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Explicit factorization of Seiberg-Witten curves with matter from random matrix models

Within the Dijkgraaf-Vafa correspondence, we study the complete factorization of the Seiberg-Witten curve for U(N_c) gauge theory with N_f<N_c massive flavors. We obtain explicit expressions, from random matrix theory, for the moduli, parametrizing the curve. These moduli characterize the submanifold of the Coulomb branch where all monopoles become massless. We find that the matrix model reveals some non-trivial structures of the gauge theory. In particular the moduli are additive with respect to adding extra matter and increasing the number of colors.


Introduction
Very recently there appeared new powerful methods of extracting effective superpotentials for a wide class of N = 1 gauge theories. The proposal by Dijkgraaf and Vafa [1,2], building upon earlier string theoretical constructions [3,4], links these superpotentials with quantities in random matrix models. The proposal have since been proven [5,6]. The proposal has been extended to theories with fundamental matter [7] and there has been significant work on studying some features of the link with Seiberg-Witten curves [8,9,10,11,12,13,14].
These curves made their appearance in the ground-breaking work of Seiberg and Witten [15,16] on the study of N = 2 supersymmetric gauge theories. It turned out that one can describe the low energy dynamics of the gauge theory in terms of geometrical properties of the Seiberg-Witten curves. Subsequently it was realized [16,19] that one could also study, within the same framework, deformations to N = 1 theories by adding a tree level superpotential. In this context one is led to the points (submanifolds) in moduli space where monopoles become massless and condense. At these points the Seiberg-Witten curve factorizes -it has only two single zeroes (branch points) in the case of complete factorization, where all monopoles condense. Once the form of the Seiberg-Witten curve at the factorization point is known one can calculate effective potentials for the deformed theory [16,19,4].
The possibility of integrating in the glueball field S was realized in [9] and used to check the matrix model result following from the Dijkgraaf-Vafa proposal for deformed pure N = 2 theories (without fundamental matter). There the explicit factorization of curves without matter of [19] was known. Factorization properties were also used in the context of SO(N c ) theories [20] and multi-trace operators [21] to obtain the appropriate effective superpotentials W ef f (S).
However once one adds fundamental matter to the theory, the factorization problem becomes exceedingly complicated (even in the case of U(2) with 1 flavour) and no explicit solutions are known. The aim of this paper is to use the Dijkgraaf-Vafa proposal linking superpotentials to matrix models and derive, from the random matrix model solution, explicit factorization of Seiberg-Witten curves of U(N c ) theory with N f < N c fundamental matter fields.
If it were not for the Dijkgraaf-Vafa correspondence, we would not expect that any analytical solutions to this problem could exist. The fact that they can be obtained in this way shows that the random matrix model with matter is intimately linked to fine details of the geometry of the appropriate Seiberg-Witten curves.
In addition we find that the matrix theory variables capture a surprising robust structure of the factorized Seiberg-Witten curve. The 'pure' N = 2 solutions and the new contributions of each flavor appear additively. All nonlinearity is concentrated in a single relation involving the scale of the gauge theory Λ and the matrix theory variables. Moreover we find that the 'integrating-out' equations of pure N = 2 theories reappear, in terms of matrix variables, in the context of N = 2 theories with fundamental flavors. This arises naturally in the matrix model but is rather unexpected from a gauge theory perspective.
In section 7 we collect the final results of the paper. The plan of this paper is as follows. In section 2 we describe the field theoretical ingredients in more detail. In section 3 we introduce the setup for the Dijkgraaf-Vafa proposal with matter fields. We then go to use the orthogonal polynomial method to rederive directly the solution of the factorization for pure N = 2 theory, which is an ingredient of the expressions for theories with matter. In sections 5 and 6 we derive the linearity in couplings of the superpotential and the expressions for factorization. Section 7 contains the main result of the paper. In section 8 we give some specific examples and close the paper with a discussion. Three appendices contain some technical details.

Field theory considerations
It has been known for a long time that N = 2 supersymmetric U(N c ) gauge theories with N f < N c massive flavors has a Coulomb branch that is not lifted by quantum corrections [15]. This quantum moduli space is N c -dimensional, parametrized for example by u k = 1 k tr Φ k with k ≤ N c . At each point of the moduli space, the low energy theory is described by an N = 2 effective abelian U(1) Nc gauge theory. All the relevant quantum corrections in the IR can be recast in terms of the period matrix of a particular meromorphic one-form of the auxilliarly complex curve -the Seiberg-Witten curve [15,15,17,18], or more precisely a family of genus N c − 1 hyperelliptic Riemann surfaces: with The coefficients s α are polynomials of the u k 's parameterizing the Coulomb branch: One can deform this N = 2 theory to a N = 1 gauge theory by adding a tree level superpotential: The classical vacuum structure is given by all possible distributions of the N c eigenvalues φ k of Φ amongst the N c critical points a j of the potential. This corresponds, at the classical level, to a breaking of the U(N c ) gauge symmetry to the gauge group where the N i 's are the nonzero multiplicities of the eigenvalues. For the purposes of this paper we can safely set g Nc+1 = 0 as we will be considering the case with no breaking of gauge symmetry (complete factorization).
Turning to the quantum picture, the presence of this superpotential will lift the quantum moduli space, characteristic of the N = 2 Coulomb phase, except for the codimension n submanifolds, where n mutually local magnetic monopoles become massless. These are the N = 1 vacua solving the F-flatness and D-flatness conditions and are characterized by a monopole condensate of the massless monopoles. This is believed to produce, by the dual Meissner effect, the expected confinement of the electric N = 1 theory. Hence, the final quantum theory is described at low energies by a N = 1 U(1) Nc−n gauge theory. These U(1)'s can be thought of as the U(1) ⊂ U(N i ) of the classical theory. The N = 1 SU(N i ) part of the theory confines, has a mass gap and is characterized by a gaugino condensate.
These N = 1 vacua, being the codimensional n submanifolds of the N = 2 Coulomb branch, where n mutually local monopoles become massless, are parameterized by the sets of moduli {u f act. k } where the Seiberg-Witten curve factorizes, i.e. the r.h.s. of (1) has n double roots 4 and 2(N c −n) single roots: Moreover it is shown in [8] that the reduced curve, captures the full quantum dynamics of the N = 1 U(1) Nc−n low energy theory.
The effective superpotential of the theory deformed by (5), where we set g Nc+1 = 0, is obtained by plugging the solutions u f act. k , parametrized by N c −n parameters into the tree level superpotential: Then (8) should be minimized with respect to the N c − n parameters.

Complete factorization
In this paper we will be interested in the case where N c − 1 mutually local monopoles condense. This corresponds to a complete factorization of the Seiberg-Witten curve: the vacua form a 1 dimensional submanifold such that the curve has only 2 single roots and N c − 1 double roots: The main goal of this paper is to find explicit expressions for the moduli u f act. k where complete factorization occurs. Let us briefly review the solution for the pure N = 2 Yang-Mills case found by Douglas and Shenker [19] some time ago using Chebyshev polynomials.
Their solution factorizing the curve P Nc (x, u k ) 2 − 4Λ 2Nc is 5 Note that the one dimensional submanifold where the U(N c ) curve completely factorized, is parametrized by u 1 = tr Φ. These results can be easily restricted to the SU(N c ) case, by putting explicitly u 1 = 0. All parameters are then uniquely fixed in terms of the only scale of the theory Λ.

Integrating in S
The quantum N = 1 effective potential generated by the tree level potential is obtained by minimizing (8) with respect to u 1 .
Here we follow an alternative route taken in [9], and integrate in S by performing a Legendre transformation with respect to log Λ 2Nc−N f . The superpotential is then given by: (Ω, u 1 ). (12) Note that the only Λ-dependence is in the linear S log Λ 2Nc−N f term. Integrating out S forces Ω = Λ and brings us back to (11). In order to get the effective potential W ef f (Λ, S) one has to integrate out both Ω and u 1 .
Our strategy for identifying the factorization parameters u f act.
p for the theory with matter is to rewrite the random matrix expression in the form given by the last two terms of (12) and to read off the appropriate gauge theoretic moduli u f act.
p . The first term in (12) could also be absorbed into the matrix model expressions if appropriate rescalings of the integration measure of the matrix model was made. We will not do this here.
For later reference let us quote the equations of motion for the pure N = 2 gauge theory: These follow directly from (12) and (10).

The Dijkgraaf-Vafa proposal with fundamental matter
According to the Dijkgraaf-Vafa prescription the the perturbative part of the superpotential can be expressed as [1] where the F χ are defined through the matrix integral [7,1] e − N 2 where we included the coupling of fundamental matter to the adjoint field in accordance with N = 2 supersymmetry. The matter fields in the matrix model (16) appear only quadratically and hence may be integrated out giving where Z is the partition function of the matrix model without matter. For the complex matrix model it is well know that Z will contribute only to F χ=2 and not to F χ=1 . Using large N factorization we also have We see that the matter determinant appears at subleading order in N w.r.t the tree level potential. This has the important consequence that the saddle point solution (eigenvalue density ρ(λ)) and hence F χ=2 will not be influenced by the presence of matter. The χ = 1 contribution is then given by In the following section we will evaluate the F χ=2 piece of the partition function Z appearing in (17) using the method of orthogonal polynomials. As a byproduct we will rederive the factorization expression for the pure gauge theory case. Then in section 5 we will start investigating the matter contribution (19).

Orthogonal polynomials and F χ=2
In this section we will use the method of the orthogonal polynomials to study thoroughly the one cut solution. A very nice introduction to this powerful method can be found in [22]. We will show that all results obtained from the field theory analysis appear naturally in this setting from random matrix computations (compare [9]).
The orthogonal polynomials associated to the matrix model with potential N S tr V (Φ) satisfy the recursion relation sP n (s) = P n+1 (s) + T n P n (s) + R n P n−1 (s) (20) In the large N limit the recursion coefficients T n , R n can be taken to be continous functions of u = S · (n/N) ≡ S · x. In addition, the equations of motion of the matrix model with general potential W tree (Φ) become purely algebraic [23,22]: When x = 1 we will denote R(S) by R and T (S) by T . These two variables are related to the endpoints a and b of the support of the matrix eigenvalue distribution through For a general matrix potential V (Φ) = g p 1 p Φ p , one obtains easily: defining u and v for later convenience 6 . In terms of these polynomials, the matrix model partition function takes a very elegant form: where h 0 is the integral 6 Apart from the present section u and v are always taken at x = 1.
From equation (27), one can easily extract the χ = 2 contribution (we will comment on the h 0 piece below) Performing an integration by parts leads to: Following the Dijkgraaf-Vafa prescription, the contribution to the gauge theory effective potential is proportional to: The remaining integral is rather tricky. The integrand is a function of two variables R and T , tied together by the highly nonlinear constraint v = 0. This complicates the integral, and moreover spoils the manifest linearity in the couplings g p , appearing in the matrix potential. It is convenient, however, to treat the one-dimensional integral as an integral of a 1-form over the path {v = 0} in the two dimensional R-T plane. Then one can rewrite the integral as an integral over a closed 1-form ω, such that the original integral remains unchanged. One can then deform the contour v = 0 into a piece with R = 0, while integrating T from T (0) to T (S) and another piece, keeping T fixed at T (S), and integrating R from 0 to R(S) (see fig. 1): R=const ω (32) An extension of the integrand of (31) to a closed one-form can be found to be Note that the extra piece doesn't give a contribution to the original integral, which is taken over v = 0. Performing the integral is now straightforward. The first piece, integrating over T and constraining R to 0 gives: The evaluation of the integral at T (0), can be cancelled (at the saddle point) against the h N 0 , appearing in the partition fuction (27). For the second piece, one fixes T at T (S), and integrates over R: From now on until the end of the paper we will denote by R and T the recursion coefficients defined at S. Bringing the two contributions together, leads to the following result: This is exactly, term by term, the field theory result for pure N = 2 U(N c ) gauge theory ( (12) and (10)), provided we identify field theory variables and matrix theory variables in the following way: The first identification is extracted from the linear term in S, the second part follows from the term linear in g 1 . As it should, the equations of motion for R and T (25)- (26), are mapped, under this identification, to the equations obtained from integrating out Ω and u 1 (13)- (14).
From (36) we may read off the coefficients of g p : Under the identification (37) and setting Ω = Λ these coincide exactly with the factorization solution (10) of Douglas and Shenker for pure N = 2 gauge theory. For theories with matter, (38) will be just a part of the final result, and an identification between R, T and gauge theoretic quantities will have to be made only after the F χ=1 contribution is evaluated in the following sections.

The F χ=1 matter contribution
In the previous section we have rederived the result that the first piece in (15) can be recast in the form and the equations of motion for R and T derived from (39) are exactly the random matrix saddle point equations The first of these can be interpreted as the formula for integrating in S. The linearity of (39) in the random matrix couplings is essential for identifying the U pure p (R, T ) with the point in N = 2 moduli space where the Seiberg-Witten curve y 2 = P Nc (x, u k ) 2 − 4Λ 2Nc factorizes.
In the following section we will recast the random matrix expression (19) for F χ=1 in the same way: Let us comment on an ambiguity, leading to a unique determination of a crucial term to obtain the correct final result. Once we start interpret this expression for F χ=1 , as a part of the effective gauge theory potential, we should be able to integrate out R and T . The equations of motion thus obtained, should be consistent with the random matrix saddle point equations. But, as explained in section 3, adding fundamental matter in the gauge theory does not modify the saddle point equations of the matrix model. So it is necessary to impose that the equations of motion for R and T , derived from the effective potential, with the F χ=1 contribution are consistent with (40) and (41). This fact is highly non-trivial and reveals an unexpected relation between the factorization of Seiberg-Witten curves with and without matter.
We will show that the addition of F χ=1 to the superpotential does not change the equations of motion for R and T provided we add an extra term proportional to v. From the matrix model perspective nothing changes, while the extra term turns out to be crucial to obtain the correct gauge theoretical interpretation of our results. We stress that this does not alter the Dijkgraaf-Vafa prescription. It merely solves an ambiguity that arises when interpreting the matrix model variables directly in terms of gauge theoretical quantities.
Let us now focus on a single summand of (19). In this case it seems to be difficult to use the orthogonal polynomial method, which was the most straightforward way of deriving the F χ=2 contribution. Here it is more convenient to use the saddle point expression for the eigenvalue density of the single-cut solution: where M(x) is a polynomial which is expressible in terms of the random matrix potential through The features of interest of the above expression are (i) it is linear in the explicit dependence on the couplings g p , (ii) the coefficients of g p are universal functions of the endpoints a, b and hence of the variables R and T , (iii) for given g p , its coefficient is a polynomial in x of order p − 2.
In order to obtain an explicit dependence on R and T let us perform the change of variables Then the contribution to F χ=1 of a single flavour is given by As noted before, one should take into account a possible addition of v multiplied by any function f (R, T ) (since v = 0 is an equation of motion of the random matrix model).

Factorization formulas
We will first derive the formulas involving g 1 and g 2 , in particular this will allow us to fix uniquely the function f (R, T ). Also this will make the general structure more transparent. Then we will derive the results for arbitrary g p .
Contribution of g 1 , g 2 To this order M(ψ) = g 2 and the formula (46) gives This can be evaluated to give (48) At this stage we should identify the coefficient of the logarithm with S (here this is trivial since to this order the equation of motion for R is just S = g 2 R, but later we will see that this property will hold in general). Thus we are left with which indeed has the form of (42). Now we require that the saddle point equations remain consistent with the F χ=1 equations of motion. This detemines uniquely the term Indeed the requirement that integrating out T from the sum of (49) and (50) gives This extra term does not change the equation for R, which is consistent with the saddle point equations. This function will stay unchanged in the general case. We may now read off the final expressions for the mass dependent contributions to u 1 and u 2 : Arbitrary g p We will now extend the previous considerations to the calculation of arbitrary U matter p . The general structure will remain unchanged. The function f (R, T ) multiplying v will remain unmodified (as it should). Also the coefficient of the logarithm will turn out to be exactly S.
In appendix A we derive the following expression for the polynomial M(ψ): So the integral (46) can be rewritten as We now have to distinguish two cases. n odd: Then the first integral vanishes and we will denote the second integral by f n (z). As discussed in appendix B, f n (z) is essentially a polynomial in z of order (n + 1)/2 (divided by (z − 1) n/2+1 ) when expressed in terms of the variable Appendix B contains a general formula for f n (z). Explicit expressions for some specific cases are shown in table 1 in section 7. n even: In this case the first integral is nonvanishing. Moreover the second integral involves a logarithm with the same coefficient as log(m + T ) in the first integral. Together they combine to give It is shown in appendix C that the coefficient in curly braces is exactly equal to S. The second integral with the logarithmic part subtracted out has again a simple polynomial structure (see appendix B for details). At this stage we arrive to the analogue of (49): Again we have to add to this the correction term We checked explicitly for some cases that this term together with (59) gives equations of motion consistent with the random matrix constraints. The sum of (59) and (60) is now of the expected form (42), thus defining U matter p (R, T, m i ).

Final results
Putting all results together ((36), (59), (60)), and putting in the term S log Λ 2Nc−N f as required from the Dijkgraaf-Vafa prescription gives a prediction from the matrix model side for the quantum effective gauge potential: This expression should be compared with the potential W ef f (S, u 1 , Ω, Λ) (see (12)), obtained from the field theory analysis. The relation between the parameters of the matrix model and the field theory is highly non-linear: In order to obtain the final factorization formulae we integrate out S from W ef f (S, T, R, Λ) : Combined with (63) this gives an expression for Λ in terms of R and T . The remaining part of (61) should then be compared with field theory result: where the u f act.
p , is the one parameter solution, factorizing completely the Seiberg-Witten curve for U(N c ) SYM with N f flavours (N f < N c ): Comparing with the results from matrix models gives an expression for the u f act.
p 's, Table 1: Examples of the functions f n (z) for small n.
in terms of two parameters R and T , tied together with the constraint: For completeness, we recall the formulas If we wanted instead to obtain the effective potential W (S, Λ), we would have to integrate out R and T from (61). On the field theory side it is very cumbersome how the structure of the Seiberg-Witten curve appears in the equations of motion for u 1 and Ω. On the matrix model, on the other hand, the equations of motion for R and T appear naturally to be the same as ones obtained in the case without flavours. It seems that the particular combinations of u 1 and Ω, embodied in R and T , captures some nontrivial structure of the Seiberg-Witten curves.

Some examples
In this section we will study some examples to verify that the u f act. p we obtained from random matrix models, do factorize the appropriate Seiberg-Witten curves with fundamental matter.

U (2) with 1 flavour
In this case we have We have verified that with the above choices, the discriminant of the Seiberg-Witten curve vanishes identically, which in the special case of 2 colours proves factorization. Note that for general N c , complete factorization is a much stronger condition than the vanishing of the discriminant.

SU (N c ) with 1 flavour
In order to consider SU(N c ) theory we have to impose the constraint that u 1 = 0. Then the parameter T can be expressed in terms of R and the mass m of the additional flavour. Namely we have which gives Now R is linked directly to the scale of the gauge theory through where the expression (77) should be used. The remaining formulas remain however quite complicated functions of R, m and N c . This is in marked contrast to the case of pure gauge theory without fundamental matter where the passage from U(N c ) to SU(N c ) is very simple, and N c enters linearly. It is interesting to look at the m → ∞ limit. Then T → 0 as expected for a pure N = 2 SU(N c ) theory, while (78) becomes Λ 2Nc−1 = R Nc /m. Recall that in pure SU(N c ) theory R had the interpretation of Λ 2 pure . Hence we obtained the correct field theoretic matching of scales Moreover it is easy to check that then the U matter p (R, T, m) give a vanishing contribution as the functions f n (z) → 0 when z → ∞ (see appendix B). The above behaviour is quite clear from the Seiberg-Witten curve perspective. It is reassuring that it could be also obtained in a simple way from the random matrix formulas.

Discussion
In this paper we used the Dijkgraaf-Vafa proposal linking random matrix models with fundamental matter and superpotentials for obtaining explicit formulas for the complete factorization of Seiberg-Witten curves for U(N c ) theories with N f < N c flavours. These points in the moduli space, forming effectively a 1-parameter manifold, correspond to condensation of all species of monopoles. As a byproduct we obtained formulas for the solution of the random matrix model with matter with an arbitrary polynomial superpotential.
In order to identify the points in moduli space where the Seiberg-Witten curve factorizes we recast the random matrix solution in a way that exhibits (i) linearity in the couplings g p of the deforming tree level superpotential, (ii) the whole dependence on the glueball superfield S could be written as a linear coupling of S to a logarithmic expression. The first property allowed us to identify the moduli space parameters of the factorized curve u p as the coefficients of g p , while the second property is exactly the one found in 'integrating-in' S and thus gave the expression for the gauge theoretic scale Λ in terms of random matrix model quantities.
The fact that the above procedure works is yet another argument for the Intriligator-Leigh-Seiberg linearity principle [24] and the validity of 'integratingin'. In addition it shows that the matrix model of the Dijkgraaf-Vafa proposal captures quite detailed properties of the field theoretical Seiberg-Witten curve. In fact we found it surprising that any analytical description of the very nonlinear complete factorization property could be found for the case with matter.
A curious feature of the random matrix formulas is that the solution for the u p for the pure N = 2 theory appears linearly in the complete expression for the theory with fundamental matter fields. The full 'nonlinearity' is encoded in the formula for Λ in terms of random matrix parameters. It would be interesting to understand this structure from the field-theoretical point of view. In addition the random matrix constraints expressed in terms of R and T don't change when adding fundamental matter. They have precisely the form of equations of motion for the pure N = 2 theory. But now the mapping between R and T and field theoretical Λ and u 1 becomes complicated. Nevertheless, the question why the pure N = 2 equations still arise in a disguised form for theories with fundamental matter poses an interesting question from the gauge theory perspective.
There are numerous issues that one could investigate further. The factorization properties of the pure N = 2 curve are linked with the concept of master field. This has been investigated in the context of associated random matrix theory in [25]. It would be interesting to study the factorization formulas obtained in this paper from a similar point of view, albeit it will surely be much more involved.
Another interesting question would be to explore the mathematical structure linking the random matrix model with matter with factorization properties of the associated curves. In this paper we relied heavily on recasting the random matrix expressions guided by field theoretic ingredients such as the ILS principle and integrating-in, for which there is no real direct proof. It would be very interesting to uncover the mathematical interrelation between such seemingly unconnected topics as the factorization of SW curves and random matrix models.
Finally we hope that the above results could be used for a more detailed investigation of the physics of U(N c ) theories with N f < N c flavours along the lines of [19,26].
it is straightforward to obtain the Laurent expansion of the function in (81): For odd n, f n (z) is expressed through elementary functions (see examples in section 7). It has the form of a polynomial in z of order (n + 1)/2 divided by (z − 1) n 2 +1 . n even The integral (86) can be again obtained using a resummation procedure. The result is I even n Changing to a new summation variable m = k + l + 1 gives the expression: This is exactly the expression for S, provided that: which can be verified.