DV and WDVV

15 Pages·0.25 MB·

by  L. Chekhov

FIAN/TD-02/03
ITEP/TH-04/03
DV and WDVV
L. Chekhov
§
Steklov Mathematics Institute, Moscow, Russia
A. Marshakov A. Mironov
¶ k
Theory Department, Lebedev Physics Institute and ITEP, Moscow, Russia
3
0
0
2 D.Vasiliev
∗∗

n
MIPT and ITEP, Moscow, Russia
a
J
We prove that the quasiclassical tau-function of the multi-support solutions to matrix models,
2
proposed recently by Dijkgraaf and Vafa to be related to the Cachazo-Intrilligator-Vafa superpo-
1
tentials of the =1 supersymmetric Yang-Mills theories, satisfies the Witten-Dijkgraaf-Verlinde-
N
1 Verlinde equations.
v
1
7
0
1
0
3
0 1 Introduction
/
h
t
- The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations [1] in the most general form can be writ-
p
ten [2] as system of algebraic relations
e
h
v: FIFJ−1FK = FKFJ−1FI, ∀ I,J,K (1)
i
X
for the third derivatives
ar ∂3
= F (2)
I JK IJK
kF k ∂T ∂T ∂T ≡ F
I J K
of some function (T). Have been appeared first in the context of topological string theories [1],
F
they were rediscovered later on in much larger class of physical theories where the exact answer for a
multidimensional theory could be expressed through a single holomorphic function of several complex
variables [2, 3, 4, 5, 6, 7, 8].
Recently, anewexampleof similar relations between thesuperpotentialsof = 1supersymmetric
N
gauge theories in four dimensions and free energies of matrix models in the planar limit was proposed
[9, 10]. It has been realized that superpotentials in some = 1 four-dimensional Yang-Mills theories
N
can be expressed through a single holomorphic function [9] that can be further identified with free
energy of the multi-support solutions to matrix models in the planar limit [10]. A natural question
which immediately arises in this context is whether these functions – the quasiclassical tau-functions,
determined by multi-support solutions to matrix models, satisfy the WDVV equations? In the case of
positive answer this is rather important, since multi-support solutions to the matrix models can play
§E-mail: [email protected]
¶E-mail: [email protected]
kE-mail: [email protected];[email protected]
∗∗E-mail: [email protected]
1
a role of “bridge” between topological string theories and Seiberg-Witten theories [11] which give rise
to two different classes of solutions to the WDVV equations (see, e.g., [12] and [13, 14]).
This question was already addressed in [15], where it was shown that the multicut solution to
one-matrix model satisfies the WDVV equations. However, this was verified only perturbatively and,
what is even more important, for a particular non-canonical! (and rather strange) choice of variables.
In this paper, we demonstrate that the quasiclassical tau-function of the multi-support solution
satisfies the WDVV equations as a function of canonical variables identified with the periods and
residues of the generating meromorphic one-form dS [16]. An exact proof of this statement consists
of two steps. The first, most difficult step is to find the residue formula for the third derivatives (2)
of the matrix model free energy. Then, using an associative algebra, we immediately prove that free
energy ofmulti-supportsolutionsatisfies WDVVequations, uponthenumberofindependentvariables
is fixed to be equal to the number of critical points in the residue formula.
Insect.2,wedefinethefreeenergyofthemulti-supportmatrix modelintermsofthequasiclassical
tau-function [16] along the line of [17, 18, 19]. In sect. 3, we derive the residue formula for the
third derivatives of the quasiclassical tau-function for the variables associated both with the periods
S = S and residues t = t of the generating differential dS. In sect. 4, we prove that the
i i
{ } { } { } { }
free energy of the multisupport solution (T) solves the WDVV equations (1) as a function of the
F
full set of variables T = S,t whose total number should be fixed to be equal to the number of
{ } { }
critical points in the residue formula for the third derivatives (2). In sect. 5, we verify this statement
explicitly for the first nontrivial case where the total number of variables is equal to four. 1 Finally
we present several concluding remarks and discuss possible generalizations.
We restrict ourselves by the = 1 supersymmetric theories without flavours originally considered
N
in [9]. The results can be easily generalized. Note that the literature on the subject is already quite
extensive [20, 21], and different interesting developments of the issues discussed in this paper can be
immediately obtained.
2 Tau-function of multi-support matrix model
We first exactly define what we call below the multi-support free energy of the matrix model in the
planar limit. We are mostly doing with one-matrix integrals 2 of the form
Z = dΦ eh¯1TrW(Φ) (3)
Z
where the potential W(Φ) is supposed to be a polynomial of a degree (n+1). The free energy in the
planar limit of (3) can be defined as the first term in the expansion
F(t,t0) = lim(logZ(t)) = ∞ ¯h2g−2Fg(t,t0) (4)
g=0
X
implying N , h¯ 0 with N¯h= t being fixed. In what follows we are only interested in the first
0
→ ∞ →
term of this expansion, F (t,t ). In fact, we deal with another quantity, (t,t ,S), where S = h¯N ,
0 0 0 i i
F
S = t are extra variables – the filling numbers of (metastable) vacua. In order to get from this
i 0
quantity F (t,t ), one needs to minimize the free energy with respect to the filling numbers, ∂ = 0.
P 0 0 ∂SFi
However, one can still preserve S as free parameters introducing the ”chemical potentials”.
i
The origin of the new variables S becomes rather transparent after one says that instead of direct
i
computation of (3)thisproblemis replaced by thesaddlepointapproximation –findingtheextremum
of the functional F [ρ(λ)] Wρ ρ(λ )log λ λ ρ(λ ) +Π ( ρ t ) where Π is just a
0 1 1 2 2 0 0 0
∝ − | − | −
1Let us point out that the WDVRV equatiRonRs (1) are nontrivial only for the functioRns of at least three independent
variables. However, as we see below, the structure of residue formula for the matrix model free energy requires the
minimal number of independent variables to be at least four! From this point of view, the origin of the “experimental
observation” of [15] valid for a function of threevariables still remains unclear to us.
2Thegeneralization to thetwo-matrix case [19] is rather straightforward and will bediscussed in thelast section.
2
λ
A
B
Figure 1: Cuts in the λ- or “eigenvalue” plane for the planar limit of 1-matrix model. The eigenvalues
are supposed to be located “on” the cuts. The distribution of eigenvalues is governed by the period
integrals S = ρ(λ)dλ along the corresponding cycles and the dependence of partition function on
A
“distributions” S is given by the quasiclassical tau-function ∂ = ρ(λ)dλ.
H i ∂FS B
H
Lagrange multiplier to fix the total normalization of the eigenvalue density. This latter condition
means the saddle point equation is non-trivial only on the support of ρ. For one matrix model, this
support can be presented as a set D of cuts in complex eigenvalue plane, see fig. 1. Then, one
i
{ }
should add to this functional the term Π ρ S , which via Lagrange multipliers, controls the
i Di − i
filling numbers at each cut, i.e. to consider (cid:16) (cid:17)
P R
(t,t ,S) Wρ ρ(λ )log λ λ ρ(λ ) +Π ρ t + ′Π ρ S (5)
0 1 1 2 2 0 0 i i
F ∝ Z −Z Z | − | (cid:18)Z − (cid:19) (cid:18)ZDi − (cid:19)
X
Then, the extra variables appear due to the extra information hidden in (5) compare to (3) – the
structure of nontrivial eigenvalue supports. It is well-known that at “critical” densities δ = 0, (5) is
δFρ
a (logarithm) of quasiclassical tau-function [16] (see, e.g., [18, 19]).
In principle, in order to compare with the matrix model quantity F , one needs to put further
0
F
restrictions to get rid of the metastable vacua. This would lead to shrinking part of the cuts into
the double points (see discussions of these issues, say, in [18]). Here we would forget this issue and
consider smooth curves (6) with only two marked points at infinities on two λ-sheets of the curve (6).
Note also that in (5) one can make two different natural choices for the set of new independent
variables: the first choice corresponds to independent filling of all cuts, then t = S , while the
0 i
second choice corresponds to choosing as independent t and all S except of corresponding to one of
0 i P
the cuts (that is why the correspondingsum in (5) is denoted as ). These choices are related by the
′
linear change of variables which does not influence the WDVV equations (see [2, 30]). The first choice
P
is more “symmetric” while the second one correspondsto the canonical choice of variables in the sense
of [16] or to the homology basis on smooth curve (6) with added marked points at two infinities. We
use both of them below depending on convenience.
The complex curve of one-matrix model “comes from” the loop equations (see, for example, [22])
and can be written in the form
y2 = W (λ)2+f(λ) R(λ) (6)
′
≡
with the matrix model potential (3) parameterized as
W(λ)= t λl+1
l+1 (7)
l 0
X≥
or
n
W′(λ) = (l+1)tl+1λl (8)
l=0
X
3
being the polynomial of n-th degree in our conventions. The coefficients of the function
n 1
f(λ)= − f λk (9)
k
k=0
X
are related to the extra data (the filling numbers) of the multicut solution. The eigenvalue density
ρ(λ)istheimaginarypartofy(λ)andvanishesoutsidethecuts. Therefore,theeigenvalue distribution
S can be fixed by the periods
Si = dS (10)
IAi
of the generating differential
dS = ydλ (11)
takenaroundtheeigenvaluesupportstobeidentified(exceptforoneofthesupports)withthecanonical
A-cycles. Then
∂dS
= dω
i
∂S
i
(12)
dω = δ
j ij
IAi
when the derivatives are taken at fixed coefficients t of the potential (8). One can show that the
l
{ }
Lagrangian multipliers in (5) are given by integrals of the same generating differential (11) over the
dual contours (see fig. 1)
Πi = dS (13)
IBi
To the set of parameters (10) one should also add 3 the total number of eigenvalues N¯h = t
0
f
n 1
res (dS) = − t0 (14)
∞ 2(n+1)tn+1 ≡
and the parameters of the potential (7), (8), which can be equivalently written as
1
t = res λ kdS
k −
k ∞ (15)
(cid:16) (cid:17)
k = 1,...,n
while the leading term coefficient t is supposed to be fixed (we will discuss this issue in detail
n+1
below). Then
∂dS λn−1dλ 1 n−2∂fk λkdλ
dΩ = = (n+1)t + (16)
0 n+1
∂t y 2 ∂t y
0 0
k=0
X
and the dependence of f with k = 0,1,...,n 2 on t is fixed by the condition
k 0
{ } −
λn−1dλ n−2∂fk λkdλ
(n+1)t + = 0 (17)
n+1
IAi y k=0 ∂t0 y !
X
which for i = 1,...,n 1 gives exactly n 1 relations on the derivatives of f ,f ,...,f w.r.t. t .
0 1 n 2 0
− − −
The bipole differential (16) can be also presented as
E(P, )
dΩ = dlog ∞ (18)
0
E(P, )
(cid:18) ∞− (cid:19)
3By the ∞-point in what follows we call for short the point ∞+ or λ = ∞ on the “upper” sheet of hyperelliptic
Riemann surface (6) corresponding to thepositive sign of thesquare root, i.e. to y=+ W′(λ)2+f(λ).
p
4
where E(P,P ) is the Prime form. Differential (18) obviously obey the properties
′
res dΩ = res dΩ = 1
∞ 0 − ∞− 0
(19)
dΩ = 0, i= 1,...,n 1
0
−
IAi
For the derivatives w.r.t. parameters of the potential (15), one gets
∂dS W′(λ)kλk−1dλ 1 n−2∂fj λkdλ
dΩ = = + (20)
k
∂t y 2 ∂t y
k j=0 k
X
obeying
W′(λ)kλk−1dλ 1 n−2∂fj λkdλ
dΩ = + = 0 (21)
k
y 2 ∂t y
IAi IAi j=0 k IAi
X
and this is again a system of linear equations on ∂fj. To complete the setup one should also add to
∂tk
(13) the following formulas 4:
+
Π0 = ∞ dS (22)
Z∞−
(we again remind that, instead of t , the parameter S = t n 1S can be used equivalently) and
0 n 0− i=−1 i
P
v = res λkdS , k > 0 (23)
k
∞
(cid:16) (cid:17)
Ongenusg = n 1smoothRiemannsurface(6),thereare2g = 2n 2independentnoncontractable
− −
contours which can be split into the so-called A A and B B , i = 1,...,g, cycles with the
i i
≡ { } ≡ { }
intersection form A B = δ . The canonical holomorphic differentials (12) are normalized to the
i j ij
◦
A-cycles, and their integrals along the B-cycles give the period matrix,
dωi = Tij (24)
IBj
To check integrability of (13) and (22) one needs to verify the symmetricity of the second derivatives.
For the part related with the derivatives only w.r.t. the variables (10), this is just a symmetricity of
theperiodmatrix of (6)andfollows fromtheRiemannbilinear relations for thecanonical holomorphic
differentials (12)
0 = dωi dωj = dωi dωj dωj dωi =Tij Tji (25)
∧ − −
ZΣ k (cid:18)IAk IBk IAk IBk (cid:19)
X
Analogously
0 = dω dΩ = dω dΩ dΩ dω +
i 0 i 0 0 i
ZΣ ∧ k (cid:18)IAk IBk −IAk IBk (cid:19)
X (26)
+ + +
+res (dω ) ∞ dΩ res (dΩ ) ∞ dω = dΩ ∞ dω
i 0 0 i 0 i
∞ Z∞− − ∞ Z∞− IBi −Z∞−
Formula (25) means that
∂Π ∂Π
i j
= T = T = (27)
ji ij
∂S ∂S
j i
4 Naively understood the integral in (22) is divergent and should be supplemented by some proper regularization.
In what follows we ignore this subtlety since it does not influence the residue formulas for the third derivatives, those
one really needs for the WDVV equations (1). The simplest way to avoid these complications is to think of the pair of
markedpoints∞and∞ asofdegeneratehandle;thentheresidue(14)comesfromdegenerationoftheextraA-period,
−
while theintegral (22) from degeneration of the extraB-period.
5
−1
A
l
B v−
l j
Σ
A v+
l j
Figure 2: Cut Riemann surface with boundary ∂Σ. The integral over the boundary can be divided
into several pieces (see formula(32)). Intheprocess of computation weusethefactthat theboundary
values of Abelian integrals v on two copies of the cut differ by period integral of the corresponding
j±
differential dω over the dual cycle.
j
while from (26) one gets
∂Π ∂Π
j 0
= (28)
∂t ∂S
0 j
This allows one to introduce the function (T) = (S,t ,t) such that
0
F F
∂ ∂ ∂
F = Π , F = Π , F = v (29)
j 0 k
∂S ∂t ∂t
j 0 k
The integrability of the last relation can be checked similar to (27), (28) with the help of Riemann
P
bilinear relations involving the Abelian integrals Ω = dΩ , for example (cf. e.g. with [23], where
k k
similar relations were used for the quasiclassical tau-function of the Seiberg-Witten theory):
R
res (Ω dω ) = Ω dω = Ω+dω Ω dω Ω+dω Ω dω =
∞ k i I∂Σ k i l (cid:18)ZAl k i−ZAl −k i(cid:19)− l (cid:18)ZBl k i−ZBl −k i(cid:19)
X X
= dΩ dω dΩ dω = dΩ
k i k i k
−
l (cid:18)IBl IAl IAl IBl (cid:19) IBi
X
(30)
where ∂Σ is the cut Riemann surface (6) (see fig. 2), and in the last equality we used (21).
3 Residue formula
3.1 Holomorphic differentials
Letusnowderivetheformulasforthethirdderivativesof ,followingthewayproposedbyI.Krichever
F
[16, 25]. We first note that the derivatives of the elements of period matrix (in this section, for
simplicity, we set the coefficients of potential (8) to be fixed) can be expressed through the integral
over the “boundary” ∂Σ of cut Riemann surface Σ (see fig. 2)
∂T
ij
∂kTij = ∂kdωi = ωj∂kdωi (31)
∂Sk ≡ ZBj Z∂Σ
6
P
where ω = dω are the Abelian integrals, whose values on two copies of cycles on the cut Riemann
j j
surface (see also fig. 2) are denoted below as ω . Indeed, the computation of the r.h.s. of (31) gives
R j±
ω ∂ dω = ω+∂ dω ω ∂ dω ω+∂ dω ω ∂ dω =
Z∂Σ j k i l (cid:18)ZBl j k i−ZBl j− k i(cid:19)− l (cid:18)ZAl j k i−ZAl j− k i(cid:19)
X X
= dω ∂ dω dω ∂ dω =
j k i j k i (32)
−
l IBl(cid:18)IAl (cid:19) l IAl(cid:18)IBl (cid:19)
X X
= dω ∂ dω dω ∂ dω = ∂ T
j k i j k i k ij
−
Xl (cid:18)IAl (cid:19)IBl Xl (cid:18)IBl (cid:19)IAl Aidωj=δij
H
One can now rewrite (31) as
∂ T = ω ∂ dω = ∂ ω dω = res (∂ ω dω ) (33)
k ij j k i k j i dλ=0 k j i
−
Z∂Σ Z∂Σ
X
where the sum is taken over all residues of the integrand, i.e. over all residues of ∂ ω since dω are
k j i
holomorphic. In order to investigate these singularities and understand the last equality in (33), we
discuss first how to take derivatives ∂ w.r.t. moduli, or introduce the corresponding connection.
k
To this end, we introduce a covariantly constant function – the hyperelliptic co-ordinate λ, i.e.
such a connection that ∂ λ = 0. Roughly speaking, the role of covariantly constant function can be
k
played by one of two co-ordinates – in the simplest possible description of complex curve by a single
equation on two complex variables. Then, using this equation, one may express the other co-ordinate
as a function of λ and moduli. Any Abelian integral ω can be then expressed in terms of λ, and
j
in the vicinity of critical points λ where dλ = 0 (for a general (non-singular) curve this is always
α
{ }
true) we get an expansion
ωj(λ) = ωjα+cjα λ λα+... (34)
λ λα −
→
p
whose derivative
c
jα
∂kωj ≡ ∂kωj|λ=const = −2√λ λ ∂kλα+regular (35)
α
−
gives first order poles at λ = λ up to regular terms which do not contribute to (33). The exact
α
coefficient in (35) can be computed for λ related with the generating differential dS = ydλ. Then,
using
y(λ) = Γα λ λα+... (36)
λ λα −
→
p
where Γ = (λ λ ) or
α β6=α α− β
q
Q
∂ Γ ∂λ
α α
y(λ) = +regular (37)
∂t −2√λ λ ∂t
k α k
−
together with
Γ
α
dy = dλ+regular (38)
2√λ λ
α
−
and
c
jα
dωj = dλ+... (39)
2√λ λ
α
−
and, following from (11) and (12)
Γ ∂ λ
α k α
dω = ∂ dS = dλ+regular (40)
k k
−2√λ λ
α
−
one finally gets for (33)
c ∂ λ dω dω dω dω
jα k α j i j k
res(∂kωjdωi)= res dωi = res dωi∂kλα = res (41)
2√λ λ dλ dλdy
X Xα (cid:18) − α (cid:19) Xα (cid:18) (cid:19) Xα (cid:18) (cid:19)
7
In hyperelliptic situation, the derivation presented above is equivalent to using the Fay formula [24]
∂T
ij
= ωˆ (λ )ωˆ (λ ) (42)
i α j α
∂λ
α
where ωˆ (λ )= dωi(λ) is the ”value” of canonical differential at a critical point.
i α d√λ−λα λ=λα
(cid:12)
(cid:12)
3.2 Meromorphic (cid:12)differentials
Almost in the same way the residue formula can be derived for the meromorphic differentials (20).
One gets
∂
F = res λkdS , k > 0 (43)
∂tk ∞
(cid:16) (cid:17)
therefore
∂2
F = res λkdΩ = res ((Ω ) dΩ ) (44)
n k + n
∂tk∂tn ∞ ∞
(cid:16) (cid:17)
where (Ω ) is the singular part of the integral of 1-form dΩ . Further
k + k
∂ ∂dΩ ∂Ω ∂Ω
res λkdΩ = res λk n = res (dΩ ) n = res dΩ n (45)
n k + k
∂tm ∞(cid:16) (cid:17) ∞(cid:18) ∂tm (cid:19) − ∞(cid:18) ∂tm(cid:19) − ∞(cid:18) ∂tm(cid:19)
The last expression can be rewritten as
∂Ω ∂Ω ∂Ω ∂Ω
n n n n
res dΩ = dΩ + res dΩ = res dΩ (46)
− ∞(cid:18) k∂tm(cid:19) I∂Σ(cid:18) k∂tm(cid:19) λα(cid:18) k∂tm(cid:19) λα(cid:18) k∂tm(cid:19)
X X
since dΩ ∂Ωn = 0 due to dΩ = 0, (cf. with (32)):
∂Σ k∂tm Ai n
(cid:16) (cid:17)
H H
∂ ∂ ∂ ∂ ∂
Ω dΩ = Ω+ dΩ Ω dΩ Ω+ dΩ Ω dΩ =
j∂t i j ∂t i− −j ∂t i − j ∂t i− −j ∂t i
Z∂Σ k l (cid:18)ZBl k ZBl k (cid:19) l (cid:18)ZAl k ZAl k (cid:19)
X X
∂ ∂
= dΩ dΩ dΩ dΩ =
j i j i
∂t − ∂t
l IBl(cid:18)IAl (cid:19) k l IAl(cid:18)IBl (cid:19) k
X X
∂ ∂
= dΩ dΩ dΩ dΩ = 0
j i j i
Xl (cid:18)IAl (cid:19)IBl ∂tk −Xl (cid:18)IBl (cid:19)IAl ∂tk AidΩj=0
(47)
H
Now, as in the holomorphic case one takes
Ωn(λ) = Ωnα+γnα λ λα+... (48)
λ λα −
→
p
and, therefore
∂ ∂ γ ∂λ
jα α
Ω Ω = +regular (49)
j j
∂t ≡ ∂t −2√λ λ ∂t
k k (cid:12)λ=const − α k
(cid:12)
Then, using (36), (37) and (38) together(cid:12)with
(cid:12)
γ
jα
dΩj = dλ+... (50)
2√λ λ
α
−
and the relation, following from (20), (37)
∂ Γ dλ ∂λ
α α
dΩ = dS = +regular (51)
k
∂t −2√λ λ ∂t
k α k
−
8
one gets for (46)
∂3 ∂Ω γ ∂λ
n nα α
F = res dΩ = res dΩ =
∂t ∂t ∂t λα k∂t − λα k2√λ λ ∂t
k n m X (cid:18) m(cid:19) X (cid:18) − α m(cid:19) (52)
dΩ ∂λ dΩ dΩ dΩ
n α k n m
= res dΩ = res
− λα k dλ ∂t λα dλdy
(cid:18) m(cid:19) (cid:18) (cid:19)
X X
In a similar way, one proves the residue formula for the mixed derivatives, so that we finally conclude
∂3 dH dH dH φ φ φ
I J K I J K
F = res = res dy =
∂T ∂T ∂T λα dλdy λα dλ/dy
I J K Xλα (cid:18) (cid:19) Xλα (cid:18) (cid:19) (53)
Hˆ (λ )Hˆ (λ )Hˆ (λ )
= Γ2φ (λ )φ (λ )φ (λ ) = I α J α K α
α I α J α K α (λ λ )2
Xλα Xλα β6=α α− β
Q
for thewhole set T = t ,t ,S and dH = dΩ ,dΩ ,dω . In formula (53) we have introduced
I k 0 i I k 0 i
{ } { } { } { }
the meromorphic functions
dH Hˆ (λ)
φ (λ) = I = I (54)
I
dy R(λ)
′
for any (meromorphic or holomorphic) differential dH = Hˆ (λ)dλ and R(λ) = W (λ)2 +f(λ). The
I I y ′
derivation of theresidueformula for theset of parameters includingt correspondingto thethird-kind
0
Abelian differential (16) can be performed in a similar way.
4 Proof of WDVV
Havingtheresidueformula(53), theproofoftheWDVVequations(1)isreducedtosolvingthesystem
of linear equations [7, 26, 14], which requires only fulfilling the two conditions:
The “matching” condition
•
#(I) = #(α) (55)
and
nondegeneracy of the matrix built from (54):
•
det φ (λ ) = 0 (56)
I α
Iα k k 6
Under these conditions, the structure constants CK of the associative algebra
IJ
K L K L
(C ) (C ) = (C ) (C )
I L J M J L I M
(57)
(C )K CK
I J IJ
≡
responsible for the WDVV equations can be found from the system of linear equations
φ (λ )φ (λ )= CKφ (λ ), λ
I α J α IJ K α α (58)
∀
K
X
with the solution
CIKJ = φI(λα)φJ(λα)(φK(λα))−1 (59)
α
X
To make it as general, as in [2, 4, 3], one may consider an associative isomorphic algebra (again λ )
α
∀
φ (λ )φ (λ ) = CK(ξ)φ (λ ) ξ(λ )
I α J α IJ K α α (60)
·
K
X
9
which instead of (59) leads to
φ (λ )φ (λ )
CIKJ(ξ) = I ξα(λ J) α (φK(λα))−1 (61)
α α
X
The rest of the proof is consistency of this algebra with relation
= CL (ξ)η (ξ)
IJK IJ KL (62)
F
L
X
with
η (ξ) = ξ
KL A KLA (63)
F
A
X
expressing structure constants in terms of the third derivatives and, thus, leading to (1). It is easy to
see that (62) is satisfied if are given by residue formula (53).
KLA
F
Indeed, requiring only matching #(α) = #(I), one gets
φ (λ )φ (λ )
CIKJ(ξ)ηKL(ξ) = I ξα(λJ) α ·(φK(λα))−1·φK(λβ)φL(λβ)ξ(λβ)Γβ (64)
α
K K,α,β
X X
and finally
φ (λ )φ (λ )
CIKJ(ξ)ηKL(ξ) = I ξα(λJ) α φL(λα)ξ(λα)Γα = ΓαφI(λα)φJ(λα)φL(λα) = FIJL (65)
K α α α
X X X
Hence, for the proof of (1) one has to adjust the number of parameters T according to (55). The
I
{ }
number of critical points #(α) = 2n since dλ = 0 for y2 = R(λ) = 0. Thus, one have to take a
codimension one subspace in the space of all parameters T , a natural choice will beto fix the eldest
I
{ }
coefficient of (8). Then the total number of parameters #(I), including the periods S, residue t and
0
the rest of the coefficients of the potential will be g+1+n= (n 1)+1+n = 2n, i.e. exactly equal
−
to #(α). In sect. 5 we present an explicit check of the WDVV equations for this choice, using the
expansion of free energy computed in [9, 31].
Notethatequations(60)arebasicallyequivalenttothealgebraofforms(ordifferentials)considered
in [3, 4]. In this particular case one may take the basis of 1-differentials dω , dΩ with multiplication
i k
given by usual (not wedge!) multiplication modulo dS = ydλ. Then, one can either directly check
that the algebra with this multiplication is associative (similar to how it was done in [4, 6]), or, using
hyperelliptic parameterization, remove the factor dλ in order to reduce the algebra to the ring of
y
polynomials with multiplication modulo the polynomial ideal y2 = W 2(λ)+f(λ), which is obviously
′
associative.
In the proof of the WDVV equations we used in this section, the associativity (57) is absolutely
evident, being associativity of the usual multiplication, and the main point to check was to derive
(62)-(63). When using instead the algebra of differentials, the main non-trivial point is to check its
associativity, while the relations (62)-(63) appear as even more transparent than above corollary of
the residue formula.
5 Explicit check of the WDVV equations
To convince ourselves that the general proof indeed works and to get some further insights, in this
section we consider the explicit check of the WDVV equations (1) perturbatively. To do this, let us
take the perturbative expansion of the prepotential (29) at small S ’s (here we take the symmetric
i
choice of variables with i= 1,...,n = g+1), keeping the coefficients of the matrix model potential t’s
arbitrary, and see if this perturbative expansion of satisfies the WDVV equations order by order.
F
A general procedure of getting such perturbative expansion was proposed in [9].
10