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Effective SO Superpotential for N=1 Theory with N


E?ective SO Superpotential for N = 1 Theory with Nf Fundamental Matter
Pravina Borhade 1 , P. Ramadevi2 Department of Physics, Indian Institute of Technology Bombay, Mumbai 400 076, India

arXiv:0704.2482v1 [hep-th] 19 Apr 2007

Abstract Motivated by the duality conjecture of Dijkgraaf and Vafa between supersymmetric gauge theories and matrix models, we derive the e?ective superpotential of N = 1 supersymmetric gauge theory with gauge group SO(Nc ) and arbitrary tree level polynomial superpotential of one chiral super?eld in the adjoint representation and Nf fundamental matter multiplets. For a special point in the classical vacuum where the gauge group is unbroken, we show that the e?ective superpotential matches with that obtained from the geometric engineering approach.

1 2

E-mail: pravina@phy.iitb.ac.in Email: ramadevi@phy.iitb.ac.in

1

Introduction

Large N topological duality relating U (N ) Chern-Simons gauge theory on S 3 to Amodel topological string [1] and its embedding in the superstring context [2] has led to interesting interconnections between geometry of Calabi-Yau three-folds (CY3 ) and N = 1 supersymmetric gauge theories. Strong coupling dynamics of supersymmetric gauge theories can be studied within the superstring duality [2] by geometrically engineering D -branes. Using the geometric considerations of dualities in IIB string theory, Cachazo et al [3] have obtained low energy e?ective superpotential for a class of CY3 geometries whose singular limit is given by W ′ (x)2 + y 2 + z 2 + w 2 = 0 , (1.1)

where W (x) is a polynomial of degree n + 1. In fact, the low energy e?ective superpotential corresponds to a N = 1 supersymmetric U (N ) Yang-Mills with adjoint +1 k scalar Φ and tree level superpotential Wtree (Φ) = n k =1 (gk /k )T r Φ . The mirror version of the large N topological duality conjecture [1] was considered in ref. [4] relating topological B strings on the CY3 geometries [3] to matrix models. The potential of the matrix model W (Φ) = (1/gs )Wtree (Φ) where Φ denotes a hermitian matrix. Further, Dijkgraaf-Vafa have conjectured that the low-energy e?ective superpotential can be obtained from the planar limit of these matrix models [4–6]. The Dijkgraaf-Vafa conjecture was later on proved by various methods: (i) by factorization of Seiberg-Witten curves [7], (ii) using perturbative ?eld theory arguments [8] and (iii) generalized Konishi anomaly approach [9]. The extension of topological string duality relating Chern-Simons theory with SO/Sp gauge groups to A-model closed string on an orientifold of the resolved conifold was studied by Sinha-Vafa [10]. Generalizing the geometric procedure considered for U (N ) [3], the e?ective superpotential for N = 1 supersymmetric theories with SO/Sp gauge groups with Wtree = k (g2k /2k )T r Φ2k where Φ is adjoint scalar super?eld were derived for the orientifolds of the CY3 geometries [11]. These e?ective superpotentials have also been computed within perturbative gauge theory [12], using matrix model techniques in [13] and using the factorization property of N = 2 Seiberg-Witten curves [14]. Related works involving second rank tensor matter ?elds have been considered in refs. [15–17]. The geometric engineering of N = 1 gauge theories with unitary gauge group and matter in the adjoint and symmetric or antisymmetric representations has been investigated in [18]. Also for the SO/Sp theory with symmetric/antisymmetric tensor, the geometric construction was studied in [19]. So far, the e?ective superpotential computation involved N = 1 supersymmetric gauge theories with either adjoint matter or second rank tensor matter. The inclusion of matter transforming in the fundamental representation of these gauge groups can also be studied within the Dijkgraaf-Vafa setup [20–26]. For U (N ) gauge group, it was shown that the e?ective superpotential gets contributions from the matrix model planar diagrams with zero or one boundary [20,22]. In [24], it has been shown 1

that the U (N ) e?ective superpotential for theories with Nf fundamental ?avors can be calculated in terms of quantities computed in the pure gauge theory. Chiral N = 1 U (N ) gauge theories with antisymmetric, conjugate symmetric, adjoint and fundamental matter have also been studied in Refs. [19, 27]. Further, geometric engineering of the supersymmetric theories with Nf fundamental ?avors was considered in [28, 29] by placing D 5 branes at locations given by the the mass ma (a = 1, 2, . . . Nf ) which are not the zeros of W ′ (x) = 0. The SO/Sp e?ective superpotential computations from matrix model approach have been presented for tree level superpotential of adjoint matter upto quartic terms [25, 26]. Though there are several indications as to how the U (N ) e?ective superpotential can be related to SO/Sp groups, it is certainly not a proof. So we need to explicitly obtain the results using an independent method as elaborated in this paper. We consider N = 1 supersymmetric SO (Nc ) gauge theory with arbitrary tree level superpotential of one chiral super?eld in the adjoint representation and Nf fundamental matter multiplets. We use the technique developed in [24] to calculate the e?ective superpotential of this theory. The organization of the paper as follows: In section 2, we brie?y discuss the relevant matrix model and its free energy. Then we discuss the SO (Nc ) e?ective superpotential for N = 1 supersymmetric theory with fundamental matter in section 3. In particular, for a point in classical vacuum where the gauge group is unbroken, we obtain a neat expression for the e?ective superpotential for a most general tree level superpotential involving one adjoint matter ?eld. We also give a formal expression for a generic vacua where the gauge group is broken. In section 4, we recapitulate the geometric considerations of dualities. Then, we evaluate the e?ective superpotential for an example involving sixth power tree level potential and show that the results agree with our expressions in section 3 for the unbroken case. We conclude with summary and discussions in section 5.

2

Relevant Matrix Model

Let us consider N = 1 supersymmetric SO (Nc ) gauge theory with one adjoint ?eld Φ and Nf ?avors of quarks QI ’s with mass mI ’s (I = 1, 2, . . . Nf ) in the vector(fundamental) representation. The tree level superpotential of this theory is given by [25]
Nf

Wtree = W (Φ) +
I =1

? ΦQ + mI QQ ? Q ,

(2.1)

where W (Φ) is a polynomial with even powers of Φ:
n+1

W (Φ) =

g2k T r Φ2k . k =1 2k

(2.2)

According to Dijkgraaf-Vafa conjecture, the e?ective superpotential of this theory can be obtained from the planar limit of the matrix model whose tree level potential 2

is proportional to Wtree . Hence the partition function of the matrix model is [25] Z = e?F =
f ? 1 ? ΦQ + mI QQ ? ? , D ΦD Q exp ? ?W (Φ) + Q ? gs ? I =1

? ?

?

N

??

(2.3)

? are M dimensional vectors. where Φ is M × M real antisymmetric matrix and Q, Q For this theory, the Dijkgraaf-Vafa conjecture can be generalized [30] to obtain e?ective superpotential as a function of glueball ?eld S = T rWα W α : ? Fs2 (2.4) + 4FRP 2 + FD2 , ?S where Fs2 is a free energy of a diagram with topology of sphere, FRP 2 is free energy of the diagram with cross-cap one (topology of RP 2) and FD2 is the free energy of the diagram with one boundary (topology of D 2 ). It is also known that [12] Wef f = Nc FRP 2 = ? Using this, Wef f becomes Wef f = (Nc ? 2) ? Fs2 + FD 2 ?S ? Fχ=2 = (Nc ? 2) + Fχ=1 ?S pert ? Fχ =2 + Fχ=1 , = WV Y + (Nc ? 2) ?S 1 ? Fs2 . 2 ?S (2.5)

(2.6)

where WV Y denotes the Veneziano-Yankielowicz potential [31]. Here we have absorbed FRP 2 in Fχ=2 and Fχ=1 contains the contribution to the free energy coming from the fundamental matter. As proposed by Dijkgraaf-Vafa, we need to take the planar limit of the matrix model. The planar limit can be obtained by taking (M, Nf → ∞) as well as gs → 0 such that S = gs M and Sf = gs Nf are ?nite. The total free energy of this matrix model can be expressed as an expansion in genus g and the number of quark loops h: F=
g,h 2g ?2 gs Sf h Fg,h (S ) .

(2.7)

Assuming that the fundamental quarks are massive compared to adjoint matter, we can integrate out the fundamental matter ?elds appearing quadratically in the partition function (2.3) to give Z = e?F =
f 1 D Φ exp ?? T r ?W (Φ) + Sf log (Φ + mI )?? . gs I =1

?

?

N

??

(2.8)

We are now in a position to calculate Fχ=1 and Fχ=2 contributions to the superpotential. In the following section we apply the method developed in [24] for the SO (Nc ) gauge theory with one adjoint matter ?eld and Nf fundamental ?avors. 3

3

E?ective Superpotential

We wish to compute the exact e?ective superpotential of N = 2 supersymmetric SO (Nc ) gauge theory with Nf ?avors of quark loops in the fundamental representation, broken to N = 1 by addition of a tree level superpotential W (Φ) given by eqn.(2.2). The supersymmetric vacua of the theory with superpotential (2.2) are obtained by diagonalizing Φ such that the eigenvalues are in the set of critical points of W (Φ) which are given by the zeros of
n

W (x) = g2n+2 x



(x2 + a2 i ).
i=1

(3.1)

Choosing all the N eigenvalues of Φ = 0 gives unbroken gauge group SO (N ). A generic gauge group SO (N0 ) × n i=1 U (Ni ) corresponds to N0 eigenvalues of Φ = 0, N1 eigenvalues of Φ = ia1 , . . .. We will now look at the e?ective superpotential computation for both unbroken and broken gauge group in the next two subsections.

3.1

Unbroken gauge Group

Following the arguments in [24], for the e?ective superpotential evaluation we can still look at a point in the quantum moduli space of N = 2 pure gauge theory where r = [Nc /2] (rank of SO (Nc )) monopoles become massless [14]. This corresponds to the point where the Seiberg-Witten curve factorizes completely. We have seen that the e?ective superpotential of this theory gets contributions from free energies Fχ=2 and Fχ=1 of the matrix model described in the previous section. We shall ?rst calculate the contribution to the superpotential coming from Fχ=2 using the moduli associated with Seiberg-Witten factorization. 3.1.1 Contribution of Fχ=2

In this subsection we compute the contribution of Fχ=2 to the e?ective superpotential. This contains free energies of the diagrams having topology of S 2 and RP 2 . Taking derivative of eqn.(2.7) with respect to gs ?F = ?gs
2g ?3 gs (Sf )h (2g ? 2)Fg,h + S g,h

? Fg,h ?S

+
g,h

2g ?3 hgs (Sf )h Fg,h (S ) .

(3.2)

According to Dijkgraaf-Vafa, one should take the planar limit on the matrix model side. Also we take the number of quark loops, h = 0 for χ = 2 free-energy computation. Planar limit of the above equation gives ?F ? Fχ=2 ?3 = gs S ? 2Fχ=2 ?gs ?S . (3.3)

4

We can also di?erentiate eqn.(2.8) with respect to gs to give ?F ?2 = ?gs T rW (Φ) . ?gs ?From the above two equations gs T rW (Φ) = 2Fχ=2 ? S ? Fχ=2 . ?S (3.5) (3.4)

The form of W (Φ) shows that the LHS contains the vacuum expectation values T r Φ2p . It is clear from the above equation that once we obtain the vevs T r Φ2p , we can easily compute Fχ=2. In the case of N = 2 SO (Nc ) gauge theory, the moduli are given by u2p = 21p T r Φ2p . We are interested in the complete factorization of the Seiberg-Witten curve. The moduli that factorizes the Seiberg-Witten curve are given by [14] Nc ? 2 p 2p u2p = C2p Λ , (3.6) 2p
! i and Λ is the scale governing the running of the gauge coupling where Cj = i!(jj? i)! constant. The matrix model calculation of the vevs of the moduli done in the context of SU (Nc ) [32, 33] can be extended to SO (Nc) giving

u2p = (Nc ? 2)

? gs T r Φ2p . ?S 2p

(3.7)

It is obvious from the above two equations that ? p 2p gs T r Φ2p = C2 . pΛ ?S (3.8)

0 We denote the e?ective superpotential of pure SO (Nc ) gauge theory by Wef f . From eqn.(2.6) we can write 0 Wef f

? Fχ=2 (Nc ? 2) S ? Fχ=2 = S ?log 3 + 1 + (Nc ? 2) . = (Nc ? 2) ? ?S 2 ?S Λ

pert

(3.9)

The ?rst term in the above equation is the Veneziano-Yankeilowicz superpotential ? 3(Nc ?2) is the strong coupling scale of the N = 1 theory. The second term [34] and Λ is perturbative in glueball super?eld S with
pert Fχ =2 = n≥1 χ=2 Once we compute the functions fn (g2p ), we will have the e?ective superpotential of SO (Nc ) pure gauge theory. In order to compute these functions we need to take the derivative of (3.5) with respect to S χ=2 fn (g2p )S n+2 .

(3.10)

? Fχ=2 ? 2 Fχ=2 ? gs T rW (Φ) = ?S . ?S ?S ?S 2 5

(3.11)

Substituting eqns.(3.9,3.10) in the above equation, we get (Nc ? 2)
0 ?Wef ? f 0 gs T rW (Φ) = Wef ? S f ?S ?S ? ? S χ=2 = (Nc ? 2) ? ? n(n + 2)fn (g2p )S n+1 ? . (3.12) 2 n≥1

0 At the critical point of the superpotential, that is when ?Wef f /?S = 0, we have 0 Wef f = (Nc ? 2)

? gs T rW (Φ) = ?S

g2p u2p .
p

(3.13)

The glueball super?eld can be obtained at the critical point by the following relation [14]: 0 ?Wef f p 2p = S= g2p C2 (3.14) pΛ . ?log ΛNc ?2 p≥1 Inserting eqn.(3.8) and eqn.(3.14) in eqn.(3.12) one gets 1 1 p p χ=2 2p 2p n(n + 2)fn (g2p )S n+1 . g2p C2 g2p C2 = pΛ ? pΛ 2 p ≥1 n≥1 p ≥1 2 p (3.15)

Substituting glueball ?eld S in terms of Λ (3.14) and equating the powers of Λ on χ=2 both sides of the above equation, we can extract the functions fn (g2p ):
χ=2 f1 = χ=2 fn ≥2

1 g4 2 8 g2
n+1 C2( n+1)

g2(n+1) n+1 + 1)(n + 2) g2 n+1 n?1 l(l + 2) χ=2 fl ? p1 ,...pl+1 =1 l=1 n(n + 2) = 2n+2 (n
p1 +...+pl+1 =n+1

p1 l+1 C2 p1 g2p . . . C2pl+1 g2pl+1 , n+1 2n+1g2

p

(3.16)

χ=2 Now that we have computed the functions fn (g2p ), the χ = 2 contribution to the e?ective superpotential of the SO (Nc ) theory with one adjoint chiral super?eld with arbitrary tree level superpotential is known exactly. From eqn.(3.9) and eqn.(3.10), it is given by

0 Wef f

In the case of quadratic tree level superpotential, that is when g2p = 0 for p ≥ 2, the χ=2 ? 3 to 2g2 Λ2 functions fn (g2p ) vanish for all n. And we can ?x the coupling scale Λ 0 0 by the requirement that Wef f satis?es equation (3.14). The Wef f for the quartic 6

S S χ=2 (n + 2)fn (g2p )S n+1 ? . ?log 3 + 1 + = (Nc ? 2) ? ? 2 Λ n≥1

?

?

(3.17)

tree level superpotential can be obtained by substituting g2p = 0 for p ≥ 3 in the above result (3.17):
0 Wef f = WV Y +(Nc ? 2)

3 2

g4 9 S2 ? 2 4g 2 2

2 g4 45 S3 + 4 8g 2 2

3 g4 S 4 + . . . . (3.18) 6 16g2

This is in perfect agreement with the result of [26] where it has been evaluated in terms of the matrix model as well as IIB closed string theory on Calabi-Yau with 0 ?uxes. Substitution of g2p = 0 for p ≥ 4 in eqn.(3.17) gives Wef f for the the theory with sixth order potential:
0 Wef f = WV Y + (Nc ? 2) 2 3 9 g4 15 g4 g6 45 g4 5 g6 3 g4 2 3 S + ? S + ? + S4 2 3 4 5 6 8 g2 12 g2 16 g2 8 g2 32 g2 (3.19)

+ . . .] .

We now compare the result (3.17) with the corresponding result in the SU (Nc ) gauge theory with one adjoint matter. The e?ective superpotential of SU (Nc ) theory has been obtained in [24]. Comparison of the e?ective superpotentials of these two theories provides the following equivalence: Wef f
0 SO (Nc )

(g2p ) =

Nc ? 2 0 SU (Nc ) ′ Wef f (g2p = 2g2p ) , 2Nc

(3.20)

which agrees with the relation obtained in [14]. We shall now address the fundamental matter contribution Fχ=1 to the e?ective potential. 3.1.2 Contribution of Fχ=1

We di?erentiate the free energy given by eqn.(2.7) with respect to Sf ?F = ?Sf
2g ?2 hgs (Sf )h?1 Fg,h (S ) . g,h

(3.21)

We are interested in genus g = 0 and one quark loop h = 1 contribution in the ?2 planar limit gs → 0. The dominant term from eqn.(3.21) is ?Sf F = gs Fχ=1 . Di?erentiation of eqn.(2.8) with respect to Sf gives
f ?F ?1 = gs T r log (Φ + mI ) , ?Sf I =1

N

(3.22)

This implies
Nf

Fχ=1 = gs
I =1

T r log (Φ + mI ) .

(3.23)

Expanding the above equation around the critical point Φ = 0, we get
Nf

Fχ=1 =
I =1

S log mI ? 7

(?1)k g T r Φk k s km I k =1



.

(3.24)

Di?erentiating with respect to S and using eqn.(3.8) we get
f ∞ 1 ? Fχ=1 C k Λ2k = log mI ? 2k 2k ?S 2 km I I =1 k =1

N

.

(3.25)

Integrating the above equation with respect to S we obtain
Nf

Fχ=1 =
I =1

S log mI ?

l k lg2l C2 l C2k Λ2(k+l) + D , 2k I =1 k,l≥1 2k (k + l)mI

Nf

(3.26)

where D is the constant of integration. We postulate
Nf

D=
I =1

Wtree (mI ) ,

(3.27)

and we will see in the next section that the result agrees with the one obtained from Calabi-Yau geometry with ?uxes. In order to write this expression in powers of S , we write Fχ=1 as
Nf Nf

Fχ=1 = S
I =1

log mI +
n≥1

χ=1 fn (g2p )S n+1

+
I =1

Wtree (mI ) .

(3.28)

Comparison with eqn.(3.26) gives the following recursive relation for the coe?cients χ=1 fn (g2p )
χ=1 f1

1 f 1 =? 4 I =1 m2 I g2
f n l k lg2l C2 1 l C2k = ? n+1 n+1 ? 2k 2 g2 I =1 k,l=1 2k (n + 1)mI

N

χ=1 fn ≥1

?

N

n?1

n+1 χ=1 fq
p1 ,...pq +1 p1 +...+pq +1 =n+1

+
q =1

p1 q +1 ? C2 p1 g2p1 . . . C2pq+1 g2pq+1 ? .

p

? ?

(3.29)

The eqn.(3.28) alongwith eqn.(3.29) gives the e?ective superpotential from fundamental matter, for the most general Wtree . If we substitute g2p = 0 for p ≥ 2 in the above result, we get Fχ=1 for the gauge theory with quadratic superpotential. It is explicitly given by,
Nf

Fχ=1 =
I =1

1 S3 5 S4 1 S2 ? ? ? ... + D. Slog mI ? 4 mI 2 g2 8 mI 4 g2 2 48 mI 6 g2 3

(3.30)

8

Also substituting g2p = 0 for p ≥ 3, we get Fχ=1 for the theory with quartic superpotential.
Nf

Fχ=1 =
I =1

Slog mI + ? ?

1 g4 1 2 S + ? + S3 4mI 2 g2 8mI 4 g2 2 4mI 2 g2 3 (3.31)

+

2 5 9g 4 9g 4 S4 + . . . + D . + ? 48mI 6 g2 3 32mI 4 g2 4 16mI 2 g2 5

If we set g2p = 0 for p ≥ 4, the resulting theory has sixth order tree level potential and the corresponding Fχ=1 is given by
Nf

Fχ=1 =
I =1

Slog mI + ? ?

1 g4 1 2 S + ? + S3 4mI 2 g2 8mI 4 g2 2 4mI 2 g2 3

+

2 5 9g 4 9g 4 15 g6 + ? ? S 4 + . . . + D(3.32) . 4 6 3 4 4 2 5 48mI g2 32mI g2 16mI g2 16 m2 g I 2

The total e?ective superpotential of the theory under consideration is
0 Wef f = Wef f + Fχ=1

S S χ=2 = (Nc ? 2) ? ?log +1 + (n + 2)fn (g2p )S n+1 ? 2 2g 2 Λ 2 n≥1
Nf Nf

?

?

+S
I =1

log mI +
n≥1

χ=1 fn (g2p )S n+1 + I =1

Wtree (mI )

(3.33)

3.2

Broken Gauge Group

In the previous subsection, we have computed the e?ective superpotential of SO (Nc ) supersymmetric gauge theory for unbroken gauge group. In this section we obtain the e?ective superpotential for broken gauge group. In particular we consider the following breaking pattern,
n

SO (N ) → SO (N0) ×
i=1

U (Ni ) ,

(3.34)

such that N = N0 + 2 n i=1 Ni and for every factor of the gauge group, there is a glueball super?eld Si . We introduce the variables, e0 = 0, ei = iai , e?i = ?iai , i = 1, 2, . . . , n. 3.2.1 Contribution of Fχ=2

Let us ?rst compute the free energy Fχ=2 for the pure gauge theory. For this case the eqn.(3.5), which has been used to evaluate Fχ=2 in the case of unbroken gauge 9

group, modi?es to
n

gs T rW (Φ) = 2Fχ=2 ?
i=?n

Si

? Fχ=2 . ?Si

(3.35)

The free energy Fχ=2 is a combination of non-perturbative part, coming from the Veneziano-Yankielowitz term [35] and a perturbative part:
n

Fχ=2 =
i=?n

Si W (ei ) ?

1 n Si ei ? ej 1 n 2 Si log ? Si Sj log + 4 i=?n αΛ?i 2 i,j =?n Λ
(m) Fχ=2 ,

Fχ=2 ,
m

(m)

(3.36) where the perturbative part is contained in which is polynomial of order m in Si . Substitution of Fχ=2 given by eqn.(3.36) in eqn.(3.35) implies
n

gs T rW (Φ) =
i=?n

W (ei )Si +

1 n (m) S2 ? (m ? 2)Fχ=2 . 4 i=?n i m≥3

(3.37)

It is clear from the above equation that, we are close to having the free energy Fχ=2 if we can compute the expectation value gs T rW (Φ) . In order to compute these expectation values, we use the following matrix model loop equation [13] : w 2(x) ? 2W ′(x)w (x) + f2n (x) = 0 , (3.38)

where w (x) is a resolvent of the matrix model and f2n (x) is an even polynomial of order 2n which can be chosen to be n ?j S , (3.39) f2n (x) = 2W ′ (x) i=?n x ? ej ?j = S ??j . The n + 1 coe?cients of the function f2n (x) can be related to the where S glueball super?elds by computing the following period integral. Si = 1 2πi
Ai

?i + w (x)dx = S
? ?m?p

m
p=0 m≥2

?ip ?j ? m+p?2 S 1 (2m ? 3)!! S ? ? m?1 m?1 p!(m ? p)! (m + p ? 2)! ?xm+p?2 g2 x ? e R ( x ) j n+2 i j =i

(3.40)
x =e i

where Ai denote the cycle enclosing the branch point centered in point ei of the spectral curve associated with the matrix model and Ri (x) = j =i (x ? ej ). Also note that Si = S?i . Using the resolvent ω (x), the expectation value gs T rW (Φ) can be calculated from the following contour integration: gs T rW (Φ) =
n i=?n m
p=0 m≥2

1 2πi

n A

W (x)w (x)dx =
i=?n

?i W (ei ) + S
? ?m?p

(3.41)

?j ?ip (2m ? 3)!! S S ? m+p?2 W (x) ? ? m?1 m?1 p!(m ? p)! (m + p ? 2)! ?xm+p?2 g2 x ? e R ( x ) j i n+2 j =i 10

x =e i

Here the contour A = n i=?n Ai . One can use eqn.(3.40) to write the above expres?i . The resulting relation can be expressed in the form sion in terms of Si instead of S (m) of eqn.(3.37). And the comparison with eqn.(3.37) gives the polynomials Fχ=2 . As an example, for m = 3 we get
(3) g2n+2 Fχ=2

1 n 1 n Si Sj Sk Si2 Sj Si2 Sj 1 n + + =? 2 i=?n j =i k=i Ri eij eik 2 i=?n j =i k=i Ri eij eik 4 i=?n j =i Ri e2 ij 1 n 1 n Si3 ? 1? Si3 + ? , 16 i=?n j =i k=i Ri eij eik 6 i=?n Ri j =i eij
? ?2

(3.42)

where eij = ei ? ej and Ri = j =i(ei ? ej ). This result matches with the one given in [25], where it has been written by using the relation between free energies of U (N ) and SO (N ) gauge theories. 3.2.2 Contribution of Fχ=1

For the computation of matter contribution, we incorporate the fact of broken gauge group in eqn.(3.23) as follows,
Nf Nf n

Fχ=1 = gs
I =1 (m)

T rlog (Φ + mI ) =
I =1 i=?n

Si log (ei + mI ) +
m≥2

Fχ=1

(m)

(3.43)

where Fχ=1 are polynomials in Si of order m. We obtain Fχ=1 by evaluating the expectation value of log (x + mI ). Fχ=1 =
Nf n

1 I =1 2πi
m
p=0 m≥2

Nf

Nf A

n

log (x + mI )w (x)dx =
I =1 i=?n

?i log (ei + mI ) + S
?

(3.44)
?m?p
x =e i

I =1 i=?n

?ip ?j (2m ? 3)!! S ? m+p?2 log (x + mI ) ? S ? m?1 m?1 p!(m ? p)! (m + p ? 2)! ?xm+p?2 g2 x ? e j n+2 Ri (x) j =i

This result when expressed in terms of Si , can be compared with eqn.(3.43) to get (m) (2) Fχ=1 . For m = 2, the expression for Fχ=1 is
(2) g2n+2 Fχ=1

Si ? = I =1 i=?n eiI Ri

Nf

n

?

j =i

S2 S 2 R′ Sj ? 2 i ? i i2 ? eij 4eiI Ri 2eiI Ri

?
(3)

(3.45)

where eiI = ei + mI . This result agrees with [25]. For m = 3, Fχ=1 takes the following form:
′′ ′2 ′′′ Si3 1 R′ Ri Ri Ri ? 3 ? 2i 2+ ? ? 2 3 2 8eiI Ri 3eiI Ri 8eiI Ri 2eiI Ri 6Ri I =1 i=?n eiI Ri Nf n

(3) 2 g2 n+2 Fχ=1

=

11

′ ′′ ′3 5 Ri 3 Ri 1 Ri + ? ? 3 4 4 Ri 2 Ri 2

j =i

+

′ Ri 2 eiI Ri j =i ′ Ri 2 2 Ri j =i Nf n

Sj 1 + eij 4eiI Ri
′ Sj Rj 2 2 eij Rj j =i

f n 1 ? Si2 ? 1 + 2 Rj e3 ij I =1 i=?n eiI Ri 2eiI Ri

?

N

?

j =i

Sj eij

j =i

Sj ? e2 ij

′′ 5 Ri 2 4 Ri j =i

Sj + eij

R′2 3 i3 Ri j =i

Sj eij Sk ? ? Rj e2 ij ejk Sj Sk eij eik
?

+

Sj ? e2 ij

+
j =i

Sj 3 1 + 3 Rj eij 2 Ri

j =i

Sj + e3 ij

j =i k=i,j

′ Sj Sk Rj Si ? + + ×? 2 j =i Rj eij ejk I =1 i=?n eiI Ri
k =j

?

j =i k =j

1 1 Sj Sk ? 2 Rj eij ejk 2 eiI Ri 1 Sk Sl Rj eij ejk ejl

j =i k =j

?

′ 3 Ri 2 2 Ri

j =i k =j

Sj Sk 1 ?2 eij eik Ri

j =i k =j

1 Sj Sk ? 2 eij eik 2

j =i k,l=i,j

+

1 4

j =i

2 ′′ Sj Rj 1 ? 2 Rj eij 2

j =i

2 ′2 Sj Rj ? 3 Rj eij

?

(3.46)

In principle, the above computation can be done to any order m. It is important to realize the power of assimilating Dijkgraaf-Vafa conjecture and the connections to factorization of Seiberg-Witten curves which led to such precise determination of SO (Nc ) e?ective superpotential for arbitrary polynomials of tree level superpotential at generic point in the classical moduli space (both unbroken and broken gauge group). In order to make sure that the results are consistent, we need to compare with other approaches. In the next section, we compare the results with explicit answers obtained from geometric approach of dualities.

4

Geometric Engineering and E?ective SO Superpotential

We will brie?y recapitulate geometric dualities leading to the computation of SO superpotential.

4.1

Geometric Transition

Consider type IIB String theory compacti?ed on an orientifold of a resolved CalabiYau geometry whose singular limit is given by eqn.(1.1). For description of SO gauge group, W (x) (1.1) must be even functions of x. Further, W ′ (x) = 0 determines the eigenvalues of Φ which can be 0, ±ia′i s. We are interested in N = 1 SO (Nc ) supersymmetric gauge theory in four dimensions. This can be realized by wrapping Nc D5branes on RP2 of the orientifolded 12

resolved geometry- i.e., we place all the Nc branes at x = 0 where eigenvalues of Φ are zero. Invoking large N duality [2,3,11], the supersymmetric gauge theory is dual to IIB string theory on a deformed Calabi-Yau geometry with ?uxes. The deformed geometry is described by k ≡ W ′ (x)2 + f2n (x) + y 2 + z 2 + w 2 = 0 , (4.1)

where f2n (x) is a n degree polynomial in x2 . The three-cycles in this geometry can be given in terms of basis cycles Ai , Bi ∈ H3 (M, Z) (i = 1, 2, . . . 2n + 1) satisfying symplectic pairing (Ai , Bj ) = ?(Bj , Ai ) = δij , (Ai , Aj ) = (Bi , Bj ) = 0 . Here the pairing (A, B ) of three-cycles A, B is de?ned as the intersection number. For the deformed Calabi-Yau (4.1), these three-cycles are constructed as P1 ?bra? tion over the line segments between two critical points x = 0+ , 0? , ±ia+ i , ±iai . . . of W ′(x)2 + f2n (x) in x-plane. In particular, A0 cycle corresponds to P1 ?bration over the line segment 0? < x < 0+ and Ai ’s to be ?bration over the line segments + ′ ia? i < x < iai . The three-cycles B0 (Bi s) are non-compact and are given by ?brations over line segments between 0 < x < Λ0 (ia+ i < x < iΛ0 ) where Λ0 is a cut-o?. The deformed geometry (4.1) has Z2 symmetry and hence we can restrict the discussion to the upper half of x-plane. The holomorphic three-form ? for the deformed geometry (4.1) is give by ?=2 dx ∧ dy ∧ dz . ?k/?ω (4.2)

The periods Si and the dual periods Πi for this deformed geometry are Si =
Ai

? , Πi =

Bi

?.

The dual periods in terms of prepotential F (Si) is Πi = ? F /?Si . Using the fact that these three cycles can be seen as P1 ?brations over appropriate segments in the x-plane, the periods can be rewritten as integral over a one-form ω in the x-plane. + 0 That is, S0 = 1/(2πi) 00? ω , Π0 = 1/(2πi) 0Λ + ω , . . . where the one-form ω is given by ω = 2dx W ′ (x)2 + f2n (x)
1 2

.

(4.3)

0 The e?ective superpotential Wef f (recall the su?x 0 denotes the contribution from adjoint matter ?eld Φ) can be obtained as follows

?

1 W0 = 2πi ef f

? ∧ (H R + τ H N S ) ,

(4.4)

where τ is the complexi?ed coupling constant of type IIB strings, the HR and HN S denotes the RR-three form and NS-NS three-form ?eld strengths. Inclusion of matter 13

in fundamental representations in the geometric framework corresponds to placing D5 branes at locations x = ma where ma ’s are the masses of Nf fundamental ?avors. These locations are not the zeros of W ′ (x) = 0. The fundamental matter contribution to the e?ective potential is given by [28]:
f lav Wef f

≡ Fχ=1

1 f = 2 a=1

N

Λ0 ma

ω.

(4.5)

For simplicity, we will con?ne to the classical solution Φ = 0 with Nc D5-branes at x = 0. The corresponding dual theory will require RR-?ux over A0 cycle alone 0 and a non-zero period S0 ≡ S . The e?ective superpotential Wef f (4.4) in terms of χ = 2 part of matrix model free energy F (S ) will be
0 Wef f =

? F (S ) Nc Nc ?1 = ?1 2 ?S 2

Λ0 0+

wdx

(4.6)

It is important to work out explicitly these formal integrals for speci?c potentials and compare with our closed form expression obtained for arbitrary potentials in subsection 3.1.

4.2

E?ective Superpotential for Sixth Order Potential

In this subsection we consider the N = 1 SO (Nc ) gauge theory with fundamental matter and the following tree level superpotential: Wtree (Φ) = m g λ T r Φ2 + T r Φ4 + T r Φ6 . 2 4 6 (4.7)

The geometry corresponding to this gauge theory is given by W ′ (x)2 + f4 (x) + y 2 + z 2 + w 2 = 0 , (4.8)

where f4 (x) is an even polynomial of degree 4. We concentrate on the special classical vacuum Φ = 0, which is sometimes called as one cut solution in the context of matrix models [26]. We require the critical points of W ′ (x)2 + f4 (x) to be 0+ , 0? . This is achieved by the following one form: ω = 2 W ′ (x)2 + f4 (x)dx = 2λ(x2 + a)(x2 + b) x2 ? 4?2 dx , (4.9)

where 0± = ±2?. Also a and b are related to the couplings of the tree level potential in the following way, g (a + b) = + 2?2 , λ m g 2 ab = + 2 ? + 6? 4 . λ λ 14

The period integral can be computed from S= 1 2πi
2? ?2?

ωdx .

(4.10)

For the sixth order Wtree , it is explicitly given by S = 2m?2 + 6g?4 + 20λ?6 . (4.11)

For the given one-form, the χ = 2 contribution to the e?ective superpotential is
0 Wef f =

Nc ? F (S ) Nc ?1 = ?1 2 ?S 2

Λ0 2?

ωdx .

(4.12)

After taking the limit Λ0 → ∞ and ignoring the Λ0 dependent terms, the above equation leads to 3 4 10 6 0 2 λ? . Wef f = (Nc ? 2) S log (2?) ? m? ? g? ? 2 3 (4.13)

0 All higher powers of ? vanish. Substitution of S from eqn.(4.11) in Wef f obtained from factorization of Sieberg-Witten curve given by eqn.(3.19), agrees with the above result. The e?ective superpotential that comes from the contribution of ?avors (4.5) is Nf f lavor Wef f

=?
I =1

2 mI m2 I ? 4?

1 1 1 1 4 2 g 2 m + gm2 + λm2 I + λmI + ? I + λ? 2 4 6 2 3

3 10 + ?2 m + g?2 + λ?4 + 2?2 m + 3g?2 + 10λ?4 log (2Λ0 ) 2 3 ? 2?2 m + 3g?2 + 10λ?4 log mI +
2 m2 I ? 4?

.

(4.14)

In obtaining the above result, we take the limit Λ0 → ∞ and ignore the Λ0 dependent terms. Substituting for S in terms of ?2 (4.11) in eqn. (3.32), the result agrees with the above expression. Substitution of λ = 0 in the above equation, we get ?avor contribution of the SO (Nc ) gauge theory with quartic tree level superpotential.
Nf f lavor Wef f

=?
I =1

2 mI m2 I ? 4?

3 1 g 2 1 + ?2 m + g?2 m + gm2 I + ? 2 4 2 2

(4.15) .

+ 2?2 m + 3g?2 log (2Λ0 ) ? 2?2 m + 3g?2 log mI + The corresponding S is given by S = 2m?2 + 6g?4 , 15

2 m2 I ? 4?

(4.16)

which is quadratic in ?2 and can be solved to give the roots. Discarding the negative root, we get m 6gS m (4.17) 1+ 2 . ?2 = ? + 6g 6g m Substituting ?2 and rewriting in powers of S agrees with our expansion (3.31). If we f lavor take g → 0 limit in the above equation, we get Wef of the SO (Nc ) gauge theory f with quadratic tree level superpotential.
Nf f lavor Wef f

=?
I =1

S M mI 2 + 2 2

1?

Λ0 2S + Slog 2 M mI mI

? Slog

1 1 + 2 2

1?

2S . M mI 2 (4.18)

If we replace M by M ′ /2, we get the A?eck-Dine-Seiberg SU (N ) superpotential [36]. Expanding the above equation in powers of S agrees with eqn. (3.30). Though we have considered in detail the sixth order potential, it is straightforward to obtain the e?ective superpotential for any polynomial potential. So far, we have discussed the results for unbroken gauge group. It will be interesting to verify the results in section 3.2 for broken gauge group also. Some work in this direction has already been reported in Ref. [37] for quartic potential without matter. Even though a formal expressions can be written in integral form for a general polynomial potential, we still have to work out the results in a certain limit to compare with the answers in section 3.2. We hope to report on these aspects in future.

5

Summary and Discussion

In this paper, we have derived SO (Nc ) e?ective superpotential for the supersymmetric theory with Nf fundamental ?avors (3.33). Using Dijkgraaf-Vafa conjecture and also the Sieberg-Witten factorization, we have obtained the e?ective superpotential for a most general tree level potential Wtree (Φ2 ). We have shown agreement with the results from the geometric considerations of superstring dualities for a sixth order tree level polynomial potential. We hope to report the explicit computation within geometric framework for the broken gauge group in future. Though we have concentrated on the SO gauge group, it appears that the fundamental matter contribution to the Sp (symplectic) e?ective superpotential will be identical (Fχ=1 ). However, one has to elaborately perform the derivation as done for SO group. The e?ective potential in the absence of matter is well-studied from various approaches which leads to the replacement of factor Nc ? 2 in eqn.(3.33) by 0 Nc + 2 to get Wef f for Sp(Nc ) gauge group. Within supersymmetric theories, the e?ective superpotentials for di?erent regimes like Nf = Nc or Nf < Nc or Nf > Nc could be addressed [38]. Some of these issues have been considered within the matrix model approach in [22, 39].

16

We have con?ned to a speci?c form of tree level potential which breaks N = 2 to N = 1. It will be interesting to look at other tree potentials involving more than one adjoint matter. We hope to report on these issues elsewhere. Acknowledgments PB would like to thank CSIR for the grant. The work of PR is supported by Department of Science and Technology grant under “ SERC FAST TRACK Scheme for Young Scientists”.

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