Table Of ContentGenus one correlation to multi-cut matrix model solutions
L. Chekhov1
Steklov Mathematical Institute,
Gubkina 8, 117966, GSP–1, Moscow, Russia
4
0 We calculate genus onecorrections to Hermitian one-matrix modelsolution with
0 arbitrary number of cuts directly from the loop equation confirming the answer
2
previously obtained from algebro-geometrical considerations and generalizing it to
n
the case of arbitrary potentials.
a
J
3
1 1 Introduction
1
v Solving matrix models using the loop equation has more than decade long history. An
9
algorithm for finding genus expansion solutions to the Hermitian one-matrix model was
8
0 elaborated (see [1] and references therein) in the case where the limiting eigenvalue distri-
1 bution spans a single interval on the real line in the large N limit (in what follows, N is
0
4 the dimension of the Hermitian matrix). Such a situation is commonly called the one-cut
0 solution. Results for multi-cut solutions were few. The problem was that it is not too
/
h difficult to find a solution for the loop equation, i.e., the genus expansion of the so-called
t
- loop mean—this problem was solved by Akemann [2]. The universal critical behavior of
p
the corresponding correlators was considered in [3]. However, a difficult problem is to
e
h integrate the obtained solution in order to produce higher genus contributions to the free
:
v energy. Akemann has succeeded in finding such a genus-one contribution in the two-cut
i
X case only.
r Recently, it was realized that multi-cut solutions to matrix models is the powerful
a
tool for studying broken vacua structures of N = 2 super Yang–Mills theory of Seiberg
and Witten [4]. Simultaneously, a new relation between the superpotentials of N = 1
supersymmetric gauge theories in four dimensions and free energies of matrix models in
the planar limit was proposed [5, 6]. The superpotentials in some N = 1 four-dimensional
Yang–Mills theories has turned out to be expressed through a single holomorphic function
[5], which was later identified with the planar limit of the free energy of the multi-support
solutions to matrix models [6].
On the other hand, there are strong indications that such an analogy goes beyond the
tree level and just the large-N limit of the matrix model partition function. The relation
between gauge theories and matrix model solutions were checked on the one-loop level
for the solution with two cuts and a cubic matrix model potential [7]. A comparison in
general case must involve the structure of the general matrix model multi-cut solution in
1E-mail: [email protected].
1
genus one approximation, which has been missed as yet. Kostov [8] had provided argu-
ments in favor of using the conformal field theory (CFT) methods for calculating such the
corrections in the general case following the way proposed by Moore [9]; this method has
been improved by Dijkgraaf, Sinkovics, and Temu¨rhan [10] who used algebro-geometrical
and CFT arguments for proposing the formula for genus one correction and comparing
it with the gauge theory answer. Note, however, that the very applicability of Kostov’s
star operator (dressing) method needs to be justified by direct calculations using the loop
equation alone and, second, the algebro-geometrical methods need to be improved in a
way to allow calculating contributions from ”non-geometrical” part of the action—from
the so-called ”moments” of the matrix model potential (such contributions always arise if
the power of the polynomial that is the derivative of the potential exceeds the number of
cuts in play; then, double points arise and the genus of the arising hyperelliptic surface
can bemuch lesser thanthe highest degree of thepotential. Inthis situation, however, the
superpotential satisfying the Whitham hierarchy equations and the WDVV equations is
still attained by the planar (large-N) limit of the free energy of the corresponding matrix
model [11], [12].
In this paper, we derive the subleading (genus one) expression for the free energy of
the generalized matrix model multi-cut solution directly from the loop equation, without
referring to geometry or to the CFT reasoning thus supporting the calculations in [8]
and [10]. In Secs. 2, 3, we closely follows the method of [1], [2]reviewing the iterative
procedure for finding the loop mean in the genus expansion; the answer for the genus-one
contribution W (p) to the loop mean is presented in Sec. 4. We integrate the obtained
1
answer in Sec. 4 thus obtaining the complete expression for the genus-one part F of
1
the free energy. As a byproduct, we obtain interesting combinatorial identities on the
quantities entering W (p).
1
2 Definitions. Loop equation
The partition function of the Hermitian one-matrix model is
Z[N,{ti}] ≡ eN2F = e ∞g=0N2−2gFg[{ti}] ≡ dΦe−NtrV(Φ), (2.1)
P Z
where the integration is over Hermitian N ×N matrices Φ. The matrix potential, which
we assume to be polynomial of degree (m+1), is given by the power series
m+1 t
V(x) = jxj. (2.2)
j
j=1
X
Derivingtheloopequationimpliesintroducingallcouplingconstantst intoplay; aspecific
j
potential of finite order can be inspected by setting the extra couplings to zero in the final
result. The coupling constants t are then the sources for polynomial expectation values
j
with the means defined customarily as
1
hQ(φ)i = dΦQ(Φ)e−NtrV(Φ).
Z
Z
2
Introducing the loop insertion operator
d ∞ j d
(p) ≡ − , (2.3)
dV pj+1dt
j=1 j
X
the one-loop mean W(p) can thus be obtained from the free energy F
1 ∞ hTrΦki 1 1 d 1
W(p) ≡ = tr = (p)F + . (2.4)
N pk+1 N * p−φ+ dV p
k=0
X
Analogously, all the multi-loop correlators can be derived by applying d (p) to F (or to
dV
W(p))
1 1
W(p ,...,p ) ≡ Nn−2 tr ···tr
1 n
* p1 −φ pn −φ+conn
d d d
= (p ) (p )··· (p )F , n ≥ 2. (2.5)
n n−1 1
dV dV dV
Here conn refers to the connected part. Because the loop correlators and the free energy
have the same genus expansion,
∞ 1
W(p ,...,p ) = W (p ,...,p ) , (2.6)
1 n N2g g 1 n
g=0
X
Eq. (2.5) is valid for each genus g ≥ 0 separately. We assume W(p) to be analytic at
infinity with the asymptotic behavior
1
lim W(p) ∼ , (2.7)
p→∞ p
i.e.,
1
W (p)| = δ +O(1/p2). (2.8)
g p→∞ g,0
p
This means that higher W (p) must be total derivatives:
g
d
W (p) = (p)F , g ≥ 1. (2.9)
g g
dV
The loop equation is the exact equation satisfied by the loop mean [13]. Deriving the
loop equation is irrelevant to the multiple cut structure and relies only on the invariance
of the partition function under a field redefinition Φ → Φ+ǫ/(p−Φ). The only effect is
that the contour C encircles now a number n of disjoint intervals on the real line:
D
dω V′(ω) 1 d
W(ω) = (W(p))2 + (p)W(p), p 6∈ D, (2.10)
2πip−ω N2dV
ICD
where V′(ω) = t ωj−1. We assume that the density ρ (λ) ≡ 1h N δ(λ − λ )i of
j j N N i i
eigenvalues of the matrices Φ has a compact support D as N → ∞. We let D to comprise
P P
n disjoint intervals:
n
D ≡ [µ ,µ ], µ < µ < ... < µ . (2.11)
2i−1 2i 1 2 2n
i=1
[
3
Presenting W(p) in terms of ρ (λ), W(p) = dλρN(λ), we find that as N → ∞, W(p) is
N p−λ
analytic outside n cuts on the real axis.
R
Inserting genus expansion (2.6) into loop equation (2.10), we obtain
dω V′(ω)
W (ω) = (W (p))2 (2.12)
0 0
2πip−ω
ICD
for genus zero and
g−1 d
(Kˆ −2W (p))W (p) = W (p)W (p)+ (p)W (p), g ≥ 1, (2.13)
0 g g′ g−g′ g−1
dV
g′=1
X
for higher genera, where Kˆ is a linear integral operator
dω V′(ω)
Kˆf(p) ≡ f(ω). (2.14)
2πip−ω
ICD
Given W (p), one can then determine W (p) for g ≥ 1 iteratively genus by genus provided
0 g
the operator (Kˆ −2W (p)) can be inverted uniquely.
0
We can solve Eq. (2.12) for the planar solution W (p) as follows. Deforming the
0
contour in Eq. (2.12) to infinity, we obtain
dω V′(ω)
(W (p))2 = V′(p)W (p)+ W (ω). (2.15)
0 0 0
2πip−ω
IC∞
The solution is then formally
1 1
W (p) = V′(p)− (V′(p))2 +4P(p),
0
2 2
q
dω V′(ω)
P(p) = W (ω), (2.16)
0
2πip−ω
IC∞
where the minus sign is chosen in order to fulfill the asymptotic Eq. (2.8) and P(p) is a
polynomial of power m − 1. If W (p) has n cuts in the complex plane, we propose the
0
ansatz for the square root,
1
W (p) = (V′(p)−y(p)), (2.17)
0
2
where
2n
y(p) ≡ M(p)y˜(p), and y˜(p) ≡ (p−µ ) (2.18)
i
i=1
r
Y
(the notation comes from [11]), and M(p) is assumed to be a polynomial of degree m−n,
which still has to be determined. By convention, we set y˜(p)| ∼ pn, and M(p) is then
p→∞
dω V′(ω)
M(p) = . (2.19)
2πi(ω −p)y˜(ω)
IC∞
4
Inserting this solution in Eq. (2.17) and deforming the contour back, we obtain the planar
one-loop correlator with an n-cut structure,
1 dω V′(ω) y˜(p)
W (p) = , p 6∈ D. (2.20)
0
2 2πip−ω y˜(ω)
ICD
Then, the planar eigenvalue density ρ(λ) ≡ lim ρ (λ) has the form
N→∞ N
1 1
ρ(λ) = lim W (λ−iǫ)−W (λ+iǫ) = Imy(λ), λ ∈ D. (2.21)
0 0
2πi ǫ→0 2π
(cid:16) (cid:17)
We still must determine the positions of branching points µ , α = 1,...,2n. Here, we
α
have two sorts of restrictions. The first set comes from asymptotic conditions (2.8) to be
satisfied by ansatz (2.20), from which we have
1 dω ωkV′(ω)
δ = , k = 0,...,n. (2.22)
k,s
2 2πi y˜(ω)
ICD
These conditions provide n+1equations for 2nconstants µ (which suffices onlyif n = 1).
α
Another bunch of relations cannot be read from the analytical structure alone. Intro-
ducing the occupation numbers
dω dω
S = M(ω)y˜(ω) ≡ y(ω), (2.23)
i
4πi 4πi
IAi IAi
where A , i = 1,...,n− 1 is the basis of A-cycles on the hyperelliptic Riemann surface
i
y˜2 = 2n (x−µ ) (we may conveniently choose them to be the first n−1 cuts), we can
α=1 α
consider these numbers as independent parameters of the theory in the Dijkgraaf–Vafa
Q
setting, thus imposing the restrictions
d
(p)S = 0, i = 1,...,n−1. (2.24)
i
dV
(Adopting a quite opposite standpoint, which is intrinsic for an original approach to
the matrix models, see [14], [2], we must look for the genuine minimum of the matrix
model action with respect to variables S as well as with respect to the times t . This
i j
means that partial derivatives of F with respect to S must vanish. One can obtain
0 i
(see [14], [6]), that these derivatives are differences of chemical potential on disjoint cuts
and those are equal to integrals over dual B-cycles on the corresponding hyperelliptic
Riemann surface, that is, this set of conditions reads
d
(p) dωy(ω) = 0. (2.25)
dV
IBi
Note that, as we shall see below, from the technical viewpoint, it does not matter which
cycle set, A- or B-cycles, we take as the basic set. That is, the higher genus corrections
become scheme dependent: choosing determining conditions (2.24) or (2.25), we obtain
different expressions for the free energy in the the genus one approximation. These ex-
pressions are however of the same type; the difference will be exactly in the choice of the
corresponding basis cycles.)
5
3 The iterative procedure
3.1 Moments of the model
We can determine higher genus contributions iteratively by inverting genus expanded loop
equation (2.13). From W (p), all multi-loop correlators of the same genus can be obtained
g
then simply by applying the loop insertion operator d (p) to it (see Eqs. (2.5) and (2.4)).
dV
Asintheone-cutsolution[1],changingvariablesfromcouplingconstantstomomentsM(k)
α
allows expressing higher genus correlators nonperturbatively in the coupling constants t ,
j
and these correlators turn out to depend only on a finite number of the moments
dω
M(k) ≡ V′(ω)φ(k)(ω), k = 1,2,..., α = 1,...,2n, (3.1)
α 2πi α
ICD
with
1 1
φ(k)(ω) ≡ φ(0)(ω), φ(0)(ω) ≡ . (3.2)
α (ω −µ )k y˜(ω)
α
Note that expression (2.19) implies
M(1) = M(µ ). (3.3)
α α
3.2 Determination of the basis
We now introduce a basis for the operator acting on W (p) in Eq. (2.13),
g
1
(Kˆ −2W (p)) χ(k)(p) ≡ , k = 1,2,..., α = 1,...,2n. (3.4)
0 α (p−µ )k
α
Given the r.h.s. of Eq. (2.13) to be a fractional rational function of p having poles at the
µ only, W (p) must have the following structure:
α g
3g−1 2n
W (p) = A(k)χ(k)(p), g ≥ 1. (3.5)
g α,g α
k=1 α=1
X X
Here A(k) are complicated functions of µ and the moments M(k). As the order of the
α,g β β
highest pole in W (p) is insensitive to a multi-cut structure, W (p) will depend on at most
g g
2n(3g −1) moments, just like the one-cut solution case [1].
A set of basis functions fulfilling Eq. (3.4) is defined by
1 k−1
χ˜(k)(p) ≡ φ(k)(p)− M(k−r+1)χ˜(r)(p) , k = 1,2,..., α = 1,...,2n, (3.6)
α (1) α α α
M !
α r=1
X
which can be proved by induction. However, this definition is not unique, as the kernel
of (Kˆ −2W (p)) is not empty,
0
Ker(Kˆ −2W (p)) = Span{plφ(0)(p); l = 0,...,n} . (3.7)
0
6
This follows from boundary conditions (2.22). Now, because of the asymptotic behavior
(2.8), only terms ∼ O( 1 ) at large p may be added to W (p) for g ≥ 1. This requirement
p2 g
reduces the set of zero modes to be
allowed zero modes : {plφ(0)(p); l = 0,...,n−2} . (3.8)
In the general case, any linear combination of n − 1 functions can be added to W (p).
g
But, besides its asymptotic behavior, W (p) must fulfill Eq. (2.9), which can be used
g
to fix the basis uniquely. The p-dependence of W (p) must be completely absorbed into
g
derivatives with respect to ddV (p). This is possible only if the basis functions χ(αk)(p) can be
expressed completely in terms of ddµVα(p) and dMdVα(k)(p) as functions of p. So, χ˜(αk)(p) must be
redefined in Eq. (3.6) in accordance with this demand, which fixes completely their zero
mode components. The derivatives of branching points µ and the moments M(k) can
α α
be obtained by applying the loop insertion operator d (p) to definitions (3.1) and using
dV
restrictions (2.24) taking into account that
d ∂ 2n dµ ∂ 2n ∞ dM(k) ∂
(p) = (p)+ α(p) + α (p) , (3.9)
dV ∂V α=1 dV ∂µα α=1k=1 dV ∂Mα(k)
X X X
Using the identity
∂ 1
(p)V′(ω) = − , (3.10)
∂V (p−ω)2
the result for the moments reads
dM(k) dµ
α (p) = (k +1/2) M(k+1) α(p)−φ(k+1)(p)
dV α dV α !
1 2n k 1 dµ
+ φ(l)(p)−M(l) β(p)
2 (µ −µ )k−l+1 α α dV
Xβ=1 Xl=1 β α (cid:16) (cid:17)
β6=α
1 2n 1 dµ
(1) β (1)
+ M (p)−φ (p) ,
2 (µ −µ )k β dV β
Xβ=1 β α (cid:16) (cid:17)
β6=α
α = 1,...,2n , k = 1,2,.... (3.11)
The quantities dµα(p) are given by the solution of the following set of linear equations,
dV
2n dµ
0 = µkM(1) α(p)−pkφ(1)(p) +2k pk−1φ(0)(p), k = 0,...,n ,
α α dV α !
α=1
X
2n dµ
0 = M(1) α(p)−φ(1)(p) K , j = 1,...,n−1 ,
α dV α ! α,j
α=1
X
y˜(λ)
K ≡ dλ , (3.12)
α,j
λ−µ
IAj α
when applying d (p) to (2.22) and using (2.24). While the first relations can be attained
dV
easily, deriving the latter can be performed in a tricky way first proposed by Akemann.
7
Indeed,
d 1 2n 1 dµ d
α
(p)y(ω) = − M(ω)y˜(ω) (p)+y˜(ω) (p)M(ω) (3.13)
dV 2 ω−µ dV dV
α=1 α
X
Calculating the second term, we find from (2.19) that
d dξ V′(ξ) 1 2n 1 dµ 1
α
(p)M(ω) = (p) − ,
dV IC∞,p 2πi "(ξ −ω)y˜(ξ) 2 α=1 ξ −µα dV ! (p−ξ)2(ξ −ω)y˜(ξ)#
X
and the first term can be evaluated by the residue at infinity replacing V′(ξ) by M(ξ)y˜(ξ)
in this limit, while the second term is given by the residue at ξ = p (as the integration
contour must encircle the infinity and p). The first term then becomes
1 2n M(ω)−M(µ ) dµ
α α
(p), (3.14)
2 ω −µ dV
α=1 α
X
and it partially cancels the first term in (3.13), the difference is just the term proportional
to M(µ ) = M(1), while evaluating the second term upon substituting it into the integral
α α
in (2.24) needs partial integration:
∂ 1 y˜(ω) 1 1 2n 1 1 y˜(ω)
dω = dω −
∂p IAj p−ω y˜(p) 2 IAj p−ω α=1 ω −µα p−µα! y˜(p)
X
1 2n
= K φ(1)(p), (3.15)
2 α,j α
α=1
X
which together with (3.13) and (3.14) gives (3.12).
Linear system of equations (3.12) implies that the solution always takes the form
dµ n−2
M(1) α(p) = φ(1)(p)+ L plφ(0)(p), α = 1,...,2n. (3.16)
α dV α α,l
l=0
X
The quantities L , α = 1,...,2n, l = 0,...,n−2, are then determined by the following
α,l
set of equations:
2n n−2 2n k−1
0 = L µk pl − µk−l−1pl +2kpk−1, k = 0,...,n,
α,l α α
α=1 l=0 α=1l=0
X X X X
2n n−2
0 = L K pl, j = 1,...,n−1. (3.17)
α,l α,j
α=1 l=0
X X
We can redefine basis (3.6) in terms of total derivatives. Solving Eq. (3.11) for φ(k+1)(p)
α
andexpressing therethep-dependencethrough d (p)-termspluszeromodes, wecanobtain
dV
a unique basis inductively from Eq. (3.6) properly subtracting the corresponding zero
modes. We then have
1 k−1
χ(k)(p) ≡ φ(k)(p) − M(k−r+1)χ(r)(p) , α = 1,...,2n, n = 1,2,... .
α Mα(1) α (cid:12)ddV−part rX=1 α α !
(cid:12) (3.18)
(cid:12)
8
Few first basis functions are
dµ
χ(1)(p) = α(p), α = 1,...,2n,
α dV
2 d 1 2n d
χ(2)(p) = − (p)log|M(1)|− (p)log|µ −µ |. (3.19)
α 3dV α 3 dV α β
β=1
X
β6=α
4 Calculations in genus one
4.1 Genus one partition function
Having determined the basis, the loop equation can now be inverted step by step in genus.
For genus g = 1, Eq. (2.13) reads
d
(Kˆ −2W (p))W (p) = (p)W (p). (4.1)
0 1 0
dV
Given W (p) (2.20) and the loop insertion operator (3.9), the r.h.s. becomes
0
d 3 2n 1 1 2n 1
(p)W (p) = − −
dV 0 16 (p−µ )2 8 (p−µ )(p−µ )
α=1 α α,β=1 α β
X X
α<β
1 1 2n 1 dµ
+ M(1) α(p)
4φ(0)(p) p−µ α dV
α=1 α
X
1 2n 1 1 2n 1 1 1
= − −
16 α=1 (p−µα)2 8 α,β=1 µα −µβ p−µα p−µβ!
X X
α<β
1 2n n−2 L µl
+ α,l α. (4.2)
4 p−µ
α=1 l=0 α
X X
Here we took into account that regular parts coming from pl , l = 1,...,n−2, vanish
p−µα
due to Eq. (3.17) in order for W0(p,p) = ddV (p)W0(p) to satisfy the correct asymptotic
behavior, and we can just replace pl by µl in numerators of such expressions. The result
α
for the one-loop correlator of genus one with n cuts can now be easily obtained using Eq.
(3.19),
1 2n 1 1
W (p) = χ(2)(p)− χ(1)(p)−χ(1)(p)
1 16 α 8 µ −µ α β
αX=1 1≤αX<β≤2n α β (cid:16) (cid:17)
1 2n n−2
+ L µl χ(1)(p)
4 α,l α α
α=1 l=0
X X
1 2n 1 1
= χ(2)(p)− χ(1)(p)−χ(1)(p)
16 α 8 µ −µ α β
αX=1 1≤αX<β≤2n α β (cid:16) (cid:17)
1 2n n−2
+ L µl χ(1)(p)
4 α,l α α
α=1 l=0
X X
9
1 2n 2 d 1 2n d
= − (p)log|M(1)|− (p)log|µ −µ |
16 3dV α 3 dV α β
α=1 β=1
X X
β6=α
1 2n 1 dµ dµ
α β
− (p)− (p)
8 α,β=1 µα −µβ dV dV !
X
α<β
1 2n dµ
α
+ L (µ ) (p), (4.3)
α α
4 dV
α=1
X
where the polynomials L (p) are, by definition, n−2L pl.
α l=0 α,l
We now integrate (4.3) in order to obtain F . While integrating first two terms is easy,
1 P
the term with zero modes is difficult. Even before presenting the result, we need more
notation. Let us introduce the quantities
λi−1
Q ≡ dλ, i = 1,2,... (4.4)
j,i
y˜(λ)
IAj
(part of them constitute the A-matrix [8], [10]) and the (polynomial) basis of holomorphic
differentials on the hyperelliptic Riemann surface
n−1
H (λ) ≡ H λl−1, k = 1,...,n−1, (4.5)
k l,k
l=1
X
c c
such that
H (λ)
k
dλ = δ , (4.6)
k,j
y˜(λ)
IAj c
whence
n−1
Q H = δ for j,k = 1,...,n−1. (4.7)
j,l l,k j,k
l=1
X
The following lemma, to bce proven in the next subsection, establishes the relation
between zero modes L and Q .
α,l j,i
Lemma 1 For quantities L determined by system (3.17) and the quantities (4.4) and
α,l
(4.5), we have the relation
n−2 n−1 H (λ)
L µl = − j dλ, α = 1,...,2n. (4.8)
α,l α y˜(λ)(λ−µ )
Xl=0 jX=1IAj c α
It follows from Lemma 1 that the last term in (4.3) is just
1 2n ∂ dµ
α
− log det Q (p),
j,i
2 α=1 ∂µα (cid:18) i,j=1,...,n−1 (cid:19) dV
X
i.e.,
1 2n
F = − log M(1) ·∆4 ·( det Q )12 , (4.9)
1 24 α i,j=1,...,n−1 j,i !
α=1
Y
where ∆ = (µ −µ ) is the Vandermonde determinant. This is our final answer
1≤α<β≤2n α β
for the genus one partition function.
Q
10
