CALT-68-2896
SPECTRAL CURVES AND THE SCHR
¨
ODINGER EQUATIONS FOR THE
EYNARD-ORANTIN RECURSION
MOTOHICO MULASE AND PIOTR SU LKOWSKI
Abstract. It is predicted that the principal specialization of the partition function of a B-model
topological string theory, that is mirror dual to an A-model enumerative geometry problem, sat-
isfies a Schr¨odinger equation, and that the characteristic variety of the Schr¨odinger operator gives
the spectral curve of the B-model theory, when an algebraic K-theory obstruction vanishes. In
this paper we present two concrete mathematical A-model examples whose mirror dual partners
exhibit these predicted features on the B-model side. The A-model examples we discuss are the
generalized Catalan numbers of an arbitrary genus and the single Hurwitz numbers. In each
case, we show that the Laplace transform of the counting functions satisfies the Eynard-Orantin
topological recursion, that the B-model partition function satisfies the KP equations, and that
the principal specialization of the partition function satisfies a Schr¨odinger equation whose total
symbol is exactly the Lagrangian immersion of the spectral curve of the Eynard-Orantin theory.
Contents
1. Introduction and the main results 1
2. The Eynard-Orantin topological recursion 6
3. The generalized Catalan numbers and the topological recursion 10
4. The partition function for the generalized Catalan numbers and the Schr¨odinger
equation 13
5. Single Hurwitz numbers 17
6. The Schur function expansion of the Hurwitz partition function 24
7. Conclusion 30
Appendix A. Proof of the Schr¨odinger equation for the Catalan case 31
Appendix B. Hierarchy of equations for S
m
34
References 37
1. Introduction and the main results
In a series of remarkable papers of Mari˜no [51] and Bouchard, Klemm, Mari˜no, and Pas-
quetti [8], these authors have developed an inductive mechanism to calculate a variety of
quantum invariants and solutions to enumerative geometry questions, based on the funda-
mental work of Eynard and Orantin [27, 28, 29]. The validity of their method, known as the
remodeled B-model based on the topological recursion of Eynard-Orantin, has been estab-
lished for many different enumerative geometry problems, such as single Hurwitz numbers
([7, 26, 60], based on the conjecture of Bouchard and Mari˜no [9]), open Gromov-Witten in-
variants of smooth toric Calabi-Yau threefolds ([29, 80], based on the remodeling conjecture
of Mari˜no [51] and Bouchard, Klemm, Mari˜no, Pasquetti [8]), and the number of lattice
2000 Mathematics Subject Classification. Primary: 14H15, 14N35, 05C30, 11P21; Secondary: 81T30.
1
arXiv:1210.3006v3 [math-ph] 28 Nov 2012
2 M. MULASE AND P. SU LKOWSKI
points on M
g,n
and its symplectic and Euclidean volumes ([11, 20, 58], based on [63, 64]).
It is expected that double Hurwitz numbers, stationary Gromov-Witten invariants of P
1
[65, 66], certain Donaldson-Thomas invariants, and many other quantum invariants would
also fall into this category.
Unlike the familiar Topological Recursion Relations (TRR) of the Gromov-Witten theory,
the Eynard-Orantin recursion is a B-model formula [8, 51]. The significant feature of the
formula is its universality: independent of the A-model problem, the B-model recursion
takes always the same form. The input data of this B-model consist of a holomorphic
Lagrangian immersion
ι : Σ −−−−→ T
∗
C
y
π
C
of an open Riemann surface Σ (called a spectral curve of the Eynard-Orantin recursion)
into the cotangent bundle T
∗
C equipped with the tautological 1-form η, and the symmetric
second derivative of the logarithm of Riemann’s prime form [30, 62] defined on Σ ×Σ. The
procedure of Eynard-Orantin [27] then defines, inductively on 2g − 2 + n, a meromorphic
symmetric differential n-form W
g,n
on Σ
n
for every g ≥ 0 and n ≥ 1 subject to 2g−2+n > 0.
A particular choice of the Lagrangian immersion gives a different W
g,n
, which then gives a
generating function of the solution to a different enumerative geometry problem.
Thus the real question is how to find the right Lagrangian immersion from a given A-
model.
Suppose we have a solution to an enumerative geometry problem (an A-model problem).
Then we know a generating function of these quantities. In [20] we proposed an idea of
identifying the spectral curve Σ, which states that the spectral curve is the Laplace transform
of the disc amplitude of the A-model problem. Here the Laplace transform plays the role
of mirror symmetry. Thus we obtain a Riemann surface Σ. Still we do not see the aspect
of the Lagrangian immersion in this manner.
Every curve in T
∗
C is trivially a Lagrangian. But not every Lagrangian is realized as
the mirror dual to an A-model problem. The obstruction seems to lie in the K-group
K
2
C(Σ)
⊗ Q. When this obstruction vanishes, we call Σ a K
2
-Lagrangian, following
Kontsevich’s terminology. For a K
2
-Lagrangian Σ, we expect the existence of a holonomic
system that characterizes the partition function of the B-model theory, and at the same
time, the characteristic variety of this holonomic system recovers the spectral curve Σ as the
Lagrangian immersion. A generator of this holonomic system is called a quantum Riemann
surface [1, 16, 17], because it is a differential operator whose total symbol is the spectral
curve realized as a Lagrangian immersion [18]. It is the work of Gukov and Su lkowski
[39] that suggested the obstruction to the existence of the holonomic system with algebraic
K-theory as an element of K
2
.
Another mysterious link of the Eynard-Orantin theory is its relation to integrable systems
of the KP/KdV type [5, 27]. We note that the partition function of the B-model is always
the principal specialization of a τ-function of the KP equations for all the examples we know
by now.
The purpose of the present paper is to give the simplest non-trivial mathematical exam-
ples of the theory that exhibit these key features mentioned above. With these examples one
can calculate all quantities involved, give proofs of the statements predicted in physics, and
examine the mathematical structure of the theory. Our examples are based on enumeration
problems of branched coverings of P
1
.
SPECTRAL CURVES AND THE SCHR
¨
ODINGER EQUATION 3
The idea of homological mirror symmetry of Kontsevich [44] allows us to talk about
the mirror symmetry without underlying spaces, because the formulation is based on the
derived equivalence of categories. Therefore, we can consider the mirror dual B-models
corresponding to the enumeration problems of branched coverings on the A-model side. At
the same time, being the derived equivalence, the homological mirror symmetry does not
tell us any direct relations between the quantum invariants on the A-model side and the
complex geometry on the B-model side. This is exactly where Mari˜no’s idea of remodeling
B-model comes in for rescue. The remodeled B-model of [8, 51] is not a derived category
of coherent sheaves. Although its applicability is restricted to the case when there is a
family of curves Σ that exhibits the geometry of the B-model, the new idea is to construct
a network of inter-related differential forms on the symmetric powers of Σ via the Eynard-
Orantin recursion, and to understand this infinite system as the B-model. The advantage
of this idea is that we can relate the solution of the geometric enumeration problem on the
A-model side and the symmetric differential forms on the B-model side through the Laplace
transform. In this sense we consider the Laplace transform as a mirror symmetry.
The first example we consider in this paper is the generalized Catalan numbers of an
arbitrary genus. This is equivalent to the “c = 1 model” of [39, Section 5]. In terms of
enumeration, we are counting the number of algebraic curves defined over Q in a systematic
way by using the dual graph of Grothendieck’s dessins d’enfants [4, 74].
Let D
g,n
(µ
1
, . . . , µ
n
) denote the automorphism-weighted count of the number of con-
nected cellular graphs on a closed oriented surface of genus g (i.e., the 1-skeleton of cell-
decompositions of the surface), with n labeled vertices of degrees (µ
1
, . . . , µ
n
). The letter
D stands for ‘dessin.’ The generalized Catalan numbers of type (g, n) are defined by
C
g,n
(µ
1
, . . . , µ
n
) = µ
1
···µ
n
D
g,n
(µ
1
, . . . , µ
n
).
While D
g,n
(~µ) is a rational number due to the graph automorphisms, the generalized Cata-
lan number C
g,n
(~µ) is always a non-negative integer. It gives the dimension of a linear skein
space. In particular, the (g, n) = (0, 1) case recovers the original Catalan numbers:
C
0,1
(2m) = C
m
=
1
m + 1
2m
m
= dim End
U
q
(s`
2
)
(T
⊗m
C
2
).
As explained in [20], the mirror dual to the Catalan numbers C
m
is the plane curve Σ
defined by
(1.1)
(
x = z +
1
z
y = −z,
where
(1.2) z(x) =
∞
X
m=0
C
m
1
x
2m+1
.
Note that (1.1) also gives a Lagrangian immersion Σ −→ T
∗
C. Let us introduce the free
energies by
(1.3) F
C
g,n
z(x
1
), . . . , z(x
n
)
=
X
~µ∈Z
n
+
D
g,n
(~µ)e
−(w
1
µ
1
+···+w
n
µ
n
)
=
X
~µ∈Z
n
+
D
g,n
(~µ)
n
Y
i=1
1
x
µ
i
i
as the Laplace transform of the number of dessins, where the coordinates are related by
(1.2) and x
i
= e
w
i
. The free energy F
C
g,n
(z
1
, . . . , z
n
) is a symmetric function in n-variables,
4 M. MULASE AND P. SU LKOWSKI
and its principal specialization is defined by F
C
g,n
(z, . . . , z). Now let
W
C
g,n
(z
1
, . . . , z
n
) = d
1
···d
n
F
g,n
(z
1
, . . . , z
n
).
It is proved in [20] that W
C
g,n
’s satisfy the Eynard-Orantin topological recursion.
The Catalan partition function is given by the formula of [27]:
(1.4) Z
C
(z, ~) = exp
∞
X
g=0
∞
X
n=1
1
n!
~
2g−2+n
F
C
g,n
(z, z, . . . , z)
.
In this paper we prove
Theorem 1.1. The Catalan partition function satisfies the Schr¨odinger equation
(1.5)
~
2
d
2
dx
2
+ ~x
d
dx
+ 1
Z
C
z(x), ~
= 0.
The characteristic variety of this ordinary differential operator, y
2
+ xy + 1 = 0 for every
fixed choice of ~, is exactly the Lagrangian immersion (1.1), where we identify the xy-plane
as the cotangent bundle T
∗
C with the fiber coordinate y = ~
d
dx
.
Remark 1.2. A purely geometric reason of our interest in the function appearing as the
principal specialization F
C
g,n
(z, . . . , z) is that, in the stable range 2g − 2 + n > 0, it is a
polynomial in
(1.6) s =
z
2
z
2
− 1
of degree 6g − 6 + 3n. It is indeed the virtual Poincar´e polynomial of M
g,n
×R
n
+
[58], and
its special value at s = 1 gives the Euler characteristic (−1)
n
χ
(M
g,n
) of the moduli space
M
g,n
of smooth n-pointed curves of genus g. Thus Z
C
(z, ~) is the exponential generating
function of the virtual Poincar´e polynomials of M
g,n
× R
n
+
.
As such, the generating function Z
C
(z, ~) is also expressible in terms of a Hermitian
matrix integral
(1.7) Z
C
(z, ~) =
Z
H
N×N
det(1 −
√
sX)
N
e
−
N
2
trace(X
2
)
dX
with the identification (1.6) and ~ = 1/N. Here dX is the normalized Lebesgue measure
on the space of N × N Hermitian matrices H
N×N
. It is a well-known fact that Eq.(1.7) is
the principal specialization of a KP τ -function (see for example, [54]).
Another example we consider in this paper is based on single Hurwitz numbers. As
a counting problem it is easier to state than the previous example, but the Lagrangian
immersion requires a transcendental function, and hence the resulting Schr¨odinger equation
exhibits a rather different nature.
Let H
g,n
(µ
1
. . . , µ
n
) be the automorphism-weighted count of the number of topological
types of Hurwitz covers f : C −→ P
1
of a connected non-singular algebraic curve C of genus
g. A holomorphic map f is a Hurwitz cover of profile (µ
1
. . . , µ
n
) if it has n labeled poles
of orders (µ
1
. . . , µ
n
) and is simply ramified otherwise. Introduce the Laplace transform of
single Hurwitz numbers by
(1.8) F
H
g,n
t(w
1
), . . . , t(w
n
)
=
X
~µ∈Z
n
+
H
g,n
(~µ)e
−(w
1
µ
1
+···w
n
µ
n
)
,
SPECTRAL CURVES AND THE SCHR
¨
ODINGER EQUATION 5
where
t(w) =
∞
X
m=0
m
m
m!
e
−mw
is the tree-function. Here again F
H
g,n
(t
1
, . . . , t
n
) is a polynomial of degree 6g − 6 + 3n if
2g − 2 + n > 0 [60]. Bouchard and Mari˜no have conjectured [9] that
W
H
g,n
(t
1
, . . . , t
n
) = d
1
···d
n
F
H
g,n
(t
1
, . . . , t
n
)
satisfy the Eynard-Orantin topological recursion, with respect to the Lagrangian immersion
(1.9)
(
x = e
−w
= ze
−z
∈ C
∗
y = z ∈ C,
where
z =
t − 1
t
and we use η = y
dx
x
as the tautological 1-form on T
∗
C
∗
. The Bouchard-Mari˜no conjecture
was proved in [7, 26, 60].
Now we define the Hurwitz partition function
(1.10) Z
H
(t, ~) = exp
∞
X
g=0
∞
X
n=1
1
n!
~
2g−2+n
F
H
g,n
(t, . . . , t)
.
Then we have the following
Theorem 1.3. The Hurwitz partition function satisfy two equations:
1
2
~
∂
2
∂w
2
+
1 +
1
2
~
∂
∂w
− ~
∂
∂~
Z
H
t(w), ~
= 0,(1.11)
~
d
dw
+ e
−w
e
−~
d
dw
Z
H
t(w), ~
= 0.(1.12)
Moreover, each of the two equations recover the Lagrangian immersion (1.9) from the as-
ymptotic analysis at ~ ∼ 0. And if we view ~ as a fixed constant in (1.12), then its total
symbol is the Lagrangian immersion (1.9) with the identification z = −~
d
dw
.
Remark 1.4. The second order equation (1.11) is a consequence of the polynomial recursion
of [60]. This situation is exactly the same as Theorem 1.1. The differential-difference
equation (1.12), or a delay differential equation, follows from the principal specialization of
the KP τ-function that gives another generating function of single Hurwitz numbers [41, 67].
We remark that (1.12) is also derived in [82]. The point of view of differential-difference
equation is further developed in [59] for the case of double Hurwitz numbers and r-spin
structures, where we generalize a result of [82].
Remark 1.5. Define two operators by
P = ~
d
dw
+ e
−w
e
−~
d
dw
and(1.13)
Q =
1
2
~
∂
2
∂w
2
+
1 +
1
2
~
∂
∂w
− ~
∂
∂~
.(1.14)
Then it is noted in [49] that
(1.15) [P, Q] = P.
