β) are numbers.The latter can be computed recursively using the holomorphic anomaly equation with boundary conditions fixed by an E 2 -independent function.We can make the following Ansatz where c n are numerical coefficients encoding the holomorphic ambiguities and depend on the phase of the Ω background.Plugging (3.10) into (2.6)leads to the confluent hypergeometric equation7 where the boundary conditions are chosen such that (3.10) has a power-like behavior for x → 0. The final solution is with U(a, b, z) the confluent U hypergeometric function8 and c 0 is an overall normalization which can be set to c 0 = 1 without loss of generality.The coefficients {c n } n≥1 are βdependent coefficients encoded in the E 2 -independent part of Z.In the next section we will derive the first few coefficients c n for the theories in Table 1, and show that they are rational numbers.The strategy will be to first turn on suitable mass or coupling deformations for such theories, in order to isolate a monopole point where the gap condition can be imposed.
The coefficients c n will then be derived by turning off the deformation. 9We check explicitly that the final result is independent of the deformation.We also note that (3.12) as it is written holds for i = 0, i.e. all phases of the Ω background except the NS phase.Indeed if we consider the NS limit there is a non-trivial re-organization of (3.12) which we discuss in Sec. 5.

Deformations
Conformal invariance can be broken by turning on masses or couplings.Here we consider the simplest deformation splitting democratically the discriminant into its N 7 roots We will refer to m generically as a mass deformation, although for the case of H 0 , where masses are not available, the dimension-one parameter m is related to the IR-relevant coupling c via m = c 5 4 .The deformed SW curves look like In all these examples, we will derive q-exact formulae for the first few F g 's.An important ingredient in our procedure will be to parametrize the holomorphic ambiguities for A and B theories respectively in the following form (g ≥ 2) where h g,i (β) are q-independent coefficients to be determined.The above expressions are dictated by the requirement that h g has modular weight 6g−6, allowing only integer powers of E 4 and E 6 , such that h g does not grow faster than its corresponding non-ambiguous part when E 4 → 0 and E 6 → 0. In App.C, we will consider an alternative deformation of the AD theory H 1 described by the SW curve where c is the IR-relevant coupling.In particular we will show that the results for the F g 's in the conformal limit are the same, independently of the deformation used to compute them.An analogous match for the theory H 2 is obtained in App.D, where we consider the N f = 3-SQCD description of this AD theory.
4 Examples: β = 1 In this section, we consider the Ω background given by 1 = 2 = , i.e. β = 1.This choice enters for example the computation of the round-sphere partition function [54,82] and of extremal correlators [45,46,[57][58][59][60]62].Despite such a large interest, the holomorphic anomaly techniques have not been explored so far for this particular phase of the Ω background.10 In the following we will compute (2.24) stopping at the first order in g in which the holomorphic ambiguity contributes in the conformal limit.This is dictated by a reason of simplicity given that the formulae become very large.In App. E e will give results up to g = 18, 7, 15 for H 0 , H 1 , H 2 respectively.

H 0 theory
The SW curve for the deformed H 0 theory is Plugging g 2 = c, g 3 = u into (2.16) and (2.21) gives In this case the holomorphic ambiguity takes the form of the first expression in (3.15).Solving recursively the holomorphic anomaly equation (2.10), one finds the first few terms: The ambiguous part is given by which has been determined by imposing the gap conditions (2.29).

The conformal limit
The theory becomes conformal in the limit c → 0 and fits into the class A according to (3.2).In this limit τ → e πi/3 .Therefore E 2 , E 6 become constants 11 and E 4 vanishes.More precisely, using (4.2), we find From the above formulae we notice that while both E 4 and ξ go to zero in the limit c → 0, their ratio stays finite and goes like Keeping only the leading terms in (4.3) and (4.4), and using (4.6), one finds (4.7) The above formulae reproduce the result (3.12), with and Higher-genus prepotentials F g can also be computed.Results for the ambiguity coefficients c n are listed in (E.2).As we can see, the growth of c n is relatively fast.It is likely that the sum over hypergeometric is divergent.However, a more detailed analysis is needed.

H 1 theory
The SW curve for the H 1 theory deformed by the second-order mass Casimir is In this case g 2 = u and g 3 = m 2 leading to Here the holomorphic ambiguity takes the form of the second expression in (3.15).Solving recursively the holomorphic anomaly equation (2.10), one finds the first few terms 12) The ambiguous part is given by which has been determined by imposing the gap conditions (2.29).

The conformal limit
The theory becomes conformal in the limit m → 0 and fits into the class B according to (3.2).In this limit τ → i. Therefore E 2 , E 4 become constants 12 and E 6 vanishes.More precisely, using (4.11), we find From the above formulae we notice that while both E 6 and ξ go to zero in the limit m → 0, their ratio stays finite and goes like Keeping only the leading terms in (4.12) and (4.13), and using (4.15), one finds The above formulae reproduce the result (3.12), with Higher-genus prepotentials F g can also be computed.Results for the ambiguity coefficients c n are listed in (E.3).

H 2 theory
The SW curve for the H 2 theory deformed by the second-order mass Casimir is In this case g 2 = m 2 and g 3 = u 2 leading to Here the holomorphic ambiguity takes again the form of the first expression in (3.15).Solving recursively (2.10) one finds the first few terms The ambiguous part is given by which has been determined by imposing the gap conditions (2.29).

The conformal limit
The theory becomes conformal in the limit m → 0 and fits into the class A according to (3.2).In this limit τ → e πi/3 .Therefore E 2 , E 6 become constants 13 and E vanishes.More precisely, using (4.20), we find From the above formulae we notice that while both E 4 and ξ go to zero in the limit m → 0, their ratio stays finite and goes like

.25)
Keeping only the leading terms in (4.21) and (4.23), and using (4.25), one finds The above formulae reproduce the result (3.12), with Higher-genus prepotentials F g can also be computed.Results for the ambiguity coefficients c n are listed in (E.4).

E 6 theory
The SW curve for the E 6 theory deformed by the eighth-order mass Casimir is In this case g 2 = m 8 and g 3 = u 4 leading to Here the holomorphic ambiguity takes again the form of the first expression in (3.15).Solving recursively (2.10) one finds the first few terms The ambiguous part is given by which has been determined by imposing the gap conditions (2.29).

The conformal limit
The theory becomes conformal in the limit m → 0 and fits into the class A according to (3.2).In this limit τ → e πi/3 .Therefore E 2 , E 6 become constants 14 and E 4 vanishes.More 14 Their values are clearly the same as in the H0 and H2 theories.
precisely, using (4.30), we find From the above formulae we notice that while both E 4 and ξ go to zero in the limit m → 0, their ratio stays finite and goes like The above formulae reproduce the result (3.12), with Higher-genus prepotentials F g can also be computed in exactly the same manner as in the previous examples.

E 7 theory
The SW curve for the E 7 theory deformed by the eighteenth-order mass Casimir is In this case g 2 = u 3 and g 3 = m 18 leading to u = 3 Here the holomorphic ambiguity takes again the form of the second expression in (3.15).Solving recursively (2.10) one finds the first few terms The ambiguous part is given by which has been determined by imposing the gap conditions (2.29).

The conformal limit
The theory becomes conformal in the limit m → 0 and fits into the class B according to (3.2).In this limit τ → i. Therefore E 2 , E 4 become constant 15  From the above formulae we notice that while both E 6 and ξ go to zero in the limit m → 0, their ratio stays finite and goes like

.46)
Keeping only the leading terms in (4.41) and (4.43), and using (4.46), one finds 15 Their values are clearly the same as in the H1 theory.
The above formulae reproduce the result (3.12), with Higher-genus prepotentials F g can also be computed in exactly the same manner as in the previous examples.

E 8 theory
The SW curve for the E 8 theory deformed by the twentyth-order mass Casimir is In this case g 2 = m 20 and g 3 = u 5 leading to u = E Here the holomorphic ambiguity takes again the form of the first expression in (3.15).Solving recursively (2.10) one finds the first few terms The ambiguous part is given by which has been determined by imposing the gap conditions (2.29).

The conformal limit
The theory becomes conformal in the limit m → 0 and fits into the class A according to (3.2).In this limit τ → e πi/3 .Therefore E 2 , E 6 become constants and E 4 vanishes.More precisely, using (4.51), we find From the above formulae we notice that while both E 4 and ξ go to zero in the limit m → 0, their ratio stays finite and goes like Keeping only the leading terms in (4.52), (4.53) and (4.54), and using (4.56), one finds The above formulae reproduce the result (3.12), with and Higher-genus prepotentials F g can also be computed in exactly the same manner as in the previous examples.
Here we are interested in the conformal limit where such deformationss are turned off.From the point of view of quantum mechanics this corresponds to having a potential with a single term of the form V (x) = x n , n ≥ 3. Parallel to (3.10) we make the following Ansatz to capture the conformal limit16 (5.7) Equation ( 5.3) can then be solved order by order in E 2δ .At zero order we have17 where the boundary conditions are chosen such that the solution does not have negative power-like behavior as x → 0. This gives At the first order in E 2δ , (5.3) gives Using (5.9) we obtain where c 1 is the integration constant.Likewise, at second order we find where c 2 is the integration constant and c 1 is as in (5.11).In principle higher-order f n terms can also be obtained in a similar manner.However, in contrast to the case of finite β, when β = 0 we do not have a general form for f n .The value of the integration constants c n is fixed by the holomorphic ambiguity.

The example of H 0
Let us spell out some detail for the case of H 0 .The starting point of the recursion is given in (5.5),where m is the relevant coupling c 5/4 .The special geometry relations are as in (4.2) and we have γ = 1/10, δ = 3.The ambiguity is of the form (3.15) (first line) and the gap condition is given in (2.30).Using these initial conditions and running the equation (5.3), we obtain (5.15) The results perfectly agree with those obtained in [74].The conformal limit c → 0 can be computed using (4.5) and (4.6), and gives

Outlook
This paper exploits the holomorphic anomaly equation to compute the partition function of intrinsically strongly coupled SCFT's with eight supercharges living on a generic Ω-background.We studied one-parameter deformations of such theories allowing for an exact integration of the anomaly equation, order by order in the expansion of the Ωdeformed prepotential.Within this framework, we observed important simplifications at the conformal point.The Ω-deformed prepotential is given by the elegant formula (1.1) in terms of hypergeometric functions, with coefficients c n determined by the gap conditions.It would be interesting to understand whether this non-trivial re-organization of the expansion of the partition function can improve the precision in the computation of extremal correlators made in [45,62], and shed light on the analytic structure of the exact answer.
We will report on this in [90].
In the NS phase of the Ω background ( 1 = 0), we show that (1.1) undergo a nontrivial re-organization in which the hypergeometric functions become simpler functions, see e.g (5.9) and (5.11).This results are relevant for the study of quantum periods of anharmonic oscillators.We will report more on this in [90].
The AD point is obtained by taking and taking u AD , c and C 2 , C 3 small The same curve is obtained starting from the standard H 2 quartic form after the identification

B Modular functions
In this section we collect some definitions and useful modular identities.The Eisenstein series are defined as A basis of modular forms is given by E 4 , E 6 and the quasi-modular form E 2 .They are related to the theta functions via We introduce the functions Under S-duality they transform as whereas under T -duality they transform as In terms of these variables the Eisenstein series read In this Appendix we work at β = 1 and consider a deformation of H 1 obtained by turning on the IR-relevant coupling c.The SW curve is now described by a cubic curve with We notice that we have now two monopole points u = 3c 2 and u = 12c 2 .It is convenient in this case to introduce the modular functions K 2 and L 2 related to E 4 and E 6 via Plugging this into (2.16) and solving for ω 1 and u one finds three inequivalent solutions These solutions correspond to three different duality frames related by S, T transformations (B.4), (B.5).In order to make contact with the results derived in Sec.4.2 for the massdeformed H 1 theory, we choose the duality frame given by the second line in (C.4).To make formulae simpler, it is convenient to define which correspond to the ST transformations of K 2 and L 2 respectively.This leads to Likewise we define u D , ω 1D = da D /du, F D g by the same formulae (C.4) replacing K 2 → K2 and L 2 → L2 .
The holomorphic ambiguity for the theory has the following form (see [74]) with coefficients h g,i determined by requiring that both F g and F D g satisfy the gap conditions when a → 0 and a D → 0 respectively, i.e. q → 0 or q D → 0. Solving recursively, the holomorphic anomaly equation (2.10) one finds the first few terms The conformal limit This is a theory of type B, hence τ * = i at the conformal point.In particular at this point both L 2 and K 2 are finite with as well as , . . .(C.14) which agrees with the results of Sec.4.2.This provides an explicitly test that the conformal limit is independent of the deformation we perform.

D SQCD N f =3 at the conformal point
The SW curve of SQCD with N f = 3 flavors of equal mass m = − Λ 8 is described by a curve in the Weirstrass form with These can be resummed using the hypergeometric functions (3.12) and in agreement with (4.28).This provides an explicitly test that the conformal limit is independent of the deformation we perform.

Z(a, β)|
In this Appendix we list the first few c n coefficients at β = 1 for the three AD theories we analyzed in this paper.These coefficients determine the behavior of f g in (3.9) up to g = 18.These coefficients determine the behavior of f g in (3.9) up to g = 15.

) and c 0 = 1 , c 1
the behavior of f g in (3.9) up to g = 7.
Prototypical examples are the AD theories H 0 , H 1 , H 2 , and the Minahan-Nemeschansky theories E 6 , E 7 , E 8 .These are all isolated, non-Lagrangian field theories and are the focus of the present work.They can be split into two classes depending on the value of the modular parameter .21) 12 4 Plugging this into the anomaly equation, one finds for the first few gravitational corrections at β = 1 This is a theory of type A, hence τ * = e