April, 2007
OCU-PHYS 262
Deformation of Dijkgraaf-Vafa Relation via
Spontaneously Broken N = 2 Supersymmetry
H. Itoyamaa,b∗ and K. Maruyoshia†
a Department of Mathematics and Physics, Graduate School of Science
Osaka City University
b Osaka City University Advanced Mathematical Institute (OCAMI)
3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Abstract
It is known that the fermionic shift symmetry of the N = 1, U(N) gauge model with
a superpotential of an adjoint chiral superfield is replaced by the second (spontaneously
broken) supersymmetry in the N = 2, U(N) gauge model with a prepotential and
Fayet-Iliopoulos parameters. Based on a diagrammatic analysis, we demonstrate how
the well-known form of the effective superpotential in the former model is modified
in the latter. A set of two equations on the one-point functions stating the Konishi
anomaly is modified accordingly.
∗e-mail: itoyama@sci.osaka-cu.ac.jp
†e-mail: maruchan@sci.osaka-cu.ac.jp
http://arxiv.org/abs/0704.1060v3
I. Introduction
For more than two decades, effective superpotential has been a central object in the
nonperturbative study of N = 1 supersymmetric theories. This object is protected from
perturbative corrections in the conventional sense [1], and yet receives important nonpertur-
bative corrections (see for example [2, 3]). In recent years, analyses from superstring theory
have revealed an interesting perturbative window into nonperturbative physics with the use
of the gluino condensate superfield variable [4, 5, 6, 7]. In [8], field theoretic discussion based
on the model with U(N) gauge group and rigid N = 1 supersymmetry (see eq. (2.2) for its
action SN=1) is given and this is in accord with the string theory based developments.
Superstring theory, on the other hand, insists upon maximally extended supersymmetry
with no adjustable parameter. A scenario that one may draw is that this extended super-
symmetry becomes spontaneously broken to N = 1. Along this vein, a field theory model
with U(N) gauge group and rigid N = 2 supersymmetry spontaneously broken to N = 1
has been introduced in [9, 10, 11] (see eq. (2.1) for its action SN=2), generalizing the abelian
counterpart of [12]. (See also [13] for N = 2 supergravity and [14] for related discussions.)
Several properties of this model have been derived.
In this letter, we make a first analysis on the interplay between the effective superpotential
and partially as well as spontaneously broken N = 2 supersymmetry, shedding a light upon
the comparison of the two models mentioned above. A key aspect of this comparison is that
the fermionic shift symmetry of SN=1 gets replaced by the second (spontaneously broken)
supersymmetry of SN=2. In fact, this is one of the original motivations/results of [9].
The fermionic shift symmetry of SN=1 supplies the well-known formula [7, 8] constraining
the form of the effective superpotential, which is originally proposed from flux compactifi-
cation of string theory [15, 16]. Based on a diagrammatic analysis [17] (for a review see
[18]), we are able to state how this form undergoes modifications in the model SN=2. After
giving a few accounts of the model in the next section, we present a diagrammatic analysis
of Weff in section III. Our final understanding is summarized in eq. (3.10). This is followed
by a computation of the two-loop contribution to Weff in section IV. In the final section, we
derive a set of two equations on the two generating functions R(z) and T (z) of the one-point
functions, generalizing the argument based on the chiral ring and the Konishi anomaly in
[8]. We observe a modification from that given in [8] here as well.
II. The U(N) gauged model with spontaneously broken N = 2
supersymmetry
Let us briefly recall a few ingredients of the model, which are needed in what follows.
The action [9] given in the Wess-Zumino gauge can be written as
SN=2 =
d4xd4θ
Φ̄eadV
∂F(Φ)
− h.c.
+ ξV 0
d4xd2θ
∂2F(Φ)
∂Φa∂Φb
WaWb + eΦ0 +m
∂F(Φ)
+ h.c.
, (2.1)
where V = V ata and W
α are the vector superfield and the gauge superfield strength re-
spectively and Φ = Φata (a = 0, 1, . . . , N
2 − 1) is the chiral superfield ∗. There are three
Fayet-Iliopoulos parameters (e,m, ξ) which are all real. For simplicity, we choose the prepo-
tential as a single trace function of degree n + 2: F(Φ) =
k=1 gkTrΦ
k+1/(k + 1)!. While
this action is shown to be invariant under the N = 2 supersymmetry transformations [9, 10],
the vacuum breaks half of the N = 2 supersymmetries. Extremizing the scalar potential,
we obtain the condition 〈 ∂
∂Φ0∂Φ0
〉 = −(e± iξ)/m, which is a polynomial of order n and this
determines the expectation value of the scalar field.
The action SN=2 in (2.1) is to be compared with that of the N = 1, U(N) gauge model
with a single trace tree level superpotential W (Φ):
SN=1 =
d4xd4θTr Φ̄eadV Φ+
d4xd2θTr (iτWW +W (Φ)) + h.c.
, (2.2)
where τ is a complex gauge coupling τ = θ/2π + 4πi/g2.
In [9], it is checked that the second supersymmetry reduces to the fermionic shift symme-
try in the limit m → ∞. The action SN=2 in fact reduces to SN=1 in the limit m, e, ξ → ∞
with mgk (k ≥ 2) fixed [19]. We show that our result reduces to that of [7, 17] in this limit.
III. Diagrammatic analysis of the effective superpotential
In this letter, we consider the matter-induced part of the effective superpotential by
integrating out the massive degrees of freedom Φ:
d4x(d2θWeff+h.c.+d
4θ(nonchiral terms)) =
DΦDΦ̄eiSN=2. (3.1)
∗a = 0 corresponds to the overall U(1) part.
Let us take Wα (or V ) as the background field †. We consider the case of unbroken U(N)
gauge group. For simplicity, we choose 〈Φ〉 = 0 by setting g1 = −(e± iξ)/m.
We are interested in the holomorphic superpotential which does not contain the anti-
holomorphic couplings ḡk. We can take ḡk = 0 for k ≥ 3 without loss of generality. Collecting
the Φ̄ dependent terms, we obtain
SΦ̄ =
d4xd4θ
Φ̄eadV
∂F(Φ)
− (ḡ1Φ̄ +
Φ̄2)eadV Φ
d4xd2θ̄
TrΦ̄2
d4xd4θTr
Φ̃ḡ2
ḡ1Φ−
. (3.2)
In the last expression, we have introduced a covariantly anti-chiral superfield Φ̃ = Φ̄eadV ,
which satisfies ∇αΦ̃ = 0 (∇α = e
−adV Dαe
adV ). Eq. (3.2) is quadratic in Φ̃ and can be
integrated straightforwardly. As a result, we obtain the following terms,
16ḡ2
ḡ1Φ−
ḡ1Φ−
(Img1)
8mḡ2
Φ∇2Φ+ . . . , (3.3)
where . . . denotes the higher order interaction terms, which we will not consider here. Indeed,
these interaction vertices are higher order in m−1 compared to the vertices which we consider
below. These contribute to our main result (3.10) as higher order corrections in m−1 and do
not spoil our conclusion that the effective superpotential is modified from the case of SN=1
(2.2).
Replacing d2θ̄ integration by −∇̄2/4 and collecting the terms which are not in SΦ̄, we
obtain an action after the Φ̄ integration ‡:
d4xd2θTr
(Img1)
32mḡ2
Φ∇̄2∇2Φ +m
(WΦsWΦk−1−s)
. (3.4)
The first two terms are already present in the integrations with regard to the action SN=1
(2.2). The last term is new and originates from the gauge kinetic term in eq. (2.1). As we
will see below, this last term does contribute to the effective superpotential and becomes
responsible for the violation of the well-known relation [7, 8] between the effective superpo-
tential of the gauge theory and the planar free energy of the matrix model having the tree
level (bare) superpotential as its potential.
†The simplest background is that consisting of a vanishing gauge field Aµ and a constant gaugino λ
which satisfies {λα, λβ} = 0 [18]. This configuration implies that traces of more than two W vanish.
‡In eq. (3.4), it is understood that the generating functional has a renormalized perturbation expansion
in which a nonvanishing tadpole is always canceled by a nonvanishing value of the source coupled to Φ. This
implies that the tadpole can in practice be ignored.
After rescaling Φ → aΦ with a2 = mḡ2/(Img1)
2, the quadratic part of the action (3.4)
reduces to
−�+m′ +
adWαDα
(2WWΦ2 +WΦWΦ),
where we have used the relation ∇̄2∇2Φ = 16(�Φ − adWαDαΦ/2) and introduced m
a2mg2 and g
3 = a
2g3/12. The propagator in the momentum space is
∆(p, π) =
dse−s(p
2+m′+ 1
adWαπα−ig
The Grassmann momentum πα is Fourier transformation of superspace coordinate θα and
the matrix M is
Mabcd = (WW)daδbc + (WW)bcδda +WdaWbc, (3.5)
where we have exhibited the gauge index dependence explicitly. This matrix is not present
in the propagator of [17]. Using eq. (3.5), we are able to insert W without involving the
momentum πα.
The interaction terms in eq. (3.4) are divided into the following two types:
type I. m
Tr Φk, k = 3, . . . , n+ 1.
type II. −
Tr(WΦsWΦk−1−s), k = 4, . . . , n+ 1.
Type I vertices are already present in [17]. Type II vertices are not present in [17]. They
insert two W in specific ways.
Before going on to consider loop diagrams, let us first demonstrate that we have only
to consider planar diagrams in our case as well [17, 18]. For a given diagram, we denote
by V the number of vertices, by P the number of propagators and by h the number of
holes (or index loops). There are V sets of chiral superspace integrations from V vertices.
One of them becomes the chiral superspace integration over the effective superpotential,
and the number of remaining πα momentum integrations is P − V + 1. These Grassmann
integrations must be saturated by 1
adWαπα terms in the propagators. Furthermore, we can
freely insert W both from the M terms in the propagators and from the type II vertices. If
we denote the number of these additional insertions by 2α, the total number of W insertions
is 2(P − V + 1 + α). On the other hand, one index loop can accommodate at most two W.
Thus we have h ≥ P − V + 1+ α. This implies that only the planar diagrams contribute to
the effective superpotential as the Euler number of the diagram is χ = V − P + h.
A planar diagram with h index loops has (h − 1) loop momenta. Let us consider the
(h− 1)-loop planar diagrams (contributing to the (h− 1)-loop vacuum amplitude) in which
all vertices are type I. Let us, for a moment, ignore the M term of (3.5). The calculation is
then the same as that of [17] which we briefly describe. Each diagram is a product of the
bosonic part obtained by integrating over the momentum p and the fermionic one coming
from the πα integrations. As we have seen in the last paragraph, we have exactly 2(h − 1)
W insertions in the fermion part. There are two possibilities for these W insertions. The
one is to keep one of the index loops empty, filling the remaining index loops with two
W. This yields NSh−1 term, where S = − 1
TrU(N)W
αWα. The other is to fill each of
two index loops chosen with single W, which yields Sh−2wαwα terms where w
α = 1
TrWα.
After calculating the both parts, we perform the Schwinger parameter integrals. Clearly this
procedure is universal to every (h− 1)-loop planar diagram up to the multiplications by the
symmetric factor and by the coupling constants. Therefore every such diagram is a product
of these factors with the following expression
{NhSh−1 + hC22S
h−2wαwα} ≡
(h−1)
0 ,(3.6)
where we have introduced A
(h−1)
0 . The factor h of the first term comes from the choice of
the empty index loop, and hC2 of the second term is the combination of inserting two W
into different index loops. The most important fact is that the dependence on Schwinger
parameters of the bosonic part is cancelled by that of the fermionic part. This explains
that the calculation of the effective superpotential of the gauge theory reduces to that of the
matrix model [17].
There are two types of corrections to A
(h−1)
0 . The one is due to the presence of the M
terms in the propagators, which we denote by A
(h−1)
1 . The other is due to the type II vertices,
which is obtained by replacing one of the type I vertices in A
(h−1)
0 by the corresponding type
II vertex and by summing over all possibilities. We denote this by A
(h−1)
2 . We consider them
in order.
Let us see the effects of the M term, namely, eq. (3.5). It plays a role of inserting two
W further. Thus we will obtain terms which are proportional to Sh. Note that we cannot
insert more than two W because, in such case, at least one of the index loops has more
than two insertions of W. For the parts contributing to NSh−1, which have an empty index
loop, we can further insert WαWα from the first two terms in (3.5). In the case in which
they are inserted in the a-th index loop, we obtain
TrWW, where ia
labels the propagators which form the a-th index loop. The absence of factor N is explained
by the absence of an empty index loop. The factor h is not present as we have so far
restricted ourselves to the a-th index loop. Summing over all index loops, we obtain the first
contribution to A
(h−1)
TrWW = 2ig′3
TrWW,
where we have used that when all index loops are summed, they pass through each double
line propagator exactly twice.
Let us note that the parts contributing to the second term of eq. (3.6) can receive further
insertions of W as well. They have two index loops with a single W insertion, for which
we can exploit the last term of M . An insertion of this term requires that two index loops
share a propagator. Let us define the index A = 1, . . . , hC2 as labeling the combinations of
such two index loops and the index Ã labeling the cases which have a common propagator
in the two index loops chosen. Let us further introduce the index iÃ labeling the common
propagator in case Ã. With these notations, we obtain the second contribution to A
(h−1)
2Sh−2
WαabWαcdW
Wβdc = ig
TrWαWα.
Putting all these together, we obtain the contributions from the vertices of type I,
(h−1)
(h−1)
1 (si))
)h−1(
16π2iPg3S
wαwα. (3.7)
It is important that the above new term has Schwinger parameter dependence aside from
the exponential factor. In [17], it was pointed out that the cancellation of this dependence
represents the reduction of the system to the matrix model. The appearance of this new
term with Schwinger parameter dependence may spoil this reduction. Note also that this new
term does not have an overall factor N , indicating the violation of the well-known relation
due to Dijkgraaf-Vafa [7].
We now turn to the vertices of type II which contain two W insertions. The ℓ-th order
vertex in Φ is
Tr(2WWΦℓ +WΦWΦℓ−1 + . . .+WΦℓ−1WΦ). (3.8)
where we have omitted the overall factors. The first term inserts two W into an index loop
while the remainder insert them into two different index loops. Having done 2(h − 1) πα
Figure 1: two-loop planar diagrams
integrations, we obtain 2(h − 1) W insertions. We can therefore use vertex (3.8) only once
in a diagram. When this is done, insertion of the M term from the propagator is disallowed.
Let us consider A
(h−1)
2 and suppose that one of the type I vertices, TrΦ
ℓ, is replaced by
the above vertex (3.8). The first term connects ℓ index loops and we can insert W2 into ℓ
different ways. Thus we obtain
2ℓTrWW as a contribution to A
(h−1)
2 . For the other
terms of eq. (3.8), there are in total ℓ(ℓ − 1) ways of inserting two W into different index
loops. These give
2Sh−2
ℓ(ℓ− 1)
WαabWαcdW
baWβdc =
ℓ(ℓ− 1)TrWW.
Summing the above two contributions, we obtain
ℓ(ℓ + 1)TrWW. Thus, in any
(h − 1)-loop diagram, changing a vertex from type I to type II is equivalent to considering
only NSh−1 terms in eq. (3.6) and changing the coupling constant by
mgℓ →
16π2igℓ+1S
, for ℓ ≥ 3. (3.9)
Therefore, we obtain a formula for the contribution from the (h− 1)-loop diagrams with
P propagators to Weff in (2.1),
(h−1)
∂F (h−1)
∂2F (h−1)
wαwα −
16π2iPmg3
∂F (h−1)
(h−1)
2 , (3.10)
where W
(h−1)
2 is defined by replacing, in the first term, one coupling constant according to
eq. (3.9) and summing over all possibilities. We have denoted by F (h−1) the (h − 1)-loop
contribution to the planar free energy of the matrix model.
IV. Example
As a sample computation, let us take the two-loop contribution to the effective superpo-
tential. There are two two-loop planar diagrams depicted in Fig.1. Collecting all possible
insertions of W, we obtain
(mg3)
32(mg2)3
NS2 −
(mg3)
16(mg2)3
Swαwα +
π2i(mg3)
2(mg2)4
π2i(mg3)(mg4)
3(mg2)3
. (4.1)
The first two terms are the ones which are present in the computation based on [7, 8] with
SN=1. The third one comes from the M term in the propagator and the last one from the
type II vertices. Note that, in the limit m → ∞ with mgk (k ≥ 2) fixed, we reproduce the
result of [17]. In an arbitrary loop amplitude, the situation is the same: new terms are of
order m−1 in this limit.
The overall U(1) part does not decouple from the SU(N) part. This can be easily seen by
translating S into the glueball superfield Ŝ = − 1
TrSU(N) W
αWα and extracting the factor
in front of wαwα. By the existence of the last two terms in eq. (3.10), it is nonvanishing.
For example, in the two-loop example, this part in (4.1) reads
πi(mg3)[2(mg2)(mg4)− 3(mg3)
2(mg2)4
wαwα 6= 0,
V. The chiral ring and the generalized Konishi anomaly
An alternative approach to the effective superpotential is to exploit and extend the prop-
erties of the N = 1 chiral ring and the generalized Konishi anomaly equations based on
reference [20, 8]. The anomalous Ward identity of our model for the general transformation
δΦ = f(Φ,W) is
= 〈TrfW ′(Φ)〉Φ −
Tr(fF ′′′(Φ)WαWα)
, (5.1)
where W ′′(Φ) = mF ′′′(Φ). In terms of the two generating functions of chiral one-point
functions
R(z) = −
TrWαWα
z − Φ
T (z) =
z − Φ
the anomalous Ward identities (5.1) are
R(z)2 = W ′(z)R(z) +
f(z),
2R(z)T (z) = W ′(z)T (z) +
c(z) + 16π2iF ′′′(z)R(z) +
c̃(z),
where f(z) and c(z) are polynomials of degree n− 1 in z and c̃(z) is a polynomial of degree
n− 2:
f(z) = −
(W ′(Φ)−W ′(z))WαWα
z − Φ
c(z) = 4
W ′(Φ)−W ′(z)
z − Φ
c̃(z) = −i
(F ′′′(Φ)− F ′′′(z))WαWα
z − Φ
The last term of eq. (5.1) does not contribute to the equation for R(z) because of the chiral
ring relation TrWαWαW
βWβ = 0. The equation for R(z) is the same as that of [8], which
is the loop equation of the matrix model. On the other hand, the equation for T (z) alters
from that of [8].
The final step of this approach is to express the effective superpotential in terms of R(z)
and T (z). Taking a variational derivative of (3.1) with respect to the coupling gk, we obtain
∂Weff
dzzkT (z) +
16π2i
(k − 1)!
dzzk−1R(z).
Hence we can determine the effective superpotential up to gk independent terms.
Acknowledgements
We thank Kazuhito Fujiwara, Yosuke Imamura, Hiroaki Kanno, Hironobu Kihara, Ya-
sunari Kurita, Kazutoshi Ohta and Makoto Sakaguchi for useful discussions. We are grateful
to Hiraku Yonemura for his collaboration at an early stage. This work is supported in part by
the Grant-in-Aid for Scientific Research (18540285) from the Ministry of Education, Science
and Culture, Japan. Support from the 21 century COE program “Constitution of wide-angle
mathematical basis focused on knots” is gratefully appreciated. The preliminary version of
this work was presented in YITP workshop “Fundamental Problems and Applications of
Quantum Field Theory”, YITP-W-06-16 in Yukawa Institute for Theoretical Physics, Kyoto
University (December 14-16 2006). We wish to acknowledge the participants for stimulating
discussions.
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	Introduction
	The U(N) gauged model with spontaneously broken N=2 supersymmetry
	Diagrammatic analysis of the effective superpotential
	Example
	The chiral ring and the generalized Konishi anomaly

  It is known that the fermionic shift symmetry of the N=1, U(N) gauge model
with a superpotential of an adjoint chiral superfield is replaced by the second
(spontaneously broken) supersymmetry in the N=2, U(N) gauge model with a
prepotential and Fayet-Iliopoulos parameters. Based on a diagrammatic analysis,
we demonstrate how the well-known form of the effective superpotential in the
former model is modified in the latter. A set of two equations on the one-point
functions stating the Konishi anomaly is modified accordingly.

Introduction
For more than two decades, effective superpotential has been a central object in the
nonperturbative study of N = 1 supersymmetric theories. This object is protected from
perturbative corrections in the conventional sense [1], and yet receives important nonpertur-
bative corrections (see for example [2, 3]). In recent years, analyses from superstring theory
have revealed an interesting perturbative window into nonperturbative physics with the use
of the gluino condensate superfield variable [4, 5, 6, 7]. In [8], field theoretic discussion based
on the model with U(N) gauge group and rigid N = 1 supersymmetry (see eq. (2.2) for its
action SN=1) is given and this is in accord with the string theory based developments.
Superstring theory, on the other hand, insists upon maximally extended supersymmetry
with no adjustable parameter. A scenario that one may draw is that this extended super-
symmetry becomes spontaneously broken to N = 1. Along this vein, a field theory model
with U(N) gauge group and rigid N = 2 supersymmetry spontaneously broken to N = 1
has been introduced in [9, 10, 11] (see eq. (2.1) for its action SN=2), generalizing the abelian
counterpart of [12]. (See also [13] for N = 2 supergravity and [14] for related discussions.)
Several properties of this model have been derived.
In this letter, we make a first analysis on the interplay between the effective superpotential
and partially as well as spontaneously broken N = 2 supersymmetry, shedding a light upon
the comparison of the two models mentioned above. A key aspect of this comparison is that
the fermionic shift symmetry of SN=1 gets replaced by the second (spontaneously broken)
supersymmetry of SN=2. In fact, this is one of the original motivations/results of [9].
The fermionic shift symmetry of SN=1 supplies the well-known formula [7, 8] constraining
the form of the effective superpotential, which is originally proposed from flux compactifi-
cation of string theory [15, 16]. Based on a diagrammatic analysis [17] (for a review see
[18]), we are able to state how this form undergoes modifications in the model SN=2. After
giving a few accounts of the model in the next section, we present a diagrammatic analysis
of Weff in section III. Our final understanding is summarized in eq. (3.10). This is followed
by a computation of the two-loop contribution to Weff in section IV. In the final section, we
derive a set of two equations on the two generating functions R(z) and T (z) of the one-point
functions, generalizing the argument based on the chiral ring and the Konishi anomaly in
[8]. We observe a modification from that given in [8] here as well.
II. The U(N) gauged model with spontaneously broken N = 2
supersymmetry
Let us briefly recall a few ingredients of the model, which are needed in what follows.
The action [9] given in the Wess-Zumino gauge can be written as
SN=2 =
d4xd4θ
Φ̄eadV
∂F(Φ)
− h.c.
+ ξV 0
d4xd2θ
∂2F(Φ)
∂Φa∂Φb
WaWb + eΦ0 +m
∂F(Φ)
+ h.c.
, (2.1)
where V = V ata and W
α are the vector superfield and the gauge superfield strength re-
spectively and Φ = Φata (a = 0, 1, . . . , N
2 − 1) is the chiral superfield ∗. There are three
Fayet-Iliopoulos parameters (e,m, ξ) which are all real. For simplicity, we choose the prepo-
tential as a single trace function of degree n + 2: F(Φ) =
k=1 gkTrΦ
k+1/(k + 1)!. While
this action is shown to be invariant under the N = 2 supersymmetry transformations [9, 10],
the vacuum breaks half of the N = 2 supersymmetries. Extremizing the scalar potential,
we obtain the condition 〈 ∂
∂Φ0∂Φ0
〉 = −(e± iξ)/m, which is a polynomial of order n and this
determines the expectation value of the scalar field.
The action SN=2 in (2.1) is to be compared with that of the N = 1, U(N) gauge model
with a single trace tree level superpotential W (Φ):
SN=1 =
d4xd4θTr Φ̄eadV Φ+
d4xd2θTr (iτWW +W (Φ)) + h.c.
, (2.2)
where τ is a complex gauge coupling τ = θ/2π + 4πi/g2.
In [9], it is checked that the second supersymmetry reduces to the fermionic shift symme-
try in the limit m → ∞. The action SN=2 in fact reduces to SN=1 in the limit m, e, ξ → ∞
with mgk (k ≥ 2) fixed [19]. We show that our result reduces to that of [7, 17] in this limit.
III. Diagrammatic analysis of the effective superpotential
In this letter, we consider the matter-induced part of the effective superpotential by
integrating out the massive degrees of freedom Φ:
d4x(d2θWeff+h.c.+d
4θ(nonchiral terms)) =
DΦDΦ̄eiSN=2. (3.1)
∗a = 0 corresponds to the overall U(1) part.
Let us take Wα (or V ) as the background field †. We consider the case of unbroken U(N)
gauge group. For simplicity, we choose 〈Φ〉 = 0 by setting g1 = −(e± iξ)/m.
We are interested in the holomorphic superpotential which does not contain the anti-
holomorphic couplings ḡk. We can take ḡk = 0 for k ≥ 3 without loss of generality. Collecting
the Φ̄ dependent terms, we obtain
SΦ̄ =
d4xd4θ
Φ̄eadV
∂F(Φ)
− (ḡ1Φ̄ +
Φ̄2)eadV Φ
d4xd2θ̄
TrΦ̄2
d4xd4θTr
Φ̃ḡ2
ḡ1Φ−
. (3.2)
In the last expression, we have introduced a covariantly anti-chiral superfield Φ̃ = Φ̄eadV ,
which satisfies ∇αΦ̃ = 0 (∇α = e
−adV Dαe
adV ). Eq. (3.2) is quadratic in Φ̃ and can be
integrated straightforwardly. As a result, we obtain the following terms,
16ḡ2
ḡ1Φ−
ḡ1Φ−
(Img1)
8mḡ2
Φ∇2Φ+ . . . , (3.3)
where . . . denotes the higher order interaction terms, which we will not consider here. Indeed,
these interaction vertices are higher order in m−1 compared to the vertices which we consider
below. These contribute to our main result (3.10) as higher order corrections in m−1 and do
not spoil our conclusion that the effective superpotential is modified from the case of SN=1
(2.2).
Replacing d2θ̄ integration by −∇̄2/4 and collecting the terms which are not in SΦ̄, we
obtain an action after the Φ̄ integration ‡:
d4xd2θTr
(Img1)
32mḡ2
Φ∇̄2∇2Φ +m
(WΦsWΦk−1−s)
. (3.4)
The first two terms are already present in the integrations with regard to the action SN=1
(2.2). The last term is new and originates from the gauge kinetic term in eq. (2.1). As we
will see below, this last term does contribute to the effective superpotential and becomes
responsible for the violation of the well-known relation [7, 8] between the effective superpo-
tential of the gauge theory and the planar free energy of the matrix model having the tree
level (bare) superpotential as its potential.
†The simplest background is that consisting of a vanishing gauge field Aµ and a constant gaugino λ
which satisfies {λα, λβ} = 0 [18]. This configuration implies that traces of more than two W vanish.
‡In eq. (3.4), it is understood that the generating functional has a renormalized perturbation expansion
in which a nonvanishing tadpole is always canceled by a nonvanishing value of the source coupled to Φ. This
implies that the tadpole can in practice be ignored.
After rescaling Φ → aΦ with a2 = mḡ2/(Img1)
2, the quadratic part of the action (3.4)
reduces to
−�+m′ +
adWαDα
(2WWΦ2 +WΦWΦ),
where we have used the relation ∇̄2∇2Φ = 16(�Φ − adWαDαΦ/2) and introduced m
a2mg2 and g
3 = a
2g3/12. The propagator in the momentum space is
∆(p, π) =
dse−s(p
2+m′+ 1
adWαπα−ig
The Grassmann momentum πα is Fourier transformation of superspace coordinate θα and
the matrix M is
Mabcd = (WW)daδbc + (WW)bcδda +WdaWbc, (3.5)
where we have exhibited the gauge index dependence explicitly. This matrix is not present
in the propagator of [17]. Using eq. (3.5), we are able to insert W without involving the
momentum πα.
The interaction terms in eq. (3.4) are divided into the following two types:
type I. m
Tr Φk, k = 3, . . . , n+ 1.
type II. −
Tr(WΦsWΦk−1−s), k = 4, . . . , n+ 1.
Type I vertices are already present in [17]. Type II vertices are not present in [17]. They
insert two W in specific ways.
Before going on to consider loop diagrams, let us first demonstrate that we have only
to consider planar diagrams in our case as well [17, 18]. For a given diagram, we denote
by V the number of vertices, by P the number of propagators and by h the number of
holes (or index loops). There are V sets of chiral superspace integrations from V vertices.
One of them becomes the chiral superspace integration over the effective superpotential,
and the number of remaining πα momentum integrations is P − V + 1. These Grassmann
integrations must be saturated by 1
adWαπα terms in the propagators. Furthermore, we can
freely insert W both from the M terms in the propagators and from the type II vertices. If
we denote the number of these additional insertions by 2α, the total number of W insertions
is 2(P − V + 1 + α). On the other hand, one index loop can accommodate at most two W.
Thus we have h ≥ P − V + 1+ α. This implies that only the planar diagrams contribute to
the effective superpotential as the Euler number of the diagram is χ = V − P + h.
A planar diagram with h index loops has (h − 1) loop momenta. Let us consider the
(h− 1)-loop planar diagrams (contributing to the (h− 1)-loop vacuum amplitude) in which
all vertices are type I. Let us, for a moment, ignore the M term of (3.5). The calculation is
then the same as that of [17] which we briefly describe. Each diagram is a product of the
bosonic part obtained by integrating over the momentum p and the fermionic one coming
from the πα integrations. As we have seen in the last paragraph, we have exactly 2(h − 1)
W insertions in the fermion part. There are two possibilities for these W insertions. The
one is to keep one of the index loops empty, filling the remaining index loops with two
W. This yields NSh−1 term, where S = − 1
TrU(N)W
αWα. The other is to fill each of
two index loops chosen with single W, which yields Sh−2wαwα terms where w
α = 1
TrWα.
After calculating the both parts, we perform the Schwinger parameter integrals. Clearly this
procedure is universal to every (h− 1)-loop planar diagram up to the multiplications by the
symmetric factor and by the coupling constants. Therefore every such diagram is a product
of these factors with the following expression
{NhSh−1 + hC22S
h−2wαwα} ≡
(h−1)
0 ,(3.6)
where we have introduced A
(h−1)
0 . The factor h of the first term comes from the choice of
the empty index loop, and hC2 of the second term is the combination of inserting two W
into different index loops. The most important fact is that the dependence on Schwinger
parameters of the bosonic part is cancelled by that of the fermionic part. This explains
that the calculation of the effective superpotential of the gauge theory reduces to that of the
matrix model [17].
There are two types of corrections to A
(h−1)
0 . The one is due to the presence of the M
terms in the propagators, which we denote by A
(h−1)
1 . The other is due to the type II vertices,
which is obtained by replacing one of the type I vertices in A
(h−1)
0 by the corresponding type
II vertex and by summing over all possibilities. We denote this by A
(h−1)
2 . We consider them
in order.
Let us see the effects of the M term, namely, eq. (3.5). It plays a role of inserting two
W further. Thus we will obtain terms which are proportional to Sh. Note that we cannot
insert more than two W because, in such case, at least one of the index loops has more
than two insertions of W. For the parts contributing to NSh−1, which have an empty index
loop, we can further insert WαWα from the first two terms in (3.5). In the case in which
they are inserted in the a-th index loop, we obtain
TrWW, where ia
labels the propagators which form the a-th index loop. The absence of factor N is explained
by the absence of an empty index loop. The factor h is not present as we have so far
restricted ourselves to the a-th index loop. Summing over all index loops, we obtain the first
contribution to A
(h−1)
TrWW = 2ig′3
TrWW,
where we have used that when all index loops are summed, they pass through each double
line propagator exactly twice.
Let us note that the parts contributing to the second term of eq. (3.6) can receive further
insertions of W as well. They have two index loops with a single W insertion, for which
we can exploit the last term of M . An insertion of this term requires that two index loops
share a propagator. Let us define the index A = 1, . . . , hC2 as labeling the combinations of
such two index loops and the index Ã labeling the cases which have a common propagator
in the two index loops chosen. Let us further introduce the index iÃ labeling the common
propagator in case Ã. With these notations, we obtain the second contribution to A
(h−1)
2Sh−2
WαabWαcdW
Wβdc = ig
TrWαWα.
Putting all these together, we obtain the contributions from the vertices of type I,
(h−1)
(h−1)
1 (si))
)h−1(
16π2iPg3S
wαwα. (3.7)
It is important that the above new term has Schwinger parameter dependence aside from
the exponential factor. In [17], it was pointed out that the cancellation of this dependence
represents the reduction of the system to the matrix model. The appearance of this new
term with Schwinger parameter dependence may spoil this reduction. Note also that this new
term does not have an overall factor N , indicating the violation of the well-known relation
due to Dijkgraaf-Vafa [7].
We now turn to the vertices of type II which contain two W insertions. The ℓ-th order
vertex in Φ is
Tr(2WWΦℓ +WΦWΦℓ−1 + . . .+WΦℓ−1WΦ). (3.8)
where we have omitted the overall factors. The first term inserts two W into an index loop
while the remainder insert them into two different index loops. Having done 2(h − 1) πα
Figure 1: two-loop planar diagrams
integrations, we obtain 2(h − 1) W insertions. We can therefore use vertex (3.8) only once
in a diagram. When this is done, insertion of the M term from the propagator is disallowed.
Let us consider A
(h−1)
2 and suppose that one of the type I vertices, TrΦ
ℓ, is replaced by
the above vertex (3.8). The first term connects ℓ index loops and we can insert W2 into ℓ
different ways. Thus we obtain
2ℓTrWW as a contribution to A
(h−1)
2 . For the other
terms of eq. (3.8), there are in total ℓ(ℓ − 1) ways of inserting two W into different index
loops. These give
2Sh−2
ℓ(ℓ− 1)
WαabWαcdW
baWβdc =
ℓ(ℓ− 1)TrWW.
Summing the above two contributions, we obtain
ℓ(ℓ + 1)TrWW. Thus, in any
(h − 1)-loop diagram, changing a vertex from type I to type II is equivalent to considering
only NSh−1 terms in eq. (3.6) and changing the coupling constant by
mgℓ →
16π2igℓ+1S
, for ℓ ≥ 3. (3.9)
Therefore, we obtain a formula for the contribution from the (h− 1)-loop diagrams with
P propagators to Weff in (2.1),
(h−1)
∂F (h−1)
∂2F (h−1)
wαwα −
16π2iPmg3
∂F (h−1)
(h−1)
2 , (3.10)
where W
(h−1)
2 is defined by replacing, in the first term, one coupling constant according to
eq. (3.9) and summing over all possibilities. We have denoted by F (h−1) the (h − 1)-loop
contribution to the planar free energy of the matrix model.
IV. Example
As a sample computation, let us take the two-loop contribution to the effective superpo-
tential. There are two two-loop planar diagrams depicted in Fig.1. Collecting all possible
insertions of W, we obtain
(mg3)
32(mg2)3
NS2 −
(mg3)
16(mg2)3
Swαwα +
π2i(mg3)
2(mg2)4
π2i(mg3)(mg4)
3(mg2)3
. (4.1)
The first two terms are the ones which are present in the computation based on [7, 8] with
SN=1. The third one comes from the M term in the propagator and the last one from the
type II vertices. Note that, in the limit m → ∞ with mgk (k ≥ 2) fixed, we reproduce the
result of [17]. In an arbitrary loop amplitude, the situation is the same: new terms are of
order m−1 in this limit.
The overall U(1) part does not decouple from the SU(N) part. This can be easily seen by
translating S into the glueball superfield Ŝ = − 1
TrSU(N) W
αWα and extracting the factor
in front of wαwα. By the existence of the last two terms in eq. (3.10), it is nonvanishing.
For example, in the two-loop example, this part in (4.1) reads
πi(mg3)[2(mg2)(mg4)− 3(mg3)
2(mg2)4
wαwα 6= 0,
V. The chiral ring and the generalized Konishi anomaly
An alternative approach to the effective superpotential is to exploit and extend the prop-
erties of the N = 1 chiral ring and the generalized Konishi anomaly equations based on
reference [20, 8]. The anomalous Ward identity of our model for the general transformation
δΦ = f(Φ,W) is
= 〈TrfW ′(Φ)〉Φ −
Tr(fF ′′′(Φ)WαWα)
, (5.1)
where W ′′(Φ) = mF ′′′(Φ). In terms of the two generating functions of chiral one-point
functions
R(z) = −
TrWαWα
z − Φ
T (z) =
z − Φ
the anomalous Ward identities (5.1) are
R(z)2 = W ′(z)R(z) +
f(z),
2R(z)T (z) = W ′(z)T (z) +
c(z) + 16π2iF ′′′(z)R(z) +
c̃(z),
where f(z) and c(z) are polynomials of degree n− 1 in z and c̃(z) is a polynomial of degree
n− 2:
f(z) = −
(W ′(Φ)−W ′(z))WαWα
z − Φ
c(z) = 4
W ′(Φ)−W ′(z)
z − Φ
c̃(z) = −i
(F ′′′(Φ)− F ′′′(z))WαWα
z − Φ
The last term of eq. (5.1) does not contribute to the equation for R(z) because of the chiral
ring relation TrWαWαW
βWβ = 0. The equation for R(z) is the same as that of [8], which
is the loop equation of the matrix model. On the other hand, the equation for T (z) alters
from that of [8].
The final step of this approach is to express the effective superpotential in terms of R(z)
and T (z). Taking a variational derivative of (3.1) with respect to the coupling gk, we obtain
∂Weff
dzzkT (z) +
16π2i
(k − 1)!
dzzk−1R(z).
Hence we can determine the effective superpotential up to gk independent terms.
Acknowledgements
We thank Kazuhito Fujiwara, Yosuke Imamura, Hiroaki Kanno, Hironobu Kihara, Ya-
sunari Kurita, Kazutoshi Ohta and Makoto Sakaguchi for useful discussions. We are grateful
to Hiraku Yonemura for his collaboration at an early stage. This work is supported in part by
the Grant-in-Aid for Scientific Research (18540285) from the Ministry of Education, Science
and Culture, Japan. Support from the 21 century COE program “Constitution of wide-angle
mathematical basis focused on knots” is gratefully appreciated. The preliminary version of
this work was presented in YITP workshop “Fundamental Problems and Applications of
Quantum Field Theory”, YITP-W-06-16 in Yukawa Institute for Theoretical Physics, Kyoto
University (December 14-16 2006). We wish to acknowledge the participants for stimulating
discussions.
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	Introduction
	The U(N) gauged model with spontaneously broken N=2 supersymmetry
	Diagrammatic analysis of the effective superpotential
	Example
	The chiral ring and the generalized Konishi anomaly
