Consistency conditions in the chiral ring of super Yang-Mills theories

0.29 MB·English

by  Frank Ferrari

LPTENS-07/48
arXiv:0710.2978 [hep-th]
Consistency conditions in the chiral ring
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0 of super Yang-Mills theories
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n
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0 Frank Ferrari and Vincent Wens
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Service de Physique Th´eorique et Math´ematique

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h Universit´e Libre de Bruxelles and International Solvay Institutes
t Campus de la Plaine, CP 231, B-1050 Bruxelles, Belgique
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[email protected], [email protected]
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Starting from the generalized Konishi anomaly equations at the non-perturbative
1
7 level, we demonstrate that the algebraic consistency of the quantum chiral ring of the
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N = 1 super Yang-Mills theory with gauge group U(N), one adjoint chiral superfield
:
v X and N ≤ 2N flavours of quarks implies that the periods of the meromorphic
i f
X one-form Tr dz must be quantized. This shows in particular that identities in the
z−X
r open string description of the theory, that follow from the fact that gauge invariant
a
observables are expressed in terms of gauge variant building blocks, are mapped onto
non-trivial dynamical equations in the closed string description.
February 2, 2008
1 Introduction
The fact that any four dimensional gauge theory has two seemingly unrelated for-
mulations, one in terms of open strings, which is equivalent to the standard field
theoretic Yang-Mills description and the other in terms of closed strings, which thus
contains quantum gravity, is an extremely deep and fascinating property. Following
[1], many successfull examples of this duality have been studied over the last decade.
Yet many questions, both technical and conceptual, remain unsolved.
A fundamental conceptual issue is to understand how the basic ingredients in
one formulation are encoded in the other formulation and vice-versa. For example,
how does the closed string gravity theory know about the Yang-Mills equations of
motion? In the closed string description, we do not see the gauge group, for only
gauge invariant quantities can be constructed. This is of course not an inconsistency,
since the gauge symmetry is really a redundancy in the description of the theory and
not a physical symmetry. However, how then can we understand charge quantization
`a la Dirac, which is usually derived from gauge invariance, in the closed string set-
up? A directly related question, which will be at the basis of the present work, is
the following. In the open string framework, gauge invariant observables are built
in terms of fields that transform non-trivially under the gauge group, and this has
some non-trivial mathematical consequences. For example, imagine that the gauge
group is U(N) and that the theory contains an adjoint field X. The gauge invariant
operators built from X are obtained by considering traces
u = TrXk (1.1)
k
or product of traces. The fact that X is a N × N matrix implies that there exists
homogeneous polynomials P of degree N + p, if the degree of homogeneity of X is
p
one, such that
u = P (u ,...,u ), p ≥ 1. (1.2)
N+p p 1 N
Thus only u ,...,u are independent. But how does the closed string theory know
1 N
about (1.2), while the matrix X does not exist in the closed string framework? In a
sense we are asking how to build the open strings starting from the closed strings,
which is a notoriously difficult question.
An extremely interesting incarnation of the open/closed string duality is ob-
tained when one focus on the chiral sector of N = 1 supersymmetric gauge theo-
ries. The closed string set-up involves a geometric transition [2] and is equivalent to
the Dijkgraaf-Vafa matrix model description [3]. On the other hand, the model has
been solved recently starting from the usual field theoretic description [4, 5, 6], using
2
Nekrasov’s instanton technology [7]. The theory is essentially reduced to a statistical
model of colored partitions which, remarkably, yields gauge theory correlators that
coincide with the matrix model predictions [5, 6]. The open/closed string duality is
thus fully understood in this case. Our aim in the present paper, which is a continua-
tion of [8], is to address some of the above conceptual questions in this well-controlled
framework. Our main result will be to show that identities like (1.2) are equivalent
to dynamical equations of motion in the closed string description.
The plan of the paper is as follows. In Section 2, we introduce some basic ideas
on a very simple example and present the model we are studying, the N = 1 super
Yang-Mills theory with gauge group U(N), one adjoint chiral superfield and N ≤ 2N
f
flavours of quarks. We also state the chiral ring consistency theorem [8]. This is our
main result and the proof of the theorem is given in Section 3. Finally in Section 4
we summarize our findings and conclude.
2 Preliminaries
2.1 A simple example: the classical limit
We can immediately give the flavour of the arguments that we are going to use by
looking at the classical limit. We consider the U(N) super Yang-Mills theory with
one adjoint chiral superfield X and tree-level superpotential TrW(X) such that
d d
W′(z) = g zk = g (z −w ). (2.1)
k d i
X Y
k=0 i=1
The equations of motion in the open string description are thus
W′(X) = 0. (2.2)
The most general solution is labeled by the positive integers N , with
i
d
N = N , (2.3)
i
X
i=1
such that the matrix X has N eigenvalues equal to w . In particular, the generating
i i
function
1 u
k
R(z) = Tr = (2.4)
z −X zk+1
X
k≥0
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is given by
N
i
R(z) = · (2.5)
z −w
X i
i
In the closed string description, we can use only the gauge invariant operators u ,
k
not the matrix X. The equations of motion (2.2) are then written as
Tr Xn+1W′(X) = 0 = g u , n ≥ −1. (2.6)
k n+k+1
(cid:0) (cid:1) Xk≥0
In terms of R(z), this is equivalent to the existence of a degree d−1 polynomial ∆
such that
W′(z)R(z) = ∆(z). (2.7)
The vanishing of the terms proportional to negative powers of z in the large z ex-
pansion of the left hand side of (2.7) is indeed equivalent to the equations (2.6). The
most general solution to (2.6) or (2.7) is given by
d
∆(z) c
i
R(z) = = · (2.8)
W′(z) z −w
X i
i=1
The constants c can be arbitrary complex numbers, with the only constraint
i
d
c = N (2.9)
i
X
i=1
that follows from the definition of R(z).
To make contact with the open string formula (2.5), we have to prove that the c
i
must be positive integers. This is obvious in the open string framework since c = N
i i
is then identified with the number of eigenvalues of the matrix X that are equal to w .
i
The question is: how can we understand this quantization condition in a formulation
where only the gauge invariant operators u are available?
k
The fundamental idea is to implement the constraints (1.2) [8]. We are going to
show the simple
Theorem. The equations (2.6) are consistent with the constraints (1.2) if and only
if the constants c s in (2.8) are positive integers. In particular, the integrals 1 Rdz
i 2iπ
over any closed contours are integers. H
This is a toy version of the chiral ring consistency theorem that we shall prove later.
Very concretely, it means that a set of variables u given by the formulas
k
d
u = c wk (2.10)
k i i
X
i=1
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can satisfy the constraints (1.2) if and only if the c s are positive integers. To prove
i
this simple algebraic result, we use the following trick. We introduce the function
F(z) defined by the conditions
F′(z)
= R(z), F(z) ∼ zN . (2.11)
F(z) z→∞
In terms of the matrix X, one would simply have F(z) = det(z −X), but we do not
want to use the matrix X here but only deal with the gauge invariant variables u .
k
The function F is expressed in terms of these variables by integrating (2.4),
u
F(z) = zN exp − k . (2.12)
(cid:16) X kzk(cid:17)
k≥1
The crucial algebraic property is that the relations (1.2) are equivalent to the fact
that F(z) is a polynomial. A very effective way to compute the polynomials P is
p
actually to write that the terms with a negative power of z in the large z expansion
of the right-hand side of (2.12) must vanish. If F is a polynomial, then of course it
is a single-valued function of z, and thus
1 1
Rdz = dlnF ∈ Z. (2.13)
2iπ I 2iπ I
In particular, the c s are integers. They are positive because F does not have poles.
i
Conversely, if the c are positive integers, then we can introduce the matrix X defined
i
to have c eigenvalues equal to w for all i. The relations (1.2) are then automatically
i i
satisfied.
2.2 The model
Our aim in the present paper is to generalize the above analysis to the full non-
perturbative quantum theory, by analysing the consistency between the quantum
versions of (2.6) and (1.2) to prove that the periods 1 Rdz must always be quan-
2iπ
tized. These quantization conditions are highly non-trivHial constraints, known to be
equivalent to a specific form of the Dijkgraaf-Vafa glueball superpotential, including
the Veneziano-Yankielowicz coupling-independent part, and to contain the crucial
information on the non-perturbative dynamics of the theory in the matrix model
formalism [9, 10, 8].
We shall focus on the U(N) theory with one adjoint chiral superfield X and N
f
flavours of fundamentals (Q˜a,Q ). We always assume that the theory is asymptoti-
b
cally free or conformal in the UV,
N ≤ 2N . (2.14)
f
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When N < 2N the instanton factor is given by
f
q = Λ2N−Nf (2.15)
in terms of the dynamically generated complex scale Λ. When N = 2N we have
f
q = e−8π2/g2+iϑ (2.16)
in terms of the Yang-Mills coupling constant g and ϑ angle. The tree-level superpo-
tential has the form
W = TrW(X)+ TQ˜am b(X)Q . (2.17)
tree a b
X
1≤a,b≤Nf
ThederivativeofW(z)isasin(2.1),andm b(z)isaN ×N matrix-valuedpolynomial,
a f f
δ
m b(z) = m b zk, (2.18)
a a,k
X
k=0
with
Nfδ
detm(z) = U(z) = U (z −b ). (2.19)
0 Q
Y
Q=1
It is useful to introduce the symmetric polynomials
σ = b ···b , 1 ≤ α ≤ N δ. (2.20)
α Q1 Qα f
X
Q1<···<Qα
We shall consider the case where m b(z) is a linear function of z,
a
δ = 1, (2.21)
in the following.1
The classical theory has a large number of vacua obtained by extremizing the
superpotential (2.17). The most general solution |N ;ν i is labeled by the numbers
i Q cl
of eigenvalues of the matrix X, N ≥ 0 and ν = 0 or 1, that are equal to w and b
i Q i Q
respectively [9]. The constraint
d Nf
N + ν = N (2.22)
i Q
X X
i=1 Q=1
1This is not strictly necessary as long as the constraint Nfδ ≤2N is satisfied.
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must be satisfied. The gauge group U(N) is broken down to U(N )×···×U(N ) in
1 d
a vacuum |N ;ν i . We shall call the number of non-zero integers N the rank r of
i Q cl i
the vacuum.
In addition to (1.1), we have other basic gauge invariant operators in the theory
that are constructed by using the vector chiral superfield Wα,
1 1
uα = TrWαXk, v = − TrWαW Xk, w b = TQ˜bXkQ . (2.23)
k 4π k 16π2 α a,k a
The associated generating functions are defined by
uα v w b
Wα(z) = k , S(z) = k , G b(z) = a,k · (2.24)
zk+1 zk+1 a zk+1
X X X
k≥0 k≥0 k≥0
The relations that replace (2.6), or equivalently (2.7), in the full quantum theory
are given by the following generalized Konishi anomaly equations [11]
NW′(z)R(z)+Nm′ b(z)G a(z)−2S(z)R(z)−2Wα(z)W (z) = ∆ (z) (2.25)
a b α R
NW′(z)Wα(z)−2S(z)Wα(z) = ∆α(z) (2.26)
NW′(z)S(z)−S(z)2 = ∆ (z) (2.27)
S
NG c(z)m b(z)−S(z)δb = ∆b(z) (2.28)
a c a a
Nm c(z)G b(z)−S(z)δb = ∆˜b(z). (2.29)
a c a a
The functions ∆ , ∆α, ∆ , ∆b and ∆˜b must be polynomials. By expanding (2.25)–
R S a a
(2.29) at large z, and writing that the terms proportional to negative powers of z
must vanish, we obtain an infinite set of constraints on the gauge invariant operators,
valid for any integer n ≥ −1,
N g u +(k +1)m b w a −2 u v +uα u = 0 (2.30)
k n+k+1 a,k+1 b,n+k+1 k1 k2 k1 k2α
Xk≥0(cid:0) (cid:1) k1+Xk2=n(cid:0) (cid:1)
N g uα −2 v uα = 0 (2.31)
k n+k+1 k1 k2
X X
k≥0 k1+k2=n
N g v − v v = 0 (2.32)
k n+k+1 k1 k2
X X
k≥0 k1+k2=n
N w c m b −v δb = 0 (2.33)
a,k+n+1 c,k n+1 a
X
k≥0
N m c w b −v δb = 0. (2.34)
a,k c,k+n+1 n+1 a
X
k≥0
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Figure 1: The hyperelliptic Riemann surface C, with the contours α and γ used in
i ij
the main text.
Equations (2.25)–(2.29)show thatthegenerating functions aremeromorphic func-
tions on a hyperelliptic Riemann surface of the form
r
C : y2 = (z −w−)(z −w+). (2.35)
r r i i
Y
i=1
The integer r, called the rank of the solution, must satisfy
r ≤ d (2.36)
and we have
W′(z)2 −4∆ (z) = φ (z)2y2 (2.37)
d−1 d−r r
for some polynomials ∆ = ∆ /N2 and φ of degrees d−1 and d−r respectively.
d−1 S d−r
The curve (2.35) with some closed contours is depicted in Figure 1. It corresponds
to the geometry of the closed string background. We shall use extensively in the
following the most general solution to (2.25)–(2.29) of rank r for the expectation
value hR(z)i . It has the form [9]
r
C 1U′ 1 (1−2ν )y (z = b )
r−1 Q r Q
R(z) = + − · (2.38)
r y 2 U 2y z −b
(cid:10) (cid:11) r r X Q
Q
The polynomialC = 1(2N−N )zr−1+··· isof degree r−1 andis aprioriunknown
r−1 2 f
except for its term of highest degree that is fixed by the large z asymptotics of R(z).
In the classical limit, the solutions (2.38) correspond to the rank r classical vacua
|N ;ν i described previously. Quantum mechanically, the anomaly equations (2.38)
i Q cl
leave 2r−1 arbitrary parameters, which are the coefficients of C and of ∆ that
r−1 d−1
are not fixed by the factorization condition (2.37). These unknown parameters are
the quantum analogues of the coefficients c in (2.8), and our main goal is to show
i
that they are fixed by a quantum version of the simple consistency proof explained
in 2.1.
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2.3 Chiral ring relations and anomaly equations
The model (2.17) has a useful SU(N ) ×SU(N ) ×U(1) ×U(1) ×U(1) global
f L f R A B R
symmetry. The charges of the various parameters and operators of the theory are
given in the following table
u uα v w b g m b σ U q
k k k a,k k a,k k 0
U(1) k k k k −1 −k −1 −k +1 k 0 2N −N
A f
U(1) 0 0 0 2 0 −2 0 −2N 2N
B f f (2.39)
U(1) 0 1 2 2 2 0 0 0 0
R
SU(N ) 1 1 1 N 1 N 1 1 1
f L f f
SU(N ) 1 1 1 N 1 N 1 1 1.
f R f f
It is useful for our purposes to consider the subring A of the chiral ring of the theory
0
that is invariant under SU(N ) ×SU(N ) ×U(1) ×U(1) . This subring isgenerated
f L f R B R
by the operators u and the parameters2 σ and
k k
q = U q. (2.40)
0
It is a simple polynomial ring given by
A = C[q,σ ,...,σ ,u ,...,u ]. (2.41)
0 1 Nf 1 N
As stressed in the Section 2 of [8], a polynomial ring has no deformation, and thus
(2.41), which is trivially valid at the classical level due to the relations (1.2), is also
valid in the full quantum theory. The meaning of this statement is simply that any
operator in A can be expressed as a finite sum of finite products of U q, σ for
0 0 k
1 ≤ k ≤ N and u for 1 ≤ k ≤ N, a rather trivial result. It is sometimes claimed
f k
in the literature that the ring A is deformed because the relations (1.2) can get
0
quantum corrections. This is not correct. In the full quantum theory, we can define
what is meant by u ∈ A for k > N by a relation of the form
k 0
u = P (u ,...,u ,σ ,...,σ ,q), p ≥ 1. (2.42)
N+p p 1 N 1 Nf
The P are chosen to be consistent with the symmetries (2.39) and the classical limit
p
(1.2), but can be completely arbitrary otherwise. It can be convenient to work with a
particular definition (2.42), and we shall see shortly that there is indeed a canonical
choice, but this remains a choice and has no physical content [8]. Let us note that
a parallel discussion applies to the variables uα, v and w b , which are independent
k k a,k
only for 0 ≤ k ≤ N −1.
2It is convenient to include the parameters,which can always be promoted to backgroundchiral
superfields, in the chiral ring.
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The equations (2.25)–(2.29) werederived inperturbationtheoryin[12,11]. At the
non-perturbative level, these equations do get quantum corrections. However, these
quantum corrections have a very special form. The general theorem is as follows:
Non-perturbative anomaly theorem [6]: The non-perturbative corrections to the
generalized anomaly equations are such that they can be absorbed in a non-perturbative
redefinition of the variables that enter the equations.
This means that there exists a canonical choice for the definitions of the variables u
k
for k > N as in (2.42), and other similar canonical definitions of the variables uα ,
k−1
v and w b for k > N, that make all the non-perturbative corrections implicit.
k−1 a,k−1
The theorem has been proven recently in the case of the theory with no flavour [6].
The arguments used in [6] can in principle be generalized straightforwardly, and we
shall take the result for granted in the theory with flavours as well.
2.4 The chiral ring consistency theorem
We can now state the quantum version of the classical problem solved in Section
2.1. On the one hand, in the closed string description, the theory is described by
the equations (2.25)–(2.29) or equivalently by (2.30)–(2.34). On the other hand, we
know that the existence of the open string formulation implies that relations of the
form (2.42) must exist. These relations imply that there are only a finite number of
independent variables. The anomaly equations (2.30)–(2.34) thus yield an infinite set
of constraints on a finite set of independent variables. Generically, such an overcon-
strained system of equations is inconsistent. The main result of the present work is
to prove the
Chiral ring consistency theorem: The system of equations (2.30)–(2.34) is con-
sistent with the existence of relations of the form (2.42) if and only if the periods of
the gauge theory resolvent 1 Rdz are integers. The relations (2.42) (and all the
2iπ
other relations amongst chiralHoperators) are then fixed in a unique way.
This theorem was conjectured in [8]. As discussed in 2.2, the equations (2.30)–(2.34)
imply that R is a meromorphic function on a hyperelliptic curve of the form (2.35).
The theorem then states that the algebraic consistency of the chiral ring implies
1
hRi dz ∈ Z (2.43)
2iπ I r
αi
1
hRi dz ∈ Z, (2.44)
2iπ I r
γij
where the contours α and γ are defined in Figure 1. Actually, the α -periods are
i ij i
automatically positive, as we shall see. Several comments on this result are in order.
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