-deformed Matrix Models and 2d/4d correspondence
Imperial Collage London, The Blackett Laboratory, Prince Concert Rd, London, SW7 2AZ, UK
We review the -deformed matrix model approach to the correspondence between four-dimensional gauge theories and two-dimensional conformal field theories. The -deformed matrix model equipped with the log-type potential is obtained as a free field (Dotsenko-Fateev) representation of the conformal block of chiral conformal algebra in two dimensions, with the precise choice of integration contours. After reviewing various matrix models related to the conformal field theories in two-dimensions, we study the large limit corresponding to turning off the Omega-background . We show that the large analysis produces the purely gauge theory results. Furthermore we discuss the Nekrasov-Shatashvili limit () by which we see the connection with the quantum integrable system. We then perform the explicit integration of the matrix model. With the precise choice of the contours we see that this reproduces the expansion of the conformal block and also the Nekrasov partition function. This is a contribution to the special volume on the 2d/4d correspondence, edited by J. Teschner.
- 1 Introduction
- 2 Integral representation of conformal block
- 3 Large limit
- 4 Nekrasov-Shatashvili limit
- 5 Finite analysis
- 6 Conclusion and discussion
- A Integral formulas
Matrix models have played a crucial role in the studies of theoretical physics. It has turned out that these models compute quantum observables or the partition function of quantum field theory  and two-dimensional gravity [2, 3] (see references therein). Rather recent examples are a one-matrix model which describes the low energy effective superpotential of four-dimensional supersymmetric gauge theory , and the exact partition functions of supersymmetric gauge theories in various dimensions [5, 6] which are itself written as matrix models (Reviews can be found in [V:5, V:9] in this volume). These have already shown the usefulness of the matrix model in theoretical physics.
This paper reviews the matrix model introduced by Dijkgraaf and Vafa  which was proposed to capture the non-perturbative dynamics of four-dimensional supersymmetric gauge theory and two-dimensional conformal field theory (CFT). This proposal is strongly related with the remarkable relation between the Nekrasov partition function  of four-dimensional supersymmetric gauge theory and the conformal block of two-dimensional Liouville/Toda field theory found by . (We refer to this relation as AGT relation [V:3].) The four-dimensional gauge theory is obtained by a partially twisted compactification of the six-dimensional theory on a Riemann surface , [V:1], and the associated conformal block is defined on the same Riemann surface where vertex operators are inserted at the punctures [V:11].
The conformal block has several different representations. The one we focus here on is the Dotsenko-Fateev integral representation [11, 12], which will be interpreted as -deformed matrix model. This integral representation has long been known, but regarded as describing degenerate conformal blocks where the degenerate field insertion restricts the internal momenta to fixed values depending on the external momenta. However the recent proposal by  is that it does describe the full conformal block. The point is the prescription of the contours of the integrations which divides integrals into sets of integral contours whose numbers are (with where is the size of the matrix.) In other words, in the large perspective, we fix the filling fractions when evaluating the matrix model. This gives additional degrees of freedom corresponding to the internal momenta.
This matrix model plays an interesting role to bridge a gap between four-dimensional gauge theory on the background and two-dimensional CFT. In addition to the correspondence with the CFT mentioned above, this is because the matrix model has a standard expansion in . The large limit in the matrix model corresponds to the limit on the gauge theory side. Therefore, the matrix model approach is suited for the expansion of the Nekrasov partition function.
In section 2, we derive the -deformed matrix model with the logarithmic potential starting from the free scalar field correlator in the presence of background charge. The case of the Lie algebra-valued scalar field is described by the -deformation of the quiver matrix model [13, 14, 15]. We further see that the similar integral representation can be obtained for the correlator on a higher genus Riemann surface. These matrix models are proposed to be identified with the Nekrasov partition functions of four-dimensional (UV) superconformal gauge theories and the conformal blocks.
In section 3 we analyze these matrix models, by taking the size of the matrix large. The leading part of the large expansion is studied by utilizing the so-called loop equation. We identify the spectral curve of the matrix model with the Seiberg-Witten curve of the corresponding four-dimensional gauge theory in the form of [16, 10]. We then see evidence of the proposal by checking that the free energy at leading order reproduces the prepotential of the gauge theory.
In section 4 another interesting limit which keeps one of the deformation parameter finite while in the four-dimensional side will be analyzed. This limit was considered in [17, 18, 19] to relate the four-dimensional gauge theory on the background with the quantization of the integrable system. We will see that the -deformation is crucial for the analysis, and that the matrix model indeed captures the quantum integrable system.
In section 5, we will perform a direct calculation of the partition function of the matrix model keeping all the parameters finite. We compare the explicit result of the direct integration with the Virasoro conformal block and with the Nekrasov partition function.
2 Integral representation of conformal block
In this section, we introduce the -deformed matrix model as a free field representation of the conformal block, and the proposal  that the matrix model is related to the four-dimensional gauge theory. In section 2.1, we see the simplest version of this proposal: the -deformed one-matrix model with the logarithmic-type potential***The matrix model with a logarithmic potential was first studied by Penner  related to the Eular characteristic of a Riemann surface. obtained from the correlator of the single-scalar field theory on a sphere corresponds to the four-dimensional linear quiver gauge theory. In section 2.2, we will introduce the quiver matrix model corresponding to the gauge theory with higher rank gauge group. We will then generalize this to the one associated with a generic Riemann surface in section 2.3.
2.1 -deformed matrix model
In , it was found that the conformal block on a sphere with punctures can be identified with the Nekrasov partition function of superconformal linear quiver gauge theory. We will first review the integral representation of the conformal block, first introduced by Dotsenko and Fateev [11, 12], and interpret it as a -deformed matrix model [21, 22]. (See  for a review of the relation between the matrix model and the CFT.) We then state the conjecture among the matrix model, the Nekrasov partition function, and the conformal block.
We start with the free scalar field
with the following commutation relations
Thus, the OPE of is
The energy-momentum tensor is given by with the central charge .
Let us introduce a background charge at the point at infinity by changing the energy-momentum tensor
The central charge with this background is .
The Fock vacuum is defined by
The energy-momentum tensor satisfies the Virasoro constraints
Now we consider the correlator , where the vertex operator is defined by with conformal dimension . This is nonzero only if the momenta satisfy the condition . To relax the condition, let us consider the following operators
Since the integrand of each operator has conformal dimension , the screening operators are dimensionless. Therefore we can insert these operators into the correlator without changing the conformal property. The insertion however changes the momentum conservation condition, thus we refer these as screening operators. By inserting screening operators in the correlator we define
The momentum conservation condition now relates the external momenta and the number of integrals as . This adds one more degree of freedom, , to the model. Nevertheless, it is important to note that the momenta (or ) cannot be completely arbitrary because is an integer. This point will be discussed in section 5.
By evaluating the OPEs, it is easy to obtain
where is of the matrix model like form
with the following potential
We have introduced the parameter by defining . (We will use parameters and interchangeably below.) We also have taken by which the corresponding term in disappeared. While the dependence on cannot be seen in the potential, this is recovered by the momentum conservation condition
Note that the hermitian matrix model corresponds to the case because the first factor in the integrand is the familiar vandermonde determinant. Also the cases with and correspond to an orthogonal matrix and a symplectic matrix respectively. However for generic choice of , there is no such expression in terms of a matrix. This integral expression is known as ensemble or -deformed matrix model with .
It is useful to rewrite the deformed matrix model (2.10) as
where we defined . Thus the insertion of (the derivative of) the scalar field in the correlator (2.13) is written as
in the matrix model average defined by
Relation to conformal block
The proposal  is that the partition function of this -deformed matrix model can be identified with the Virasoro conformal block, and the Nekrasov partition function of four-dimensional linear quiver gauge theory. The relation to the former is
where is defined such that the is expanded in as . Here is the Virasoro -point conformal block on the sphere and defined such that . We will review this in section 5.1. The momenta are identified with the external momenta of the conformal block, as it should be. The parameters and are defined in the conformal block side in the same way as the free field theory. Thus, the only nontrivial point is the identification of the internal momenta ().
At the first sight there is no parameter corresponding to the internal momenta in the matrix model. However the prescription to identify them was established by [24, 25, 26, 27]: as we will see in section 5.1, the conformal block can be computed from the three-point functions, denoted by the trivalent vertices, and the propagators, denoted by the lines connecting the vertices, as in figure 1. The idea is that there are screening operators inserted at each vertex, with , where the momentum conservation is satisfied as
In the last equality we used the momentum conservation (2.12). This means that in the integral representation we have sets of integrals, each number of the integrals is .
The precise choice of the integration contours will be seen in section 5. Here let us see a rationale of this identification by considering the large limit shortly. The critical points of the eigenvalues are obtained from the equations of motion
Focusing on the first term, when the parameters are generic enough there are critical points. Let be the number of the matrix eigenvalues which are at the -th critical point. These critical points are diffused to form line segments by the second term. The integrals are defined such that they include these segments. Now we introduce the filling fractions , and consider the matrix model by fixing these values in the large limit. Because of the momentum conservation, we have independent degrees of freedom.
Relation to Nekrasov partition function
The relation to the Nekrasov partition function is as follows:
under the following identification of the parameters. We choose three insertion points as , and . The remaining parameters are identified with the gauge theory coupling constants () as follows:
We denote the gauge group whose gauge coupling constant is as . Let , and , be the mass parameters of hypermultiplets in the fundamental representation of the and those of the respectively. Let also () be the mass parameter of the hypermultiplet in the representation of . Then the mass parameters and the external momenta are identified as
The identification of the parameter with the -deformation parameters is given by
Note that the case corresponds to the self-dual background . Finally, the vacuum expectation values of the scalar fields in the vector multiplets are identified as
for . By using the momentum conservation, can also be written as .
The first factor in the right hand side of (2.19) is the so-called factor corresponding to the part of the gauge theory, which is, e.g., given by
for the case†††This is slightly different from the one in . This is because we consider the Nekrasov partition function where the hypermultiplets are in the fundamental representation of the gauge group. Changing the representation to the anti-fundamental one leads to in this case, then we recover the factor in .
2.2 Quiver matrix model and higher rank gauge theory
In this section, we briefly review the -deformation of the ADE quiver matrix model [13, 14, 15, 28]. We then see that the matrix model can be obtained from the CFT of a free chiral boson valued in Lie algebra. A review of the undeformed quiver matrix model can be found in .
Let be a finite dimensional Lie algebra of ADE type with rank , the Cartan subalgebra of , and its dual. We denote the natural pairings between and by :
Let be simple roots of and is the inner product on . Our normalization is chosen as . The fundamental weights are denoted by
In the Dynkin diagram of we associate Hermitian matrices with vertices for simple roots , and complex matrices and their Hermitian conjugate with links connecting vertices and . We label links of the Dynkin diagram by pairs of nodes with an ordering . Let and be the set of “edges” (with ) and the set of “arrows” respectively:
The partition function of the quiver matrix model associated with is given by
with real constants obeying the conditions . Note that
The integration measures and are defined by using the metrics and respectively.
Integrations over are easily performed:
where is the identity matrix and denotes transposition. For simplicity we have chosen the normalization of the measure to set the proportional constant in the right hand side of (2.32) to be unity. Now the integrand depends only on the eigenvalues of Hermitian matrices . Let us denote them by ( and ). The partition function of the quiver matrix model reduces to the form of integrations over the eigenvalues of
where is a potential and
We then define the deformation of the above quiver matrix model (with ) by
At , it reduces to the original quiver matrix model (2.33).
The partition function (2.35) can be rewritten in terms of CFT operators. Let be -valued massless chiral field and . Their correlators are given by
obey the commutation relations
The Fock vacuum is given by
It is convenient to introduce the -valued potential by
Note that .
As in the previous subsection, we put the background charge which leads to the energy-momentum tensor
where is the Killing form and is the Weyl vector of , half the sum of the positive roots. Let () be an orthonormal basis of the Cartan subalgebra with respect to the Killing form: . In this basis, the components of the -valued chiral boson are just independent free chiral bosons:
and the energy-momentum tensor in this basis is given by
The central charge is given by
Here is the Coxeter number of the simply-laced Lie algebra whose rank is . Explicitly, (with ), , , and .
Note that for a root , with . Then, the bosons associated with the simple roots are expressed in this basis as follows:
For roots and , the inner product on the root space is expressed in their components as . Here and .
Let us now consider the four-point correlator of this theory. The vertex operator is defined by
where . As in the one-matrix case, we introduce the screening operators associated with the simple roots are defined by
We define the chiral four-point correlation function
For later convenience, we set . The momentum conservation condition is required
Using this four-point function, we define the partition function of the deformed quiver matrix model by sending (2.35) with the potential :
We will set , and . Using these definitions, the partition function (2.33) can be written as follows
2.3 Higher genus case
A generalization of the matrix model to a higher genus Riemann surface has also been considered in . The integral representation is basically obtained by changing the two-point function of the free field on a sphere to the one on a Riemann surface, which can be written in terms of the prime form, and by adding a term to the action which is the integral of the holomorphic differentials on the Riemann surface. For the conformal block on a torus with punctures, for instance, the two-point function is proportional to the theta function and the integral representation is given by [7, 30]
where , , and
The last term in is the integral of the holomorphic differential on the torus, , as mentioned above. Since the factor can be regarded as the generalization of the Vandermonde determinant, we refer to the integral (2.53) as “generalized matrix model”. In , the potential (2.54) of the generalized matrix model was expected from the geometrical argument of topological string theory.
In the following, we explain how the generalized matrix model is obtained from the full Liouville correlation function  for the torus case and  for the generic Riemann surface, based on the perturbative argument of . This method is different from the one seen in the previous subsection, although the both use the free field formalism.
The -point function of the Liouville theory on a genus Riemann surface is formally given by the following path integral
where the Liouville action is given by
Here is Ricci scalar and is a constant. We divide the Liouville field into the zero mode and the fluctuation . By integrating over , we obtain
and is the free scalar field action. When , the correlator diverges due to the factor . The residues at these poles are evaluated in the perturbation theory in around the free field action:
That is integer ensures the momentum conservation in the free theory.
Now let us focus on the torus case which simplifies the expression. The -point function of the free theory on a torus is written in terms of the factorized expression by introducing an additional integral as [33, 34, 35]
where , is the moduli of the torus and . By using the explicit expression (2.61), we find that the -point function of the Liouville theory reduces to the following integral
where , and we have chosen the insertion points such that they satisfy . The factor in front of the integral is irrelevant for the analysis below.
The discussion above is valid even for finite . However, it is not straightforward to divide the integral over the torus into the product of the holomorphic and the anti-holomorphic pieces for generic . In order to proceed, we evaluate the integral (2.62) in the large limit. We see that all the three terms in the exponent in (2.62) are . Thus, the integral (2.62) is evaluated at the critical points of the exponent of the integrand. The conditions for the criticality of the exponent are factorized into holomorphic equations and anti-holomorphic equations, which indicates that the integral over the torus in (2.62) can be replaced by the product of the holomorphic and the anti-holomorphic integrals in the large limit. Thus we define the holomorphic part of the correlation function as in (2.53) after introduction of by .
Relation to conformal block and gauge theory
We propose that this generalized matrix model (2.53) reproduces the full conformal block on the punctured torus (not only in the large limit), and also the Nekrasov partition function of the elliptic quiver gauge theory which is obtained from two M5-branes on the same torus.
Let us shortly see the relation of the parameters in the conformal block and the generalized matrix model. In the toric conformal block with punctures, we have external and internal momenta, giving parameters in total. The parameters are directly identified with the external momenta. Then, the potential (2.54) has critical points for each variable , assuming that the parameters are generic. Similar to the case in subsection 2.1, we expect that the critical points are “diffused” to form line segments due to the “determinant” factor. Then, the partition function is labelled by the filling fractions , in which out of variables take the value on the -th line segment. Due to the momentum conservation condition the sum of all is not independent degree of freedom. Thus we have independent filling fractions. These and the parameter in the potential are mapped to the internal momenta. (See  for the precise identification in the case.)
The relation to the gauge theory is stated as follows: the gauge theory coupling constants () are identified with the moduli of the torus as
The parameters are directly identified with the mass parameters of the bifundamentals. The filling fractions and the parameters in the potential are mapped to the vevs of the scalars in the vector multiplets.
Finally, let us quickly consider the case of the genus Riemann surface with puncture. As stated above the two-point function is written in terms of the prime form, and the generalized matrix model is the one in (2.54) where the theta function is replaced by the prime form and the last term in the potential is the integral of the holomorphic differential, with some additional terms. The precise form is presented in . The parameters are identified as follows [36, 31]: the conformal block is parameterized by parameters, where the first factor is from the external momenta and the second from the internal ones. In general the generalized matrix model corresponding to this Riemann surface has parameters and parameters including in the term involving the integrals of the holomorphic differentials. Since critical points of the potential lead to filling fraction (where comes from the momentum conservation), we have the same number of the parameters as the conformal block.
3 Large limit
Let us start an analysis of the matrix models introduced in section 2, focusing on the relation with four-dimensional gauge theory. One way to study a hermitian matrix model is to make use of the loop equation [37, 38, 39], and take the limit where the size of matrix, , is large. By this we can calculate the partition function of the matrix model in the iterative way as in [40, 41] (see e.g.  for a review). The systematic study of this method, so-called topological recursion has been performed in [43, 44, 45], and in [22, 46, 47] for the -deformed case. An advantage of considering the large limit (while kept fixed) of the matrix model introduced above is that the limit nicely corresponds to the one where and go to zero in the four-dimensional side, as can be seen from (2.22). Thus, this section is devoted to study this limit and see the correspondence with the four-dimensional gauge theory.
In section 3.1, we derive the loop equation of the -deformed matrix model. We see that this equation can be interpreted as the Virasoro constraints in the conformal field theory. Then we show in section 3.2 that in the large limit the spectral curve obtained from the loop equation can be identified with the Seiberg-Witten curve of the corresponding gauge theory. The free energy of the matrix model can also be computed and agrees with the prepotential of the gauge theory. In section 3.3, we turn to the generalized matrix model on torus, and consider the large limit.
3.1 Loop equation
Let us define the generator of the multi-trace operators as
When this is simply the generator of the single trace operators. First of all, we consider the Schwinger-Dyson equation associated to the transformation , keeping the potential arbitrary