1 Introduction

October, 2007

OCU-PHYS 278



Deformation of Dijkgraaf-Vafa Relation via

Spontaneously Broken Supersymmetry II


H. Itoyama111e-mail: itoyama@sci.osaka-cu.ac.jp  and  K. Maruyoshi222e-mail: maruchan@sci.osaka-cu.ac.jp


Department of Mathematics and Physics, Graduate School of Science

Osaka City University

Osaka City University Advanced Mathematical Institute (OCAMI)


3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan


We consider the matter induced part of the effective superpotential of , gauge model in which supersymmetry is spontaneously broken to , by using the properties of the chiral ring and the generalized Konishi anomaly equations derived in our previous paper arXiv:0704.1060. It is shown that the effective superpotential is related to the planar free energy of the matrix model by a formula which consists of two parts — the well-known part due to Dijkgraaf-Vafa and the part that acts as a deformation of the couplings. These couplings are those of the original bare prepotential in the action and at the same time matrix model couplings.

1 Introduction

In the last two decades, various investigations have been made on the low energy effective action of supersymmetric gauge theory. It has been shown that the low energy effective action of supersymmetric gauge theory, which is governed by the effective prepotential, can be explicitly calculated, by exploiting its powerful constraints associated with holomorphy [1] and instanton calculation [2]. In contrast to the fact that supersymmetric Yang-Mills theories are in the Coulomb phase, supersymmetric gauge theories offer a wealth of vacua. Physically interesting phenomena, such as confinement and mass gap occur in low energy. It has been conjectured in the context of the topological string theory and the gauge/gravity correspondence [3, 4, 5] that the effective superpotential is related to the matrix model free energy [6], which we refer to as Dijkgraaf-Vafa relation. This relation has been shown in [7, 8, 9] by the purely field theoretical argument. (For subsequent developments on the calculus associated with the matrix model curve as algebraic integrable systems, see [10].)

More recently, a supersymmetric gauge model, in which supersymmetry is spontaneously broken to , has been found in [11, 12], and this model is the non-Abelian generalization of the Abelian model [13]. (See also [14, 15] for the cases with hypermultiplet, [16] for supergravity and [17] for related discussions.) It is not difficult to imagine that this model connects the above and theories. On the one hand, supersymmetry is restored in the small Fayet-Iliopoulos parameters limit. To be precise, in this limit, the action of the model [11, 12] reduces to that of the extended supersymmetric Yang-Mills theory whose effective superpotential has been discussed in the literature [18, 19]. On the other hand, the action of the model reduces to that of the supersymmetric theory with an adjoint chiral superfield and a tree level superpotential , which has been considered by [6, 7, 8], in the limit where the Fayet-Iliopoulos parameters are taken to be infinite [20]. Therefore, we can regard, at the classical level, the above two different theories as the particular limits of the model. We illustrate this in Figure 1.

So it is quite interesting to consider the quantum structure of this model: how is the effective superpotential? and how is the Dijkgraaf-Vafa relation deformed? In [21], we have started an analysis on the matter induced part of the effective superpotential of the model by computing the loop diagrams, following the spirit of [7] and have shown that the Dijkgraaf-Vafa relation is deformed in the region of large Fayet-Iliopoulos parameters. We have determined the leading term of this deviation from the Dijkgraaf-Vafa relation. In this computation, however, we have to treat many interaction terms and it is technically difficult to calculate all the contributions to the effective superpotential. We have also derived a set of two generalized Konishi anomaly equations on the two one-point functions and .

The aim of this paper is to obtain an exact expression which relates the effective superpotential with the planar free energy of the matrix model. For this purpose, we use an alternative method which is based on the properties of the chiral ring and the Konishi anomaly [22, 8]. In this approach, we do not need to take the Fayet-Iliopoulos parameters to be large. The effective superpotential consists of two parts both of which are written as operators acting on the planar free energy of the bosonic one-matrix model. The first part is well-known from the case of [6] while the second part acts as a (Whitham) deformation111See, for example, [18]. of the couplings.

In [23], the effective superpotential of a generic gauge model containing the non-canonical gauge kinetic term has been derived, so as to justify the important assumptions of the matrix model and the generalized Konishi anomaly equations. The model we consider has been studied for a while [11, 12], as an non-Abelian generalization of [13] emphasizing the nature of partially and spontaneously broken supersymmetry, and can be regarded as a distinguished class of a generic model. This paper is a sequel to our previous paper [21], where the generalized Konishi anomaly equations were already derived.

Figure 1: Interpolation by Fayet-Iliopoulos parameters and . At the energy , the action of the model [11, 12] reduces to the action in [18] and the action in [6] in the small and large Fayet-Iliopoulos parameters limits respectively.

The organization of this paper is as follows. In section 2, we review results of [11, 12, 20, 21]. Using the generalized Konishi anomaly equations [21], we obtain an explicit expression of the generating function of the one-point function in section 3. Also, by making use of the solution of the generalized Konishi anomaly equation for the generating function , we obtain the relation in section 4. Finally, we compare this result with the one derived from the diagrammatical computation [21] in section 5.

2 Preliminaries

In this section, we collect some known facts which are important for the analysis of this paper. In subsection 2.1, we introduce the bare action of the model we study and discuss the partial breaking of supersymmetry. In subsection 2.2 and 2.3, we briefly review the results of our previous paper [21]. We explain the result from our diagrammatical computation in subsection 2.2 and derive the generalized Konishi anomaly equations in subsection 2.3.

2.1 The gauged model with spontaneously broken supersymmetry

The bare action of the model we study in this paper is 222In [11, 12], the action (2.1) is constructed, following the gauging procedure of the general Kähler potential in [24], and restricting itself to be the one dictated by the special Kähler geometry. For the sake of completeness, we show the equivalence of (2.1) with the action in [11, 12] in appendix A.

(2.1)

where and are the vector and chiral superfields whose on-shell components are (, ) and (, ) respectively. In terms of generators , ( refers to the overall generator), the superfield is . (We normalize the generators as .) Theoretical inputs are the electric and magnetic Fayet-Iliopoulos terms which are two vectors or a rank two symmetric tensor in the isospin space and are parameterized by the three real parameters in the superspace formalism we employ. In addition, the model contains an arbitrary input function , which we refer to as a bare prepotential. Its prototypical form is a single trace function of a polynomial in :

(2.2)

While this action is shown to be invariant under the supersymmetry transformations [11, 12], the vacuum breaks half of the supersymmetries. Extremizing the scalar potential, we obtain the condition

(2.3)

The left hand side is a polynomial of order and determines the expectation value of the scalar field. In these vacua, the combination of the fermions, , becomes massive, while is massless, whose overall component is the Nambu-Goldstone fermion. In order to obtain the action on the vacua, we, therefore, have to redefine the superfields and such that the fermionic components of them mix as . In [20], the action on the vacua has been obtained by taking this point into account and that the Fayet-Iliopoulos D-term can be included in the superpotential;

(2.4)

where

(2.5)

is the single trace function of degree and is given by (2.2). In (2.5), we have redefined such that they include the factor which comes from the overall generator . Also, it is understood that and have been redefined as mentioned above.

The action (2.4) is to be compared with that of the , gauge model with a single trace tree level superpotential :

(2.6)

where is a complex gauge coupling . In [11], it is checked that the second supersymmetry reduces to the fermionic shift symmetry in the limit . The action in fact reduces to in the limit with () fixed [20]. We refer to this limit as limit.

In this paper, we consider the matter-induced part of the effective superpotential only by integrating out the massive degrees of freedom :

(2.7)

2.2 Diagrammatic analysis of the effective superpotential

Here, we review the diagrammatical computation of the effective superpotential [21]. For simplicity, we in this subsection consider the classical vacuum where , by setting the coupling constant as . In this case, the unbroken gauge group is still . Also, we take (or ) as the background field 333The simplest background is that consisting of a vanishing gauge field and a constant gaugino , which satisfies [25, 21]. This configuration implies that traces of more than two vanish.. Therefore, the result of the diagrammatical computation can be written in terms of the coupling constants , (), the Fayet-Iliopoulos parameter , the glueball superfield and the overall field strength . (The other Fayet-Iliopoulos parameters are always translated into and by .)

Due to the diagrammatical computation, we can obtain the following formula [21]: the contribution from the -loop diagrams which has propagators to the effective superpotential is, up to terms including the overall field strength ,

(2.8)

where can be written as

(2.9)

In (2.9), is defined by replacing, in the first term of r.h.s. of (2.8), one coupling constant according to

(2.10)

and summing over all possibilities. Also, denotes the terms which include the higher order contributions in . As discussed in [21], in (2.8) can be identified with -loop contribution to the planar free energy of the matrix model. Since and are , we can see that, in limit, we recover the result of [6, 7].

Although in (2.8) has been computed in [21], it is hard to obtain explicitly. In order to see this, we briefly recall some details of the computation. First of all, we start from (2.7) and integrate . This is easily done by setting the anti-holomorphic couplings for . With this choice, the -integral becomes a Gaussian integral and we are left with the holomorphic part of the action

(2.11)

The first line is from -term in and the second line is due to the Gaussian integration of . The latter can be expanded as

(2.12)

where denotes the higher order interaction terms, which is not considered in [21]. Note that is .

Secondly, we read off the Feynman rule from (2.11) and (2.12). Collecting the quadratic terms we can determine the propagator. Because of the second term of (2.11) which does not exist in , the propagator is modified compared with that [7] of . The higher order interaction terms in the first term in (2.11) are same as that [7] in . On the other hand, the interaction terms in the second term in (2.11) do not exist in . In addition, there are a lot of interaction terms in .

Finally, we compute the amplitude of the loop diagram. The amplitude of the non-planar diagram is exactly zero because of our choice of the background. (The detailed argument is found in [21].) Therefore, we only have to consider the planar diagrams. From the contributions of the -loop diagrams with propagators, we obtain (2.8) and (2.9). The first term of (2.9) is due to the fact that the propagator of the model is modified. Also, the second term of (2.9) arises by considering the set of new vertices which are seen in the first line of (2.11). The residual interaction is too complicated to compute its contribution to the effective superpotential explicitly. We have denoted it as in (2.8). The result of the diagrammatical computation (2.8) is to be compared with the effective superpotential which will be derived in section 4, by making use of the generalized Konishi anomaly equations. Actually, as we will show in section 5, exactly vanishes.

2.3 Generalized Konishi anomaly equations

An alternative approach to the effective superpotential is to exploit and extend the properties of the chiral ring and the generalized Konishi anomaly equations based on [22, 8]. We will mainly use this approach in the rest of this paper. In this subsection, we derive the generalized Konishi anomaly equations with respect to the chiral one-point functions [21].

The anomalous Ward identity of our model for the general transformation is

(2.13)

in the chiral ring. The second term in r.h.s. is due to the fact that the coefficient of -term in is function of , rather than the constant . Note that and are related as . In terms of the two generating functions of the chiral one-point functions

(2.14)

the anomalous Ward identities (2.13) are

(2.15)
(2.16)

where and are polynomials of degree and

(2.17)

Since the explicit forms of and are not needed in the analysis of the subsequent sections, we will not write it here. Note that the second term of r.h.s. of (2.13) does not contribute to the equation for because of the chiral ring relation . The equation for is, therefore, the same as that of [8], which is identified with the loop equation of the matrix model. On the other hand, the equation for alters from that of [8]. This leads to the deformation of our effective superpotential from the well-known form in the theory [6].

3 Solution of the anomaly equation for

By solving the generalized Konishi anomaly equations (2.15) and (2.16), we can obtain the explicit form of and . In this section, we focus on .

The classical vacua are determined by the condition (2.3) which is a polynomial of order . If we denote the roots of (2.3) by (), the vacuum expectation value of the scalar field is

(3.1)

Note that can be less than . Let us denote the number of appearing in (3.1) by . If , corresponding () are zero. We use indices () rather than when we refer only to nonvanishing ’s. In this notation, the gauge symmetry is broken to and .

Let us first consider (2.15). Its solution is

(3.2)

The sign of square root is determined by the asymptotics at large . From the above form, we can see that has cuts in the complex plane and is a meromorphic function on a Riemann surface of genus

(3.3)

Let us denote by -cycles of . In the semiclassical approximation where is small, to each cycle one can associate a zero of , . Also, if we denote by () the contours which circle around with , these contours are trivial. Therefore, we have

(3.4)

where we have defined the contour integral to include a factor of . Also, we define . (3.4) means that factorizes as

(3.5)

and are, respectively, polynomials of degree and . We obtain a reduced Riemann surface of genus

(3.6)

Since is a polynomial of degree , a priori, has undetermined coefficients. However, (3.5) produces constraints on the coefficients. Furthermore, the remaining undetermined coefficients are completely fixed by the first equation of (3.4). Therefore, we can fix and completely.

For future reference, we consider the derivative of with respect to . From (3.2), we obtain

(3.7)

Also, by taking a derivative of (3.5), we can see that are proportional to and therefore we can write where are polynomials of degree . Hence, (3.7) can be written as

(3.8)

where we have used the factorization condition (3.5) in the denominator. It is easy to see that () is a set of normalized holomorphic differentials on the reduced Riemann surface (3.6). In fact, taking the derivative of (3.4) with respect to , we obtain

(3.9)

Multiplying and summing over , we obtain

(3.10)

4 Effective superpotential

In this section, we first state our formula for the effective superpotential and make a comment on this. In subsection 4.1, we provide a derivation of the formula.

Let us define the one point functions as

(4.1)

In terms of , we define as

(4.2)

Since can be evaluated from which has been fixed completely as we have seen in section 3, we can compute up to -independent terms. Using , the formula for the effective superpotential is given by

(4.3)

up to -independent terms. Indeed, the quantity can be identified with the free energy of the bosonic one matrix model as we will see in section 5.1. Hence we find that -dependent part of the effective superpotential of our model can be obtained from the matrix model computation by the simple formula (4.3). In contrast to the case of [6], we have the new term, the second term in (4.3). Because of its dependence (and since we can see in section 5 that depends only on and not on ), the second term disappears in limit where with (for ) fixed. Therefore, we obtain Dijkgraaf-Vafa formula as a particular limit of (4.3).

In the theory , it is known that the full effective superpotential has the non-perturbative correction [26] which is called Veneziano-Yankielowicz term and do not depend on the coupling . In [6], it has been suggested that the effective superpotential of the theory can be computed from the matrix model including Veneziano-Yankielowicz term. The free energy of the matrix model in fact has -independent term by taking into account the volume of group rotating the hermitian matrix . From this term of the free energy, we can obtain the well-known Veneziano-Yankielowicz term of the effective superpotential.

In [23], it has been shown that the -independent term is same as the well-known Veneziano-Yankielowicz term using the instanton calculation [19], for a generic gauge model. Here, however, we focus on only -dependent part.

4.1 Proof of the formula

Let us show the formula for the effective superpotential up to -independent terms. To begin with, we take a derivative of (4.3) with respect to the coupling ,

(4.4)

Also, by taking a variational derivative of (2.7) with respect to the coupling , we obtain

(4.5)

By comparing (4.4) and (4.5), we obtain

(4.6)

Hence, once we prove the equation

(4.7)

the formula (4.3) follows as a truncation of (4.7) up to the first terms in the expansion.

For this purpose, we start by solving the remaining generalized Konishi anomaly equation (2.16). By substituting (3.2) into (2.16), we obtain

(4.8)

Recall that satisfies the following conditions;

(4.9)

Let us show that the right hand side of (4.7) is equal to the right hand side of (4.8). As we have already observed in (3.8), provides a set of normalized holomorphic differentials on the reduced curve. (4.9) is, therefore, saturated by

(4.10)

with

(4.11)

Introducing

(4.12)

we obtain

(4.13)

On the other hand, the derivatives of with respect to are

(4.14)

Recalling (2.17) as well as the definition of and hence , we obtain

(4.15)

Our proof becomes complete as soon as we obtain

(4.16)

Observing , we obtain

(4.17)

Eq. (4.13) and (4.17) give

(4.18)

Expanding the integrand by a set of holomorphic differentials of the original curve, we deduce (4.16).

5 Comparison with diagrammatical computation

The effective superpotential (4.3) should be obtained from computing all the possible planar diagrams based on the procedure in the subsection 2.2. From (4.3), the -loop contribution to the effective superpotential can be written as

(5.1)

In this section, we compare this expression with the result of diagrammatical computation (2.8). At first sight, it seems that (5.1) is different from (2.8): while the latter contains which contains in general higher order terms in in limit, the former does not contain such terms. In section 5.1, we will show that the first terms in two expressions (5.1) and (2.8) are equal, which needs the consideration of the matrix model. Then, we show that the second term in (5.1) are equivalent to in (2.8) in section 5.2. This leads to that vanishes.

5.1 Comparison with the matrix model

As discussed in [21], in (2.8) is the -loop contribution to the free energy of the matrix model. Therefore, in this subsection, let us show that in (4.2) or (4.3) is identified with the free energy of the matrix model except for -independent terms, which leads to the identification in (5.1) and in (2.8). The argument here is the same as that of [8].

The bosonic one matrix model is defined by integral of hermitian matrix . The definition of the free energy is

(5.2)

where

(5.3)

Note that the matrix size is not related with the rank of the gauge group .

Let us define the matrix model resolvent as . With this, the loop equation reduces, in the planar limit, that is, the large limit, to whose form is the same as that of the generalized Konishi anomaly equation (2.15). A polynomial is determined by the condition , where is the number of the eigenvalues of near the -th critical point and each contour is defined to cycle the -th critical point. If we identified the filling fraction with the glueball superfield , we can see that the polynomial is equal to in the gauge theory. Therefore, by the identification , we can conclude .

As a final step, by taking a variational derivative of the partition function (5.2) with respect to , we obtain

(5.4)

In the last equality, we have used . This is the same equation as the definition of (4.2). Hence, we conclude that in the effective superpotential (4.3) is the free energy of the matrix model up to -independent terms.

5.2 Comparison with the result of diagrammatical computation

In the last section, we have established the equivalence of and . Here, we show the second term in (5.1) is equal to in (2.8).

Let us first consider the coupling dependence of the . is the contribution from the -loop diagrams to the matrix model free energy. From the form of the action of the matrix model (5.2), we can read off the propagator which is proportional to and the vertices which are proportional to . Therefore, the amplitude of the -loop diagrams with propagators and vertices is

(5.5)

where is a function of of degree . is the number of the index loops and the factor is due to the traces of the index loops. The function is determined by calculating the symmetric factor and the coupling constants of each diagram we consider. Since we take the planar limit, the diagrams which should be considered have the topology of sphere . By taking account of the factor in front of in (5.2), we obtain the contribution of the -loop planar diagrams

(5.6)

We have used the identification in the case of unbroken . Hence, if we use , we have

(5.7)

Now, we are ready to show the second term in (5.1) is in (2.8). From (2.8) and (2.9), can be written as,

(5.8)

where means the procedure of changing the coupling constant by for each coupling in and summing over all possibilities. The forms of (5.6) and (5.7) lead to . Therefore, we derive