Hamburger Beiträge zur Mathematik 497
Mayer-Cluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz
and Gang Yang
Fachbereich Mathematik, Universität Hamburg
Bundesstrae 55, 20146 Hamburg
Theory Group, DESY
Notkestrae 85, 22603 Hamburg, Germany
Institut für Physik, Humboldt-Universitüt zu Berlin
Newtonstrae 15, 12489 Berlin, Germany
In  Nekrasov and Shatashvili pointed out that the instanton partition function in a special limit of the -deformation parameters is characterized by certain thermodynamic Bethe ansatz (TBA) like equations. In this work we present an explicit derivation of this fact as well as generalizations to quiver gauge theories. To do so we combine various techniques like the iterated Mayer expansion, the method of expansion by regions, and the path integral tricks for non-perturbative summation. The TBA equations derived entirely within gauge theory have been proposed to encode the spectrum of a large class of quantum integrable systems. We hope that the derivation presented in this paper elucidates further this completely new point of view on the origin, as well as on the structure, of TBA equations in integrable models.
- 1 Introduction
2 Mayer-Cluster expansion
- 2.1 Only long range interactions
- 2.2 Only short range interactions
- 2.3 Expansion of the instanton partition function
- 3 Derivation of the TBA
- 4 Generalization to quiver gauge theories
- 5 Conclusion and Discussion
- A On the integral representation form of the Nekrasov partition function
- B Useful formulas
- C Expansion by regions for instanton partition functions
- D Alternative derivation of (2.43)
A number of remarkable connections have been observed between gauge theories and integrable systems. They appear to be useful to increase our understanding of both subjects. On the one hand, using powerful integrability techniques one may hope to solve certain gauge theories non-perturbatively. On the other hand, gauge theory can help to formulate and solve integrable system. A spectacular example is given by planar super Yang-Mills theory, for which in the last ten years or so tremendous progress has been achieved in solving the theory based on the underlying integrability and on the AdS/CFT correspondence, see  for a recent review. There exists another amazing connection between gauge theories and integrable models. The gauge theories in this case are supersymmetric gauge theories. They do not need to be planar but the connection with integrable models is restricted to a special class of supersymmetric observables. In this paper we focus on an important object in this class, the so-called instanton partition function and its relation with quantum integrable systems.
In the groundbreaking work of Seiberg and Witten [2, 3], the exact solution for the low energy effective action of certain gauge theories was proposed based on holomorphicity properties and electromagnetic duality. The low energy dynamics are encoded in a single object, called prepotential . It is a holomorphic function on the Coulomb moduli space, with coordinates , and can be reconstructed from the so-called Seiberg-Witten (SW) curve and SW differential. Shortly after, it was realized that this description provides a direct connection between gauge theories and classical algebraically integrable systems [4, 5, 6], see e.g.  for a pedagogical introduction. These are essentially a complex analogue of integrable systems in the sense of Liouville.
The challenging program of obtaining the Seiberg-Witten prepotential by a direct gauge theory calculation, developed on [8, 9, 10], was finalized in . This problem was solved using powerful localization techniques. Interestingly, this calculation produced a two-parameter, called and , deformation of the prepotential. The SW prepotential can be obtained as
where we add the explicit dependence on the coupling constant , but suppress dependence on further parameters such as masses. The partition function receives tree level, one loop and instanton contributions. The latter part is usually referred to as Nekrasov instanton partition function.
The parameters correspond to a Lorentz rotation , thus the name -deformation, that encodes certain twisted boundary conditions for the four dimensional gauge theory. It was first introduced in [8, 9] in order to regularize the volume of the instanton moduli space. The -deformation can be understood in a simple way by considering the five dimensional lift of the theory, further compactified on a circle . In this set-up it is interpreted as twisting of by a Lorentz rotation while going around the circle. Introducing a two-parameter generalization of the prepotential triggered a huge progress. An important example is the connection with topological strings. Upon taking , the gauge theory partition function reproduces the topological string partition function with as genus parameter [13, 14]. The question of what is the topological string theory analog of the case leads to the definition of so-called “refined” topological strings [15, 16]. Another spectacular example of progress driven by the calculation of the Nekrasov partition function is given by the Alday-Gaiotto-Tachikawa (AGT) correspondence .
More recently, Nekrasov and Shatashvili  proposed that upon taking the limit and interpreting as Plank constant, one obtains a correspondence between supersymmetric vacua of a given gauge theory and eigenstates of the corresponding quantum integrable model. The relation between the SW prepotential and classical integrable systems is thus quantized. This is usually called Nekrasov-Shatashvili (NS) limit and will be the main focus of this paper. The central role in this correspondence is played by the so-called twisted superpotential defined as
The Nekrasov-Shatashvili proposal is that, once this function is known, the eigenstates of the quantum integrable system are classified by solutions of the following quantization condition
where is the rank of the gauge group. These equations identify the twisted superpotential with the so-called Yang-Yang (YY) function  of the quantum system. The proposed correspondence provides an efficient general mechanism to define and solve quantum integral models. Remarkably, it can be argued that the instanton part of the prepotential , defined via (1.2), can be characterized as the solution of certain non-linear integral equation of the Thermodynamic Bethe Ansatz (TBA) type . The main goal of this paper is to develope some of the ideas presented in  to give a more explicit derivation, as well as some generalizations, of such TBA equations.
The proposal above originates as some sort of extension of the so-called Bethe/gauge correspondence [21, 22]. The latter is based on the observation that the vacuum equations of two dimensional gauge theories, broken to by twisted masses, coincides with Bethe equations for integrable models. The two dimensional twisted superpotential is equal to the YY function that encodes the Bethe equations and Coulomb parameters correspond to the Bethe roots. The generators of chiral ring of the gauge theory  are mapped to Hamiltonians of the integrable model, while their expectation values mapped to the corresponding eigenvalues. In this way one obtaines a large class of integrable models whose spectrum is characterized by traditional, possibly nested, Bethe equations. Many integrable models do not belong to this class. The simplest example is given by the quantum Toda chain, see e.g. , whose classical limit is connected to four dimensional pure SYM. From the insight of the Bethe/gauge correspondence it is then natural to propose  the two dimensional twisted superpotential which corresponds to such integrable models. It is the effective low energy action for the four dimensional gauge theory subject to an -background that preserves two dimensional super-Poincaré symmetry, namely . This observation provides a stong motivation for the above correspondence. An interpretation of this correspondence was given using brane constructions in . A further essential step in understanding the nature of the relation between quantum integrable systems and gauge theories has been presented in .
The proposal of Nekrasov and Shatashvili has inspired many other studies. Let us mention a few. In [27, 28] it was shown that, similarly to the prepotential , the twisted superpotential can be obtained by calculating period integrals of a suitably deformed SW differential. This analysis is also inspired by the AGT correspondence, by which the NS limit corresponds to the semiclassical limit of Liouville CFT [29, 30].
In our work the Coulomb parameters will be assumed to be in generic positions. Extra considerations are needed if they take special values. For example in the conformal SYM with , if the Coulomb parameters are set to be equal to , where , one can quantize the corresponding integrable system to obtain a lenght spin-chain with infinite dimensional heighest weigth representations of at each site [31, 32, 33]. Its spectrum is described in terms of traditional Bethe Ansatz equations. Such developments triggered the discovery of a number of new dualities between various integrable models [34, 35, 36, 37, 38]. The NS proposal has also inspired various studies in (refined topological) string theories [39, 40, 41, 42, 43, 44, 45, 46] where the the general -background plays a crucial role.
Despite the importance of this correspondence, the precise mechanism by which the instanton part of the twisted superpotential defined in (1.2) turns out to be characterized as the solution of TBA equations is still to be elucidated. In this paper, we will fill this gap. In order to fully prove the NS’s proposal, at least in some example, one should be able to show that the same TBA equation characterizes the spectrum of the corresponding integrable model. In the case of pure SYM, which corresponds to the periodic Toda chain with sites, this was achieved in . This interesting problem will be studied elsewhere . In the following we briefly outline the main ingredients used in our derivation of the TBA equations for (1.2), as well as the structure of the paper.
As pointed out in , it is convenient to start with the contour integral form of the instanton partition function. In this representation the instanton partition function can be interpreted as the partition function of a non-ideal gas of particles. The particular structure of the two-particle interaction potential makes the study of the limit rather subtle. More precisely, this potential is the sum of a short-range (of order ) strongly attractive piece and a long-range interaction part. In Sections 2.1 and 2.2, we consider the simplified situation in which either the long- or short-range part is set to zero. In order to study these simplified partition functions in the limit, we combine a number of techniques like Mayer expansion  (a standard method in statistical mechanics) and the method of expansion by regions  (a powerful method to compute Feynman integrals in small parameter expansions). For the case with only long-range interactions, the limit turns the logarithm of the partition function into a sum over certain tree graphs. On the other hand, the free energy corresponding to only short-range interactions gives rise to the dilogarithm function Li, which can be shown either by direct residue calculation of relevant integrals or via Mayer expansion together with the method of expansion by regions.
In order to study the full partition function we find it convenient to use an iterated version of Mayer expansion, see . This expansion effectively creates a new partition function whose “new particles” are clusters of the original particles. The interaction within each cluster is governed by the short-range interaction, the one between different clusters by the long-range part. The iterated Mayer expansion thus produces an expression for the twisted superpotential as a sum over tree graphs, with vertices given by clusters. This expansion is carried over in some details in Section 2.3. The expression can be compared to high order in the instanton number with the expression coming from the solution to the TBA equation as discussed in Section 3.1, providing direct non-trivial check of the equality.
There is an elegant way to prove that this equality holds to all orders in the instanton counting parameter . It is based on rewriting the grand canonical partition function of the non-ideal gas in terms of a -dimensional path integral. The analysis needs some special care as the potential has an unusual feature of depending in a singular way on , which is identified with in this case. A slight modification of the argument in , together with the calculation of the contribution from the short range interactions corresponding to the dilogarithm, shows that the instanton partition function in the limit is obtained by the saddle point evaluation of the path integral. The saddle point equations are nothing but the TBA equations. This is explained in details in Section 3.2.
In Section 4 we present a generalization of the TBA equations corresponding to quiver gauge theories. More precisely, we consider quivers characterized by a Dynkin diagram of , or type. The twisted superpotential for such theories is shown to satisfy a set of coupled TBA equations with one equation for each node of the quiver and couplings corresponding to edges in the quiver. The derivation is a simple extension of the one for the single gauge group case. This is so as the short range interaction, responsible for the clustering of particle, is non-vanishing only for particles corresponding to the same gauge group factor in the quiver.
In order not to overload the main text, in Appendices we include some review material together with a few technical points concerning the derivation. A review of the contour integral form of the instanton partition function is given in Appendix A. Some useful formulas are collected in Appendix B. A discussion of the method of expansion by regions is given in Appendix C. In Appendix D, we present an alternative derivation of a tree graphs expansion of the instanton partition function.
The full partition function is a product of three terms . In this paper we will be only concerned with the study of the instanton part . For this reason from now on it will be simply denoted by .
1.1 NS’s correspondence
To complete the introduction, we present the integral representation of the instanton partition function for pure super-Yang-Mills and the corresponding TBA equation. The derivation of the TBA starting from the gauge theory expression of the instanton partition function is the main goal of the paper.
Instanton partition function
The instanton partition function can be written in a contour integral representation as
where , , and
The origin of this expression is reviewed in Appendix A. The parameters entering this integrals, namely and are taken to be real with a small positive imaginary part . The domain of integration above should be understood either as a real slice integration or equivalently, upon closing the integration in the upper-half plane, as a multiple contour integral. In Appendix A.4 we review how the residue evaluation of (1.4) reproduces the representation of the instanton partition function as sum over -tuples of Young diagrams.
We emphasize that the precise form of and does not affect the derivation presented in this paper. This is the main reason why the generalization to quiver gauge theories is rather straightforward. On the other hand the presence of the factor in , which have a particularly singular limit for small, will play a crucial role and will be responsible for the appearance of the dilogarithm function in the TBA.
2 Mayer-Cluster expansion
As mentioned in , the contour integral form of the instanton partition function (1.4) can be interpreted (for each ) as the partition function of a one dimensional non-ideal gas of particles , subject to an external potential and a pair-wise interaction potential respectively given by
Upon summing over the number of particles , the instanton partition function takes the form of a grand canonical partition function. The free energy of this gas can be studied by Mayer expansion techniques  (see for example  for a nice introduction), as pointed out in . In this section, we will perform such kind of expansion in full details. The limit of appears to be rather subtle. In order to perform this limit we need to face the problem of studying the leading behavior of some multiple integral where a parameter is small. It turns out that this can be conveniently studied by employing the method of expansion by regions  discussed in Appendix C. We introduce this method to provide a unified framework to study certain integrals, but we stress that all the result of this section are also obtained without employing such technique.
The reason of such decomposition is as follows. The factor corresponds to a pair-wise interaction which is strong and attractive at distances of order and rapidly decreases at large distances. The remaining factor corresponds to a pair-wise interaction which is different from zero only at distances of order . Thus (2.2) corresponds to splitting the potential into short- and long-range parts. The natural candidate to study such kind of potentials is the so-called iterated Mayer expansion . As we will see in Section 2.3, this effectively creates a new grand canonical partition function whose “new particles” correspond to clusters of the original particles. We will start by considering some simplified situation.
2.1 Only long range interactions
Let us first consider a simplified version of (1.4) without the factors
The basic idea of Mayer expansion, see e.g. , is to introduce the function as
and expand the interaction products as
Each monomial in the right hand side of this equation can be visualized as a graph (not necessarily connected) on the set . More precisely, each particle in corresponds to a vertex, and to each factor we associate an edge between particle and in the corresponding graph. More explicitly,
As an example for the case, we have the expansion in terms of graphs shown in Figure 1. It is clear from the left hand side of (2.5) that there are no multiple edges between two vertices, or edges connecting one vertex to itself.
This expansion is particularly useful, as the logarithm of the grand canonical partition function can be formally given as a sum over connected graphs  (see for example Appendix A of  for a simple derivation):
Here denotes the collection of connected graphs on the set and belongs to the graph if there is an edge between the vertices and . Graphs up to four points are shown in Figure 2.
We stress that (2.7) is an exact relation as a formal power series
where is defined in (1.11). As each factor of contributes one power of , the leading contribution to the sum in (2.7) is given only by connected tree graphs . Such graphs have the minimal number of edges () among the connected graphs. The sum of these tree graphs can be shown to be convergent. Collecting the powers of we conclude that
Graphs up to five points are shown in Figure 3. Notice that the leading behaviour of the free energy (2.9) is at order . This fact may be not obvious from the definition (2.3) where for each the leading contribution is proportional to . We will see that the similar behaviour applies to the more complicated situations analized in the following.
2.2 Only short range interactions
The main achievement of this subsection is to show that in the limit of small the logarithm of (2.10) is given by
We will see that, as opposed to the case of long range interactions considered in the previous subsection, the right hand side of (2.11) is not given by summing over tree graphs but by a single local term. As will be explained in details in Section 2.3, this is the underlying mechanism by which the short-range interactions turn a group of particles into a single effective particle. The result (2.11) explains why the dilogarithm function appears in the TBA. It will also be an essential input in the all order proof presented in Section 3.2. Considering its importance, we will prove (2.11) in two different ways.
Dilogarithm from a sum over residues
In this subsection we will prove (2.11) by direct evaluating (2.10) and (2.11) as a sum over residues. We find it instructive to first consider the partition function for the gauge theory, i.e. we evaluate (2.10) for
compare to (1.6). In (2.12) and are real and we wrote explicitely the prescription. For each in (2.10), there is only one residue (up to permutation of the integration variables) in the upper half plane, compare to the general discussion in Appendix A.4. It is given by . It follows that
One can also directly calculate the one dimensional integral
For the more general theory, it is easy to classify the poles contributing to (2.10) following the same reasoning as in Appendix A.4. For fixed , residues are classified, up to permutation of the particles, by a set of integers such that . The corresponding pole is given by
and . As opposed to the full partition function (1.4), for which residues are classified by -tuple of Young tableaux with a total number of boxes equal to , in the simplified integrals (2.10) only Young tableaux with one row, whose length is denoted by , contribute. It is straigthforward to calculate and collect all residues, see Appendix B, to obtain
and . The structure of the result (2.16)-(2.17) represents a simple generalization of the computation (2.13). The limit of the logarithm of (2.16) can be readily obtained using the relation (B.7). One recognizes that
By Mayer expansion and separation of regions
We now calculate (2.10) for small by first applying Mayer expansion and then the so-called method of expansion by regions. The first step is to decompose the interaction factor as
To show that this gives (2.11), we need to evaluate the integrals entering (2.20) as a Laurant series in . We are actually interested only in the terms with leading negative powers of . We use a powerful method, usually applied to the evaluation of Feynman integrals, called expansion by regions introduced in . The method goes as follows: divide the integration domain into regions and expand the integrand in a Taylor series in small parameters, extend the integration to the full domain of integration, set to zero scaleless integrals. This method turns out to be particularly efficient to calculate the leading term for the integrals (2.20). Concerning these contributions, we do not need to apply the somewhat subtle point above. A more detailed discussion of this method will be given in Appendix C.
To identify the set of relevant regions, we note that the Taylor expansion of the interaction term starts at order except for the region in which is of order . It is then natural to expect that regions are classified as follows. Let denotes the set of grouping of labeled particles in clusters, see figure 4 for the case. For each grouping we define the corresponding region as
The next step is to Taylor expand the integrand in each region. We denote by the operation of Taylor expaning in the region corresponding to the grouping . We have
the product on the right hand side runs over the clusters in the grouping and denotes the number of particles in the cluster . From (2.22)-(2.23) it is clear that, for each , the leading contribution to (2.20) comes from the region in which all are in the same cluster. Indeed, if there were two or more clusters, the factor, with in different clusters, would decrease the power of , compare to (2.22). An explicit example of this expansion of is given in appendix C. As we will shortly see, the leading contribution to (2.20) starts at order for each . This fact is not obvious from the form (2.20).
By further separating the integration over the “center of cluster” coordinate as
where we applied the change of variables
one can rewrite (2.20) as
Notice that in (2.27) all connected graphs, independently on the number of edges, contribute to the leading term. The next observation to be made is that
The validity of this statement can be easily argued as follows. For each connected component in we can define its center as the average of the in that connected component. The integrand in (2.28) does not depend on the distance between the centers. As the integral is calculated by residue, it trivially vanishes in this case as, after integrating trivially the delta function, the -dimensional residue is absent. The identity (2.28) implies that in (2.27), we can replace the sum over connected graphs with the sum over all graphs. Finally using the relation
where is the set of all graphs on , we conclude that , where, using (2.19),
2.3 Expansion of the instanton partition function
We are now ready to consider the full instanton partition function (1.4). We will combine the considerations from the previous sections. In Section 2.1 we learned that the limit singles out certain tree level graphs. In Section 2.2 we learned that the factors produce the effect of combining particles into clusters. To exploit the combination of these two mechanisms in the most transparent way we find it convenient to use the so-called iterated Mayer expansion advocated at the beginning of our analysis. We start by reviewing this expansion. Based on this iterative expansion, by combining the discussion from the two previous sections, one obtains a tree graph expansion for the full instanton pationtion as given in Section 2.3.2. In Appendix D we show that the same result can be obtained by first applying the method of expansion by regions to the original partition function and then exploiting some combinatorics to conclude that only certain connected graphs contribute to the free energy.
Iterated Mayer expansion
We start with a review of the iterated Mayer expansion . Consider the partition function
where, as in (2.2), we split the pair-wise interaction potential in a short and long range part. Next we introduce
where denotes a set of particles or, in other words, a cluster. Iterated Mayer expansion is the statement that the free enery can be expanded as
The sum in (2.35) is taken over : groupings of (labeled) particles into clusters. As the notation may need some time to be digested, in Appendix B.3 we spell out the definitions for . Note that although the cluster here is in a different context, the picture of grouping is similar to that used before in expansion by regions, see for example figure 4 for case.
Iterated Mayer expansion for the Nekrasov partition function
We can apply the iterated Mayer expansion reviewed in Section 2.3.1
to the full partition function, compare (2.32) to (1.4), (2.2).
Once this is done we need to evaluate the leading contribution for small to
integrals of the type (2.35).
The crucial observation is that the Taylor expansion
is valid everywhere in the domain of integration
where refers to next to leading contributions in . We point out that, in analogy with (2.9), only connected tree graphs contributes to (2.38). As opposed to (2.9), now they are tree graphs on the set of clusters rather then the set of fundamental particles. Notice that we did not expand the factor. The integration in (2.35) is still over variables.
The next step is to explicitly perform the integration over the distances of particles within the same cluster. This turns out to be essentially the same as in Section 2.2.2. For each cluster we introduce a “center of cluster” coordinate as in (2.24). It is clear that the entering the factors , see (2.37), are independent of the “center of cluster” coordinates . Using this observation and (2.38) we write
As extensively discussed in the previous sections, the leading contribution in of integrals of the type (2.39), can be obtained by neglecting the deviation from the center of cluster coordinate in the functions and . Doing so, (2.39) becomes
In the limit of small , the summands in (2.35) depend only on the sizes of the clusters corresponding to . Converting the sum in (2.35) from a sum over groupings of particles to a sum over the number of clusters and their sizes produces a factor . Assembling the pieces together we finally arrive at the main result
where we replaced with . We emphasize once again that the sum has been rearranged from a sum over particles, weighted by , to a sum over clusters. More precisely, for each there are clusters and -factors. For one immediatly recovers the dilogarithm
For one has
The result (2.43) can be visualized diagrammatically in a simple way. For each in the sum one draws all three graphs on the set . To each node of the graph is associated an integer . Once the diagrams are drawn, the corresponding integrals can be written using the following “Feynann rules”
Here we used a dashed square to indicate that the propagator is associated to the edge only.
The graphs contributing to are given by
The first two graphs correspond to (2.44) and (2.45) respectively. Notice that each integral obtained by applying the Feynman rules, should be multiplied by an overall factor, which is explicit in (2.43). Graphs with more vertices, upon dressing the vertices with positive integers, are as in Figure 3.
3 Derivation of the TBA
In this section we show that the logarithm of the instanton partition function in the limit, whose structure have been studied in last section, satisfies TBA equations. We will first provide a perturbative check to a very high order in the instanton counting parameter , and then present an all order proof based on rewriting the instanton partition function as a -dimensional path integral.
3.1 Perturbative expansion
Let us recall the expression of the superpotential coming from the TBA
compare to (1.9) and (1.10). The basic observation is that TBA equation has a natural expansion in terms of tree graphs, therefore one can compare it directly with the Mayer expansion of the instanton partition function (2.43).
One writes and as
can be solved recursively via the TBA equation, for example up to the first two orders
Substituting the expression for in and collecting the terms at a given order of , one obtains, here up to order