Explicit factorization of Seiberg-Witten curves with matter from random matrix models. 1

Transcription

1 KUL-TF NORDITA-HE Exlicit factorization of Seiberg-Witten curves with matter from random matrix models. 1 Yves Demasure a,b and Romuald A. Janik c,d3 a Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 00D, B-3001 Leuven, Belgium b NORDITA, Blegdamsvej 17, DK-100 Coenhagen, Denmark c The Niels Bohr Institute, Blegdamsvej 17, DK-100 Coenhagen, Denmark d Jagellonian University, Reymonta 4, Krakow, Poland Abstract Within the Dijkgraaf-Vafa corresondence, we study the comlete factorization of the Seiberg-Witten curve for UN c gauge theory with N f <N c massive flavors. We obtain exlicit exressions, from random matrix theory, for the moduli, arametrizing the curve. These moduli characterize the submanifold of the Coulomb branch where all monooles become massless. We find that the matrix model reveals some nontrivial structures of the gauge theory. In articular the moduli are additive with resect to adding extra matter and increasing the number of colors. 1 Work artially suorted by the Euroean Commission RTN rogramme HPRN-CT demasure@nordita.dk 3 janik@nbi.dk

2 Contents 1 Introduction 1 Field theory considerations 3 The Dijkgraaf-Vafa roosal with fundamental matter 6 4 Orthogonal olynomials and F χ= 7 5 The F χ=1 matter contribution 11 6 Factorization formulas 13 7 Final results 15 8 Some examles 17 9 Discussion 19 Aendices 1 A Formula for Mψ 1 B Logarithmic integrals f n z C Coefficient of the logarithmic term in?? 3 1 Introduction Very recently there aeared new owerful methods of extracting effective suerotentials for a wide class of N = 1 gauge theories. The roosal by Dijkgraaf and Vafa [1, ], building uon earlier string theoretical constructions [3, 4], links these suerotentials with quantities in random matrix models. The roosal have since been roven [5, 6]. The roosal 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, 1, 13, 14]. 1

3 These curves made their aearance in the ground-breaking work of Seiberg and Witten [15, 16] on the study of N = suersymmetric gauge theories. It turned out that one can describe the low energy dynamics of the gauge theory in terms of geometrical roerties 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 suerotential. In this context one is led to the oints submanifolds in moduli sace where monooles become massless and condense. At these oints the Seiberg-Witten curve factorizes it has only two single zeroes branch oints in the case of comlete factorization, where all monooles condense. Once the form of the Seiberg-Witten curve at the factorization oint is known one can calculate effective otentials for the deformed theory [16, 19, 4]. The ossibility of integrating in the glueball field S was realized in [9] and used to check the matrix model result following from the Dijkgraaf-Vafa roosal for deformed ure N = theories without fundamental matter. There the exlicit factorization of curves without matter of [19] was known. Factorization roerties were also used in the context of SON c theories [0] and multi-trace oerators [1] to obtain the aroriate effective suerotentials W eff S. However once one adds fundamental matter to the theory, the factorization roblem becomes exceedingly comlicated even in the case of U with 1 flavour and no exlicit solutions are known. The aim of this aer is to use the Dijkgraaf-Vafa roosal linking suerotentials to matrix models and derive, from the random matrix model solution, exlicit factorization of Seiberg-Witten curves of UN c theorywithn f <N c fundamental matter fields. If it were not for the Dijkgraaf-Vafa corresondence, we would not exect that any analytical solutions to this roblem 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 aroriate Seiberg- Witten curves. In addition we find that the matrix theory variables cature a surrising robust structure of the factorized Seiberg-Witten curve. The ure N = solutions and the new contributions of each flavor aear 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 ure N = theories reaear, in terms of matrix variables, in the context of N = theories with fundamental flavors.

4 This arises naturally in the matrix model but is rather unexected from a gauge theory ersective. In section 7 we collect the final results of the aer. The lan of this aer is as follows. In section we describe the field theoretical ingredients in more detail. In section 3 we introduce the setu for the Dijkgraaf-Vafa roosal with matter fields. We then go to use the orthogonal olynomial method to rederive directly the solution of the factorization for ure N = theory, which is an ingredient of the exressions for theories with matter. In sections 5 and 6 we derive the linearity in coulings of the suerotential and the exressions for factorization. Section 7 contains the main result of the aer. In section 8 we give some secific examles and close the aer with a discussion. Three aendices contain some technical details. Field theory considerations It has been known for a long time that N = suersymmetric UN 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 sace is N c -dimensional, arametrized for examle by u k = 1 tr k Φk with k N c. At each oint of the moduli sace, the low energy theory is described by an N = effective abelian U1 Nc gauge theory. All the relevant quantum corrections in the IR can be recast in terms of the eriod matrix of a articular meromorhic one-form of the auxilliarly comlex curve the Seiberg-Witten curve [15, 15, 17, 18], or more recisely a family of genus N c 1 hyerellitic Riemann surfaces: N f y = P Nc x, u k 4Λ Nc N f x + m i, 1 with P Nc x, u k = det xi Φ = i=1 N c α=0 s α x Nc α. The coefficients s α are olynomials of the u k s arameterizing the Coulomb branch: α αs α + ks α k u k =0 3 k=0 s 0 =1, u 0 =0 4 3

5 One can deform this N = theory to a N = 1 gauge theory by adding a tree level suerotential: W tree = N c+1 =1 g 1 tr Φ. 5 The classical vacuum structure is given by all ossible distributions of the N c eigenvalues φ k of Φ amongst the N c critical oints a j of the otential. This corresonds, at the classical level, to a breaking of the UN c gauge symmetry to the gauge grou N c n i=1 UN i with N c n i=1 N i = N c, where the N i s are the nonzero multilicities of the eigenvalues. For the uroses of this aer we can safely set g Nc+1 = 0 as we will be considering the case with no breaking of gauge symmetry comlete factorization. Turning to the quantum icture, the resence of this suerotential will lift the quantum moduli sace, characteristic of the N = Coulomb hase, excet for the codimension n submanifolds, where n mutually local magnetic monooles become massless. These are the N = 1vacuasolvingthe F-flatness and D-flatness conditions and are characterized by a monoole condensate of the massless monooles. This is believed to roduce, by the dual Meissner effect, the exected confinement of the electric N = 1 theory. Hence, the final quantum theory is described at low energies by a N =1 U1 Nc n gauge theory. These U1 s can be thought of as the U1 UN i of the classical theory. The N =1SUN i art of the theory confines, has a mass ga and is characterized by a gaugino condensate. These N = 1 vacua, being the codimensional n submanifolds of the N = Coulomb branch, where n mutually local monooles become massless, are arameterized by the sets of moduli {u fact. k } where the Seiberg-Witten curve factorizes, i.e. the r.h.s. of 1 has n double roots 4 and N c n singleroots: N f P Nc x, u fact. k 4Λ Nc N f x + m i =F Nc nxhnx. 6 i=1 Moreover it is shown in [8] that the reduced curve, y = F Nc n 7 catures the full quantum dynamics of the N = 1 U1 Nc n low energy theory. 4 For simlicity we are assuming that higher order roots are not occuring. 4

6 The effective suerotential of the theory deformed by 5, where we set g Nc+1 = 0, is obtained by lugging the solutions u fact. k, arametrized by N c n arameters into the tree level suerotential: N c W eff = g u fact., 8 Then 8 should be minimized with resect to the N c n arameters. Comlete factorization =1 In this aer we will be interested in the case where N c 1 mutually local monooles condense. This corresonds to a comlete factorization of the Seiberg-Witten curve: the vacua form a 1 dimensional submanifold such that the curve has only single roots and N c 1 double roots: N f P Nc x, u fact. k 4Λ Nc N f x + m i =x ax bhn c 1 x 9 i=1 The main goal of this aer is to find exlicit exressions for the moduli u fact. k where comlete factorization occurs. Let us briefly review the solution for the ure N = Yang-Mills case found by Douglas and Shenker [19] some time ago using Chebyshev olynomials. Their solution factorizing the curve P Nc x, u k 4Λ Nc is 5 u fact. Λ,u 1 = N c [/] q=0 q q Λ q u1 q. 10 q N c Note that the one dimensional submanifold where the UN c curvecomletely factorized, is arametrized by u 1 = tr Φ. These results can be easily restricted to the SUN c case, by utting exlicitly u 1 = 0. All arameters are then uniquely fixed in terms of the only scale of the theory Λ. Integrating in S The quantum N = 1 effective otential generated by the tree level otential is obtained by minimizing 8 N c W eff Λ,u 1 = 5 Adated to the UN c gauge grou [9]. =1 g u fact. Λ,u

7 with resect to u 1. Here we follow an alternative route taken in [9], and integrate in S by erforming a Legendre transformation with resect to log Λ Nc N f.thesuerotential is then given by: W eff S, u 1, Ω, Λ = S log Λ Nc N f + W eff S, u 1, Ω = = S log Λ Nc N f S log Ω Nc N N c f + =1 g u fact. Ω,u 1. 1 Note that the only Λ-deendence is in the linear S log Λ Nc N f term. Integrating out S forces Ω = Λ and brings us back to 11. In order to get the effective otential W eff Λ,S one has to integrate out both Ω and u 1. Our strategy for identifying the factorization arameters u fact. for the theory with matter is to rewrite the random matrix exression in the form given by the last two terms of 1 and to read off the aroriate gauge theoretic moduli u fact.. The first term in 1 could also be absorbed into the matrix model exressions if aroriate 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 ure N = gauge theory: W eff S, u 1, Ω log Ω Nc W eff S, u 1, Ω u 1 = = 1 [/] g q=0 [/] g q=0 q q q These follow directly from 1 and 10. q Ω q u1 q q N c q Ω q u1 q N c q S = 0 13 q 1= The Dijkgraaf-Vafa roosal with fundamental matter According to the Dijkgraaf-Vafa rescrition the the erturbative art of the suerotential can be exressed as [1] W eff S =N c F χ= S S 6 + F χ=1 S 15

8 where the F χ are defined through the matrix integral [7, 1] e N S F χ=s N S Fχ=1S+... = DΦDQ i D Q i e N S tr V Φ mq i Qi Q i Φ Q i 16 where we included the couling of fundamental matter to the adjoint field in accordance with N = suersymmetry. The matter fields in the matrix model 16 aear only quadratically and hence may be integrated out giving DΦdetm +Φ 1 e N S tr V Φ = Z det m +Φ 1 17 where Z is the artition function of the matrix model without matter. For the comlex matrix model it is well know that Z will contribute only to F χ= and not to F χ=1.usinglargen factorization we also have detm +Φ 1 = e log detm+φ = e tr logm+φ = e tr logm+φ 18 We see that the matter determinant aears at subleading order in N w.r.t the tree level otential. This has the imortant consequence that the saddle oint solution eigenvalue density ρλ and hence F χ= will not be influenced by the resence of matter. The χ = 1 contribution is then given by N f F χ=1 = dλρλlogm i + λ 19 i=1 In the following section we will evaluate the F χ= iece of the artition function Z aearing in 17 using the method of orthogonal olynomials. As a byroduct we will rederive the factorization exression for the ure gauge theory case. Then in section 5 we will start investigating the matter contribution Orthogonal olynomials and F χ= In this section we will use the method of the orthogonal olynomials to study thoroughly the one cut solution. A very nice introduction to this owerful method can be found in []. We will show that all results obtained from the field theory analysis aear naturally in this setting from random matrix comutations comare [9]. 7

9 The orthogonal olynomials associated to the matrix model with otential N tr V Φ satisfy the recursion relation S sp n s =P n+1 s+t n P n s+r n P n 1 s 0 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 otential W tree Φ become urely algebraic [3, ]: dz πi V z + RSx + T Sx = Sx 1 z dz 1 πi z V z + RSx z + T Sx = 0 When x = 1 we will denote RS byr and T S byt. These two variables are related to the endoints a and b of the suort of the matrix eigenvalue distribution through T = a + b 3 a b R = For a general matrix otential V Φ = 1 g Φ, one obtains easily: u Sx = v 1 [/] g q=0 [/] g q=0 q q q q q RSx q T Sx q 5 q q RSx q T Sx q 1 =0, 6 q defining u and v for later convenience 6. In terms of these olynomials, the matrix model artition function takes a very elegant form: where h 0 is the integral Z = N!h N 0 en 1 0 dx1 xlnrsx. 7 h 0 = ds e N S V s 8 6 Aart from the resent section u and v are always taken at x =1. 8

10 From equation 7, one can easily extract the χ = contribution we will comment on the h 0 iece below 1 F χ= = S dx1 xlnrsx 9 Performing an integration by arts leads to: S S 0 0 du ln Ru+ S 0 duu ln Ru 30 Following the Dijkgraaf-Vafa rescrition, the contribution to the gauge theory effective otential is roortional to: F χ= S = S ln RS+ RS R0=0 ur, T dr 31 R 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 comlicates the integral, and moreover soils the manifest linearity in the coulings g, aearing in the matrix otential. It is convenient, however, to treat the one-dimensional integral as an integral of a 1-form over the ath {v = 0} in the two dimensional R-T lane. 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 iece with R =0, while integrating T from T 0 to T S and another iece, keeing T fixed at T S, and integrating R from 0 to RS see fig. 1: RS ur, T R0=0 R v=0 dr ur, T R v=0 dr = ω = ω + ω T =const R=const 3 An extension of the integrand of 31 to a closed one-form can be found to be ω = u dr + vdt. 33 R Note that the extra iece doesn t give a contribution to the original integral, which is taken over v =0. Performing the integral is now straightforward. The first iece, integrating over T and constraining R to0gives: T S T 0 vdt = 1 g T S T

11 T v=0 Figure 1: The dotted line reresents the original contour of integration, while the solid lines show the deformed contour in the T -R lane. R The evaluation of the integral at T 0, can be cancelled at the saddle oint against the h N 0, aearing in the artition fuction 7. For the second iece, one fixes T at T S, and integrates over R: RS 0 u R dr = g [/] q=1 q q R q T q 35 q From now on until the end of the aer we will denote by R and T the recursion coefficients defined at S. Bringing the two contributions together, leads to the following result: W eff = N c S ln R + N c 1 g [/] q=0 q q R q T q 36 q This is exactly, term by term, the field theory result for ure N =UN c gauge theory 1 and 10, rovided we identify field theory variables and matrix theory variables in the following way: R =Ω, T = u 1 N c 37 The first identification is extracted from the linear term in S, the second art follows from the term linear in g 1. As it should, the equations of motion for R and T 5-6, are maed, under this identification, to the equations obtained from integrating out Ω and u

12 From 36 we may read off the coefficients of g : U ure R, T = 1 [/] q=0 q q R q T q 38 q Under the identification 37 and setting Ω = Λ these coincide exactly with the factorization solution 10 of Douglas and Shenker for ure N = gauge theory. For theories with matter, 38 will be just a art 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. 5 The F χ=1 matter contribution In the revious section we have rederived the result that the first iece in 15 can be recast in the form N c F χ= S = N c [ S log R + g U ure R, T ] 39 and the equations of motion for R and T derived from 39 are exactly the random matrix saddle oint equations S = u 0 = v 1 [/] g q=0 [/] g q=0 q q q q R q T q 40 q q q R q T q 1 41 q The first of these can be interreted as the formula for integrating in S. The linearity of 39 in the random matrix coulings is essential for identifying the U ure R, T with the oint in N = moduli sace where the Seiberg-Witten curve y = P Nc x, u k 4Λ Nc factorizes. In the following section we will recast the random matrix exression 19 for F χ=1 inthesameway: N f F χ=1 = i=1 [ S log LR, T, mi + g U matter R, T, m i ] 4 11

13 Let us comment on an ambiguity, leading to a unique determination of a crucial term to obtain the correct final result. Once we start interret this exression for F χ=1, as a art of the effective gauge theory otential, we should be able to integrate out R and T. The equations of motion thus obtained, should be consistent with the random matrix saddle oint equations. But, as exlained in section 3, adding fundamental matter in the gauge theory does not modify the saddle oint equations of the matrix model. So it is necessary to imose that the equations of motion for R and T, derived from the effective otential, with the F χ=1 contribution are consistent with 40 and 41. This fact is highly non-trivial and reveals an unexected relation between the factorization of Seiberg-Witten curves with and without matter. We will show that the addition of F χ=1 to the suerotential does not change the equations of motion for R and T rovided we add an extra term roortional to v. From the matrix model ersective nothing changes, while the extra term turns out to be crucial to obtain the correct gauge theoretical interretation of our results. We stress that this does not alter the Dijkgraaf- Vafa rescrition. It merely solves an ambiguity that arises when interreting 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 olynomial method, which was the most straightforward way of deriving the F χ= contribution. Here it is more convenient to use the saddle oint exression for the eigenvalue density of the single-cut solution: ρx = 1 π Mx b xx a 43 where Mx is a olynomial which is exressible in terms of the random matrix otential through dw V w Mx = 44 C πi w x w aw b The features of interest of the above exression are i it is linear in the exlicit deendence on the coulings g, ii the coefficients of g are universal functions of the endoints a, b and hence of the variables R and T, iii for given g, its coefficient is a olynomial in x of order. In order to obtain an exlicit deendence on R and T let us erform the 1

14 change of variables x = 1 a a + b+b ψ T + Rψ 45 Then the contribution to F χ=1 of a single flavour is given by R 1 dψ 1 ψ π Mψ log m + T + Rψ + vfr, T 46 1 As noted before, one should take into account a ossible addition of v multilied by any function fr, T sincev = 0 is an equation of motion of the random matrix model. 6 Factorization formulas We will first derive the formulas involving g 1 and g, in articular this will allow us to fix uniquely the function fr, T. Also this will make the general structure more transarent. Then we will derive the results for arbitrary g. Contribution of g 1, g To this order Mψ =g and the formula 46 gives [ g R logm + T + 1 ] R dψ 1 ψ π log 1+ 1 m + T ψ 47 This can be evaluated to give g R log m + T + m + T 4R +m + T m + T m + T 4R 1 4R 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 R, but later we will see that this roerty will hold in general. Thus we are left with S log m + T + m + T 4R [ m + T +g m + T m + T 4 4R R ] 49 13

15 which indeed has the form of 4. Now we require that the saddle oint equations remain consistent with the F χ=1 equations of motion. This detemines uniquely the term vg 1,g,R,T fr, T =g 1 + g T fr, T 50 Indeed the requirement that integrating out T from the sum of 49 and 50 gives v = g 1 + g T =0fixesfR, T uniquely to be fr, T = 1 m + T m + T 4R 51 This extra term does not change the equation for R, which is consistent with the saddle oint equations. This function will stay unchanged in the general case. We may now read off the final exressions for the mass deendent contributions to u 1 and u : U1 matter R, T, m = 1 m + T m + T 4R 5 U matter R, T, m = m + T m + T m + T 4 4R R + T m + T m + T 4R 53 Arbitrary g We will now extend the revious considerations to the calculation of arbitrary U matter. The general structure will remain unchanged. The function fr, T multilying v will remain unmodified as it should. Also the coefficient of the logarithm will turn out to be exactly S. In aendix A we derive the following exression for the olynomial Mψ: Mψ = g c,n ψ n 54 n=0 where [ n c,n = n R n ] k 1 R k T n k 55 k k + n +1 k=0 14

16 So the integral 46 can be rewritten as > g n=0 1 + π [ c,n R logm + T 1 dψ 1 ψ π ψ n + 1 R 1 dψ 1 ψ ψ n log ] 1+ m + T ψ 56 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 aendix B, f n z is essentially a olynomial in z of order n +1/ divided by z 1 n/+1 when exressed in terms of the variable z = m + T R m + T + m + T 4R 57 Aendix B contains a general formula for f n z. Exlicit exressions for some secific 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 logm + T in the first integral. Together they combine to give [ ] g c,l R l l l +1 l log m + T + m + T 4R 58 l=0 It is shown in aendix C that the coefficient in curly braces is exactly equal to S. The second integral with the logarithmic art subtracted out has again a simle olynomial structure see aendix B for details. At this stage we arrive to the analogue of 49: S log m + T + m + T 4R + g c,n Rf n z 59 > Again we have to add to this the correction term m + T v fr, T 1 g v 1 n=0 m + T 4R 60 We checked exlicitly 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 exected form 4, thus defining U matter R, T, m i. 15

17 7 Final results Putting all results together 36, 59, 60, and utting in the term S log Λ Nc N f as required from the Dijkgraaf-Vafa rescrition gives a rediction from the matrix model side for the quantum effective gauge otential: W eff S, T, R, Λ = S log ΛNc N f N f i= g mi + T + m i + T 4R R Nc N c U ure R, T + N f R, T, m i. 61 U matter i=1 This exression should be comared with the otential W eff S, u 1, Ω, Λ see 1, obtained from the field theory analysis. The relation between the arameters of the matrix model and the field theory is highly non-linear: N f u 1 = N c T 1 m i + T Ω Nc N f = i=1 Nf i=1 1 R Nc m i + T 4R 6 mi + T + m i + T 4R 63 In order to obtain the final factorization formulae we integrate out S from W eff S, T, R, Λ : Ω Nc N f =Λ Nc N f, 64 Combined with 63 this gives an exression for Λ in terms of R and T.The remaining art of 61 should then be comared with field theory result: W eff Λ,u 1 = g u fact. Λ,u 1, 65 1 where the u fact., is the one arameter solution, factorizing comletely the Seiberg-Witten curve for UN c SYMwithN f flavours N f <N c : N f y = P Nc x, u k 4Λ Nc N f x + m i 66 Comaring with the results from matrix models gives an exression for the u fact. s, N f u fact. = N c U ure R, T + U matter R, T, m i 67 i=1 16 i=1

18 f 0 z = 1 z 1 f 1 z = 3z 4 6z 1 3/ f z = 1 16z 1 f 4 z = 3z +9z 5 96z 1 3 f 6 z = 48z3 +168z 176z z 1 4 f 3 z = 30z 65z+3 10z 1 5/ f 5 z = 55z3 1610z +158z z 1 7/ f 7 z = 4410z z z 16857z z 1 9/ Table 1: Examles of the functions f n z for small n. in terms of two arameters R and T, tied together with the constraint: Λ Nc N f = Nf i=1 1 R Nc mi + T + m i + T 4R 68 For comleteness, we recall the formulas U matter U ure R, T = 1 [/] q=0 1 R, T, m = 1 U matter R, T, m = n=0 q m + T q R q T q 69 q m + T 4R 70 1 c,n Rf n z v m + T m + T 4R 71 In the above formula the coefficients c,n are defined in 55, while v is just the coefficient of g in the constraint v = 0 see eq. 41. Finally the functions f n z are comuted in aendix B and deend on the variable zr, T, given by 57. In table 1 we resent the exlicit forms of the functions f n z for n 7. In the final result 67 we see that both the N c and N f deendence is very simle. Moreover the contribution of each extra flavor enters additively the exression for the moduli. Another curious feature is the aearance of the original factorization solutions for the ure N =UN c gauge theory. Increasing the number of colors does not change the exressions for U matter in terms of R and T. However the only nontrivial change is encoded in the exression for Λ in terms of R and T. If we wanted instead to obtain the effective otential 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 aears in the equations of motion for u 1 andω. Onthematrixmodel,ontheother 17

19 hand, the equations of motion for R and T aear naturally to be the same as ones obtained in the case without flavours. It seems that the articular combinations of u 1 and Ω, embodied in R and T, catures some nontrivial structure of the Seiberg-Witten curves. 8 Some examles In this section we will study some examles to verify that the u fact. we obtained from random matrix models, do factorize the aroriate Seiberg- Witten curves with fundamental matter. U with 1 flavour In this case we have u fact. 1 =T 1 m + T u fact. = R + T Λ 3 R = m + T + m + T 4R m R T +T m m + T 4R 7 m + T 4R We have verified that with the above choices, the discriminant of the Seiberg- Witten curve y = x u 1 x u u 4Λ 3 x + m 75 vanishes identically, which in the secial case of colours roves factorization. Note that for general N c, comlete factorization is a much stronger condition than the vanishing of the discriminant. SUN c with 1 flavour In order to consider SUN c theory we have to imose the constraint that u 1 = 0. Then the arameter T can be exressed in terms of R and the mass m of the additional flavour. Namely we have u 1 N c T 1 m + T m + T 4R =

20 which gives T = m m 4R +4 R N c N c 1 Now R is linked directly to the scale of the gauge theory through 77 Λ Nc 1 = R Nc m + T + m + T 4R / 78 where the exression 77 should be used. The remaining formulas remain however quite comlicated functions of R, m and N c. This is in marked contrast to the case of ure gauge theory without fundamental matter where the assage from UN c tosun c is very simle, and N c enters linearly. It is interesting to look at the m limit. Then T 0 as exected for a ure N =SUN c theory, while 78 becomes Λ Nc 1 = R Nc /m. Recall that in ure SUN c theoryr had the interretation of Λ ure. Hence we obtained the correct field theoretic matching of scales Λ Nc 1 m =Λ Nc ure 79 Moreover it is easy to check that then the U matter R, T, m give a vanishing contribution as the functions f n z 0whenz see aendix B. The above behaviour is quite clear from the Seiberg-Witten curve ersective. It is reassuring that it could be also obtained in a simle way from the random matrix formulas. U7 with 3 flavours As a final check of the formulas we considered U7 theory with 3 flavours with masses m 1 =1,m =,m 3 = 3. For the random choice of arameters T =0andR =0. we find Λ = , while the olynomial P 7 x = x x x x x x 0.045x For the curve y = P 7 x 4Λ 11 x +1x +x we find single zeroes at x = ± and a series of 6 double zeroes in between. 19

21 9 Discussion In this aer we used the Dijkgraaf-Vafa roosal linking random matrix models with fundamental matter and suerotentials for obtaining exlicit formulas for the comlete factorization of Seiberg-Witten curves for UN c theories with N f <N c flavours. These oints in the moduli sace, forming effectively a 1-arameter manifold, corresond to condensation of all secies of monooles. As a byroduct we obtained formulas for the solution of the random matrix model with matter with an arbitrary olynomial suerotential. In order to identify the oints in moduli sace where the Seiberg-Witten curve factorizes we recast the random matrix solution in a way that exhibits i linearity in the coulings g of the deforming tree level suerotential, ii the whole deendence on the glueball suerfield S could be written as a linear couling of S to a logarithmic exression. The first roerty allowed us to identify the moduli sace arameters of the factorized curve u as the coefficients of g, while the second roerty is exactly the one found in integrating-in S and thus gave the exression for the gauge theoretic scale Λ in terms of random matrix model quantities. The fact that the above rocedure works is yet another argument for the Intriligator-Leigh-Seiberg linearity rincile [4] and the validity of integratingin. In addition it shows that the matrix model of the Dijkgraaf-Vafa roosal catures quite detailed roerties of the field theoretical Seiberg-Witten curve. In fact we found it surrising that any analytical descrition of the very nonlinear comlete factorization roerty could be found for the case with matter. A curious feature of the random matrix formulas is that the solution for the u for the ure N = theory aears linearly in the comlete exression for the theory with fundamental matter fields. The full nonlinearity is encoded in the formula for Λ in terms of random matrix arameters. It would be interesting to understand this structure from the field-theoretical oint of view. In addition the random matrix constraints exressed in terms of R and T don t change when adding fundamental matter. They have recisely the form of equations of motion for the ure N = theory. But now the maing between R and T and field theoretical Λ and u 1 becomes comlicated. Nevertheless, the question why the ure N = equations still arise in a disguised form for theories with fundamental matter oses an interesting question from the gauge theory ersective. 0

22 There are numerous issues that one could investigate further. The factorization roerties of the ure N = curve are linked with the concet of master field. This has been investigated in the context of associated random matrix theory in [5]. It would be interesting to study the factorization formulas obtained in this aer from a similar oint of view, albeit it will surely be much more involved. Another interesting question would be to exlore the mathematical structure linking the random matrix model with matter with factorization roerties of the associated curves. In this aer we relied heavily on recasting the random matrix exressions guided by field theoretic ingredients such as the ILS rincile and integrating-in, for which there is no real direct roof. It would be very interesting to uncover the mathematical interrelation between such seemingly unconnected toics as the factorization of SW curves and random matrix models. Finally we hoe that the above results could be used for a more detailed investigation of the hysics of UN c theories with N f <N c flavours along the lines of [19, 6]. Acknowledgments RJ was suorted by the EU network on Discrete Random Geometry and KBN grant P03B096. YD was suorted by an EC Marie Curie Training site Fellowshi at Nordita, under contract number HPMT-CT A Formula for Mψ Since we use exlicitly the form of the eigenvalue density in the variable ψ, let us erform the changes of variables x = T + Rψ, w = T + Rφ in the definition 44 of Mx: Mψ = 1 R Res 1 φ= φ ψ Using the ower series exansion of the square root V T + Rφ φ φ = k=0 a k 1 φ k k=0 k k k 1 φ k 8 1

23 it is straightforward to obtain the Laurent exansion of the function in 81: 1 ψ n 1 φ a 1 l 1 k+n+ k R T 1 l φ l 83 l g n=0 k=0 l=0 From this exression we may isolate the coefficient of 1/φ giving the result quoted in the text: Mψ = c,n = n R n g n=0 [ n ] k=0 c,n ψ n 84 k k 1 k + n +1 R k T n k 85 B Logarithmic integrals f n z Here we will derive the exlicit form of the functions f n z related to the logarithmic integrals I n = 1 dψ 1 ψ π ψ n log 1 + xψ 86 1 where x = R/m + T. In fact the results simlify significantly if one reexresses everything in terms of the variable z z = m + T m + T + m + T R 4R 87 x is exressed in terms of z as z 1/z. We have to distinguish two cases: n odd The integral 86 can be erformed using a series exansion of the logarithm, integrating it term by term using 1 dψ 1 ψ π ψ n = n+1 n 88 1 n + n and resumming. The result is fn odd z = z 1Γ 1+ n 1 πzγ 5+n 3F, 1, 1+n ; 3, 5+n 4z 1 89 z

24 For odd n, f n z is exressed through elementary functions see examles in section 7. It has the form of a olynomial in z of order n +1/ divided by z 1 n +1. n even The integral 86 can be again obtained using a resummation rocedure. The result is z 1Γ 3+n In even = πz Γ 3+ n 3 F 1, 1, 3+n ;, 3+ n 4z 1 90 z In this case the integral 86 involves a logarithm, which together with the logm + T forms the logarithmic function multilying S in 59. Thus to define the function f n z for even n we have to subtract from In even this logarithm: fn even z In even n+1 n z 1 n log 91 n + z Its general form turns out to be a olynomial of order n/ divided by z 1 n +1. In table 1 in section 7, for comleteness we resent the exlicit forms of the functions f n z for n 7. C Coefficient of the logarithmic term in 58 In this aendix, we identify the coefficient of the logarithmic iece of the χ = 1 contribution with S, as exected from field theory. From 46 one can easily read off the coefficient of logm + T : 1 dψ 4R 1 ψ 1 π Mψ 9 Using the elementary integral 88 one obtains the exression: 4R l l g c l,l 93 l=0 l +1 Inserting the exlicit exression for the coefficient c,l,leadsto: [ g l=0 ] [ l ] k=0 k k l l 1 l +k l +1 Rl+k+1 T k l. 94

25 Changing to a new summation variable m = k + l + 1 gives the exression: [ ] g m 1 m=1 l=0 m l +1 m l l m l R m T m. 95 m l 1 This is exactly the exression for S, rovided that: m 1 l=0 1 l +1 l l m l = m l 1 m m 96 which can be verified. References [1] R. Dijkgraaf and C. Vafa, A erturbative window into non-erturbative hysics, arxiv:he-th/ [] R. Dijkgraaf and C. Vafa, Matrix models, toological strings, and suersymmetric gauge theories, Nucl. Phys. B [arxiv:heth/00655]. [3] C. Vafa, Suerstrings and toological strings at large N, J. Math. Phys [arxiv:he-th/000814]. [4] F. Cachazo, K. A. Intriligator and C. Vafa, A large N duality via a geometric transition, Nucl. Phys. B [arxiv:he-th/ ]. [5] R. Dijkgraaf, M. T. Grisaru, C. S. Lam, C. Vafa and D. Zanon, Perturbative comutation of glueball suerotentials, arxiv:he-th/ [6] F. Cachazo, M. R. Douglas, N. Seiberg and E. Witten, Chiral rings and anomalies in suersymmetric gauge theory, arxiv:he-th/ [7] R. Argurio, V. L. Camos, G. Ferretti and R. Heise, Exact suerotentials for theories with flavors via a matrix integral, arxiv:heth/ J. McGreevy, Adding flavor to Dijkgraaf-Vafa, arxiv:he-th/ H. Suzuki, Perturbative derivation of exact suerotential for meson fields from matrix theories with one flavour, arxiv:he-th/

26 I. Bena and R. Roiban, Exact suerotentials in N = 1 theories with flavor and their matrix model formulation, arxiv:he-th/ Y. Demasure and R. A. Janik, Effective matter suerotentials from Wishart random matrices, arxiv:he-th/ B. Feng, Seiberg duality in matrix model, arxiv:he-th/0110. B. Feng and Y. H. He, Seiberg duality in matrix models. II, arxiv:heth/ R. Argurio, V. L. Camos, G. Ferretti and R. Heise, Baryonic corrections to suerotentials from erturbation theory, arxiv:heth/ I. Bena, R. Roiban and R. Tatar, Baryons, boundaries and matrix models, arxiv:he-th/ Y. Ookouchi, N = 1 gauge theory with flavor from fluxes, arxiv:heth/ K. Ohta, Exact mesonic vacua from matrix models, arxiv:heth/0105. I. Bena, S. de Haro and R. Roiban, Generalized Yukawa coulings and matrix models, arxiv:he-th/ H. Suzuki, Mean-field aroach to the derivation of baryon suerotential from matrix model, arxiv:he-th/0111. C. Hofman, Suer Yang-Mills with flavors from large Nf matrix models, arxiv:he-th/ [8] F. Cachazo and C. Vafa, N = 1 and N = geometry from fluxes, arxiv:he-th/ [9] F. Ferrari, On exact suerotentials in confining vacua, arxiv:heth/ [10] R. Dijkgraaf, S. Gukov, V. A. Kazakov and C. Vafa, Perturbative analysis of gauged matrix models, arxiv:he-th/ [11] S. G. Naculich, H. J. Schnitzer and N. Wyllard, The N = UN gauge theory reotential and eriods from a erturbative matrix model calculation, arxiv:he-th/

27 [1] S. G. Naculich, H. J. Schnitzer and N. Wyllard, Matrix model aroach to the N = UN gauge theory with matter in the fundamental reresentation, arxiv:he-th/ [13] H. Itoyama and A. Morozov, The Dijkgraaf-Vafa reotential in the context of general Seiberg-Witten theory, arxiv:he-th/ [14] B. Feng, Geometric dual and matrix theory for SO/S gauge theories, arxiv:he-th/ [15] N. Seiberg and E. Witten, Electric - magnetic duality, monoole condensation, and confinement in N= suersymmetric Yang-Mills theory, Nucl. Phys. B [Erratum-ibid. B ] [arxiv:he-th/ ]. [16] N. Seiberg and E. Witten, Monooles, duality and chiral symmetry breaking in N= suersymmetric QCD, Nucl. Phys. B [arxiv:he-th/ ]. [17] P. C. Argyres and A. E. Faraggi, The vacuum structure and sectrum of N= suersymmetric SUn gauge theory, Phys. Rev. Lett [arxiv:he-th/ ]. [18] A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, Simle singularities and N= suersymmetric Yang-Mills theory, Phys. Lett. B [arxiv:he-th/ ]. [19] M. R. Douglas and S. H. Shenker, Dynamics of SUN suersymmetric gauge theory, Nucl. Phys. B [arxiv:he-th/ ]. [0] R. A. Janik and N. A. Obers, SON suerotential, Seiberg-Witten curves and loo equations, arxiv:he-th/ [1] V. Balasubramanian, J. de Boer, B. Feng, Y. H. He, M. x. Huang, V. Jejjala and A. Naqvi, Multi-trace suerotentials vs. Matrix models, arxiv:he-th/0108. [] P. Di Francesco, P. Ginsarg and J. Zinn-Justin, -D Gravity and random matrices, Phys. Ret [arxiv:he-th/ ]. [3] D. J. Gross and A. A. Migdal, A Nonerturbative Treatment Of Two- Dimensional Quantum Gravity, Nucl. Phys. B

28 [4] K. A. Intriligator, R. G. Leigh and N. Seiberg, Exact suerotentials in four-dimensions, Phys. Rev. D [arxiv:he-th/ ]. [5] R. Goakumar, N = 1 theories and a geometric master field, arxiv:he-th/ [6] F. Ferrari, Quantum arameter sace and double scaling limits in N = 1 suer Yang-Mills theory, arxiv:he-th/