Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals
We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coin...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
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Інститут математики НАН України
2022
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| Цитувати: | Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals. Si-Qi Liu, Zhe Wang and Youjin Zhang. SIGMA 18 (2022), 037, 18 pages |
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| author | Liu, Si-Qi Wang, Zhe Zhang, Youjin |
| author_facet | Liu, Si-Qi Wang, Zhe Zhang, Youjin |
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| description | We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy, which controls the special cubic Hodge integrals.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 037, 18 pages
Reduction of the 2D Toda Hierarchy
and Linear Hodge Integrals
Si-Qi LIU, Zhe WANG and Youjin ZHANG
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
E-mail: liusq@tsinghua.edu.cn, zhe-wang17@mails.tsinghua.edu.cn, youjin@tsinghua.edu.cn
Received October 28, 2021, in final form May 15, 2022; Published online May 18, 2022
https://doi.org/10.3842/SIGMA.2022.037
Abstract. We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-
symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the
linear Hodge integrals in the way that one part of its flows yields the intermediate long wave
hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional
Volterra hierarchy which controls the special cubic Hodge integrals.
Key words: integrable hierarchy; limit fractional Volterra hierarchy; intermediate long wave
hierarchy
2020 Mathematics Subject Classification: 53D45; 37K10; 37K25
1 Introduction
Let Mg,n be the moduli space of stable curves of genus g with n marked points, and Li be the
i-th tautological line bundle of Mg,n whose first Chern class is denoted by ψi for i = 1, . . . , n.
Let Eg,n be the Hodge bundle of Mg,n and γj ∈ Hj(Mg,n) be the degree j component of its
Chern character. In [6], the following generating function of Hodge integrals is studied:
H(t; s; ε) =
∑
g,n,m≥0
k1,...,kn≥0
l1,...,lm≥1
ε2g−2 tk1 · · · tkn
n!
sl1 · · · slm
m!
∫
Mg,n
ψk1
1 · · ·ψkn
n γ2l1−1 · · · γ2lm−1. (1.1)
Note that only odd components of the Chern character are considered due to the vanishing of
even components γ2j by Mumford’s relation [15]. It is proved that the evolutions of the two-point
function
w = ε2
∂2
∂t20
H(t; s; ε)
along the time variables tn form a tau-symmetric Hamiltonian integrable hierarchy which is
called the Hodge hierarchy.
When the parameters sk are taken to be equal to some special values, the Hodge hierarchy
degenerates to some well-known integrable hierarchies. For example, by taking sk = 0, we
recover the Korteweg–de Vries (KdV) hierarchy. When sk = (2k − 2)!s2k−1, (1.1) reduces to
a generating function of linear Hodge integrals and the corresponding integrable hierarchy is
proved to be the intermediate long wave (ILW) hierarchy in [3]. For arbitrary given non-zero
numbers p, q, r satisfying the local Calabi–Yau condition
1
p
+
1
q
+
1
r
= 0,
mailto:liusq@tsinghua.edu.cn
mailto:zhe-wang17@mails.tsinghua.edu.cn
mailto:youjin@tsinghua.edu.cn
https://doi.org/10.3842/SIGMA.2022.037
2 S.-Q. Liu, Z. Wang and Y. Zhang
we obtain the generating function of the special cubic Hodge integrals
H(t; p, q, r; ε) =
∑
g≥0
ε2g−2
∑
n≥0
tk1 · · · tkn
n!
∫
Mg,n
ψk1
1 · · ·ψkn
n Cg(−p)Cg(−q)Cg(−r) (1.2)
from (1.1) by setting
sj = −(2j − 2)!
(
p2j−1 + q2j−1 + r2j−1
)
,
where Cg(z) is the Chern polynomial of the Hodge bundle Eg,n. In [11], it is proved that the
corresponding integrable hierarchy is the fractional Volterra (FV) hierarchy which is constructed
in [12]. This fact is called the Hodge-FVH correspondence.
For a fixed parameter p ̸= 0, we see from the local Calabi–Yau condition pq + qr + rp = 0
that when q tends to zero so does r = −pq/(p+ q). Hence by taking such a limit, the generating
function (1.2) becomes a generating function of linear Hodge integrals. A natural question is
whether we can also take the limit of the FV hierarchy to obtain the ILW hierarchy? The answer
to this question is not straightforward due to the fact that the construction of the FV hierarchy
involves a complicated infinite linear combination of the time variables [11]. It turns out that
one family of the flows of the FV hierarchy does not admit a limit when q, r → 0 (see Section 3
for details) and the other family does have a limit, and this limit can be viewed as infinite linear
combinations of the flows of the ILW hierarchy.
To illustrate the limit procedure more explicitly, let us consider the following equation which
is one of the flows of the FV hierarchy:
∂u
∂t
=
(
Λ−1/r − 1
)(
1− Λ−1/q
)
ε
(
Λ1/p − 1
) eu,
where Λ = exp(ε∂x) is the shift operator. After the rescaling ε 7→ qε, t 7→ q2t, we can take the
limit q → 0 of the above equation, and obtain the following equation:
∂u
∂t
= p
(
1− Λ−1
)
(Λ− 1)
ε2∂x
eu = peuux + p
ε2
12
eu
(
uxxx + 3uxuxx + u3x
)
+O
(
ε4
)
. (1.3)
On the other hand, from the Lax equations of the ILW hierarchy [4] it follows that the first
nontrivial flow of the ILW hierarchy reads
∂w
∂s1
= wwx + ε
τ∂2x
2
Λ + 1
Λ− 1
w − τwx,
and all other flows have the form
∂w
∂sn
=
wn
n!
wx + ε2τ
(
anw
n−1wxxx + bnw
n−2wxxwx + cnw
n−3w3
x
)
+O
(
ε4
)
,
an =
(n− 1)!
12
, bn =
(n− 2)!
6
, cn =
(n− 3)!
24
, n ≥ 1,
here τ is a parameter of the ILW hierarchy and we assume n! = 0 for n < 0. Let us define a flow
by an infinite linear combination of the flows of the ILW hierarchy:
∂
∂s
:=
∑
n≥1
∂
∂sn
,
then we obtain the following expression for this flow
∂w
∂s
= ewwx +
ε2
24
ew
(
2wxxx + 4wxwxx + w3
x
)
+O
(
ε4
)
,
Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals 3
here we take τ = 1 for simplicity. It is straightforward to verify that after a Miura type
transformation
u = w +
ε2
24
wxx +O
(
ε4
)
,
we arrive at
p
∂u
∂s
= peuux + p
ε2
12
eu
(
uxxx + 3uxuxx + u3x
)
+O
(
ε4
)
,
which coincides with the flow (1.3). Therefore we see that the limit of the FV hierarchy is related
to the ILW hierarchy by infinite linear combinations of flows.
Since both the ILW hierarchy and the limit of the FV hierarchy correspond to the linear
Hodge integrals, we expect that these two integrable hierarchies are compatible with each other
and that there exists an integrable hierarchy that contains these two hierarchies. It turns out
that such an integrable hierarchy indeed exists and it is given by a certain reduction of the 2D
Toda hierarchy.
The 2D Toda hierarchy [19, 20] is one of the central objects of study in the theory of integrable
systems and its various reductions play important roles in the field of mathematical physics.
For example, the 1D Toda hierarchy (and its extension), the equivariant Toda hierarchy and
the Ablowitz–Ladik hierarchy control the Gromov–Witten theory of P1 [5, 7], the equivariant
Gromov–Witten theory of P1 [16] and the Gromov–Witten theory of local P1 [2] respectively.
For more applications of the 2D Toda hierarchy, one may refer to [17] and the references therein.
Recall that the 2D Toda hierarchy can be described in terms of the operators [20]
L = Λ+
∑
n≥0
unΛ
−n, L = ū−1Λ
−1 +
∑
n≥0
ūnΛ
n
by the following Lax equations:
∂L
∂t1,n
=
[
(Ln)+, L
]
,
∂L
∂t2,n
= −
[(
L
n)
−, L
]
,
∂L
∂t1,n
=
[
(Ln)+, L
]
,
∂L
∂t2,n
= −
[(
L
n)
−, L
]
.
Let us consider the following reduction of the 2D Toda hierarchy
logL = pL− logL = pK,
where the precise definition of the logarithm of L and L will be given in Section 2,
K =
1
p
ε∂x + euΛ−1, (1.4)
and p is a parameter. Then we define the reduction of the flows by
ε
∂K
∂t1,n
= [(Ln)+,K], ε
∂K
∂t2,n
= −
[(
L
n)
−,K
]
, n ≥ 1.
This is a well-defined reduction of the 2D Toda hierarchy, and due to the reason that the construc-
tion of this reduction is inspired by taking the limit of the FV hierarchy, we call this integrable
hierarchy the limit fractional Volterra (LFV) hierarcy. We summarize its main properties in the
following theorem.
4 S.-Q. Liu, Z. Wang and Y. Zhang
Theorem 1.1. The LFV hierarchy is a tau-symmetric Hamiltonian integrable hierarchy with
hydrodynamic limit. Moreover, the flows ∂
∂t1,n
of the LFV hierarchy are certain limits of the
flows of the FV hierarchy and the flows ∂
∂t2,n
is equivalent to the ILW hierarchy under a certain
Miura-type transformation.
The paper is organized as follows. In Section 2, we give the definition of the LFV hierarchy
and prove that it is a tau-symmetric Hamiltonian integrable hierarchy. In Section 3, we explain
how the construction of the LFV hierarchy is inspired by taking a certain limit of the FV
hierarchy, and we also obtain a limit of the Hodge-FVH correspondence. In Section 4, we relate
the LFV hierarchy to the ILW hierarchy. Finally in Section 5, we give some concluding remarks
about the relation between our work and the Gromov–Witten/Hurwitz theory.
2 The LFV hierarchy and its properties
Throughout this paper, we work with the ring of differential polynomials R(u). It consists
of formal power series in ε with coefficients being elements in the polynomial ring C∞(u) ⊗
C
[
u(k) : k ≥ 1
]
. Let us define a derivation ∂x and an automorphism Λ on R(u) by
∂x =
∑
k≥0
u(k+1) ∂
∂u(k)
, Λ = exp(ε∂x).
If we view u = u(x) as a function of the spatial variable x, then it is easy to see that u(k) = ∂kxu(x)
and Λu(x) = u(x + ε). For this reason, the operator Λ is called the shift operator. Note that
the ring R(u) is graded with respect to the differential degree degx given by degx u
(k) = k.
Let us consider the following two difference operators given by
L = Λ+ a0 + a1Λ
−1 + · · · , (2.1)
L = euΛ−1 + b0 + b1Λ + · · · , (2.2)
where ai, bi ∈ R(u) for i ≥ 0. In [20], the dressing operators P , Q for these difference operators
are defined by
L = PΛP−1, P = 1 +
∑
k≥1
pkΛ
−k, (2.3)
L = QΛ−1Q−1, Q =
∑
k≥0
qkΛ
k. (2.4)
The coefficients pk and qk of the dressing operators P and Q are not in the ring R(u) but only
exist in a certain extension of R(u) (for details, see [20]). Note that the choice of P and Q is
unique up to the right multiplication by difference operators with constant coefficients. Follo-
wing [5], we define the logarithm of the operators L, L as follows:
logL := P (ε∂x)P
−1 = ε∂x − εPxP
−1,
logL := Q(−ε∂x)Q−1 = −ε∂x + εQxQ
−1,
where Px =
∑
k≥1(∂xpk)Λ
−k and Qx =
∑
k≥0(∂xqk)Λ
k. The ambiguities of choices of P and Q
are canceled in the operators PxP
−1 and QxQ
−1, and they are difference operators with coef-
ficients belonging to R(u) [5]. Before giving the definition of the LFV hierarchy, let us make
some preparations by proving the following lemmas.
Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals 5
Lemma 2.1. There exist unique differential polynomials ak such that
lim
ε→0
ak =
pk+1
(k + 1)!
e(k+1)u (2.5)
and the difference operator L defined in (2.1) satisfies the relation
1
p
logL = K, (2.6)
here p is a formal parameter and K is the differential-difference operator defined by (1.4).
Proof. Let us first find ai ∈ R(u) such that the operator L defined in (2.1) satisfies the identities
res
[
Ln, peuΛ−1
]
= ε∂x resL
n, n ≥ 1, (2.7)
here and henceforth, for any difference operator D =
∑
k fkΛ
k with fk ∈ R(u), we define its
residue by resD = f0. By taking n = 1 in the equation (2.7), we arrive at
p(Λ− 1)eu = ε∂xa0.
Then the above equation for a0 has a unique solution by taking the integral constant to be zero,
i.e.,
a0 = p
Λ− 1
ε∂x
eu. (2.8)
For general n ≥ 1, one can show by induction that if we represent Ln =
∑
k f
n
k Λ
k, then the
differential polynomials fnk can be viewed as functions in ai and we have
fnk = fnk (a0, . . . , an−1−k), k ≤ n− 1,
fn0 =
(
1 + Λ + · · ·+ Λn−1
)
an−1 + gn(a0, . . . , an−2).
Therefore we see that the differential polynomials can be found recursively by taking n = 2, 3, . . .
in the equation (2.7). More explicitly, if we have found differential polynomials a0, . . . , an−1, to
determine an, we consider the equation
res
[
Ln+1, peuΛ−1
]
= ε∂x resL
n+1
and obtain that
p
(
1− Λ−1
) (
fn+1
1 (a0, . . . , an−1)Λe
u
)
= (1 + Λ + · · ·+ Λn) ε∂xan + ε∂xgn+1(a0, . . . , an−1).
Hence we can solve an uniquely by taking the integral constant to be zero.
Next we show that the difference operator L determined by the equations (2.7) satisfies the
relation (2.6). Indeed, by using the results given in [5], we see that the equations (2.7) imply
that
εPxP
−1 = −peuΛ−1,
and therefore the relation 1
p logL = K holds true.
Finally we show that the differential polynomials ai determined above satisfy the rela-
tion (2.5). From the relation (2.6) it follows that [L, pK] = 0, which is equivalent to the
following recursion relation:
1
p
ε∂xak+1 = akΛ
−keu − euΛ−1ak, k ≥ 0. (2.9)
Then (2.5) can be verified using this recursion relation and the initial condition (2.8). The
lemma is proved. ■
6 S.-Q. Liu, Z. Wang and Y. Zhang
Lemma 2.2. There exist unique differential polynomials bk such that
lim
ε→0
bk =
e−ku
pk+1
βk
and the difference operator L defined in (2.2) satisfies the relations
L− 1
p
logL = K,
where K is the differential-difference operator given by (1.4) and βk are polynomials in u satis-
fying the recursion relations
βk+1 = βk −
∫ u
0
kβk du, k ≥ 0,
with the initial condition β0 = u.
Proof. The lemma can be proved by using a similar method that is used in the proof of
Lemma 2.1, so we omit the details here. For later use, we write down the following recursion
relations satisfied by bk:
1
p
ε∂xbk = bk+1Λ
k+1eu − euΛ−1bk+1, k ≥ −1, (2.10)
with b−1 = eu. ■
Definition 2.3. The limit fractional Volterra (LFV) hierarchy consists of the flows
ε
∂K
∂t1,n
= [(Ln)+,K], ε
∂K
∂t2,n
= −
[(
L
n)
−,K
]
, n ≥ 1, (2.11)
where the operator K is given by (1.4), and the opertors L and L are determined by Lemma 2.1
and Lemma 2.2 respectively. Here and in what follows, for a difference operator
∑
k fkΛ
k, we
define its positive part and negative part by(∑
k
fkΛ
k
)
+
=
∑
k≥0
fkΛ
k,
(∑
k
fkΛ
k
)
−
=
∑
k<0
fkΛ
k.
It follows from Lemmas 2.1 and 2.2 that the operators L and L commute with K, therefore
we have the following identities:
[(Ln)+,K] = −[(Ln)−,K],
[(
L
n)
+
,K
]
= −
[(
L
n)
−,K
]
.
Since [(Ln)+,K] is a difference operator of the form
∑
j≥−1 fjΛ
j and [(Ln)−,K] is of the form∑
j≤−1 gjΛ
j , it follows that the first set of equations given in (2.11) yields a hierarchy of well-
defined equations of u. Similarly the second set of equations given in (2.11) is also well-defined.
Moreover, it is easy to see that the flows of the LFV hierarchy are given by differential polyno-
mials with hydrodynamic limits.
Example 2.4. The simplest flows of (2.11) read
∂u
∂t1,1
= p
(
1− Λ−1
)
(Λ− 1)
ε2∂x
eu,
∂u
∂t2,1
=
1
p
ux,
∂u
∂t2,2
=
ux
p2
ε∂x
Λ + 1
Λ− 1
u+
1
p2
ε∂2x
Λ + 1
Λ− 1
u.
Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals 7
Let us proceed to present some basic properties of the LFV hierarchy. We will show that
it is a tau-symmetric Hamiltonian integrable hierarchy. We start with proving that the flows
defined in (2.11) mutually commute. The following lemma is standard in the theory of integrable
hierarchies.
Lemma 2.5. The following equations are satisfied by the operators L and L:
ε
∂L
∂t1,n
= [(Ln)+, L], ε
∂L
∂t2,n
= −
[(
L
n)
−, L
]
,
ε
∂L
∂t1,n
=
[
(Ln)+, L
]
, ε
∂L
∂t2,n
= −[
(
L
n)
−, L
]
.
From Lemma 2.5, it is straightforward to derive the commutation relations of the flows (2.11).
Theorem 2.6. The flows (2.11) of the LFV hierarchy mutually commute, i.e.,[
∂
∂t1,n
,
∂
∂t1,m
]
=
[
∂
∂t2,n
,
∂
∂t2,m
]
=
[
∂
∂t1,n
,
∂
∂t2,m
]
= 0, n,m ≥ 1.
To derive the Hamiltonian formalism of the LFV hierarchy, we need to compute the variational
derivatives of the local functionals of the form∫
resLn,
∫
resL
n
.
Let us give a general description on how such a variational derivative can be computed (see [5]).
Consider the 1-form∑
k≥0
fkdu
(k), fk ∈ R(u),
where d is the natural exterior differential operator on R(u). For two 1-forms
∑
k≥0 fkdu
(k)
and
∑
k≥0 gkdu
(k), we denote
∑
k≥0 fkdu
(k) ∼
∑
k≥0 gkdu
(k) if there exists another 1-form∑
k≥0 hkdu
(k) such that:
∑
k≥0
fkdu
(k) −
∑
k≥0
gkdu
(k) = ∂x
(∑
k≥0
hkdu
(k)
)
,
here the derivation ∂x acts on the 1-form by ∂xdu
(k) = du(k+1). Now for a local functional
∫
h,
h ∈ R(u), we can compute its variational derivative as follows:
dh =
∑
k≥0
∂h
∂u(k)
du(k) ∼
(
δ
δu
∫
h
)
du. (2.12)
Lemma 2.7. Consider the local functionals defined by
Hn =
1
np
∫
resLn, n ≥ 1.
Then their variational derivatives have the expressions
δHn
δu
=
1
p
ε∂x
Λ− 1
resLn, n ≥ 1. (2.13)
8 S.-Q. Liu, Z. Wang and Y. Zhang
Proof. We need the following identity whose proof can be found in [5]:
resLn−1dL ∼ − resLnd
(
εPxP
−1
)
, (2.14)
where the operator P is defined in (2.3). From this identity and the relation (2.6) it follows that
d
(
1
np
resLn
)
∼ 1
p
resLn−1dL ∼ −1
p
resLnd
(
εPxP
−1
)
= resLn
(
euduΛ−1
)
.
Here and henceforth, by abusing the notations we denote
d
(∑
n
fkΛ
k
)
=
∑
k
(dfk)Λ
k
for a difference operator
∑
k fkΛ
k.
Let us represent Ln in the form Ln =
∑
k ak,nΛ
k, we then arrive at the following relations:
d
(
1
np
resLn
)
∼ a1,nΛ
(
eudu
)
∼
(
euΛ−1a1,n
)
du.
Hence by using the formula (2.12) we obtain
δHn
δu
= euΛ−1a1,n, n ≥ 1.
Finally by using the identity res[Ln, peuΛ−1] = res ε∂xL
n obtained from Lemma 2.1, we prove
the desired identity (2.13). The lemma is proved. ■
Lemma 2.8. The variational derivatives of the local functionals
Hn =
∫ (
res
L
n+1
n+ 1
− res
L
n
np
)
, n ≥ 1
can be represented as
δHn
δu
=
1
p
ε∂x
Λ− 1
resL
n
, n ≥ 1. (2.15)
Proof. Similar to the identity(2.14) we have
resL
n−1
dL ∼ resL
n
d
(
εQxQ
−1
)
.
By using Lemma 2.2 we obtain the relations
resL
n
dK = resL
n
dL− 1
p
resL
n
d
(
εQxQ
−1
)
∼ resL
n
dL− 1
p
resL
n−1
dL
∼ res d
(
L
n+1
n+ 1
− L
n
np
)
.
We then arrive at (2.15) by applying the calculation similar to the one we do in the proof of
Lemma 2.7. The lemma is proved. ■
Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals 9
Theorem 2.9. The flows (2.11) can be represented as Hamiltonian systems as follows:
ε
∂u
∂t1,n
= {u(x), Hn}, ε
∂u
∂t2,n
=
{
u(x), Hn
}
, n ≥ 1,
where the Poisson bracket {−,−} is defined by the Hamiltonian operator
P = p
(
1− Λ−1
)
(Λ− 1)
ε2∂x
.
Proof. From the definition (2.11) it is straightforward to see that
ε
∂u
∂t1,n
=
(
1− Λ−1
)
resLn = εP
δHn
δu
= ε{u(x), Hn}.
As for the flows ∂u
∂t2,n
, let us denote L
n
=
∑
k bk,nΛ
k, then by combining the definition (2.11)
and the definitions of L and L, we arrive at
εeu
∂u
∂t2,n
=
1
p
ε∂xb−1,n = eu
(
1− Λ−1
)
resL
n
,
which implies the Hamiltonian formalism of the flows ∂u
∂t2,n
due to Lemma 2.8. Finally it is
obvious that P is indeed a Hamiltonian operator and thus the theorem is proved. ■
Finally we are going to consider the tau-structure of the LFV hierarchy. The constructions
and proofs are standard and very similar to those presented in [12].
Lemma 2.10. We define the following functions for k, l ≥ 1:
Ω1,k;1,l =
l∑
n=1
Λn − 1
Λ− 1
(
res
(
Λ−nLl
)
res
(
LkΛn
))
, (2.16)
Ω2,k;1,l = Ω1,l;2,k =
l∑
n=1
Λn − 1
Λ− 1
(
res
(
Λ−nLl
)
res
(
L
k
Λn
))
, (2.17)
Ω2,k;2,l =
l∑
n=1
Λn − 1
Λ− 1
(
res
(
Λ−nL
l)
res
(
L
k
Λn
))
. (2.18)
Then the following relations hold true:
ε
∂
∂t1,l
resLk = ε
∂
∂t1,k
resLl = (Λ− 1)Ω1,k;1,l,
ε
∂
∂t1,l
resL
k
= ε
∂
∂t2,k
resLl = (Λ− 1)Ω2,k;1,l,
ε
∂
∂t2,l
resL
k
= ε
∂
∂t2,k
resLl = (Λ− 1)Ω2,k;2,l.
Lemma 2.11. The functions defined in (2.16)–(2.18) satisfy the following identities:
Ωi,k;j,l = Ωj,l;i,k,
∂Ωj,l;m,n
∂ti,k
=
∂Ωi,k;m,n
∂tj,l
, i, j,m = 1, 2, k, l, n ≥ 1.
10 S.-Q. Liu, Z. Wang and Y. Zhang
Proof. Let us start by proving the first set of identities. It follows from Lemma 2.10 that
(Λ− 1)Ωi,k;j,l = (Λ− 1)Ωj,l;i,k.
Therefore there exist constants ci,k;j,l such that
Ωi,k;j,l − Ωj,l;i,k = ci,k;j,l.
To verify that the constants cj,k;j,l vanish, we may compute the limits of the differential poly-
nomials Ωi,k;j,l by setting u = 0 and u(k) = 0. It is easy to see from Lemmas 2.1 and 2.2 that
the only non-trivial case is i = j = 1. By using Lemma 2.1, it follows from a straightforward
computation that
Ω1,k;1,l|u=u(i)=0 = pk+lkkll
l∑
n=1
(
k
l
)n n
(l − n)!(k + n)!
.
The summation on the right hand side of the above identity is evaluated using Gosper’s algo-
rithm [8].1 Let us denote
yn =
(
k
l
)n n
(l − n)!(k + n)!
, 1 ≤ n ≤ l,
and
zn = −
(
k
l
)n l
(k + l)(l − n)!(k + n− 1)!
, 1 ≤ n ≤ l.
Then it is straightforward to verify that
zn+1 − zn = yn, 1 ≤ n ≤ l − 1, zl = −yl.
Hence we conclude that
l∑
n=1
(
k
l
)n n
(l − n)!(k + n)!
= −z1 =
1
(k + l)(k − 1)!(l − 1)!
.
So we see that Ω1,k;1,l|u=u(i)=0 is symmetric with respect to the indices k and l and therefore we
see that Ω1,k;1,l = Ω1,l;1,k.
On the other hand, it follows from Lemma 2.10 that
(Λ− 1)
∂Ωj,l;m,n
∂ti,k
= (Λ− 1)
∂Ωi,k;m,n
∂tj,l
. (2.19)
Since the differential polynomials
∂Ωj,l;m,n
∂ti,k
have differential degrees greater than 1, from (2.19)
we arrive at the validity of the second set of identities of the lemma. The lemma is proved. ■
Theorem 2.12. For any solution u(x; t) of the LFV hierarchy, there exists a tau-function τ(x; t)
such that:
(Λ− 1)
(
1− Λ−1
)
log τ = u, (2.20)
ε(Λ− 1)
∂ log τ
∂t1,k
= resLk, (2.21)
ε(Λ− 1)
∂ log τ
∂t2,k
= resL
k
, (2.22)
ε2
∂2 log τ
∂ti,k∂tj,l
= Ωi,k;j,l. (2.23)
1Here we use a Mathematica package provided on the website https://www2.math.upenn.edu/~wilf/progs.
html.
https://www2.math.upenn.edu/~wilf/progs.html
https://www2.math.upenn.edu/~wilf/progs.html
Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals 11
Proof. The compatibility of the equations (2.20)–(2.23) is given by the definition of LFV hie-
rarchy (2.11), Lemmas 2.10 and 2.11. Hence such a tau-function exists and we prove the theo-
rem. ■
3 The LFV hierarchy as a limit of the FV hierarchy
Let us explain how the LFV hierarchy can be viewed as a certain limit of the FV hierarchy
given in [12]. Let p, q, r be any given non-zero complex numbers satisfying the condition
pq + qr + rp = 0. We introduce the shift operators
Λ1 = Λ1/q, Λ2 = Λ1/p, Λ3 = Λ1Λ2 = Λ−1/r, Λ = eε∂x ,
and consider the following Lax operator
LFV = Λ2 + evΛ−1
1
with v = v(x, ε). The fractional powers
A =
(
LFV
)−p/r
= Λ3 +
∑
k≥0
fkΛ
−k
3 , B =
(
LFV
)−q/r
= g−1Λ
−1
3 +
∑
k≥0
gkΛ
k
3
are well-defined, and their coefficients fk, gk belong to the ringR(v). It follows from the relations[
A, LFV
]
=
[
B, LFV
]
= 0
that the coefficients fn and gn satisfy the following recursion relations:
(Λ2 − 1)fk+1 = fkΛ
−k
3 ev − evΛ−1
1 fk, k ≥ 0, f0 =
Λ3 − 1
Λ2 − 1
ev, (3.1)
(Λ2 − 1)gk = gk+1Λ
k+1
3 ev − evΛ−1
1 gk+1, k ≥ −1, g−1 = e
1−Λ−1
3
1−Λ−1
1
v
. (3.2)
The fractional Volterra hierarchy is defined by the following Lax equations:
ε
∂LFV
∂T1,n
=
[
An
+, L
FV
]
, ε
∂LFV
∂T2,n
= −
[
Bn
−, L
FV
]
. (3.3)
Here for an operator C of the form
∑
ckΛ
k
3, we denote
C+ =
∑
k≥0
ckΛ
k
3, C− =
∑
k<0
ckΛ
k
3.
In order to take a certain limit of the FV hierarchy, we first introduce a new dispersion parameter
ε̃ = ε/q and the associated shift operator Λ̃ = eε̃∂x . We have the relations
Λ1 = Λ̃, Λ3 = Λ̃
p+q
p ,
and we can rewrite the recursion relations (3.1) and (3.2) as follows:∑
i≥1
ε̃i
i!
qi−1
pi
∂ixf̃k+1 = f̃kΛ̃
− k(p+q)
p ev − evΛ̃−1f̃k, k ≥ 0, (3.4)
∑
i≥1
ε̃i
i!
qi−1
pi
∂ixg̃k = g̃k+1Λ̃
(k+1)(p+q)
p ev − evΛ̃−1g̃k+1, k ≥ −1, (3.5)
12 S.-Q. Liu, Z. Wang and Y. Zhang
here
f̃k = qk+1fk, g̃k =
gk
qk+1
.
Then it is easy to see that the recursion relations (3.4) and (3.5) become the relations (2.9)
and (2.10) after taking the limit q → 0.
Let us look at the relations (3.1) and (3.4) more carefully. The coefficients fi of the operator A
can be uniquely determined from (3.1) by requiring that
lim
ε→0
fk =
pk+1ekv
(k + 1)!qk+1
k+1∏
i=1
(
1 +
(i− k)q
p
)
.
Therefore if we assume the limit
u(x, ε̃) = lim
q→0
v(x, qε̃)
exists, then it is easy to see from (3.4) that the limits
ak(x, ε) = lim
q→0
qk+1fk(x, qε̃)|ε̃7→ε = lim
q→0
f̃k(x, qε̃)|ε̃ 7→ε, k ≥ 0 (3.6)
satisfy the relations (2.8) and (2.9), and are exactly the coefficients of the operator L described
in Lemma 2.1.
By using the above-mentioned observation, we can relate the flows ∂u
∂t1,n
of the LFV hierarchy
to the flows ∂v
∂T1,n
of the FV hierarchy as follows. From the definition (2.11) and (3.3), we can
write these flows as follows:
ε
∂u
∂t1,n
=
(
1− Λ−1
)
resLn, ε
∂v
∂T1,n
=
(
1− Λ−1
1
)
resAn.
By using the relations (3.6), one can prove that
resLn = lim
q→0
(
res qnAn|ε 7→qε
)
.
Thus we arrive at the relation
∂u
∂t1,n
= lim
q→0
(
qn+1 ∂v
∂T1,n
∣∣∣∣
ε 7→qε
)
.
Moreover, the Hamiltonian operator of the FV hierarchy is given by
PFV =
(
1− Λ−1
1
)
(Λ3 − 1)
Λ2 − 1
,
and it is easy to verify that the Hamiltonian operator of the LFV hierarchy can be obtained by
taking the following limit:
P = lim
q→0
(
qPFV|ε→qε
)
.
We note that the flows ∂
∂t2,n
and ∂
∂T 2,n are not related by means of taking the limit q → 0 even
though after taking such a limit the recursion relations (3.5) coincide with the relations (2.10).
Indeed, let us write down the flows
∂u
∂T2,1
=
1
p
ux,
∂v
∂T2,1
=
Λ2 − Λ−1
1
ε
exp
(
1− Λ−1
2
1− Λ−1
1
v
)
=
p+ q
pq
evvx +O(ε).
We conclude that we cannot obtain the flow ∂u
∂t2,1
by taking the limit of the flow ∂v
∂T2,1
.
Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals 13
The fact that the limits of the flows ∂v
∂T2,n
do not exist can also be obtained in view of the
Hodge-FVH correspondence [11]. It is proved in [11] that the following function gives a solution
of the FVH hierarchy:
v(x;T; ε) =
(
Λ
1/2
1 − Λ
−1/2
1
)(
Λ
1/2
3 − Λ
−1/2
3
)
H
(
t(x;T); p, q, r;
√
p+ q
pq
ε
)
,
where the variables ti and Tα,k are related by
ti(x;T) = xδi,0 + δi,1 − 1 +
1
pq
∑
k>0
(kp)i+1
(
k(p+ q)/q
k
)
T1,k + (kq)i+1
(
k(p+ q)/p
k
)
T2,k,
and the generating function H (t(x;T); p, q, r; ε) of the special cubic Hodge integral is defined
in (1.2). Therefore it follows from the relations between ti and Tα,k that there does not exist
a way to rescale T2,k by multiplying a suitable power of q such that after taking the limit q → 0
of ti the variables T2,k are still preserved.
However, by setting T2,k = 0 and by performing the change of variables T1,k 7→ qk+1t1,k
and ε 7→ qε, we can obtain the following corollary which can be viewed as a limit of the Hodge-
FVH correspondence.
Corollary 3.1. Let us denote by H(t; p; ε) the following generating function of the linear Hodge
integrals:
H(t; p; ε) =
∑
g≥0
ε2g−2
∑
n≥0
tk1 · · · tkn
n!
∫
Mg,n
ψk1
1 · · ·ψkn
n Cg(−p)
and denote by u(x; t1,k; ε) the function
u(x; t1,k; ε) =
(
Λ1/2 − Λ−1/2
)2H(t(x; t1,k); p; ε),
where the relations between the variables ti and t1,k are given by
ti =
∑
k≥0
ki+1+k
k!
pk+it1,k − 1 + xδi,0 + δi,1.
Then the function u(x; t1,k; ε) satisfies the equations which form a part of the LFV hierarchy:
ε
∂K
∂t1,n
= [(Ln)+,K],
where the differential-difference operator K and the difference operator L are given by
K =
1
p
ε∂x + euΛ−1, logL = pK.
4 Relation between the ILW hierarchy and the LFV hierarchy
Due to the discussion given in the last section, we expect that the LFV hierarchy should control
a certain generating function of the linear Hodge integrals over the moduli space of stable curves.
It is proved in [3] that the integrable hierarchy corresponds to the linear Hodge integral is the
intermediate-long wave (ILW) hierarchy. Therefore it is natural to establish relations between
the LFV hierarchy and the ILW hierarchy.
14 S.-Q. Liu, Z. Wang and Y. Zhang
We start by reviewing the Lax pair formalism of the ILW hierarchy given in [4]. Compared to
the original convention used in [4], we do some rescalings for the convenience of our presentation
given below. Consider the following operators K and L defined by
L − 1
p
logL = K, K = Λ+ w − 1
p
ε∂x, (4.1)
where the operator L can be written as
L = Λ+ w + c1Λ
−1 + c2Λ
−2 + · · · , ck ∈ R(w).
Similar as before, we denote by R(w) the ring of differential polynomials of w. The logarithm
of the operator L is defined, similar to logL, via the dressing operator as follows:
L = PΛP−1, logL = ε∂x − εPxP−1.
Using the operators defined above, the ILW hierarchy can be represented as
ε
∂K
∂s1,n
= [Ln
+,K], n ≥ 1. (4.2)
The differential polynomials ck are uniquely determined by the recursion relations
ck
(
1− Λ−k
)
w + (Λ− 1)ck+1 =
1
p
ε∂xck, k ≥ 0, c0 = w, (4.3)
and by the condition that the dispersionless limits of ck are given by limε→0 ck = γk, where γk
are polynomials given by
γk+1 =
1
p
γk −
∫ w
0
γk dw, k ≥ 0, γ0 = w.
The following theorem gives a direct relation between the ILW hierarchy and the LFV hier-
archy.
Theorem 4.1. The operator L defined in Lemma 2.2 and the operator L defined in (4.1) are
related via the following relation:
Λ1/2 resLk = resL
k
, k ≥ 1, (4.4)
where the functions u(x) and w(x) in the above identity are identified by the Miura-type trans-
formation
u(x) = p
Λ1/2 − Λ−1/2
ε∂x
w(x). (4.5)
Proof. We start with considering the dressing operator Q of the operator L described in
Lemma 2.2. From the definition of Q we see that its residue q0 satisfies the following equa-
tion:
q0 = euΛ−1q0. (4.6)
Taking derivatives of both sides of the above equation with respect to x, we arrive at
∂xq0
q0
=
∂x
1− Λ−1
u. (4.7)
Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals 15
Now let us perform a gauge transformation on L by the adjoint action of q0 to obtain
L = q−1
0 Lq0 = QΛ−1Q−1, Q = q−1
0 Q.
By using the relation
L− 1
p
logL = K
together with (4.6) and (4.7), we can check the validity of the following relation:
L − 1
p
logL = Λ−1 +
1
p
ε∂x
1− Λ−1
u+
1
p
ε∂x, (4.8)
here the logarithm of L is defined by Q as follows:
logL = −ε∂x + εQxQ
−1.
Denote L = Λ−1 +
∑
k≥0 fkΛ
k, then it follows from the identity[
L ,Λ−1 +
1
p
ε∂x
1− Λ−1
u+
1
p
ε∂x
]
= 0
that we can rewrite (4.8) as the following recursion relations for the coefficients fk:
fk
(
Λk − 1
)
Λ1/2w +
(
1− Λ−1
)
fk+1 =
1
p
ε∂xfk, k ≥ 0, f0 = Λ1/2w. (4.9)
Here we have already identified u(x) and w(x) via the relation (4.5). The dispersionless limits
of fk can be computed by using Lemma 2.2 and we obtain that limε→0 fk = δk, where δk are
polynomials given by
δk+1 =
1
p
δk −
∫ w
0
kδk dw, k ≥ 0, δ0 = w.
Recall that for a difference operator
∑
k gkΛ
k, its adjoint is defined to be
(∑
k gkΛ
k
)∗
=∑
k
(
Λ−kgk
)
Λ−k. Therefore if we denote L
∗
= Λ +
∑
k≥0 f
∗
kΛ
−k, we easily obtain from (4.9)
the following recursion relations:
f∗k
(
1− Λ−k
)
Λ1/2w + (Λ− 1)fk+1 =
1
p
ε∂xf
∗
k , k ≥ 0, f∗0 = Λ1/2w. (4.10)
Finally by comparing (4.10) with (4.3), we arrive at
Λ1/2LΛ−1/2 = L
∗
,
which implies (4.4). The theorem is proved. ■
The following corollary is straightforward and gives the exact correspondence between the
flows of the LFV hierarchy and the ILW hierarchy:
Corollary 4.2. The flows ∂w
∂s1,n
of the ILW hierarchy defined in (4.2) coincide with the flows
∂u
∂t2,n
of the LFV hierarchy given in (2.11) after identifying w(x) and u(x) by the Miura-type
transformation (4.5).
16 S.-Q. Liu, Z. Wang and Y. Zhang
Proof. From the definition (4.2) of the ILW flows it follows that
ε
∂w
∂s1,n
=
1
p
ε∂x resLn.
By using the Miura-type transformation (4.5) we arrive at
ε
∂u
∂s1,n
=
(
Λ1/2 − Λ−1/2
)
resLn.
On the other hand, from the definition (2.11) of the LFV flows we know that
ε
∂u
∂t2,n
=
(
1− Λ−1
)
resL
k
,
hence we complete the proof of the corollary by using the identity (4.4). ■
From Corollary 4.2 we conclude that after the Miura-type transformation (4.5), the flows
∂u
∂t1,n
of the LFV hierarchy are transformed to symmetries of the ILW hierarchy, which has a Lax
pair description given by the following constructions. The proofs of what follows are similar to
the ones for the LFV hierarchy and we omit the details.
Lemma 4.3. There exists a difference operator
L = d−1Λ
−1 +
∑
k≥0
dkΛ
k = QΛ−1Q−1, dk ∈ R(w),
such that the following relation holds true
1
p
logL = K, (4.11)
here K is defined in (4.1) and the logarithm of L is defined by
logL = −ε∂x + εQxQ−1.
Definition 4.4. We define the following symmetries for the ILW hierarchy:
ε
∂K
∂s2,n
= −
[
Ln
−,K
]
, (4.12)
where L is introduced in Lemma 4.3.
Example 4.5. The first flow defined in (4.12) reads
ε
∂w
∂s2,1
= (Λ− 1) exp
(
p
1− Λ−1
ε∂x
w
)
.
In a similar way as we prove Theorem 4.1, we can prove the following theorem.
Theorem 4.6. The operator L defined in Lemma 2.1 and the operator L defined in (4.11) satisfy
the following identity:
Λ1/2 resLk
= resLk, k ≥ 1,
where the functions u(x) and w(x) in the above identity are related by (4.5).
Corollary 4.7. The flows ∂w
∂s2,n
defined in (4.12) coincide with the flows ∂u
∂t1,n
of the LFV
hierarchy (2.11) after identifying w(x) and u(x) by (4.5). In particular, we have[
∂
∂s1,n
,
∂
∂s2,m
]
=
[
∂
∂s2,n
,
∂
∂s2,m
]
= 0, n,m ≥ 1.
Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals 17
5 Concluding remarks
In this paper, we consider the limiting procedure from the special cubic Hodge integrals to the
linear Hodge integrals in view of the theory of integrable hierarchies. By taking a certain limit
of the FV hierarchy, we obtain an integrable hierarchy which, together with the ILW hierarchy,
forms a reduction of the 2D Toda hierarchy. We call the resulting hierarchy the LFV hierarchy,
which is a Hamiltonian tau-symmetric integrable hierarchy with hydrodynamic limit.
This limiting procedure is quite natural in geometric setting, for example in the Gromov–
Witten theory or the Hurwitz theory. Our result can be viewed as an integrable hierarchy
theoretical interpretation of the Bouchard–Mariño conjecture [1] which is proved in [14]. In [1],
it is conjectured that the generating function of linear Hodge integrals can be computed in the
scheme of Eynard and Orantin topological recursion associated with the spectral curve
C =
{
x = ye−y
}
,
which is related to the symbol of the constraint (1.4). Their conjecture is based on a limiting
procedure of the Mariño–Vafa formula which relates the open amplitude of the A-model topo-
logical string on C3 with a framed brane on one leg of the toric diagram to the cubic Hodge
integrals [9, 10, 13]. The Mariño–Vafa formula can be used to derive a special case of Hodge-FVH
correspondence [18], and the results of the present paper explain the above limiting procedure
in view of the theory of integrable hierarchies. We thank the anonymous referee for pointing
out this relation to us.
Acknowledgements
This work is supported by NSFC no. 12171268, no. 11725104 and no. 11771238. We thank the
anonymous referees for helpful comments and suggestions to improve the presentation of the
paper.
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1 Introduction
2 The LFV hierarchy and its properties
3 The LFV hierarchy as a limit of the FV hierarchy
4 Relation between the ILW hierarchy and the LFV hierarchy
5 Concluding remarks
References
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| id | nasplib_isofts_kiev_ua-123456789-211631 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T16:30:27Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Liu, Si-Qi Wang, Zhe Zhang, Youjin 2026-01-07T13:41:29Z 2022 Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals. Si-Qi Liu, Zhe Wang and Youjin Zhang. SIGMA 18 (2022), 037, 18 pages 1815-0659 2020 Mathematics Subject Classification: 53D45; 37K10; 37K25 arXiv:2110.03317 https://nasplib.isofts.kiev.ua/handle/123456789/211631 https://doi.org/10.3842/SIGMA.2022.037 We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy, which controls the special cubic Hodge integrals. This work is supported by NSFC no. 12171268, no. 11725104 and no. 11771238. We thank the anonymous referees for helpful comments and suggestions to improve the presentation of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals Article published earlier |
| spellingShingle | Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals Liu, Si-Qi Wang, Zhe Zhang, Youjin |
| title | Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals |
| title_full | Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals |
| title_fullStr | Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals |
| title_full_unstemmed | Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals |
| title_short | Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals |
| title_sort | reduction of the 2d toda hierarchy and linear hodge integrals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211631 |
| work_keys_str_mv | AT liusiqi reductionofthe2dtodahierarchyandlinearhodgeintegrals AT wangzhe reductionofthe2dtodahierarchyandlinearhodgeintegrals AT zhangyoujin reductionofthe2dtodahierarchyandlinearhodgeintegrals |