arXiv:1606.03720v1 [math-ph] 12 Jun 2016
On cubic Hodge integrals and random matrices
Boris Dubrovin, Di Yang
SISSA, via Bonomea 265, Trieste 34136, Italy Abstract
A conjectural relationship between the GUE partition function with even couplings and certain special cubic Hodge integrals over the moduli spaces of stable algebraic curves is under considera-tion.
Contents
1 Introduction 1
1.1 Cubic Hodge partition function . . . 1
1.2 GUE partition function with even couplings . . . 3
1.3 From cubic Hodge integrals to random matrices. Main Conjecture. . . 4
1.4 Computational aspects of the Main Conjecture: how do we verify it? . . . 4
1.5 An explicit expression for Fg . . . 7
2 GUE free energy with even valencies 8 2.1 Calculating the GUE free energy from Frobenius manifold of P1 topological σ-model . 8 2.2 Proof of Prop.1.4.1, Thm.1.4.3 . . . 10
3 Verification of the Main Conjecture for low genera 10 3.1 Genus 0 . . . 10
3.2 Genus 1, 2 . . . 13
3.3 Genus 3, 4 . . . 13
A Explicit formula for F5 15
1
Introduction
1.1 Cubic Hodge partition function
Let Mg,kdenote the Deligne–Mumford moduli space of stable curves of genus g with k distinct marked
Let ψi:= c1(Li), i = 1, . . . , k, and let λi:= ci(Eg,k), i = 0, . . . , g. Recall that the Hodge integrals over
Mg,k, aka the intersection numbers of ψ- and λ-classes, are integrals of the form
Z Mg,k ψi1 1 · · · ψ ik k · λ j1 1 · · · λ jg g , i1, . . . , ik, j1, . . . , jg ≥ 0.
Note that the dimension-degree matching implies that the above integrals vanish unless 3g − 3 + k = (i1+ i2+ · · · + ik) + (j1+ 2 j2+ 3 j3+ · · · + g jg).
The particular case of cubic Hodge integrals of the form Z Mg,k Λg(p) Λg(q) Λg(r) ψ1i1· · · ψkik, 1 p + 1 q + 1 r = 0 (1.1.1)
was intensively studied after the formulation of the celebrated R. Gopakumar–M. Mari˜no–C. Vafa con-jecture [17,24] regarding the Chern–Simons/string duality. Here we denote
Λg(z) = g
X
i=0
λizi
the Chern polynomial of Eg,k. A remarkable expression for the cubic Hodge integrals of the form
Z
Mg,k
Λg(p)Λg(q)Λg(r)
(1 − x1ψ1) . . . (1 − xkψk)
, k ≥ 0
conjectured in [24] was proven in [18,25]; for more about cubic Hodge integrals see in the subsequent papers [19,20,28,12].
In the present paper we will deal with the specific case of Hodge integrals (1.1.1) with a pair of equal parameters among p, q, r; without loss of generality one can assume that p = q = −1, r = 1/2. So, the special cubic Hodge integrals of the form
Z Mg,k Λg(−1) Λg(−1) Λg 1 2 ψi1 1 · · · ψ ik k (1.1.2)
will be considered. Denote H(t; ǫ) =X g≥0 ǫ2g−2X k≥0 1 k! X i1,...,ik≥0 ti1· · · tik Z Mg,k Λg(−1) Λg(−1) Λg 1 2 ψi1 1 · · · ψ ik k (1.1.3)
the generating function of these integrals. Here and below t = (t0, t1, . . . ) are independent variables,
ǫ is a parameter. The exponential eH =: ZE is called the cubic Hodge partition function while H(t; ǫ)
is the cubic Hodge free energy. It can be written in the form of genus expansion H(t; ǫ) =X
g≥0
ǫ2g−2Hg(t) (1.1.4)
where Hg(t) is called the genus g part of the cubic Hodge free energy, g ≥ 0. Clearly H0(t) coincides
with the Witten–Kontsevich generating function of genus zero intersection numbers of ψ-classes H0(t) = X k≥0 1 k! X i1,...,ik≥0 ti1· · · tik Z M0,k ψi1 1 · · · ψ ik k = X k≥3 1 k(k − 1)(k − 2) X i1+···+ik=k−3 ti1 i1!· · · tik ik! . (1.1.5) We note that an efficient algorithm for computing Hg(t), g ≥ 1 was recently proposed in [12].
1.2 GUE partition function with even couplings
Let H(N) denote the space of N × N Hermitean matrices. Denote dM = N Y i=1 dMii Y i<j dReMijdImMij
the standard unitary invariant volume element on H(N). The most studied Hermitean random matrix model is governed by the following GUE partition function with even couplings
ZN(s) = (2π)−N V ol(N ) Z H(N ) e−N tr V (M ; s)dM. (1.2.1) Here, V (M ; s) is an even polynomial of M
V (M ; s) = 1 2M
2−X
j≥1
sjM2j, (1.2.2)
or, more generally, a power series, by s = (s1, s2, s3, . . . ) we denote the collection of coefficients1 of
V (M ), and by V ol(N ) the volume of the quotient of the unitary group over the maximal torus [U (1)]N V ol(N ) = V olU (N )/ [U (1)]N= π N(N−1) 2 G(N + 1), G(N + 1) = N−1 Y n=1 n!. (1.2.3)
The integral will be considered as a formal saddle point expansion with respect to the small parameter ǫ. Introduce the ’t Hooft coupling parameter x by
x := N ǫ.
Reexpanding the free energy FN(s) := log ZN(s) in powers of ǫ and replacing the Barnes G-function
by its asymptotic expansion yields2
F(x, s; ǫ) := FN(s)|N=x ǫ − 1 12 log ǫ = X g≥0 ǫ2g−2Fg(x, s). (1.2.4)
The GUE free energy F(x, s; ǫ) can be represented [21,22,1,23] in the form F(x, s; ǫ) = x 2 2ǫ2 log x −3 2 − 1 12log x + ζ ′(−1) +X g≥2 ǫ2g−2 B2g 4g(g − 1)x2g−2 +X g≥0 ǫ2g−2X k≥0 X i1,...,ik≥1 ag(i1, . . . , ik) si1. . . sikx 2−2g−(k−|i|), (1.2.5) ag(i1, . . . , ik) = X Γ 1 # Sym Γ (1.2.6)
where the last summation is taken over all connected oriented ribbon graphs Γ of genus g with k vertices of valencies 2i1, . . . , 2ik, # Sym Γ is the order of the symmetry group of Γ, and |i| := i1+ · · · + ik.
Our goal is to compare the expansions (1.1.3) and (1.2.5).
1The notation here is slightly different from that of [7,8] where the coefficient of M2j was denoted by s 2j. 2It is often called 1/N -expansion as ǫ ∼ 1/N.
1.3 From cubic Hodge integrals to random matrices. Main Conjecture.
It was already observed by E. Witten [27] that the GUE partition function with an even polynomial V (M ) is tau-function of a particular solution to the Volterra (also called the discrete KdV) hierarchy. Recall that the first equation of the hierarchy (the Volterra lattice equation) reads
˙ wn= wn(wn+1− wn−1) where wn= Zn+1Zn−1 Z2 n ,
the time derivative is with respect to the variable t = N s1. Other couplings sk are identified with
the time variables of higher flows of the hierarchy. On another side, the study [12] of integrable systems associated with the Hodge integrals3 suggested the following conjectural statement: the Hodge partition function ZE= eH of the form (1.1.3) as function of independent parameters ti is also a
tau-function of the Volterra hierarchy. This observation provides a motivation for the main conjecture of the present paper.
It will be convenient to change normalisation of the GUE couplings. Put ¯ sk:= 2k k sk.
Conjecture 1.3.1 (Main Conjecture) The following formula holds true
∞ X g=0 ǫ2g−2Fg(x, s) + ǫ−2 −12 X k1,k2≥1 k1k2 k1+ k2 ¯ sk1s¯k2+ X k≥1 k 1 + k¯sk− x X k≥1 ¯ sk− 1 4+ x = cosh ǫ ∂x 2 ∞ X g=0 ǫ2g−22gHg(t(x, s)) . (1.3.1) where ti(x, s) := X k≥1
ki+1¯sk− 1 + δi,1+ x · δi,0, i ≥ 0. (1.3.2)
1.4 Computational aspects of the Main Conjecture: how do we verify it?
We will check validity of the Main Conjecture for small genera. Begin with g = 0. Let us start with H0(t). Instead of the explicit expansion (1.1.5) we use the following well known representation
H0 = v3 6 − X i≥0 ti vi+2 i!(i + 2) + 1 2 X i, j≥0 titj vi+j+1 (i + j + 1) i! j! (1.4.1)
where v = v(t) = t0+ . . . is the unique series solution to the equation
v =X
i≥0
ti
vi
i!. (1.4.2)
3The first example of an integrable system associated with linear Hodge integrals was investigated by A. Buryak. In
Here we recall that v = ∂ 2H 0(t) ∂t2 0 = ∞ X k=1 1 k X i1+···+ik=k−1 ti1 i1! . . .tik ik! (1.4.3) is a particular solution to the Riemann–Hopf hierarchy
∂v ∂tk = v k k! ∂v ∂t0 , k = 0, 1, 2, . . . .
For the genus zero GUE free energy F0 = F0(x, s) one has a similar representation. Like above,
introduce u(x, s) = ∂ 2F 0(x, s) ∂x2 (1.4.4) and put w(x, s) = eu(x,s). (1.4.5)
Proposition 1.4.1 The function w = w(x, s) is the unique series solution to the equation w = x +X k≥1 k ¯skwk, ¯sk:= 2k k sk, w(x, s) = x + . . . . (1.4.6)
The genus zero GUE free energy F0 with even couplings has the following expression
F0 = w2 4 − x w + X k≥1 ¯ sk x wk− k k + 1w k+1 +1 2 X k1,k2≥1 k1k2 k1+ k2 ¯ sk1s¯k2w k1+k2 +x 2 2 log w. (1.4.7) Clearly w also satisfies the Riemann–Hopf hierarchy in a different normalization
∂w ∂¯sk
= k wk∂w
∂x, k ≥ 1.
The solution can be written explicitly in the form essentially equivalent to (1.4.3) w = ∞ X n=1 1 n X i1+···+in=n−1 wt(i1) . . . wt(in) ¯si1. . . ¯sin
where we put ¯s0 = x and denote
wt(i) = 1, i = 0 i, otherwise. .
It is now straightforward to verify that the substitution (1.3.2) yields
ev(t(x,s))= w(x, s), i.e. v (t(x, s)) = u(x, s) (1.4.8) and H0(t(x, s)) = F0(x, s) − 1 2 X k1,k2≥1 k1k2 k1+ k2 ¯ sk1s¯k2 + X k≥1 k 1 + ks¯k− x X k≥1 ¯ sk− 1 4+ x. (1.4.9)
See in Sect.2for the details of this computation.
In order to proceed to higher genera we will use the method that goes back to the paper [5] by R. Dijkgraaf and E. Witten. The idea of this method is to express the positive genus free energy terms via the genus zero. Let us first explain this method for the Hodge free energy.
Theorem 1.4.2 ([12]) There exist functions Hg(v, v1, v2, . . . , v3g−2), g ≥ 1 of independent variables
v, v1, v2, . . . such that Hg(t) = Hg v(t), ∂v(t) ∂t0 , . . . ,∂ 3g−2v(t) ∂t3g−20 ! , g ≥ 1. (1.4.10)
Here v(t) is given by eq. (1.4.3). Moreover, for any g ≥ 2 the function Hg is a polynomial in the
variables v2, . . . , v3g−2 with coefficients in Qv1, v1−1 (independent of v).
Explicitly, H1(v, v1) = − 1 16v + 1 24log v1 (1.4.11) H2(v1, v2, v3, v4) = 7v2 2560 − v12 11520+ v4 1152v21 − v3 320v1 + v 3 2 360v41 + 11v22 3840v21 − 7v3v2 1920v13, (1.4.12) etc. The algorithm for computing the functions Hg can be found in [12]. They were used in the
construction of the associated integrable hierarchy via the quasi-triviality transformation approach [10].
Let us now proceed to the higher genus terms for the random matrix free energy (recall that only even couplings are allowed).
Theorem 1.4.3 There exist functions Fg(v, v1, . . . , v3g−2), g ≥ 1 of independent variables v, v1, v2,
. . . such that Fg(x, s) = Fg u(x, s), ∂u(x, s) ∂x , . . . , ∂3g−2u(x, s) ∂x3g−2 , g ≥ 1. (1.4.13) Here u(x, s) = ∂ 2F 0(x, s) ∂x2 = log w(x, s).
Recall that the function w(x, s) is determined from eq. (1.4.6). Explicitly
F1(v, v1) =
1
12log v1+ const (1.4.14)
with const=iπ24+ ζ′(−1), F2(v1, v2, v3, v4) = − v2 480 − v2 1 2880 + v4 288 v2 1 − v3 480 v1 + v 3 2 90 v4 1 + v 2 2 960 v2 1 − 7v3v2 480 v3 1 (1.4.15) etc. For any g ≥ 2 the function Fg is a polynomial in the variables v2, . . . , v3g−2 with coefficients in
Using the fact that ∂t0 = ∂x (see Section 3.2 below) along with the standard expansion cosh ǫ ∂x 2 = 1 +X n≥1 1 (2n)! ǫ 2 2n ∂x2n
we recast the Main Conjecture for g ≥ 1 into a sequence of the following relationships between the functions Fg and Hg F1 = 2H1+ v 8 + const (1.4.16) and, for g ≥ 2 Fg(v1, . . . , v3g−2) = v2g−2 22g(2g)!+ D2g−20 H1(v; v1) 22g−3(2g − 2)! + g X m=2 23m−2g (2g − 2m)!D 2(g−m) 0 Hm(v1, . . . , v3m−2) (1.4.17)
where the operator D0 is defined by
D0= v1 ∂ ∂v + X k≥1 vk+1 ∂ ∂vk . For example, F2(v1, v2, v3, v4) = 4H2(v1, v2, v3, v4) +1 4D 2 0H1+ 1 384v2. (1.4.18)
Eqs. (1.4.16), (1.4.18) can be easily verified (see below). In order to verify validity of eqs. (1.4.17) for any g ≥ 2 we write a conjectural explicit expression for the functions Fg(v1, . . . , v3g−2) responsible for
the genus g random matrix free energies. This will be done in the next subsection.
1.5 An explicit expression for Fg
We first recall some notations. Y will denote the set of all partitions. For any partition λ ∈ Y denote by ℓ(λ) the length of λ, by λ1, λ2, . . . , λℓ(λ) the non-zero components, |λ| = λ1+ · · · + λℓ(λ) the weight,
and by mi(λ) the multiplicity of i in λ. Put m(λ)! :=Qi≥1mi(λ)!. The set of all partitions of weight
k will be denoted by Yk. For an arbitrary sequence of variables v1, v2, . . . , denote vλ= vλ1· · · vλℓ(λ).
Conjecture 1.5.1 For any g ≥ 2, the genus g GUE free energy Fg has the following expression
Fg(v1, . . . , v3g−2) = v2g−2 22g(2g)!+ 1 22g−3(2g − 2)!D 2g−2 0 −1 16v + 1 24log v1 + g X m=2 23m−2g (2g − 2m)! 3m−3 X k=0 X k1+k2+k3=k 0≤k1,k2,k3≤m (−1)k2+k3 2k1 X ρ,µ∈Y3m−3−k hλk1λk2λk3τρ+1ig m(ρ)! Q ρµD2g−2m 0 vµ+1 v1ℓ(µ)+m−1−k ! (1.5.1) where for a partition µ = (µ1, . . . , µℓ), µ + 1 denotes the partition (µ1+ 1, . . . , µℓ+ 1), Qρµ is the
so-called Q-matrix defined by
Qρµ = (−1)ℓ(ρ) X µ1∈Yλ1,...,µℓ(ρ)∈Yλℓ(ρ) ∪ℓ(ρ)q=1µq=µ ℓ(ρ) Y q=1 (ρq+ ℓ(µq))! (−1)ℓ(µ q) m(µq)!Q∞ j=1(j + 1)!mj(µ q).
In this formula we have used the notation hλk1λk2λk3τνig := Z Mg,ℓ λk1λk2λk3ψ ν1 1 . . . ψ νℓ ℓ , ∀ ν = (ν1, . . . , νℓ) ∈ Y.
Details about Q-matrix can be found in [9]. Conj.1.5.1 indicates that the the special cubic Hodge integrals (1.1.2) naturally appear in the expressions for the higher genus terms of GUE free energy. Organization of the paper In Sect.2 we review the approach of [10, 7] to the GUE free energy, and prove Prop.1.4.1 and Thm.1.4.3. In Sect.3we verify Conj.1.3.1 and Conj.1.5.1 up to the genus 2 approximation, and give explicit formulae of Fg for g = 3, 4, 5.
Acknowledgements We wish to thank Si-Qi Liu and Youjin Zhang for helpful discussions.
2
GUE free energy with even valencies
2.1 Calculating the GUE free energy from Frobenius manifold of P1 topological σ-model
It is known that the GUE partition function ZN (with even and odd couplings) is the tau-function of a
particular solution to the Toda lattice hierarchy. Using this fact, one of the authors in [7] developed an efficient algorithm of calculating of GUE free energy, which is an application of the general approach of [10,6] for the particular example of the two-dimensional Frobenius manifold with potential
F = 1 2u v
2+ eu.
(Warning: only in this section, the notation v is different from that of the Introduction.) In this section, we give a brief reminder of this approach referring the readers to [10,11,7] for details.
Introduce two analytic functions θ1(u, v; z), θ2(u, v; z) as follows
θ1(u, v; z) = −2 ezv ∞ X m=0 −12u + cm emuz 2m m!2 =: X p≥0 θ1,p(u, v) zp (2.1.1) θ2(u, v; z) = z−1 X m≥0 emu+zv z 2m (m!)2 − 1 =: X p≥0 θ2,p(u, v) zp. (2.1.2)
Here cm=Pmk=11k denotes the m-th harmonic number.
Note that, as in the Introduction, we will only consider the GUE partition function with even couplings. The corresponding Euler–Lagrange equation [7,6,10] reads
x − w +X k≥1 (2k)! sk k X m=1 m wm v 2k−2m (2k − 2m)! m!2 = 0 (2.1.3) −v +X k≥1 (2k)! sk k−1 X m=0 wm v 2k−1−2m (2k − 1 − 2m)! m!2 = 0 (2.1.4)
where w = eu (as in the Introduction). Note that we are only interested in the unique series solution (v(x, s), w(x, s)) of (2.1.3), (2.1.4) such that v(x, 0) = 0, w(x, 0) = x. It is then easy to see from eq. (2.1.4) that
v = v(x, s) ≡ 0. And eq. (2.1.3) becomes
x − w + X
m≥1
s2mm wm
(2m)!
m!2 = 0. (2.1.5)
Define a family of analytic functions Ωα,p;β,q(u, v) by the following generating formula
X p,q≥0 Ωα,p;β,qzpyq= 1 z + y ∂θα(z) ∂v ∂θβ(y) ∂u + ∂θα(z) ∂u ∂θβ(y) ∂v − δα+β,3 , α, β = 1, 2. (2.1.6) The genus zero GUE free energy F0(x, s) then has the following expression
F0 = 1 2 X p,q≥2 (2p)!(2q)! spsqΩ2,2p−1;2,2q−1+ x X q≥1 (2q)! sqΩ1,0;2,2q−1− x Ω1,0;2,1 +1 2(1 − 2s1) 2Ω 2,1;2,1+ X q≥2 (2s1− 1) (2q)! sqΩ2,1;2,2q−1+ 1 2x 2Ω 1,0;1,0. (2.1.7)
The higher genus terms in the 1/N expansion of the GUE free energy can be determined recursively from the loop equation [10,7] for a sequence of functions
Fg = Fg(u, v, u1, v1, . . . , v3g−2, u3g−2), g ≥ 1.
This equation has the following form X r≥0 ∂∆F ∂vr v − λ D r − 2∂∆F∂u r 1 D r +X r≥1 r X k=1 r k 1 √ D k−1 ∂∆F ∂vr v − λ √ D r−k+1 − 2∂∆F∂u r 1 √ D r−k+1 = D−3eu 4 eu+ (v − λ)2 − ǫ2X k,l 1 4S(∆F, vk, vl) v − λ √ D k+1 v − λ √ D l+1 −S(∆F, vk, ul) v − λ√ D k+1 1 √ D l+1 +S(∆F, uk, ul) 1 √ D k+1 1 √ D l+1 −ǫ 2 2 X k ∂∆F ∂vk 4 eu(v − λ) u 1− T v1 D3 + ∂∆F ∂uk 4 (v − λ) v1− T u1 D3 (eu)k+1 (2.1.8) where △F = P g≥1ǫ2gFg, D = (v − λ)2 − 4 eu, T = (v − λ)2+ 4 eu, S(f, a, b) := ∂ 2f ∂a∂b + ∂f ∂a ∂f ∂b, and
fr stands for ∂xr(f ). Solution ∆F of (2.1.8) exists and is unique up to an additive constant. Fg is
a polynomial in u2, v2, . . . , u3g−2, v3g−2. For g ≥ 2, Fg is a rational function of u1, v1. Then [10] the
genus g term in the expansion (1.2.4), in the particular case of even couplings only, reads Fg(x, s) = Fg u(x, s), v = 0,∂u(x, s) ∂x , vx = 0, . . . , ∂3g−2u(x, s) ∂x3g−2 , v3g−2= 0 , g ≥ 1
2.2 Proof of Prop.1.4.1, Thm.1.4.3 Proof of Prop.1.4.1. Noting that
θ2(u, 0, z) = z−1 X m≥0 wm z 2m (m!)2 − 1 , ∂vθ2(u, 0, z) = X m≥0 wm z 2m (m!)2, ∂uθ2(u, 0, z) = X m≥0 mwmz 2m−1 (m!)2
and using (2.1.6) we have X p,q≥0 Ω2,p;2,qzpyq= P m≥0wm z 2m (m!)2 P m≥0mwm y 2m−1 (m!)2 + P m≥0mwm z 2m−1 (m!)2 P m≥0wm y 2m (m!)2 z + y .
It follows that if p + q is odd then Ω2,p;2,q vanishes; otherwise, we have
Ω2,p;2,q = wp+q2 +1 1 +p+q2 p 2! 2 q 2!
2, p, q are both even; (2.2.1)
Ω2,p;2,q = p+1 2 q+1 2 w p+q 2 +1 1 +p+q2 hp+1 2 !i2 hq+12 !i2
, p, q are both odd. (2.2.2)
Substituting these expressions in (2.1.7) we obtain F0= 1 2x 2u +1 2 X k1,k2≥0 (2k1+ 2)!(2k2+ 2)!sk1+1sk2+1 (k1+ 1)(k2+ 1) wk1+k2+2 (k1+ k2+ 2) [(k1+ 1)!]2 [(k2+ 1)!]2 +xX k≥0 (2k + 2)!sk+1 wk+1 (k + 1)!2 − x w + (1 − 4s1) w2 4 − X k≥1 (2k + 2)!sk+1 (k + 1) wk+2 (k + 2) [(k + 1)!]2. Equation (1.4.6) is already proved in (2.1.5). The proposition is proved.
Proof of Theorem 1.4.3. For g = 1, 2, taking v = v1 = v2 = · · · = 0 in the general expressions of
Fg(u, v, u1, v1, . . . , u3g−2, v3g−2) [10,11] one obtains (1.4.14) and (1.4.15). For any g ≥ 1, the existence
of Fg(u, u1, . . . , u3g−2) such that
Fg(x, s) = Fg u(x, s),∂u(x, s) ∂x , . . . , ∂3g−2u(x, s) ∂x3g−2
is a direct result of [10,11] when taking v = v1 = v2 = · · · = 0 in Fg(u, v, u1, v1, . . . , u3g−2, v3g−2).
3
Verification of the Main Conjecture for low genera
3.1 Genus 0Recall that the genus zero cubic Hodge free energy can be expressed as H0(t) = 1 2 X i,j≥0 ˜ ti˜tjΩi;j(v(t)).
where ˜ti = ti− δi,1, t = (t0, t1, t2, . . . ), Ωi;j are polynomials in v given by
Ωi;j(v) =
vi+j+1
(i + j + 1) i! j!,
and v(t) is the unique series solution to the following Euler–Lagrange equation of the one-dimensional Frobenius manifold v =X i≥0 ti vi i!.
(Warning: the above v is the flat coordinate of the one-dimensional Frobenius manifold; avoid confusing with v in Section 2 where (u, v) are flat coordinates of the two-dimensional Frobenius manifold of P1 topological σ-model.)
Let us consider the following substitution of time variables ti=
X
k≥1
ki+1s¯k− 1 + δi,1+ x · δi,0, i ≥ 0.
Note that with this substitution the cubic Hodge free energies will be considered to be expanded at x = 1. We have ˜ti =Pk≥1ki+1s¯k− 1 + x · δi,0, and so
H0 = 1 2 X i,j≥0 ˜ tit˜jΩi;j(v(t)) = 1 2 X i,j≥0 X k1≥1 k1i+1s¯k1− 1 + x · δi,0 X k2≥1 k2j+1¯sk2 − 1 + x · δj,0 vi+j+1 (i + j + 1) i! j! = 1 2 X i,j≥0 X k1,k2≥1 k1i+1k2j+1s¯k1s¯k2 vi+j+1 (i + j + 1) i! j! − X i,j≥0 X k1≥1 k1i+1¯sk1 vi+j+1 (i + j + 1) i! j! +x X i,j≥0 X k1≥1 k1i+1¯sk1δj,0 vi+j+1 (i + j + 1) i! j! + 1 2 X i,j≥0 vi+j+1 (i + j + 1) i! j! −x X i,j≥0 δj,0 vi+j+1 (i + j + 1) i! j! + x2 2 X i,j≥0 δi,0δj,0 vi+j+1 (i + j + 1) i! j!. (3.1.1)
We simplify it term by term: x2 2 X i,j≥0 δi,0δj,0 vi+j+1 (i + j + 1) i! j! = x2 2 v, x X i,j≥0 δj,0 v i+j+1 (i + j + 1) i! j! = x (e v − 1), 1 2 X i,j≥0 vi+j+1 (i + j + 1) i! j! = 1 2 X ℓ≥0 ℓ X i=0 vℓ+1ℓ! (ℓ + 1)! i! (ℓ − i)! = 1 2 X ℓ≥0 vℓ+12ℓ (ℓ + 1)! = 1 4(e 2v − 1), x X i,j≥0 X k1≥1 k1i+1s¯k1δj,0 vi+j+1 (i + j + 1) i! j! = x X i≥0 X k1≥1 ki+11 s¯k1 vi+1 (i + 1)! = x X k≥1 ¯ sk(ekv − 1),
X i,j≥0 X k1≥1 k1i+1s¯k1 vi+j+1 (i + j + 1) i! j! = X k≥1 k ¯sk X ℓ≥0 (1 + k)ℓ v ℓ+1 (ℓ + 1)! = X k≥1 k 1 + ks¯k e(1+k)v− 1, 1 2 X i,j≥0 X k1,k2≥1 k1i+1k2j+1s¯k1¯sk2 vi+j+1 (i + j + 1) i! j! = 1 2 X k1,k2≥1 k1k2s¯k1s¯k2 X ℓ≥0 vℓ+1 (ℓ + 1)!(k1+ k2) ℓ = 1 2 X k1,k2≥1 k1k2 k1+ k2 ¯ sk1s¯k2 e(k1+k2)v− 1. Let w = ev. We have H0 = 1 2 X k1,k2≥1 k1k2 k1+ k2 ¯ sk1¯sk2 wk1+k2− 1−X k≥1 k 1 + k¯sk w1+k− 1+ xX k≥1 ¯ sk(wk− 1) +1 4(w 2− 1) − x (w − 1) + x2 2 log w.
On the other hand, recall from Prop.1.4.1that the genus zero GUE free energy with even couplings has the form
F0= w2 4 − x w + X k≥1 ¯ sk x wk− k k + 1w k+1 +1 2 X k1,k2≥1 k1k2 k1+ k2 ¯ sk1¯sk2w k1+k2+x 2 2 log w. Here w is the power series solution to
w = x +X k≥1 k ¯skwk. Recall that w = eu; so eu = x +X k≥1 k ¯skeku. Namely, 1 +X j≥1 uj j! = x + X k≥1 k ¯sk 1 +X j≥1 kjuj j! . It follows that u(x, s) = v(t(x, s)). (3.1.2) We conclude that H0(t(x, s)) − F0(x, s) = − 1 2 X k1,k2≥1 k1k2 k1+ k2 ¯ sk1¯sk2+ X k≥1 k 1 + k¯sk− x X k≥1 ¯ sk− 1 4 + x. (3.1.3) This finishes the proof of the genus zero part of the Main Conjecture.
3.2 Genus 1, 2
Note that the substitution (1.3.2)
(t0, t1, t2, . . . ) 7→ (x, ¯s1, ¯s2, . . . ) satisfies that ∂ ∂x = ∂ ∂t0 , (3.2.1) ∂ ∂¯sk =X i≥0 ki+1 ∂ ∂ti , k ≥ 1. (3.2.2) In particular, we have ∂v ∂t0 (t(x, s)) = ∂v(t(x, s)) ∂x = ∂u(x, s) ∂x . The last equality is due to (3.1.2).
Recall, from the algorithm of [12], that the genus 1 special cubic Hodge free energy is given by H1(v; v1) = 1 24log v1− v 16. So 2H1(v; v1) + v 8 + iπ 24 + ζ ′(−1) = 1 12log v1+ iπ 24 + ζ ′(−1).
This proves the genus 1 part of the Main Conjecture.
The genus 2 term of the special cubic Hodge free energy is given by H2(v1, v2, v3, v4) = 7 v2 2560 − v2 1 11520 + v4 1152 v2 1 − v3 320 v1 + v 3 2 360 v4 1 + 11 v 2 2 3840 v2 1 − 7 v3v2 1920 v3 1 . So 4H2+ 1 4D 2 0H1+ 1 384v2 = 4H2(v1, v2, v3, v4) − 5 384v2+ 1 96 " v3 v1 − v2 v1 2# = −480v2 − v 2 1 2880 + v4 288 v2 1 − v3 480 v1 + v 3 2 90 v4 1 + v 2 2 960 v2 1 − 7v3v2 480 v3 1 = F2(v1, v2, v3, v4).
This proves the genus 2 part of the Main Conjecture.
3.3 Genus 3, 4
Conjecture 3.3.1 The genus 3 GUE free energy is given by F3(u1, . . . , u7) = 13u4 120960+ u22 24192 − u41 725760+ u7 10368u31 − u6 5760u21 − u5 13440u1 − 103u 2 4 60480u4 1 + 59u 3 3 8064u5 1 + u 2 3 2688u2 1 + u3u1 12096− 5u62 81u8 1 − 13u52 1890u6 1 + 5u 4 2 5376u4 1 − u32 9072u2 1 − 7u6u2 5760u4 1 − 53u3u5 20160u4 1 +353u5u 2 2 40320u5 1 + u5u2 840u3 1 +89u3u4 40320u3 1 − 83u4u32 1890u6 1 − 211u4u22 40320u4 1 + u4u2 2016u2 1 +59u3u 4 2 378u7 1 +1993u3u 3 2 120960u5 1 − u3u22 576u3 1 − 83u23u22 896u6 1 + 19u3u2 120960u1 − 17u23u2 2240u4 1 +1273u3u4u2 40320u5 1 . (3.3.1) Conjecture 3.3.2 The genus 4 GUE free energy is given by
F4(u1, . . . , u10) = 1852u 9 2 1215u12 1 + 151u 8 2 675u10 1 − 101u72 12600u8 1 − 772u3u72 135u11 1 +9904u4u 6 2 6075u10 1 − 1165u62 1161216u6 1 − 2851u3u62 3600u9 1 +14903u 2 3u52 2160u10 1 +70261u3u 5 2 3225600u7 1 +2573u4u 5 2 10800u8 1 + u 5 2 7200u4 1 − 2243u5u52 6480u9 1 +195677u 2 3u42 230400u8 1 + 3197u3u 4 2 967680u5 1 +12907u6u 4 2 226800u8 1 −10259u4u 4 2 1935360u6 1 − 22153u5u 4 2 414720u7 1 −101503u3u4u 4 2 32400u9 1 +1823u 2 4u32 5670u8 1 +415273u3u5u 3 2 829440u8 1 + 97u5u 3 2 120960u5 1 +26879u6u 3 2 2903040u6 1 + u 3 2 7257600 − 49u3u32 138240u3 1 − 5137u4u 3 2 4354560u41 − 877u2 3u32 57600u61 − 812729u3u4u32 2073600u71 − 212267u7u32 29030400u71 − 305129u3 3u32 103680u91 + u 2 1u22 460800+ 1379u2 4u22 34560u61 + 13138507u2 3u4u22 9676800u81 + 2417u3u4u22 537600u51 + 17u4u22 138240u21 +2143u3u5u 2 2 34560u61 + 449u5u22 1451520u31 + 2323u8u22 3225600u61 − 2623u23u22 967680u41 − 443u6u22 9676800u41 − 667u7u 2 2 537600u51 − 192983u33u22 691200u71 − 60941u3u6u22 1075200u71 − 171343u4u5u22 1935360u71 + 22809u43u2 71680u81 +1747u 3 3u2 806400u5 1 + 7u 2 3u2 38400u2 1 + 9221u 2 5u2 1935360u6 1 +17u1u3u2 3225600 + 78533u23u4u2 691200u6 1 +18713u3u4u2 14515200u3 1 + 15179u4u6u2 1935360u6 1 + 20639u3u7u2 4838400u6 1 + 37u8u2 302400u4 1 − u4u2 86400 −362880u11u5u2 1 − 923u6u2 14515200u2 1 − 113u7u2 9676800u3 1 − 55u24u2 387072u4 1 − 419u3u5u2 1935360u4 1
−1411u138240u4u5u52 1 − 7u9u2 138240u5 1 − 1751u3u6u2 268800u5 1 − 12035u23u5u2 96768u7 1 − 44201u3u24u2 276480u7 1 + 1549u 4 3 115200u6 1 + 937u 3 3 2903040u3 1 + 229u 3 4 62208u6 1 + 19u 2 5 46080u4 1 + u 3 1u3 691200 + 949u3u4u5 55296u6 1 + 59u 2 3u6 10752u6 1 + 73u4u6 107520u4 1 + 1777u3u7 4838400u4 1 + 143u7 14515200u1 + 31u8 9676800u2 1 + u10 497664u4 1 − u 2 3 115200− u1u5 138240 − 73u6 29030400 − u61 43545600 − 19u21u4 87091200 − 137u3u4 2073600u1 − 239u3u5 1451520u21 − 661u24 5806080u21 − u9 138240u31 − 17u4u5 387072u31 − 89u3u6 3225600u31 − 709u23u4 3225600u41 −1291u3u 2 4 138240u51 − 1001u23u5 138240u51 − 197u5u6 387072u51 − 163u3u8 967680u51 − 2069u4u7 5806080u51 − 2153u33u4 28800u71 . (3.3.2) We also computed the genus 5 free energy; see in Appendix A.
For the particular examples of enumerating squares, hexagons, octagons on a genus g surface, one can use (3.3.1), (3.3.2), (A.0.3) to obtain the combinatorial numbers. We checked that these numbers agree with those in [8]. This gives some evidences of validity of the Main Conjecture for g = 3, 4, 5. Remark 3.3.3 The genus 1, 2, 3 terms of the GUE free energy with even couplings were also derived in [13,14, 26] for the particular case of only one nonzero coupling (i.e., in the framework of enumeration of 2m-gons). To the best of our knowledge, explicit formulae for higher genus (g ≥ 4) terms, even in the case of the particular examples, were not available in the literature.
A
Explicit formula for F
5 F5(u1, . . . , u13) = −109514u 12 2 1215u161 − 1352u11 2 81u141 + 181628u3u102 405u151 − 2593u10 2 9450u121 + 42691u3u92 540u131 + 16091u 9 2 187110u101 − 93460u4u92 729u141 + 600763u3u82 415800u111 + 134599u5u82 4860u131 − 274289u82 255467520u81 −567199u4u 8 2 24300u121 − 1322159u23u82 1620u141 + 391519u3u4u72 972u131 + 2471441u5u72 475200u111 − 151u72 399168u61 −999473u3u 7 2 2838528u91 − 2825u4u72 5544u101 − 1149739u23u72 8640u121 − 161353u6u72 34020u121 + 12916717u33u62 19440u131 +319877u3u 6 2 159667200u7 1 +31781177u3u4u 6 2 475200u11 1 +4549471u4u 6 2 42577920u8 1 +515032871u5u 6 2 3832012800u9 1 +8477461u7u 6 2 12830400u11 1 + 263u 6 2 23950080u4 1 − 27484783u6u62 29937600u10 1 − 1098923921u23u62 425779200u10 1 −5713573u3u5u 6 2 77760u12 1 − 12051881u24u62 255150u12 1 + 173831501u 3 3u52 1824768u11 1 +6013615u 2 3u52 12773376u8 1+28957u3u 5 2 21288960u5 1 +276125491u3u4u 5 2 182476800u9 1 + 10243u4u 5 2 239500800u6 1 +53761259u4u5u 5 2 3326400u11 1 +102985067u3u6u 5 2 9979200u11 1 +84091529u7u 5 2 638668800u9 1 + u 5 2 633600u2 1 − 11532541u5u52 479001600u7 1 − 5117003u6u52 182476800u8 1 −19847231u 2 4u52 2494800u10 1 − 2268769u8u52 29937600u10 1 − 1125727817u3u5u52 91238400u10 1 − 14531719u23u4u52 36288u12 1 +2322129119u 3 3u42 1277337600u9 1 + 1211u 2 3u42 2027520u6 1 +87822499u3u 2 4u42 1197504u11 1 + 728671781u 2 3u5u42 12773376u11 1 +1046529431u4u5u 4 2 383201280u9 1 +1108533611u3u6u 4 2 638668800u9 1 +231617u6u 4 2 54743040u6 1 + 1215667u7u 4 2 255467520u7 1 +1096859u9u 4 2 153280512u9 1 − 29u 4 2 127733760 − 47u3u42 1596672u3 1 − 29783u4u42 63866880u4 1 − 24971u5u42 127733760u5 1 − 443511251u3u4u42 1916006400u7 1 −68301953u3u5u 4 2 212889600u8 1 −5935327u8u 4 2 383201280u8 1 − 133150489u 2 4u42 638668800u8 1 −5682077u4u6u 4 2 2721600u10 1 −1095679399u 2 3u4u42 19353600u10 1 −154991051u3u7u 4 2 136857600u10 1 −4872743377u 2 5u42 3832012800u10 1 −547560589u 4 3u42 2322432u12 1 + u 2 1u32 3991680 + 69761129u3u24u32 6967296u91 + 75292039u2 4u32 2874009600u61 + 8817996169u3 3u4u32 63866880u111 + 1049u4u 3 2 119750400u21 + 2479931059u2 3u5u32 319334400u91 + 38370113u3u5u32 958003200u61 + 51887531u4u5u32 638668800u71 + 2279u5u 3 2 19160064u31 + 16316161u3u6u32 319334400u71 + 25527371u5u6u32 87091200u91 + 6169u6u32 76640256u41 + 791123u4u7u32 3870720u91 +369691943u3u8u 3 2 3832012800u91 + 284539u9u32 191600640u71 − 767u3u32 212889600u1 − 2141u23u32 1520640u41 − 42149u7u32 71850240u51 −1229917u3u4u 3 2 638668800u51 − 52781u8u32 79833600u61 − 10864759u33u32 47900160u71 − 2883761u4u6u32 8294400u81 −764936639u 2 3u4u32 638668800u8 1 − 179669059u3u7u32 958003200u8 1 − 573019u10u32 1045094400u8 1 − 406451147u25u32 1916006400u8 1 −1848263u 3 4u32 467775u10 1 − 626921467u23u6u32 106444800u10 1 − 3430962641u43u32 127733760u10 1 − 35338084651u3u4u5u32 1916006400u10 1 +419193u 5 3u22 14336u11 1 + u 4 1u22 6386688 + 23669u23u22 1277337600u2 1 +1783923u3u 2 4u22 7884800u7 1 + 50047u 2 4u22 106444800u4 1 +1238492531u3u 2 5u22 1277337600u9 1 +181880015u 3 3u4u22 12773376u9 1 +73978651u 2 3u4u22 638668800u6 1 +3337u3u4u 2 2 4561920u3 1 +5302619u 2 3u5u22 30412800u7 1 + 1591687897u 2 4u5u22 1277337600u9 1 +30953u3u5u 2 2 45619200u4 1 +203383903u3u4u6u 2 2 127733760u9 1 +29758507u5u6u 2 2 638668800u7 1 +275874283u 2 3u7u22 638668800u9 1 +2069131u4u7u 2 2 63866880u7 1 +289u3u8u 2 2 19008u7 1 + 60127u8u 2 2 958003200u4 1 + 3373u9u 2 2 45619200u5 1 + 11437u11u 2 2 348364800u7 1 − 5u1u3u 2 2 2128896 − 19u4u22 54743040 − 7753u5u22 3832012800u1 − 223u6u22 9123840u2 1
− 247u7u 2 2 11612160u3 1 − 317077u3u6u22 63866880u5 1 − 998201u33u22 638668800u5 1 − 5091301u4u5u22 638668800u5 1 − 742733u25u22 106444800u6 1 − 19631u10u 2 2 174182400u6 1 − 1295627u3u7u22 212889600u6 1 − 7277719u4u6u22 638668800u6 1 − 20664649u3u4u5u22 8870400u8 1 − 9854357u23u6u22 13305600u8 1 −13316641u 3 4u22 26611200u8 1 − 1293697u5u7u22 53222400u8 1 − 328171u3u9u22 53222400u8 1 − 45787691u43u22 106444800u8 1 − 6093371u26u22 425779200u8 1 −18837199u4u8u 2 2 1277337600u8 1 − 2912894597u23u24u22 116121600u10 1 − 1661611993u33u5u22 127733760u10 1 +14367497u 5 3u2 7096320u9 1 +3625841u 4 3u2 127733760u6 1 + 4721u 3 3u2 12773376u3 1 +274614007u3u 3 4u2 182476800u9 1 +7368997u3u 2 5u2 70963200u7 1 + 927517u 2 5u2 1916006400u4 1 +17u 3 1u3u2 53222400+ 2872733u33u4u2 13305600u7 1 +20527u 2 3u4u2 14192640u4 1 +85455011u 2 4u5u2 638668800u7 1 +1121508611u 2 3u4u5u2 319334400u9 1 + 159183659u 3 3u6u2 212889600u9 1 +15535133u3u4u6u2 91238400u7 1 +1516703u4u6u2 1916006400u4 1 +2867u5u6u2 1814400u5 1 + 5833u6u2 3832012800 + 916913u2 3u7u2 19958400u7 1 + 49661u3u7u2 119750400u4 1 +698809u4u7u2 638668800u5 1 +82861u6u7u2 45619200u71 + 373u7u2 91238400u1 +650429u3u8u2 1277337600u51 + 122413u5u8u2 91238400u71 + 41u8u2 9580032u21 +38377u4u9u2 53222400u71 + 3929u3u10u2 14515200u71 + 1091u11u2 174182400u51 − 19u1u5u2 13305600 − 23u21u4u2 15966720 − 23u23u2 21288960 −766402560u4801u3u4u2 1 − 5497u24u2 63866880u21 − 26833u3u5u2 212889600u21 − 323u9u2 68428800u31 − 10961u3u6u2 79833600u31 − 64907u4u5u2 273715200u31 −116121600u727u10u2 4 1 − 1883363u23u5u2 159667200u5 1 − 3705967u3u24u2 239500800u5 1 − 7u12u2 4976640u6 1 − 14227u26u2 7096320u6 1 − 16217u4u8u2 7884800u6 1 −13001u15206400u3u9u62 1 − 864373u23u6u2 53222400u6 1 − 1184501u34u2 106444800u6 1 − 542981u5u7u2 159667200u6 1 − 32913731u3u4u5u2 638668800u6 1 −1144789u4u 2 5u2 11612160u8 1 − 946477u23u8u2 45619200u8 1 − 176093453u33u5u2 159667200u8 1 − 454075417u23u24u2 212889600u8 1 − 28069627u3u4u7u2 319334400u8 1 −80480347u638668800u3u5u86u2 1 − 103432013u24u6u2 1277337600u8 1 − 18599541u43u4u2 1576960u10 1 + 10673u 5 3 691200u7 1 + 16649u 4 3 91238400u4 1 +275599u3u 3 4 3379200u7 1 + 800453u 3 4 1916006400u4 1 + 571213u 3 5 383201280u7 1 + 64482661u 3 3u24 60825600u9 1 + 1217u 2 4 1149603840 + 2264879u3u25 1277337600u5 1 + 7003u 2 5 348364800u2 1 + 66653u3u 2 6 28385280u7 1 + u 5 1u3 15966720+ 35113699u43u5 85155840u9 1 + 2932793u 2 4u5 1277337600u5 1 + 4643u3u5 1916006400 +30303149u 2 3u4u5 159667200u7 1 +18211u3u4u5 9580032u4 1 + 1867u4u5 79833600u1 +570989u 3 3u6 14192640u7 1 + 13u 2 1u6 174182400 + 62701u2 3u6 106444800u4 1 + 8669u3u6 638668800u1 +102911u3u4u6 35481600u5 1 + 20521u4u6 638668800u2 1 +104281u4u5u6 14192640u7 1 + 4717u 2 3u7 6082560u5 1 +155959u 2 4u7 60825600u71 + 6911u1u7 11496038400 + 29611u3u7 1916006400u21 + 318517u3u5u7 79833600u71 + 299u6u7 1689600u51 + 1030877u3u4u8 425779200u71
+ 31u5u8 237600u5 1 + 7697u 2 3u9 15206400u7 1 + 44647u4u9 638668800u5 1 + 71u3u10 2721600u5 1 + 13u10 74649600u2 1 + 113u11 348364800u3 1 + u13 29859840u5 1 − 23u21u23 21288960 − u41u4 130636800 − 2011u1u3u4 638668800 − 79u8 638668800 − 241u31u5 958003200 − u81 1277337600 −547430400u361u9 1 − 5707u33 3832012800u1 − 2099u23u4 15966720u2 1 − 931u4u7 22809600u3 1 − 13261u3u24 54743040u3 1 − 10277u23u5 58060800u3 1 −958003200u17527u3u83 1 − 113089u5u6 1916006400u3 1 − u12 4976640u4 1 − 2533u3u9 106444800u4 1 − 8017u26 141926400u4 1 − 8189u4u8 141926400u4 1 −319334400u30553u5u74 1 − 809239u33u4 106444800u5 1 − 11u3u11 1741824u6 1 − 22349u4u25 3193344u6 1 − 701u5u9 18247680u6 1 − 2143u6u8 36495360u6 1 −41803776u775u4u106 1 − 520931u33u5 42577920u6 1 − 244457u24u6 42577920u6 1 − 659977u3u4u7 106444800u6 1 − 3697u27 109486080u6 1 − 1138241u3u5u6 127733760u6 1 −185851u 2 3u8 127733760u6 1 −10135361u 2 3u24 425779200u6 1 −336827u 2 3u4u6 2956800u8 1 −3980637u 4 3u4 7884800u8 1 − 653701u 4 4 34214400u8 1 −877403u 3 3u7 42577920u8 1 −8134913u3u 2 4u5 45619200u8 1 −22151509u 2 3u25 319334400u8 1 − 4543u 6 3 8192u10 1 . (A.0.3)
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