Let collectively denote equivariant parameters of the torus action on the moduli space of matrix equations of ADHM type. In general, they consist of the equivariant parameters of the torus action on the (flat) space-time coordinate (the background parameters of Nekrasov), the Cartan subgroup of the gauge symmetry (the Coulomb moduli parameters) and of the flavor symmetry (mass parameters). We can identify the equivariant group of a point with the ring of Laurent polynomials in the equivariant parameters . Hence, the equivariant character at the fixed points takes a value in .

Recall that a partition 0)$]]> is a non-increasing sequence of positive integers such that for finite . It is useful to represent in terms of the Young diagram. We denote , which is the total number of boxes (cells) in the corresponding Young diagram. We can consider a higher-dimensional generalization or a -partition , 0}^{d-1}$]]>, which is an array of positive integers with the (obvious) higher-dimensional generalization of the non-increasing condition; for example, and when , and for only a finite set of (see Fig. 1). When and , this is usually called a plane and a solid partition, respectively. We define , which means the volume (the number of boxes, cubes, ) of the -partition . It turns out that the fixed points of the toric action on are isolated and in one-to-one correspondence with the set of -tuples of -partitions , and that is identified with the instanton number . Thus, in the sector of instanton number , we can reduce the quiver matrix model to a statistical model with the configuration space , where the equivariant character at each fixed point gives the Boltzmann weight of the model.

It is interesting that the Boltzmann weight derived from the ADHM matrix model takes the form of the plethystic exponential (see Sect. 2 for a definition) . We define the topological partition function by a weighted sum over the total configuration space where, introducing the box-counting parameter , we multiply the volume (the number of boxes, cubes, ) of -partitions as the additional Boltzman weight:

That is, if we identify the instanton number as the particle number, the topological partition function corresponds to the grand canonical ensemble in statistical mechanics. The phenomena on which we will focus in this article is that in the computation of the topological partition function, the expectation value of the plethystic exponential is again expressed by the plethystic exponential:

Since the plethystic exponential can be regarded as the character of the symmetric algebra of a -module , this is an example of “super”-integrability that the expectation value of the character gives another character, which we encounter typically in the matrix model and plays an important role for extending the realm of symmetric functions [11,12].

When , with the computation of the topological partition function we may associate equivariant (or -theory) vertices [13], which are generalizations of the refined topological vertex [14,15]. In fact, in an appropriate limit of the background parameters , the equivariant vertex reduces to the refined topological vertex. Since the refined topological vertex is characterized as the intertwining operator of the quantum toroidal algebra of [16], it is tempting to expect some quantum algebras behind the “super”-integrability of Eq. (1.2). Furthermore, since physically the partition function in Eq. (1.1) is nothing but the generation function of the numbers of BPS states, this seems to be along the same line of BPS / vertex operator algebra correspondence, the correspondence of the algebra of BPS states with the chiral algebra of some two-dimensional CFT. It may be interesting to look at a cohomological Hall algebra associated with the quiver of ADHM type [17].

The paper is organized as follows. In the next section we introduce the plethystic exponential. We can regard it as a character of the symmetric algebra and hence it plays a significant role in this article. After presenting ADHM-type matrix model equations coming from the BPS condition of the -brane system in Sect. 3, we discuss a matrix model formulation or the measure for eigenvalues of matrices in Sect 4. The measure is given in terms of the plethystic exponential, and hence is naturally expressed by power sum functions of eigenvalues. In Sect. 5 we compute the equivariant character of the tangent space at the fixed points. Finally, we present the plethystic forms of the partition function in Sect. 6. In each section after Sect. 4 we first review the well-established case of (the original ADHM equation) and then try to generalize it to higher dimensions. From the viewpoint of mathematics, one of the crucial points is that though the fixed points of the torus action are still isolated and labeled by a higher-dimensional generalization of the partition, the tangent space at each fixed point is not smooth any more and it is defined only virtually.

I would like to dedicate this article to the memory of Prof. Tohru Eguchi, who passed away last year. My collaboration with him started when both of us participated in the inaugural project at the Newton Institute in the summer of 1992. Our interest was in the interplay of a topological string as a two-dimensional topological quantum field theory and integrable systems such as algebra, the Toda lattice hierarchy [18–20], which became one of the main themes in my research afterwards. After almost a decade I had a second chance of collaboration on the five-dimensional lift of Seiberg–Witten theory, the Nekrasov partition function, and topological strings [21–23], which are closely related to the subject reviewed in the present paper. I am very grateful to Eguchi-san for these fruitful and inspiring collaborations. Though I was not his student, I learned how to enjoy research through collaboration with Eguchi-san.

Let be the Cartan variables of or the eigenvalues of matrices. The equivariant integration over the instanton moduli space is reduced to a contour integral,

where we have divided the integral by the order of the Weyl group (we will order the eigenvalues) and is the Cartan variables for symmetry coming from -branes. The parameters are -background parameters or the equivariant parameters of the torus action on . The full partition function is

and we will see, by introducing the power sum function of the eigenvalues, that the integrand allows a plethystic expression. Note that we may identify as an effective action of the matrix model. The contributions to are evaluated as follows:²

Let us rescale the variable to make the last two contributions symmetric:

In terms of the function

we can write the integrand as follows:

Now, in terms of the power sum variables , we can rewrite the measure for the contour integral in a plethystic form:

Using the holonomy variables

of gauge fields, we have

where we have introduced . By the change of variables

we can eliminate linear terms in to obtain

Thus, we have

We may eliminate by the renormalization of the instanton expansion parameter . The universal factor shoud be identified with the perturbative factor.

In the Abelian case the contour integral in Eq. (4.15) is related to the inner product for Macdonald polynomials [39]. To see this, we should note that the poles of the contour integral are labeled by partitions with , and the positions of the poles are given by

where , and we have for the Abelian case. Hence, the power sum takes the following values at the poles:

Thus, we recover the topological locus:

This also explains the implication of the change of variables in Eq. (4.13). In summary, after the contour integration we have

Note that the measure factor coincides with the -deformed Vandermonde determinant

with . This is employed to define the inner product on the space of symmetric polynomials that leads to Macdonald polynomials [40].

The integrand of the residue integral can be expressed in terms of the plethystic exponential, and taking the logarithm we may recognize the “effective” action for the eigenvalues, which is in turn expressed by the power sum. Then, the integral can be related to the inner product for the Macdonald polynomials. This also means that the effective action is bilinear in the power sums (the free boson operators). To construct a refined topological vertex we have to insert a vertex operator. It is curious that the insertion induces the interaction term in the effective action.

From the ADHM-type conditions, we can similarly obtain a contour integral representation of the partition function with instanton number :

where

It is curious to see the role of the pole at in the contour integral. An analogous computation to the case of leads to the following plethystic form of the measure:

The crucial change here is the relative sign in the linear terms, which prevents us making a complete square by a change of variable, as in Eq. (4.13). The flip of the relative sign causes an asymmetry in exchanging the positive modes and the negative modes. As discussed in the next section, this seems to be related to the fact that, in contrast to the case of , we do not have a hyperKähler (holomorphic symplectic) structure any more when .

We obtain

As argued in the next section, we have to choose a “chiral-half” of the full Euler character, which depends on the ordering of the set . Here we choose by taking as a “preferred” direction. As argued in Ref. [33], due to the choice of the ordering, we should be careful with the order of the contour integral.

Using the Calabi–Yau condition , we can obtain a plethystic form of as follows:

Remarkably, this is quite close to Eq. (4.24). Since this is a “chiral-half” of the full Euler character, only the negative modes appear.

The fixed points are labeled by an -tuple of partitions (colored Young diagrams) and

where the positive contributions and come from and , from , while the negative ones from the -term constraint and from the gauge symmetry. The difference in the numbers of positive coefficients and negative coefficients is , which is exactly the (complex) dimensions of the tangent space. After cancellations only positive terms survive, and when it has a nice combinatorial formula [24]:

where and are the arm and the leg length of the box in the Young diagram . Note that in the Abelian case the dependence on disappears. In the non-Abelian 1)$]]> case the fixed points are labeled by an -tuple of Young diagrams , and we need the arm and leg length of the box with respect to a second Young diagram :

Then, an explicit formula for the equivariant character is

Introducing the polarization⁴

and the notation , we can express the character as follows:

Note that is the scaling factor of the symplectic form . This decomposition of the equivariant character reflects the fact that the moduli space of the ADHM matrix model is an example of Nakajima quiver varieties, which is defined as a hyperKähler quotient. The relevant quiver is called the Jordan quiver, which consists of a single vertex with a single loop (Fig. 2). More precisely, it is the framed Jordan quiver with a framing of . When or in gauge theory, the associated quiver variety is nothing but the Hilbert scheme of -points on , where is physically the number of -branes or the instanton number of the anti-self-dual connection. From this viewpoint the moduli space has the structure of a cotangent bundle and the polarization represents the contribution of the base space described by the Jordan quiver, where we subtract coming from the gauge symmetry. Then the second term corresponds to the fiber of the cotangent bundle, and the multiplication of the weight is necessary.

The equivariant character in Eq. (5.3) is also derived from the equivariant Chern character of the universal bundle [41,42]. The virtue of this derivation is that it is applicable for more general gauge groups of type and [43]. To construct the universal bundle , let be local coordinates on the moduli space of instantons. The tangent space of the moduli space is spanned by solutions to the linearized equations with a gauge-fixing condition. Let denote a basis of the tangent space at . For a family of instantons parametrized by , we have

Since the derivative of does not necessarily satisfy the gauge-fixing condition we need a compensating gauge transformation . With an appropriate choice of the gauge-fixing condition, for example , we can find a unique . Combining with the parameter of the compensating gauge transformation , we can define a one-form which can be regarded as a connection of the universal bundle on whose fiber is the fundamental representation of . In the following we fix a complex structure of the space-time and identify . Then the spinor bundle on is naturally identified with the space of -forms on .⁵ With this identification the Dirac operator is translated to the operator.

The equivariant Chern character of the universal bundle is computed as the Euler character of the complex

where

and the ADHM condition guarantees that Eq. (5.10) is a complex: . One can also check that [24]. Taking the alternating sum, we obtain

Now, the equivariant version of the index theorem tells us that the equivariant index of the Dirac operator coupled with the adjoint bundle is⁶

where the equivariant version of the Todd class is

where

are the equivariant Chern roots of the tangent bundle to , given by equivariantly closed two-forms. We should use the Chern class of , because we consider the adjoint bundle whose fiber is the adjoint representation of . It shoud be easy to generalize the computation to the bi-fundamental representation. The integration over the space-time corresponds to the push-forward for the projection and can be evaluated by the localization by the torus action , whose unique fixed point is the origin . The Hamiltonian of the torus action is , and the localization theorem for the equivariant closed forms gives

where the first term, which survives even , is regarded as a perturbative part.

The fixed points are labeled by an -tuple of plane partitions and

where , , and come from dynamical matrix variables , , and , while comes from the -term constraints and from the gauge symmetry. Finally, comes from the constraint . When we impose the Calabi–Yau condition , the character is anti-self-dual, , which is a consequence of the Serre duality. By the anti-self-duality, the measure on the space of plane partitions becomes uniform (up to sign); in fact it is . Hence the partition function reduces to the MacMahon function.

Now, the analogue of the polarization in Eq. (5.7) in is

and we set ; then, we have

Note that the relative sign between and should be negative for odd . Consequently, the interpretation of the decomposition in Eq. (5.19) is rather different from the case . That is, Eq. (5.19) reflects what is called symmetric obstruction theory in mathematics, where the first term corresponds to the deformation space of matrix variables coming from the framed quiver with a single vertex and three loops with the subtraction of gauge symmetry, while the second term is the contributions from the obstruction space, which are represented by anti-ghosts for constraints and the secondary ghost for the gauge symmetry. The symmetric obstruction theory gives a moduli space of virtual dimension zero.

The fixed points are labeled by an -tuple of solid partitions and

where and come from and , while from the -term constraints and from the gauge symmetry. Note that what we called the polarization in lower-dimensional cases is obtained as the character of the deformation complex of the ADHM-type condition. In other words, Eq. (5.20) is a “chiral-half” of the full Euler character :

where we have used . Contrary to the odd-dimensional case, the full character is self-dual, . To define a “chiral-half” of the full Euler character, we use the real structure of , which is allowed by the Serre duality of . This seems consistent with the idea that theory is a holomorphic version of the Donaldson theory [10]. By taking a “chiral-half” of the full Euler character we consider a square root of the tangent space and hence there is an ambiguity of the choice of sign. We can fix it locally, but the global consistency is a non-trivial issue. Mathematically this is the problem of the orientability of the moduli space.

Using the formula in Eq. (5.4) the partition function of theory with adjoint matter is

where the parameter is the equivariant (mass) parameter for the flavor symmetry of the adjoint matter. Thus, physically is the Nekrasov partition function of theory. We can show that it has the following plethystic form [39,44–46]:

There are two natural limits for : the decoupling limit and the massless () limit . In the latter case the measure on the space of partitions is uniform and we obtain the generating function of the counting of partitions:

On the other hand, in the former limit the measure becomes the (refined) Plancherel measure and corresponds to the pure theory, which is geometrically engineered by the conifold geometry:

When it gives a generalized McMahon function,

where plays the role of the Kähler parameter of the conifold. Thus, this example gives a kind of interpolation between the counting of partitions and plane partitions.

Equation (6.2) can be deduced as follows.⁷ First we note the “removable” boxes of a non-empty partition have vanishing leg and arm length , because if they have non-empty leg or arm, we cannot remove them from the diagram. From the measure factor in Eq. (6.1) we see that the measure on the non-empty partition has zeros at and . Note that these zeros are preserved under . Thus, we conclude that has the factor . Moreover, when the measure is independent of and . Hence, we arrive at

Now let us specialize and take the limit . Then

where is the hook length. On the other hand,

in this limit. Hence, we find that

We can also prove Eq. (6.2) by assuming the invariance of the topological string amplitudes under the change of the preferred direction of the refined topological vertex [46]. When we deduce the refined topological vertex from the equivariant vertex, the preferred direction is related to the ways the limit can keep the Calabi–Yau combination of constant (see the next subsection). This reminds us of the fact that the perturbative string theory can be obtained by taking appropriate limits of theory.

The theory can be regarded as quiver gauge theory. As we have seen in the last section, the quiver for the ADHM equation is the double of the framed quiver, and when the framing is the Nakajima variety is nothing but the Hilbert scheme of a point on . This coincidence seems to be the origin of the symmetric property of the topological partition function derived above.

For the Abelian case , by the localization theorem the partition function is given by the summation over the plane partitions:⁸

where is defined by

Here, is the multiplicity of the character (weight) of the torus that acts on the moduli space. Note that gives the character of symmetrized symmetric products:

where .

It turns out that the partition function allows a plethystic expression [25,36]:

where

with . It is tempting to identify the parameter with the mass parameter in the previous example. In fact, both are related to the weight of the line bundle over and , where the total space is six and ten dimensions, respectively. However, an important difference here is that the decoupling limit is not well defined, while the Calabi–Yau limit is still well defined. It seems this is related to the fact that the tangent space is smooth in the case, but it is singular (the tangent space only has a meaning as a virtual bundle) for 2$]]>. In the Calabi–Yau limit the partition function reduces to the MacMahon function,

Another interesting limit is the “refined topological vertex limit,” where we take with and , keeping constant. In such a limit, corresponds to a preferred direction of the refined topological vertex. From the relation

we find

which can be identified with the refined conifold amplitude with the Kähler parameter .

When 1$]]> (the non-Abelian case) the fixed points are labeled by an -tuple of plane partitions (colored partitions) and the topological partition function is

where

One of the significant properties of so defined is that it is completely independent of the equivariant parameters for the framing torus, or the Coulomb branch moduli, which physically means the distance of the -branes. For lower instanton numbers this crucial property in proved in Ref. [47] by checking the vanishing of residues at the possible poles of as a rational function in the equivariant parameters . Quite recently, it was proved for arbitrary by examining the contour integral representation of the partition function discussed in the last section [49]. Once we know that is independent of , we can evaluate the partition function in a judicious scaling of , for example with [47]. Then we can see that, for [49],

which implies

Hence, we have a factorization of the partition function in Eq. (6.18) into a product of partition functions with shifted instanton number counting parameters [29,48]. This factorization surely relies on the independence of of the Coulomb moduli, and is related to the orbifold action of on the transverse direction to the -branes.

In fact, we can use the following identity for generic variables to derive the partition function in Eq. (6.13) from Eq. (6.21):

where is an th root of unity, . The second equality of Eq. (6.22) is just a simple consequence of

But, as pointed out in Ref. [29], it is amusing to note that the first equality of Eq. (6.22) allows a geometric interpretation, though it can also be checked by induction on . To see the geometric meaning let us consider the asymptotically locally Euclidean (ALE) resolution of the -orbifold of , where acts by . We can compare the equivariant index of the Dirac operator before and after the resolution. Recall that the equivariant index of the Dirac operator on is simply

Then, we can recognize the middle of Eq. (6.22) as the orbifold version of the equivariant index. On the other hand, on the ALE space there appear fixed points over the origin, which is the original fixed point of the torus action. For example, when , the ALE space is nothing but the Eguchi–Hanson space [50], which is isomorphic to the cotangent bundle of . Thus we find two fixed points: the north and the south poles of . The weights at these fixed points are exactly those that appear on the left-hand side of Eq. (6.22). Hence, it is the blow-up version of the index. Mathematically, the equality of the two versions follows from the fact the the fiber of the resolution is compact [29].

Since we already know that the partition function has the plethystic form in Eq. (6.13), we can compute a plethystic form of the partition function in Eq. (6.13) as follows:

where

By applying Eq. (6.22) with , , we finally obtain

In Ref. [13] the equivariant (-theory or -theory) vertex is defined by

where the summation is taken for the plane partitions with a fixed asymptotic condition . Note that and have to be regularized by taking the edge contributions into account. It was recently shown that if one of three partitions is empty, has a plethystic expression [51]. It will be very interesting to see if this property holds for the full vertex.

The following plethystic form of the partition function is conjectured in Refs. [28,29]:

where

Note that due to the Calabi–Yau condition . are the Coulomb branch parameters (associated with the position of the -branes) for the gauge symmetry , and is the mass parameter (associated with the position of the -branes) for the flavor symmetry . It is remarkable that the final result only depends on the ratio , which is comparable to the fact that the partition function is independent of the Coulomb moduli in six dimensions.

For theory we can take by choosing the position of a single brane as the origin. With we find

It seems that we cannot produce the uniform measure on the space of solid partitions by tuning parameters. This is consistent with the fact that there is no known plethystic formula for the generating function of the counting of solid partitions. For example, the massless limit gives the trivial result .

Let us put and , and take the limit ; then

It is the MacMahon function that appears in this limit. The generating function of the counting of the solid partition never appears.

It is interesting that a reduction to six dimensions is achieved by tuning the mass parameters (the positions of the -branes) which triggers a tachyon condensation of the – system to -branes [29]. The condition is , which gives , and we obtain

where . Up to sign, this agrees with Eq. (6.27).