Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models

This is a semi-pedagogical review of a medium size on the exact determination of and the role played by the low energy effective prepotential ${\cal F}$ in QFT with (broken) extended supersymmetry, which began with the work of Seiberg and Witten in 1994. While paying an attention to an overall view of this subject lasting long over the two decades, we probe several corners marked in the three major stages of the developments, emphasizing uses of the deformation theory on the attendant Riemann surface as well as its close relation to matrix models. Examples picked here in different contexts tell us that the effective prepotential is to be identified as the suitably defined free energy $F$ of a matrix model: ${\cal F} = F$. To be submitted to PTEP as an invited review article and based in part on the talk delivered by one of the authors (H.I.) in the workshop held at Shizuoka University, Shizuoka, Japan, on December 5, 2014.


Introduction
The notion of effective action plays a vital role in the modern treatment of quantum field theory. (See, for instance, [1,2].) In this review article, we deal with a special class of low energy effective actions that are controlled by (broken) extended rigid supersymmetry in four spacetime dimensions and permit exact determination exploiting integrals on a Riemann surface in question. A main object in such study is the low energy effective prepotential to be denoted by F generically in this paper, which has proven to be central not only in the original case of unbroken N = 2 supersymmetry initiated by the work of Seiberg-Witten [3,4] but also in the case where this symmetry is broken by the vacuum or by the superpotential. The review will be presented basically in a chronological order, following the three major stages of the developments that took place during the periods 1994 ∼, 2002 ∼ and 2009 ∼. Each of the three subsequent sections will explain pieces of work done in its respective period.
An emphasis will be put on the deformation theory of the effective prepotential on the Riemann surface as an extension of the Seiberg-Witten system consisting of the curve, the meromorphic differential and the period as well as its close relation to matrix models.
We conclude from the examples taken here in the different contexts that the effective prepotential is in fact identified as the suitably defined free energy F of a matrix model: F = F . While this is hardly a surprising conclusion from the point of view of mathematics of integrable systems and soliton hierarchies, the number of examples in QFT where this is explicitly materialized is not large enough. This note may serve to improve the situation.
In the next section, after presenting the curve for N = 2, SU(N) pure super Yang-Mills theory as a spectral curve of the periodic Toda chain, we discuss the deformation of the effective prepotential by placing higher order poles to the original meromorphic differential. We give a derivation of the formula which the meromorphic differential extended this way obeys.
In section three, we discuss the degeneration phenomenon of the Riemann surface necessary to describe the N = 1 vacua that lie in the confining phase and introduce the prepotential having gluino condensates as variables. We apply the formalism in section 2 here, and describe the situation by the use of mixed second derivatives. After discussing the emergence of the matrix model curve and giving sample calculation, we finish the section with the case of spontaneously broken N = 2 supersymmetry in order to illustrate the role played by the two distinct singlet operators one of which is the QFT counterpart of the matrix model resolvent.
In section four, we go back to the situation of N = 2 and discuss the developments associated with the AGT relation and the upgraded treatment of the all-genus instanton partition function and therefore the deformation of the Seiberg-Witten curve to its noncommutative counterpart. A finite N and β-deformed matrix model with filling fractions specified emerge as an integral representation of the conformal/W block and we discuss the direct evaluation of its q-expansion as the Selberg integral. We finish the section with mentioning some of the more recent developments.
Please note that the model or theory hops from one to the other as the sections proceed and that each section has its open ending, indicating calls for further developments of this long lasting subject.

curves, periods and meromorphic differentials
The list of papers which discuss subjects closely related to that of this subsection include [3-10, 14-25, 31-57].
Let us recall the most typical situation and consider the low energy effective action (LEEA) for N = 2, SU(N) pure super Yang-Mills theory. The symmetry of LEEA at the scale much smaller than that of the W boson mass is U(1) N −1 . The relevant curve is a hyperelliptic Riemann surface of genus N − 1 described as Here, and s k (h ℓ ) are the appropriate Schur polynomials. Introducing the spectral parameter z, we write the curve as that of the periodic Toda chain: The distinguished meromorphic differential for the construction of the effective prepotential is given by The characteristic feature of this is the existence of double poles at ∞ ± . Later in this section, we interpret this to be the case where only T 1 has been turned on.
The defining property is that the moduli derivatives are holomorphic: The prepotential F SW is introduced implicitly by the A cycle and B cycle integrations on the Riemann surface: going to be coordinate independent. This is supported by the pieces of evidence we present here that the effective prepotential is identified as the free energy of a matrix model.
We would now like to review the deformation of the effective prepotential above which we have denoted by F SW . The basic idea of this extended theory of effective prepotential often referred to as Whitham deformation is to deform both moduli of the Riemann surface and the meromorphic differential above consistently without losing the defining properties: We have adopted the choice that z is fixed when the moduli derivatives are taken. We carry out the deformation by adding higher order poles to the original meromorphic differential containing the double poles. Let us denote the local coordinates in their neighborhood generically by ξ and In order to describe the deformation, let us introduce a set of meromorphic differentials dΩ ℓ that satisfy dΩ ℓ = ξ −ℓ−1 dξ + non-singular part ℓ = 1.2, 3, · · · . (2.11) We are still left with the ambiguities that any linear combination of the canonical holomorphic differentials dω i can be added to the right hand side. In order to remove these, let us require a set of conditions The ones which are not subject to the conditions eq.(2.12) are denoted by d Ω ℓ .
Let us first state the formula and outline its derivation below. As before, a i are defined to be the local coordinates in the moduli space while T ℓ , referred to as time variables or T moduli, are given by once eq.(2.13) is established. One then regards a i and T ℓ as independent, taking h k dependent: The derivation of eq.(2.13) begins with the introduction of the time variables T ℓ via a solution dŜ(T ℓ |h) to eq.(2.9), namely, ∂dŜ ∂T ℓ = dΩ ℓ , and hence In terms of our intermediate bases d Ω ℓ , eq.(2.9) reads

connection with the planar free energy of matrix models
Already at this stage of the developments, a keen connection of the extended Seiberg-Witten system with the construction of matrix models in general, or more specifically, the similarity of the effective prepotentials with the (planar) free energy of matrix models was visible. In fact, starting from the homogeneity of the moduli and the prepotential, it is possible to derive an integral expression for F which resembles that of matrix model planar free energy in terms of the density one-form on the eigenvalue coordinate. See, eq. (4.12) of [24]. Also [14,16,18].
One of the goals of the present review is to put together subsequent several developments that took place and have made this phenomenon more prominent. These are presented in the next two sections.

Gluino condensate prepotential
One major use of the deformation theory of the effective prepotential presented above took place in the context of gluino condensate prepotential built on various N = 1 vacua in contrast to F SW and its extension in section 2. We first consider the case in which the breaking to N = 1 from N = 2 supersymmetry is caused by the superpotential in the action. Later we will contrast this with the case in which N = 2 is broken spontaneously to N = 1 at the tree level [70][71][72][73][74]. 1

degeneration phenomenon and mixed second derivatives
The list of papers which discuss subjects closely related to that of this subsection include [30,.
Let's fix an action to work with: it is a U(N) gauge theory consisting of adjoint vector superfields and chiral superfields with canonical kinematic factors and the superpotential turned on in the N = 2 action drives the system to its N = 1 vacua.
As a phenomenon occurring on a Riemann surface, we consider the situation where a degeneration takes place and some of the cycles coalesce to form a new set of cycles. As for the description of the low energy effective action (LEEA), some of the original Coulomb moduli disappear and the product of these U(1) s gets replaced by non-Abelian gauge symmetry n i=1 SU(N i ). We tabulate these pictures below.
1 Actually, supersymmetry is broken dynamically in the metastable vacua in both cases as was demonstrated in [75,76] in the Hartree-Fock approximation.
The N = 1 vacua are labelled by the set of order parameters representing gluino condensates: The proportionality constant will be fixed in subsequent subsections.
We now review, following the observation made in [105] that the condition for a curve to degenerate or factorize is given by that the kernel of the matrix made of the mixed second derivatives of the deformed prepotential be nontrivial.
Continuing with the general discussion of subsection 2.2, let us first note that we obtain two different expressions for the mixed second derivatives from eq.(2.16): We impose the condition Eq.(3.3) has following straightforward implications: i) there exists a nonvanishing column vector c 1 , Here, we have exploited eq. (2.17) in the second equality and eq. (2.10) in the third equality.
The former equality implies that d Ω ≡ ℓ c ℓ dΩ ℓ has vanishing periods over all A i & B i cycles.Then one can integrate this form along any path ending with a point z to define a function holomorphic except at punctures. As for the order of the poles at the punctures, it is generically arbitrary according to the construction. But this is contradictory to the Weierstrass gap theorem 2 derived from the Riemann-Roch theorem. To avoid a contradiction, we must have a degeneration.
ii) there exists a nonvanishing row vector c 1 , c 2 , · · · , c N −1 such that in accordance with the second formula of eq. (2.16). Eq. (3.7) follows from which is regarded as the statement of the vanishing discriminant. The moduli depend actually on less than N − 1 arguments.
Once we are convinced of the degeneration of the surface, we can proceed further by factorizing the original curve, which, in the current example, is the hyperelliptic one.
Let n − 1 be the number of genus after the degeneration. Following [88,89], we state     Finally let us examine the last equality of eq.(3.4). Let The Weierstrass gap theorem states that for a given Riemann surface M, with genus g, and a point P ∈ M, (3.5) and g integers satisfying 1 = n 1 < n 2 < · · · < n g < 2g, (3.6) there does NOT exist a function f holomorphic on M \{P } with a pole of order n j at P . and x j−1 √ F 2n serve as bases of the holomorphic differentials of the reduced Riemann surface. Actually, only the j = 1 ∼ n − 1 differentials are holomorphic and the j = n one has been added through the blow-up process, which physically implies that the overall U(1) fails to decouple.
We obtain and therefore Here, f k−1 is a polynomial of degree k − 1. This is the curve appearing in the k-cut solution of the matrix model.
We still need to see that W k+1 (x) introduced above is in fact a tree level superpotential. This is easily done by taking the classical limit Λ = 0: 14) The original Seiberg-Witten differential becomes Here, we have used that the canonical holomorphic differential becomes in this limit. The period integrals over the A i cycles just pick up the residues at the poles p i : The degeneration in this limit is described as In fact, the N j poles coalesce at β j , j = 1, · · · , n and the canonical holomorphic differentials on the degenerate curve are The condition eq. (3.11) becomes which tells us that β j must coincide with one of the roots α j of W ′ k+1 . The vev's of the adjoint scalar fields are thus constrained to the extrema of W k+1 .
Let us set k = n for simplicity. We have the reduced curve of g = n − 1:
Let us now proceed to discuss the use of this machinery in calculation. As the condensates S i are quantum mechanical in nature, one can develop loop expansion using these, including the Veneziano-Yankielowicz term which contains the logarithmic singularity [77]. The first question to be raised is what the distinguished meromorphic differential is to be used for such calculation. It must be "almost" holomorphic after the b ℓ derivatives are taken. Recall that the bases of the "holomorphic" differentials are taken as x j−1 y , j = 1, · · · , n − 1, n. Rather obviously, such differential is found as As before, the effective prepotential is introduced through the period integrals  and We have, however, no reason to set equal to zero. This tells us the presence of the cutoff at the infinities of the surface.
The expansion of F in S i was done in [97], exploiting eq.  .38)). Yet, there exists a simpler procedure, namely, a calculus from T moduli thanks to the machinery discussed in the present review. The T moduli are easily identified as The dependence of the prepotential on the T moduli is determined by the equations Here Λ ℓ+1 is the term introduced in [ The differential dŜ mat of eq. (3.24) has a straightforward expansion inS i . Therefore, A i cycle integrations followed by the inversion provide an expansion ofS i in S j Here, we have introduced α ij = α i − α j , and ∆ i = j =i α ij . Another useful machinery is the T m moduli derivatives of the roots α i of the superpotential, which read Using these, the right hand side of eq. (3.29) is evaluated as which is trivially integrated in u m to provide an answer. Let us mention that this procedure is straightforwardly generalizable to higher order contributions in S i and that the terms independent of α i can be easily obtained by several other methods.
The expansion form of F (S|α) which we managed to have proposed in [97] is 3 Here, we have denoted by F k+2 (S|α) the contributions of the k + 2 order polynomials in S i .
The explicit answer for F 3 (S|α) is For the computation of higher orders as well as the inclusion of matter, see, for instance, [128,[131][132][133].

case of spontaneously broken N = 2 supersymmetry and Konishi anomaly equation
The list of papers which discuss subjects closely related to that of this subsection include [70-74, 87, 134-174].
The N = 2 effective action is completely characterized by the effective prepotential while, in the N = 1 case, a typical observable is (the matter induced part of) the effective superpotential.
The interplay of these two upon the degeneration of the original Riemann surface is most clearly seen by dealing with the case of spontaneously broken N = 2 supersymmetry. This case accomplishes a continuous deformation from one to the other by tuning the electric and magnetic Fayet-Iliopoulos parameters. The action S F in N =2 realizing this is given by Here, ξ, e, m are the electric and magnetic F-I terms and we vary these to interpolate the two ends, keepingg ℓ = mg ℓ (ℓ ≥ 2) fixed: large (ξ, e, m) small (ξ, e, m) In this subsection, we have denoted by the symbol F in an input function in the effective action eq. (3.39). For definiteness, we let the function F in be a single trace function of a The left-hand side is the contribution of the Konishi anomaly [80], which arises from the behavior of the functional integral measure under the transformation [175,176]. Introducing the two generating functions, we recast this into the following set of equations [161]: where f (z) and c(z) are polynomials of degree n − 1 and, with some abuse in notation, The explicit form of f (z) and that of c(z) are not really needed in what follows.
Let us make a few comments on this set of equations. The equation for R(z) is identical in form to that of the planar loop equation of the one-matrix model for the resolvent. This fact is shared by the theory in the large FI term limit, namely, N = 1 theory of adjoint vector superfields and chiral superfields with a general superpotential [145]. The equation for T (z), on the other hand, contains the cubic derivatives in F in and is distinct from that in the large FI term limit. This, in fact, leads us to the deformation of the formula connecting the effective superpotential with the object identified as the matrix model free energy from its well-known expression [90][91][92] in S N =1 , namely, the one in the large FI term limit.
Our final goal in this subsection is to derive a formula for the effective superpotential. Let us define the one point functions as In terms of v ℓ we define F as Using F , we can state the relation to be proven: Before proceeding to the proof of this relation, let us go back to eqs. (3.44) and (3.45) to obtain the complete information. We consider the most general case that the gauge symmetry U(N) The indices i, j, · · · run from 1 to k while the indices I, J, · · · run from 1 to N. Of course, N I = 0 (I = k + 1, · · · , n). Solving eq. (3.44), we obtain where the Riemann surface Σ is genus n − 1 but its A I cycles for I = k + 1, · · · , n are vanishing.
We conclude that the meromorphic function lives on a factorized curve Here N n−k (z), F 2k (z) are polynomials of degree n − k and 2k respectively. On the other hand, substituting eq. (3.50) into eq. (3.45), we obtain . (3.53) Let us list a few formulas that are obtained from eq. (3.50) directly. The first set is [117] ∂R(z) , i = 1, · · · , n is a set of normalized holomorphic functions, as is easily seen by taking the derivatives of the A cycle integrations. Also, define h(z) = − i N i g i (z). The second one is where we have used eq. (3.46).
The proof eq. (3.49) goes by observing that it is equivalent to the truncation of the following equation up to the first n + 1 terms in the 1/z expansion, complete as soon as we obtain Observe that there are two expressions for N i : and therefore Another consistency condition is in the integrand of eq. (3.60) and that of eq. , ℓ = 0, · · · , n − 1 of the original curve, we deduce eq. (3.57).

AGT relation and 2d-4d connection via matrices
The contents of the two preceding sections later had the upgraded treatments mentioned in the introduction. In this section we outline these developments triggered by the work [177].
Let us recall that the low energy effective action (LEEA) of N = 2 SU(N c ) SUSY gauge theory is specified by the effective prepotential denoted in this section by F SW (a i ) and that it has undetermined VEV called Coulomb moduli a i = φ i . The bare gauge coupling and the θ parameter are grouped into and F SW (a i ) consists of the one-loop contribution and the instanton sum It was shown in [181] that F (SW) inst is microscopically calculable in the presence of Ω background equipped with the deformation parameters ǫ 1 and ǫ 2 as The corrections to the original F (SW) inst are regarded as higher orders in the genus expansion with g 2 s = −ǫ 1 ǫ 2 . Its expansion in q is computable by the localization technique with ǫ 1 , ǫ 2 acting as Gaussian cutoffs.
is the "volume" of the k-instanton moduli space.
Let T Nc−1 be the maximal torus of the gauge group SU(N c ). Since we also have the maximal torus T 2 of SO(4), namely, the global symmetry of R 4 , the T = T 2 ×T Nc−1 action can be defined on the instanton moduli space. Then the integral in eq. (4.5) are computed T -equivariantly and consequently we obtain the regularized results. According to the localization formula, eq. (4.5) is reduced to the summation of the contribution from the fixed points which are parametrized by N c Young diagrams Y = (Y (1) , · · · , Y (Nc) ), where | Y | = Nc i=i |Y (i) | is the total number of boxes. Each Z Y is provided through a combinatorial method.

β-ensemble of quiver matrix model and noncommutative curve
The list of papers which discuss subjects closely related to that of this subsection include .
In this subsection, we give a general discussion of β-deformed matrix models at finite N (size of matrices) and with generic potentials and the attendant noncommutative curve. The curve at the planar level, which the original S-W curve for SU(N c ) gauge group with 2N c flavours are relevant to, turn out to come out in a relatively transparent way in the limit.
Let us begin with the β-deformed (β-ensemble of) one-matrix model : is the van der monde determinant.
The Virasoro constraints [192][193][194]197], namely the Schwinger-Dyson equations of this model for the resolvent, are obtained by inserting Adopting the operator notation of conformal field theory, Eq. (4.11) can, therefore, be written as Quite separately, let us introduce the "curve" (x, z) = (y(z), z) by (4.14) Two remarks are in order. First of all, in order for the first equality to be true, x and z must satisfy the noncommutative algebra: Second, in order for eq. (4.14) to be algebraic, the singularities in T (z) must be absent. This condition is ensured by the Schwinger-Dyson equation eq. (4.11).
Let us turn to the A Nc−1 quiver matrix model (β deformed) which the effective prepotential for the SU(N c ) gauge theory with 2N c flavours are relevant to. This matrix model has been constructed [203] such that it automatically obeys the W Nc constraints at finite N a , a = 1, · · · r, We follow the logic of β-deformed one-matrix model at finite N a . In this model, there exists N c spin 1 currents that satisfy Nc i=1 J i (z) = 0: The curve Σ (x = y i (z), z) that we postulate in [233] is The isomorphism with the Witten-Gaiotto curve has been established by taking the planar limit of this construction as we will see in the next subsection. In fact, the planar limit implies the singlet factorization which assigns the c number value to the operator ∂φ(z) and the curve factorizes as

Gaiotto curve
The list of papers which discuss subjects closely related to that of this subsection include [177,.
Let us specialize our discussion to the three Penner model. Choose the potential as The matrix integrals of this case realize the integral representation of the conformal block and the size of each matrix corresponds with the number of screening charges we have to insert to built the block. As is clear from the discussion above, the planar spectral curve of the A Nc−1 quiver matrix model takes the form for some polynomials Q k (z) in z.
On the other hand, the Seiberg-Witten curve for the case of SU(N c ) gauge theory with 2N c massive flavour multiplets, originally proposed in [236], can get converted into the Gaiotto form [238] by where P (k) 2k (t) are degree 2k polynomials in t. The two curves eq. (4.26) and eq. (4.27) are evidently similar. We can also see that the residues of y i (z)dz (i = 1, · · · , N c ) at z = 1, q, 0, ∞ and those of xdt at t = 1, q bare , 0, ∞ on the i-th sheet can be equated.
For general N c , these residues in fact match if the weights of the vertex operators are identified with the mass parameters of the gauge theory by the following relations [233] : ( m a − m a+1 )Λ a , (4.28) The matrix model potentials W a (z) (a = 1, 2, . . . , N c − 1) are fixed as With this choice of the multi-log potentials, the A Nc−1 quiver matrix model curve in the planar limit coincides with the SU(N c ) Seiberg-Witten curve with 2N c massive hypermultiplets.

direct evaluation of the matrix integral as Selberg integral
The list of papers which discuss subjects closely related to that of this subsection include .
In this subsection, we consider 2-d conformal field theory which has the Virasoro symmetry with the central charge c. The correlation functions for primary operators Φ ∆ (z,z) with the conformal weight ∆ are strongly constrained by this symmetry. We are interested in the fourpoint functions which can be expressed as (4.31) The sum on I is taken over all possible internal states. Here K ∆ and C ∆ 3 ∆ 1 ∆ 2 are the modeldependent factors. In contrast, the conformal block 4 denoted by F (q|c; ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 , ∆ I ) is a model-independent and purely representation theoretic quantity, . . , k ℓ ) and Let us consider the four-point conformal block on sphere, The parameter α 4 is determined by the following momentum conservation condition which comes from the zero-mode part: The internal momentum α I is given by Eq. (4.34) has an integral representation as a version of β-deformed matrix model. Actually, the Dotsenko-Fateev multiple integrals, are regarded as a free field representation of eq. (4.34).
In order to develop its q-expansion, it is more convenient to interpret this multiple integrals as perturbation of the products of the two Selberg integrals.
We have the following expression of the perturbed double-Selbarg model: Here S N L and S N R are the celebrated Selberg integral 43) and the averaging · · · N L ,N R is taken with respect to the unperturbed Selberg matrix model, (4.44) Below we also use · · · N L and · · · N R which imply the averaging with respect to Z Selberg (N L ) and to Z Selberg (N R ), respectively.

It takes form
Note that a pair of partitions (Y 1 , Y 2 ) naturally appears.
In order to apply this to eq. (4.39), let us set γ = b 2 E and for the "left" part. Similar replacement yields the expression for the "right" part. We obtain .

(4.59)
From the explicit form of Jack polynomials for |λ| < 2 listed in eq. (4.55), we obtain [298] b E For definiteness, let us consider the left-part, into the integrand, we obtain the loop equation at finite N, The expectation value of w N L (z) is the finite N resolvent (4.69) The first one agrees with eq. (4.60).
Now, let us determine the 0d-4d dictionary. In the matrix model (0d side), we have seven parameters with one constraint eq.  The first two formulas tell us clearly the necessity that the filling fractions of the β-deformed matrix model must be explicitly specified at finite N in order to exhibit the Coulomb moduli.
In the next order, the expansion coefficients A 2 are rearranged as where We illustrate our discussion in this section by Fig. 3.

more recent developments
The list of papers which discuss subjects closely related to that of this subsection include [262,301,.
We have reviewed the 2d-4d connection from a view point of the matrix model. In this subsection, we comment on some of the more recent developments.
In the last subsection, we have presented the connection between the Virasoro conformal blocks and the four-dimensional SU(2) instanton partition functions via the matrix model and the Selberg integral. This discussion has been generalized in part to that between the W N blocks and the SU(N) partition functions [376].
The both sides also have a natural generalization as a q-lift [364]. The Virasoro/W N symmetry in the two-dimensional CFT side is deformed to the q-deformed Virasoro/W N symmetry while the four-dimensional SU(N) gauge theory is lifted to the five-dimensional theory. It is interesting to consider the root of unity limit q → e 2πi r of the q-Virasoro/W N algebras. The appropriate limiting procedure [386,391] to the root of unity exhibits the connection between the super Virasoro (r = 2) or the Z r -parafermionic CFT and the gauge theory on R 4 /Z r [368,370].
There are several pieces of work [301,363,379,384] which prove the 2d-4d connection. The explicit identification can be established in the case of β = 1 [366,367]. In order to apply to the β = 1 case, the conformal blocks have to be expanded by the generalized Jack polynomial [385] that modifies the standard one. For some lower rank cases, this has been explicitly constructed [388].