Exact partition functions for the -deformed gauge theory
We study the low energy effective action of the -deformed gauge theory. It depends on the deformation parameters , the scalar field expectation value , and the hypermultiplet mass . We explore the plane looking for special features in the multi-instanton contributions to the prepotential, motivated by what happens in the Nekrasov-Shatashvili limit . We propose a simple condition on the structure of poles of the -instanton prepotential and show that it is admissible at a finite set of points in the above plane. At these special points, the prepotential has poles at fixed positions independent on the instanton number. Besides and remarkably, both the instanton partition function and the full prepotential, including the perturbative contribution, may be given in closed form as functions of the scalar expectation value and the modular parameter appearing in special combinations of Eisenstein series and Dedekind function. As a byproduct, the modular anomaly equation can be tested at all orders at these points. We discuss these special features from the point of view of the AGT correspondence and provide explicit toroidal 1-blocks in non-trivial closed form. The full list of solutions with 1, 2, 3, and 4 poles is determined and described in details.
\affiliation[a]Dipartimento di Matematica e Fisica Ennio De Giorgi,
Università del Salento, Via Arnesano, 73100 Lecce, Italy \affiliation[b]INFN, Via Arnesano, 73100 Lecce, Italy \emailAddmatteo.firstname.lastname@example.org
1 Introduction and results
In this paper we consider the -deformed gauge theory in four dimensions and present novel closed expressions for its low energy effective action at special values of the deformation parameters. On general grounds, before deformation, the effective action of theories is computed by the Seiberg-Witten (SW) curve [1, 2]. It is the sum of a 1-loop perturbative correction and an infinite series of non-perturbative instantonic contributions that are weighted by the instanton counting parameter where is the complexified gauge coupling constant at low energy. Due to supersymmetry, the full effective action may be expressed in terms of the analytic prepotential depending on the vacuum expectation value of the scalar in the adjoint gauge multiplet and on the mass of the adjoint hypermultiplet .
Instead of applying the SW machinery, one may compute the effective action by topological twisting the theory and exploiting localization on the many-instanton moduli spaces [4, 5, 6]. Technically, this is made feasible by introducing the so-called -deformation of the theory, i.e. a modification breaking 4d Poincaré invariance and depending on two parameters such that the initial theory is recovered when . The role of the -deformation is that of a complete regulator for the instanton moduli space integration [7, 8, 9, 10, 11, 12, 13, 14]. In this approach, it is natural to introduce a well defined partition function and its associated non-perturbative -deformed prepotential by means of
It is well established that the quantity in (1) is interesting at finite values of the deformation parameters , i.e. taking seriously the deformed theory. This is because the amplitudes appearing in the expansion are related to the genus partition function of the topological string [15, 16, 17, 18, 19, 20, 21] and satisfy a powerful holomorphic anomaly equation [22, 23, 24, 25]. Actually, understanding the exact dependence on the deformation parameters is an interesting topic if one wants to resum the above expansion in higher genus amplitudes. Clearly, this issue is closely related to the Alday-Gaiotto-Tachikawa (AGT) correspondence  mapping deformed instanton partition functions to conformal blocks of a suitable CFT with assigned worldsheet genus and operator insertions. AGT correspondence may be checked by working order by order in the number of instantons [27, 28, 29]. For the -deformed gauge theory the relevant CFT quantity is the one-point conformal block on the torus, a deceptively simple object of great interest [30, 27, 31, 28, 32, 33, 34, 35, 36, 37].
The AGT interpretation emphasizes the importance of modular properties in the deformed gauge theory. Indeed, it is known that SW methods can be extended to the case of non-vanishing deformation parameters [38, 39] and modular properties have been clarified in the undeformed case [40, 41] as well as in presence of the deformation [42, 43]. The major outcome of these studies are explicit resummations of the instanton expansion order by order in the large regime. The coefficients of the powers are expressed in terms of quasi-modular functions of the torus nome . This approach can be pursued in the gauge theory [44, 45, 46, 47, 48, 49, 50], in CFT language by AGT correspondence [34, 35, 51, 36], and also in the framework of the semiclassical WKB analysis [52, 53, 52, 54, 55, 56, 57].
An important simpler setup where these problems may be addressed is the so-called Nekrasov-Shatashvili (NS) limit  where one of the two parameters vanishes. In this case, the deformed theory has an unbroken two dimensional super-Poincaré invariance and its supersymmetric vacua are related to the eigenstates of a quantum integrable system. Under this Bethe/gauge map, the non-zero deformation parameter plays the role of in the quantization of a classically integrable system. Saddle point methods allow to derive a deformed SW curve [59, 60] that can also be analyzed by matrix model methods [33, 52, 53, 61, 62]. In the specific case of the theory, the relevant integrable system is the elliptic Calogero-Moser system  and the associated spectral problem reduces to the study of the celebrated Lamé equation. Besides, if the hypermultiplet mass is taken to be proportional to with definite special ratios , where , the spectral problem is -gap. Remarkable simplifications occur in the -instanton prepotential contributions  that may be obtained by expanding the eigenvalues of a Lamé equation in terms of its Floquet exponent. As a byproduct of this approach, it is possible to clarify the meaning of the poles that appear in the -instanton prepotential at special values of the vacuum expectation value . Indeed, the pole singularities turn out to be an artifact of the instanton expansion.
In this paper, we inquire into similar problems when both the deformation parameters are switched on, i.e. by going beyond the Nekrasov-Shatashvili limit. In particular, we explore the plane where are real parameters entering the scaling relation
In other words, we keep the hypermultiplet mass to be proportional to one deformation parameter with
ratio , but are generic ( is just a convenient
replacement of ). By dimensional scaling, the prepotential is a function
of the combination at the fixed point .
There exists a finite set of -poles points such that the -instanton prepotential is a rational function of with poles at a fixed set of positions independent on .
This claim is motivated by our previous analysis in the restricted Nekrasov-Shatashvili limit  and is far from obvious. Most important, it has far reaching consequences. At the special -poles points, we show that the instanton partition function and the perturbative part of the prepotential take the exact form
where is a polynomial in the Eisenstein series with total modular degree with coefficients depending on , and . The total prepotential is thus remarkably simple and reads
These explicit expression satisfy the modular anomaly equation expressing S-duality discussed in [44, 45, 46, 47, 48, 49]. By applying the AGT dictionary, (3) and (4) predict toroidal blocks in closed form at very specific values of the central charge and of the inserted operator conformal dimension – the perturbative part providing interesting special instances of the 3-point DOZZ Liouville correlation function. These results are derived and tested by giving a complete list of all the poles points. These turns out to be 4, 7, 12, and 11 at respectively.
The plan of the paper is the following. In Sec. (2) we determine the 1-pole points by a direct inspection of the instanton prepotential contributions. In Sec. (2.1) we discuss the special features of the instanton partition function at the 1-pole points. The AGT interpretation is analyzed in Sec. (3) where we also provide various explicit CFT tests of the proposed partition functions. In Sec. (4) we discuss the perturbative part of the prepotential at the 1-pole points. In Sec. (5) the analysis is extended to -poles points and the cases are fully classified. Finally, Sec. (6) presents a list of special toroidal blocks. Various appendices are devoted to additional comments.
2 Looking for simplicity beyond the Nekrasov-Shatashvili limit
As discussed in the Introduction, we are interested in the scaling limit (2). The instanton partition function is and it is convenient to introduce
where we used dimensional scaling independence to remove . Similarly, we define
We shall omit the explicit index when obvious. Besides, the partition function is even in and we shall always consider .
According the the claim presented in the Introduction, we now look for special points such that the -instanton Nekrasov function takes the form
with a polynomial and a single pole in the variable .
The Ansatz (7) is a non-trivial requirement. It is motivated by the analysis in ,
but its admissibility is actually one of the results of our investigation. To explore the constraints that
(7) imposes, we begin by looking at the simple one-instanton case.
and there is a simple pole . At the two-instanton level, , the denominator of turns out to vanish at
Special cases occur when one of the poles coincides with those at . This happens for
These values must be analyzed separately. Looking at higher values of we
identify the only non-trivial cases consistent with (7)
2.1 Back to the instanton partition functions
We could analyze further the structure of the prepotential in (7) at the special points in (13) by looking for regularities in the polynomials . However, it is much more convenient to go back to the instanton partition function. To see why, let us consider as a first illustration. We find indeed the simple expansion
After some educated trial and error, we recognize that (14) is the expansion of the following expression
and is an Eisenstein series.
The associated all-instanton Nekrasov functions are
Equations (2.1) and (2.1) are already remarkable because they are non-trivial closed expressions for the instanton partition function, or prepotential, at all instanton numbers. It is clear that it would be nice to provide some clarifying interpretation for this features at the special points . In the next section, we shall examine the clues coming from AGT correspondence.
3 AGT interpretation
where is the 1-point toroidal block of the Virasoro algebra of central charge on a torus whose modulus is , with one operator of dimension inserted and a primary of dimension in the intermediate channel. The precise dictionary in terms of the deformation parameters is
with central charge
In particular, at the four points we obtain the following values for
Of course, points appear in pairs with the same central charge and values related by . More remarkably, the associated values of the parameter is always such that is a positive integer. The toroidal block has a universal prefactor that is its value at . Comparing (19) with (2.1) we can write the general form for all four points as
Correspondingly, the net prediction for the toroidal block at the above central charge and insertion dimension is
We remark that the above may well be pathological for a physical CFT. Nevertheless, the toroidal block is defined by the Virasoro algebra for abritrary values of , and . Eq. (25) must be taken in this sense. We checked (25) against Zamolodchikov recursive determination of the toroidal block [68, 69, 70] with perfect agreement. Of course, by AGT, this is same as Nekrasov calculation. The remarkably simple form (25) is clearly consistent with general results for the torus block. For instance, at leading and next-to-leading order and generic operator dimensions we have 
where are the coefficients of the expansion of , i.e. the number of unrestricted partitions of ( means copies of )
3.1 Explicit CFT computations
The conformal block
It is instructive to derive the result (25) at from a direct CFT calculation.
where we used (29). The toroidal block is obtained as
where the trace is over the descendants of . The starting point is thus conformal descendant decomposition of the diagonal part of the OPE
where denotes a unrestricted partition of and are Virasoro generators
The coefficients in (33) are determined by conformal symmetry and are functions of . As a consequence of unbroken conformal symmetry () and of the fact that is the same dimension as that of the energy momentum tensor, one finds that the only vanishing coefficients are those associated with simple descendants. Besides they are all equal
Just to give an example, the explicit coefficients at level 2 are
Computing them at we see that indeed
Hence, if we apply (33) to the vacuum, we get
Now, to get the torus block, we need to evaluate the diagonal matrix elements of . Using
we obtain with one index
where we have shown only the diagonal entry. Adding one index each time, a similar calculation shows that for any number of indices
Thus the diagonal matrix element of associated with the descendent depends only on . The number of with fixed is the number of unrestricted partitions of . Summing over with we prove (31).
The conformal block
A similar computation for and is apparently quite less trivial. The main reason is that the coefficients in the conformal decomposition (33) do not trivialize in this case. This complication forbids us to prove the wanted result in general. Nevertheless, we provide an explicit check at level 4. Of course, one could simply use the recursive definition of the toroidal block, but our brute force calculation is perhaps more transparent. Besides, it emphasizes the difference compared with the previous case. The starting point is again the OPE (33) that now takes the following form up to level 4 descendants
with the simple but non trivial coefficients
As in (41), we write
for certain coefficients functions of . At level 1, we have only
At level 2,
At level 3,
At level 4
4 Perturbative part of the prepotential at the points
The prepotential has also a perturbative part, related by AGT to the DOZZ 3-point function in the Liouville theory [75, 69, 76]. The general expression for the perturbative part of the prepotential is () [4, 5]
and is a renormalization scale. Evaluating by expanding at small and resumming, we find that at all points it is possible to write
Again, this appears to be a special feature of the points because it is not possible to give such a simple expression for at generic from (48). With a redefinition of the UV cutoff, this may be written in the following suggestive form that we shall generalize later
5 Full prepotential and generalization to -poles points
with a certain coefficient . This suggests that it is convenient to organize the total prepotential in the form
where is a polynomial in of (quasi-) modular degree . We emphasize again that the Ansatz (53) is non trivial because is a combination of the perturbative and instanton contributions. Our claim is that (53) can be truncated at maximum degree for a special set of points . At such points, the instanton partition function takes the special form, see (