Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 102, 22 pages
Old and New Reductions
of Dispersionless Toda Hierarchy?
Kanehisa TAKASAKI
Graduate School of Human and Environmental Studies, Kyoto University,
Yoshida, Sakyo, Kyoto, 606-8501, Japan
E-mail: takasaki@math.h.kyoto-u.ac.jp
URL: http://www.math.h.kyoto-u.ac.jp/~takasaki/
Received June 06, 2012, in final form December 15, 2012; Published online December 19, 2012
http://dx.doi.org/10.3842/SIGMA.2012.102
Abstract. This paper is focused on geometric aspects of two particular types of finite-
variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in
terms of “Landau–Ginzburg potentials” that play the role of reduced Lax functions. One
of them is a generalization of Dubrovin and Zhang’s trigonometric polynomial. The other
is a transcendental function, the logarithm of which resembles the waterbag models of the
dispersionless KP hierarchy. They both satisfy a radial version of the Lo¨wner equations.
Consistency of these Lo¨wner equations yields a radial version of the Gibbons–Tsarev equa-
tions. These equations are used to formulate hodograph solutions of the reduced hierarchy.
Geometric aspects of the Gibbons–Tsarev equations are explained in the language of classi-
cal differential geometry (Darboux equations, Egorov metrics and Combescure transforma-
tions). Flat coordinates of the underlying Egorov metrics are presented.
Key words: dispersionless Toda hierarchy; finite-variable reduction; waterbag model; Lan-
dau–Ginzburg potential; Lo¨wner equations; Gibbons–Tsarev equations; hodograph solution;
Darboux equations; Egorov metric; Combescure transformation; Frobenius manifold; flat
coordinates
2010 Mathematics Subject Classification: 35Q99; 37K10; 53B50; 53D45
1 Introduction
The notion of finite-variable reductions in dispersionless integrable hierarchies [24] has rich ge-
ometric contents that range from the classical differential geometry of orthogonal curvilinear
coordinates [10] to the modern theory of Frobenius manifolds [12]. Moreover, the Lo¨wner equa-
tions [25], first introduced to solve the Bieberbach conjecture in the univalent function theory,
also play a fundamental role in this issue. In this paper, we consider the dispersionless Toda
hierarchy [31]. Although a general scheme of finite-variable reductions in this case is already
established [6, 26, 33, 34], finding interesting examples is another issue. We here report two
examples, one being a generalization of an “old” example, and the other being a “new” one.
These examples turn out to fit well into the aforementioned geometric perspectives. We believe
that these examples have their own interesting features, part of which will be presented later on.
As in the case of other dispersionless integrable systems [24], a finite-variable reduction
of the dispersionless Toda hierarchy can be characterized by a globally defined reduced Lax
function λ(p). Borrowing the terminology of topological field theories [11], let us call this
reduced Lax function a “Landau–Ginzburg potential”1. The two examples addressed in this
paper have Landau–Ginzburg potentials of the following form:
?This paper is a contribution to the Special Issue “Geometrical Methods in Mathematical Physics”. The full
collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html
1This is somewhat problematical, because there is no guarantee that the reduced Lax function has an associated
topological field theory or a Frobenius manifold.
2 K. Takasaki
Case I:
λ(p) = p−N
M∏
i=1
(p− bi)
κi , where
M∑
i=1
κi −N > 0, κi 6= 0, N 6= 0. (1.1)
Case II:
λ(p) =
M∏
i=1
(p− bi)
κi exp
(
N∑
k=1
ckp
−k
)
, where
M∑
i=1
κi > 0, κi 6= 0, N > 0. (1.2)
(1.1) is a generalization of the well known trigonometric Landau–Ginzburg potential studied
by Dubrovin and Zhang [16]. Dubrovin and Zhang’s Landau–Ginzburg potential corresponds
to the case where N > 0 and κ1 = · · · = κM = 1. This case contains, as particular examples,
the dispersionless limit of the 1D Toda and bigraded Toda hierarchies [5]. By allowing κi’s
and N to take negative values, we can include therein, for example, the dispersionless limit of
the Ablowitz–Ladik hierarchy [1] and its possible generalizations as well. Thus this apparently
“old” case itself deserves to be studied in detail.
(1.2) is presumably a new example that has never been studied in the literature. This Landau–
Ginzburg potential is not a rational function of p. In this respect, (1.2) is conceptually similar
to the so called “waterbag” models that, too, have irrational Landau–Ginzburg potentials. The
waterbag models were first presented in the celebrated work of Gibbons and Tsarev [21, 22]
on reductions of the Benney hierarchy, and have been further studied in the dispersionless KP
hierarchy from various points of view [2, 7, 8, 20, 28]. In the most general formulation presented
by Ferguson and Strachan [20], the Landau–Ginzburg potential is a sum of a polynomial and
logarithmic terms of the following form
λ(p) = pN +
N∑
k=2
ckp
N−2 +
M∑
i=1
κi log(p− bi). (1.3)
Analogy with the waterbag models becomes more manifest in the logarithmic form
log λ(p) =
M∑
i=1
κi log(p− bi) +
N∑
k=1
ckp
−k (1.4)
of (1.2), which takes almost the same form as (1.3) except that the (Laurent) polynomial part
of the latter is a polynomial in p−1.2 In this sense, one may think of (1.2) as a variation of the
waterbag models.
Let us mention that there are a few proposals of waterbag models for the dispersionless
Toda hierarchy. The earliest one is Yu’s model [37]3. Seemingly unaware of Yu’s work, Chang
proposed three types of waterbag models [9]. (1.1) is a slight generalization of Chang’s first
model (which amounts to the case where N = −1, and for which Chang considered a Frobenius
structure). Chang’s second and third models are actually a generalization of Yu’s model. The
logarithmic expression (1.4) of our second case resembles Chang’s second and third models. We,
however, feel that the exponentiated form (1.2) is more natural than the logarithmic expression.
2Another delicate difference is that the polynomial part of (1.3) is monic (namely, the coefficient of the highest
degree term is 1) and has no next-highest degree term. In contrast, the (Laurent) polynomial part of (1.4) is not
monic, and has no constant term.
3Actually, Yu considered a dispersionless limit of the discrete KP hierarchy. This hierarchy is a subsystem
of the dispersionless Toda hierarchy, and one can readily translate Yu’s waterbag model to the language of the
latter.
Old and New Reductions of Dispersionless Toda Hierarchy 3
This paper is organized as follows. Section 2 is devoted to the Lax formalism. We show, by
a direct method, that (1.1) and (1.2) give consistent reductions of the Lax equations of the dis-
persionless Toda hierarchy. The two-variable cases turn out to contain some well known (as well
as new) dispersionless integrable hierarchies. Section 3 is focused on the Lo¨wner equations and
the associated Gibbons–Tsarev equations. The Lo¨wner equations relevant to the dispersionless
Toda hierarchy are a “radial” version of the “chordal” Lo¨wner equations in the aforementioned
work of Gibbons and Tsarev [21, 22]. We show that (1.1) and (1.2) satisfy the radial Lo¨wner
equations, and use these equations to formulate the generalized hodograph method [35, 36] for
these reductions. Section 4 presents some geometric implications of the Gibbons–Tsarev equa-
tions in the language of Darboux equations, Egorov metrics and Combescure transformations.
We show that three particular Egorov metrics underlie the radial Gibbons–Tsarev equations.
Section 5 deals with flat coordinates of the Egorov metrics. Results in this section is somewhat
restrictive. We encounter difficulties in generalizing the dual pair of Frobenius structures of
Dubrovin and Zhang [16] to the more general Landau–Ginzburg potential (1.1). Moreover, we
can construct only a single Frobenius structure without a dual for the Landau–Ginzburg poten-
tial (1.2). The method and the result for the latter, however, exhibits remarkable similarities
with the work of Ferguson and Strachan [20].
2 Lax equations
2.1 Lax formalism of dispersionless Toda hierarchy
The Lax equations of the dispersionless Toda hierarchy [31] are formulated by two Lax func-
tions4 z(p), z¯(p) of a spatial variable s, a momentum variable p, and two sets of time variables
t = (t1, t2, . . .) and t¯ = (t¯1, t¯2, . . .).5 p is a “classical limit” of the shift operator e∂/∂s that
satisfies the twisted canonical commutation relation
[
e∂/∂s, s
]
= e∂/∂s.
In the classical (or “long-wave”) limit, this commutation relation turns into the Poisson com-
mutation relation
{p, s} = p.
This Poisson bracket can be extended to arbitrary functions of s and p as
{f, g} = p
(
∂f
∂p
∂g
∂s
−
∂f
∂s
∂g
∂p
)
.
In the most general formulation, the Lax functions z(p) and z¯(p) are understood to be
mutually independent formal Laurent series of p of the form
z(p) = p+ u1 + u2p
−1 + · · · , z¯(p) = u¯0p
−1 + u¯1 + u¯2p+ · · · .
The coefficients un = un(s, t, t¯) and u¯n = u¯n(s, t, t¯) are dynamical variables. The leading
coefficient u¯0 is assumed to take the exponential form
u¯0 = e
φ, φ = φ(s, t, t¯).
4z and z¯ amount to L and L¯−1 in our previous notations [31].
5Throughout this paper, the overline “ ¯ ” does not mean complex conjugation. For example, tn and t¯n are
independent variables.
4 K. Takasaki
Let us define the polynomials Bn(p) and B¯n(p) in p, p−1 as
Bn(p) = (z(p)
n)≥0, B¯n(p) = (z¯(p)
n)<0,
where ( )≥0 and ( )<0 are projection operators acting on the linear space of Laurent series as
(
∞∑
n=−∞
anp
n
)
≥0
=
∑
n≥0
anp
n,
(
∞∑
n=−∞
anp
n
)
<0
=
∑
n<0
anp
n.
Time evolutions are generated by the Lax equations
∂z(p)
∂tn
= {Bn(p), z(p)},
∂z(p)
∂t¯n
= {B¯n(p), z(p)},
∂z¯(p)
∂tn
= {Bn(p), z¯(p)},
∂z¯(p)
∂t¯n
= {B¯n(p), z¯(p)}. (2.1)
It is convenient to introduce the complementary generators
Bcn(p) = (z(p)
n)<0, B¯
c
n(p) = (z¯(p)
n)≥0
as well. The Lax equations can be thereby rewritten as
∂z(p)
∂tn
= {z(p), Bcn(p)},
∂z(p)
∂t¯n
= {z(p), B¯cn(p)},
∂z¯(p)
∂tn
= {z¯(p), Bcn(p)},
∂z¯(p)
∂t¯n
= {z¯(p), B¯cn(p)}. (2.2)
2.2 Landau–Ginzburg potential as reduced Lax function
We now specialize the Lax equations to the case where the formal (or local) Lax functions z(p)
and z¯(p) are linked with the globally defined Landau–Ginzburg potential λ(p) as follows:
(I) For the Landau–Ginzburg potential (1.1),
z(p) = λ(p)1/M˜ as p→∞, z¯(p) = λ(p)1/N as p→ 0,
where M˜ =
M∑
i=1
κi −N . Recall that M˜ is assumed to be positive. N can be both positive
and negative.
(II) For the Landau–Ginzburg potential (1.2),
z(p) = λ(p)1/M˜ as p→∞, z¯(p) = (log λ(p))1/N as p→ 0,
where M˜ =
M∑
i=1
κi. Recall that M˜ and N is assumed to be positive.
The Lax equations (2.1) and its complementary form (2.2) thus reduce to the Lax equations
∂λ(p)
∂tn
= {Bn(p), λ(p)} = {λ(p), B
c
n(p)},
∂λ(p)
∂t¯n
= {B¯n(p), λ(p)} = {λ(p), B¯
c
n(p)} (2.3)
for λ(p). We can confirm that this is a consistent reduction procedure in the following sense.
Old and New Reductions of Dispersionless Toda Hierarchy 5
Theorem 1. The Lax equations (2.3) are equivalent to a system of first order evolutionary
equations of the form
∂bi
∂tn
= Fin,
∂bi
∂t¯n
= F¯in
for the Landau–Ginzburg potential (1.1) and
∂bi
∂tn
= Fin,
∂ck
∂tn
= Gkn,
∂bi
∂t¯n
= F¯in,
∂ck
∂t¯n
= G¯kn
for the Landau–Ginzburg potential (1.2), where Fin, F¯in, etc. are functions of b1, . . . , bM , c1,
. . . , cN and their first order derivatives with respect to s.
Proof. Since the two cases can be treated in the same way, let us consider the Case II only.
We can rewrite the Lax equations (2.3) in terms of log λ(p) as
∂ log λ(p)
∂tn
= {Bn(p), log λ(p)} = {log λ(p), B
c
n(p)},
∂ log λ(p)
∂t¯n
= {B¯n(p), log λ(p)} = {log λ(p), B¯
c
n(p)}.
The left hand side are linear combinations of 1/(p− bi)’s and p−k’s:
∂ log λ(p)
∂tn
= −
M∑
i=1
∂bi
∂tn
κi
p− bi
+
N∑
k=1
∂ck
∂tn
p−k,
∂ log λ(p)
∂t¯n
= −
M∑
i=1
∂bi
∂t¯n
κi
p− bi
+
N∑
k=1
∂ck
∂t¯n
p−k.
Since Bn(p) and B¯n(p) are polynomials in p and p−1, the right hand sides are rational functions
of p. Actually, because of the two complementary expressions, they turn out to be linear
combinations of 1/(p− bi)’s and p−k’s. For example, {Bn(p), log λ(p)} can be expanded as
{Bn(p), log λ(p)} =
M∑
i=1
{Bn(p), κi log(p− bi)}+
N∑
k=1
{
Bn(p), ckp
−k}
=
M∑
i=1
p
(
−
∂Bn(p)
∂p
∂bj
∂s
κi
p− bi
−
∂Bn(p)
∂s
κi
p− bi
)
+
N∑
k=1
p
(
∂Bn(p)
∂p
∂ck
∂s
p−k +
∂Bn(p)
∂s
kckp
−k−1
)
,
which has first order poles at p = bi, an N -th order pole at p = 0 and no poles in the finite part
of the Riemann sphere. On the other hand, since Bcn(p) = O(p
−1) as p → ∞, the complemen-
tary expression {log λ(p), Bcn(p)} is also O(p
−1). Consequently, the right hand side of the Lax
equation is a rational function of the form
{Bn(p), log λ(p)} = {log λ(p), B
c
n(p)} =
M∑
i=1
κiFin
p− bi
+
N∑
k=1
Gknp
−k,
where Fin’s and Gin’s are functions of bi’s, ck’s and their first order derivatives with respect
to s. Thus the Lax equations reduce to evolution equations as stated in the theorem. 
6 K. Takasaki
2.3 Examples: two-variable reductions
The simplest nontrivial cases are two-variable reductions. They contain the following old and
new examples of integrable hierarchies.
(i) Landau–Ginzburg potential (1.1) with N = 1, M = 2, κ1 = κ2 = 1:
λ(p) = p−1(p− b1)(p− b2) = p+ b+ cp
−1.
This is the reduced Lax function of the dispersionless 1D Toda hierarchy. It is well known
that this hierarchy plays a fundamental role in a wide range of issues of mathematical
physics. In a literal sense, this Laurent polynomial is used for the Landau–Ginzburg (or
“mirror”) description of the topological sigma model of CP1 [17, 18]. The associated
Frobenius structure is also a significant example of Dubrovin’s duality [13, 29].
(ii) Landau–Ginzburg potential (1.1) with N = −1, M = 2, κ1 = 1, κ2 = −1:
λ(p) = p
p− b
p− c
. (2.4)
This is the reduced Lax function of the dispersionless Ablowitz–Ladik hierarchy. This
hierarchy and the dispersive version have found new applications in universality classes of
nonlinear waves [15] and local Gromov–Witten invariants of the resolved conifold [3, 4].
(iii) Landau–Ginzburg potential (1.2) with N = M = 1, κ1 = 1:
λ(p) = (p− b) exp
(
cp−1
)
.
This seems to give a new dispersionless integrable hierarchy.
As regards the third example, one can further impose the condition b = 0 and obtain a hier-
archy with the reduced Lax function
λ(p) = p exp
(
cp−1
)
.
This is an interesting case in itself, because the inverse function of λ(p) is Lambert’s W -function
that plays a role in the theory of Hurwitz numbers (see our previous paper [30] and references
cited therein). This “one-variable reduction” can be generalized to Landau–Ginzburg potentials
of the form
λ(p) = pM exp
(
N∑
k=1
ckp
−k
)
, (2.5)
where M is an arbitrary positive integer. They should be classified as “Case III”, though we
shall not pursue this case in this paper.
3 Lo¨wner equations
3.1 General scheme of finite-variable reductions
According to the general scheme [6, 26, 33, 34], finite-variable reductions of the dispersionless
Toda hierarchy are characterized by the equations
∂z(p)
∂λn
=
αnp
p− γn
∂z(p)
∂p
,
∂z¯(p)
∂λn
=
αnp
p− γn
∂z¯(p)
∂p
(3.1)
Old and New Reductions of Dispersionless Toda Hierarchy 7
(referred to as “Lo¨wner equations” in the following) for the reduced Lax functions
z(p) = z(p;λ1, . . . , λK), z¯(p) = z¯(p;λ1, . . . , λK). (3.2)
These equations are a variant of the radial Lo¨wner equations that were first introduced by
Lo¨wner [25].
The reduced Lax functions depend on the space-time variables through the reduced dynamical
variables λn = λn(s, t, t¯). The reduced dynamical variables, in turn, are required to satisfy the
“hydrodynamic system”
∂λn
∂tk
= Vkn
∂λn
∂s
,
∂λn
∂t¯k
= V¯kn
∂λn
∂s
. (3.3)
The “characteristic speeds” Vkn = Vkn(λ1, . . . , λK) and V¯kn = V¯kn(λ1, . . . , λK) are defined as
Vkn = γnB
′
k(γn), V¯kn = γnB¯
′
k(γn),
where the prime denotes the derivative, i.e., B′k(p) = ∂Bk(p)/∂p and B¯
′
k(p) = ∂B¯k(p)/∂p. Since
B1(p) = p+ u1, the characteristic speeds for the t1-flow coincides with γn:
V1n = γn.
Thus, once the reduced Lax functions are given as a solution of the Lo¨wner equations, the
Lax equations are transformed to the hydrodynamic system (3.3) for the “Riemann invariants”
λ1, . . . , λK . A precise statement of this fact reads as follows [6, 26, 33, 34]:
Theorem 2. If z(p) and z¯(p) satisfy the Lo¨wner equations (3.6) with respect to λn’s, and λn’s
satisfy the hydrodynamic system (3.3) with respect to the space-time variables, then z(p) and z¯(p)
satisfy the Lax equations (2.1) with respect to the space-time variables.
αn’s and γn’s in (3.1) are functions of (λ1, . . . , λK) to be determined in the reduction pro-
cedure. Note that they are not arbitrary functions. Consistency of (3.1) yield the differential
equations
∂γn
∂λm
=
αmγn
γm − γn
,
∂αn
∂λm
=
αmαn(γm + γn)
(γm − γn)2
, m 6= n, (3.4)
which are referred as “Gibbons–Tsarev equations” or, more precisely, “radial Gibbons–Tsarev
equations”. These equations are a radial version of the celebrated Gibbons–Tsarev equa-
tions [21, 22] (see the remarks below).
It is interesting that the product and quotient of αn, γn satisfy the following equations
∂
∂λm
(αnγn) =
2(αmγm)(αnγn)
(γm − γn)2
,
∂
∂λm
(
αn
γn
)
=
2γmγn
(γm − γn)2
αm
γm
αn
γn
.
The right hand side of these equations are symmetric with respect to m and n. This implies the
existence of potentials.
It is easy to identify these potentials. Expands both hand sides of (3.1) into Laurent series
at p = ∞ and pick out the p1 and p2 terms from the first equation and the p0 terms from the
second equation. One can thus find the relations
αn =
∂u1
∂λn
, αnγn =
∂u2
∂λn
,
αn
γn
=
∂φ
∂λn
, (3.5)
8 K. Takasaki
which show that u1, u2 and φ = log u¯0 play the role of potentials. By the first expression of αn
in (3.5), one can rewrite (3.1) as
∂z(p)
∂λn
=
pz′(p)
p− γn
∂u1
∂λn
,
∂z¯(p)
∂λn
=
pz¯′(p)
p− γn
∂u1
∂λn
. (3.6)
It is these equations that can be obtained directly from the Lax equations under the finite-
variable ansatz (3.2).
The problem is now converted to solving (3.3). This problem can be treated by the generalized
hodograph method [35, 36].
Remark 1. The original form of the Gibbons–Tsarev equations [21, 22] read
∂γn
∂λm
=
αn
γm − γn
,
∂αn
∂λm
=
2αmαn
(γm − γn)2
. (3.7)
They are integrability conditions of the chordal Lo¨wner equations
∂z(p)
∂λn
=
αn
p− γn
∂z(p)
∂p
for the Lax function
z(p) = p+ u2p
−1 + · · ·
of the Benney hierarchy or, more generally, of the dispersionless KP hierarchy [27]. Note that
the u1-term is absent here, and u2 plays the role of a potential for the coefficients αn
αn =
∂u2
∂λn
.
Remark 2. Ferapontov et al. [19] presented the radial Gibbons–Tsarev equations (3.4) (written
in a trigonometric form) in their work on finite-variable reductions of the Boyer–Finley equation.
The Boyer–Finley equation is the lowest 2D part of the dispersionless Toda hierarchy.
3.2 Hodograph solutions
The generalized hodograph method [35, 36] is based on the fact that the characteristic speeds
satisfy the equations
1
Vkm − Vkn
∂Vkn
∂λm
=
1
V¯km − V¯kn
∂V¯kn
∂λm
=
αmγn
(γm − γn)2
(3.8)
for k = 1, 2, . . .. Note that these equations include the special case
1
γm − γn
∂γn
∂λm
=
αmγn
(γm − γn)2
(3.9)
associated with V1n = γn. One can derive these equations directly from the definition of Bk(p)
and B¯k(p) [26] (see the remarks below) or by generating functions of these polynomials [6, 33].
Having these equations, one can readily apply the generalized hodograph method to the hydro-
dynamic system (3.3) as follows [26, 33, 34]:
Old and New Reductions of Dispersionless Toda Hierarchy 9
Theorem 3. If a set of functions Fn = Fn(λ1, . . . , λK) satisfy the equations
1
Fm − Fn
∂Fn
∂λm
=
αmγn
(γm − γn)2
(3.10)
and the non-degeneracy condition
det
(
∂Fn
∂λm
)
m,n=1,...,K
6= 0,
then a solution of the hydrodynamic system (3.3) can be obtained from the hodograph relations
s+
∑
k≥1
tkVkn +
∑
k≥1
t¯kV¯kn = Fn, n = 1, . . . ,K,
as a K-tuple of implicit functions λn = λn(s, t, t¯), n = 1, . . . ,K, in a neighborhood of (t, t¯) =
(0,0).
Remark 3. The characteristic speeds Vkn and V¯kn have the contour integral representation
Vkn = −γn
∮
z(p)n
(p− γn)2
dp
2pii
, V¯kn = −γn
∮
z¯(p)n
(p− γn)2
dp
2pii
,
where the contours encircle p =∞ and p = 0, respectively, leaving γn outside. One can use the
Lo¨wner equations and the Gibbons–Tsarev equations to differentiate these contour integrals.
Thus, after some algebra, one can derive (3.8).
Remark 4. Solutions of (3.10), too, can be obtained as contour integrals. For example, arbitrary
linear combinations of Vkn and V¯kn, which are obvious solutions of (3.10), can be cast into
a contour integral of the form
Fn = γn
∮
F1(z(p))
(p− γn)2
dp
2pii
+ γn
∮
F2(z¯(p))
(p− γn)2
dp
2pii
,
where F1(p) and F2(p) are arbitrary (analytic) functions. If z(p) and z¯(p) are obtained from
a globally defined Landau–Ginzburg potential λ(p), one can unify the two integrals to a single
integral
Fn = γn
∮
C
F (λ(p))
(p− γn)2
dp
2pii
along a general cycle C in the domain of definition of F (λ(p)). This gives a more general solution
of (3.10) as Ferapontov et al. [19] pointed out in their formulation.
3.3 Lo¨wner equations for Landau–Ginzburg potentials
If the Lax functions z(p) and z¯(p) are reduced to a single Landau–Ginzburg potential λ(p), the
Lo¨wner equations (3.1) for the Lax functions, too, are reduced to the equations
∂λ(p)
∂λn
=
αnp
p− γn
∂λ(p)
∂p
(3.11)
for λ(p).
We show below that the Landau–Ginzburg potentials (1.1) and (1.2) do satisfy these equa-
tions. The relevant variables λn are the critical values of λ(p), i.e., the values of λ(p) at the
critical points γn’s,
λn = λ(γn), λ
′(γn) = 0, n = 1, . . . ,K, (3.12)
10 K. Takasaki
where K = M in the case of (1.1) and K = M +N in the case of (1.2). We choose these λn’s as
new coordinates on the parameter space of λ(p), and treat λ(p) as a function λ(p;λ1, . . . , λK)
of p and λn’s.
Let us show a few technical remarks.
Lemma 1.
∂λ(p)
∂λm
∣
∣
∣
∣
p=γn
= δmn. (3.13)
Proof. By the definition (3.12) of λn’s and the chain rule of differentiation,
δmn =
∂λn
∂λm
=
∂λ(p)
∂λm
∣
∣
∣
∣
p=γn
+ λ′(γm) =
∂λ(p)
∂λm
∣
∣
∣
∣
p=γn
. 
Lemma 2. If the Lo¨wner equations (3.11) are satisfied, the coefficients αn are uniquely deter-
mined by the equations themselves as
αn =
1
γnλ′′(γn)
. (3.14)
Proof. Let p → γn in (3.11). By (3.13), the left hand side tends to 1. As regards the right
hand side,
lim
p→γn
αnp
p− γn
∂λ(p)
∂p
= αnγn lim
p→γn
λ′(p)
p− γn
= αnγnλ
′′(γn). 
Bearing these technical remarks in mind, let us examine the two cases separately.
Case I. It is convenient to consider the logarithmic derivative, rather than the derivative, of
the Landau–Ginzburg potential (1.1):
∂ log λ(p)
∂p
= −
N
p
+
M∑
i=1
κi
p− bi
=
Q(p)
p
∏M
i=1(p− bi)
. (3.15)
The numerator Q(p) is a polynomial of the form
Q(p) = M˜pM + · · · .
We assume that Q(p) has M distinct zeroes γn, n = 1, . . . ,M ,
Q(p) = M˜
M∏
n=1
(p− γn), γm 6= γn for m 6= n.
From now on, λ(p) is understood to be a function of p and λn’s. The parameters bi’s, too,
become functions of λn’s. The derivative of log λ(p) with respect to λm can be expressed as
∂ log λ(p)
∂λm
= −
M∑
i=1
κi
p− bi
∂bi
∂λm
=
Qm(p)
∏M
i=1(p− bi)
, (3.16)
where Qm(p) is a polynomial of degree less than M . (3.15) and (3.16) imply the equality
∂ log λ(p)
∂λm
=
Qm(p)p
Q(p)
∂ log λ(p)
∂p
.
Old and New Reductions of Dispersionless Toda Hierarchy 11
The problem is to find an explicit form of the pre-factor Qm(p)p/Q(p).
Since (3.13) implies that
∂ log λ(p)
∂λm
∣
∣
∣
∣
p=γn
=
1
λ(p)
∂λ(p)
∂λm
∣
∣
∣
∣
p=γn
=
δmn
λn
,
letting p→ γn in (3.16) yields that
Qm(γn) = 0 for n 6= m.
Therefore, by the Lagrange interpolation formula, Qm(p)/Q(p) can be expressed as
Qm(p)
Q(p)
=
M∑
n=1
Qm(γn)
Q′(γn)(p− γn)
=
Qm(γm)
Q′(γn)(p− γm)
.
Thus, defining αm as
αm = Qm(γm)/Q
′(γm),
we obtain the Lo¨wner equations
∂ log λ(p)
∂λm
=
αmp
p− γm
∂ log λ(p)
∂p
for log λ(p). Of course, they are equivalent to the Lo¨wner equations (3.11) for λ(p). By the
second lemma above, αm turns out to have another expression (3.14).
Case II. The logarithmic derivatives of the Landau–Ginzburg potential (1.2) can be expresses
as
∂ log λ(p)
∂p
=
M∑
i=1
κi
p− bi
−
N∑
k=1
kckp
−k−1 =
Q(p)
pN+1
∏M
i=1(p− bi)
(3.17)
and
∂ log λ(p)
∂λm
= −
M∑
i=1
κi
p− bi
∂bi
∂λm
+
N∑
k=1
∂ck
∂λm
p−k =
Qm(p)
pN
∏M
i=1(p− bi)
,
where Q(p) is a polynomial of the form
Q(p) = M˜
M+N∏
n=1
(p− γn), γm 6= γn for m 6= n.
Qm(p) is a polynomial of degree less than M + N , and the roots γn of Q(p) are assumed to
be distinct. Starting from these data, one can derive the Lo¨wner equations (3.11) in much the
same way as in the Case I.
We thus obtain the following result:
Theorem 4. λ(p) = λ(p;λ1, . . . , λK) satisfies the Lo¨wner equations (3.11) with the coeffi-
cients αn defined by (3.14).
On the basis of this result, we can apply the foregoing scheme of finite-variable reductions to
the Landau–Ginzburg potentials (1.1) and (1.2).
12 K. Takasaki
4 Darboux equations
4.1 Basic notions in classical differential geometry
Given a diagonal metric ds2 =
K∑
n=1
(hndλn)2,6 one can define the rotation coefficients βmn,
m 6= n, as
βmn =
1
hm
∂hn
∂λm
.
hn’s are called “Lame´ coefficients” in the theory of orthogonal curvilinear coordinate sys-
tems [10, 35]. The Riemann curvature of this metric vanishes if and only if the following
equations are satisfied
∂βmn
∂λk
= βmkβkn for k 6= m,n, (4.1)
∂βmn
∂λm
+
∂βmn
∂λn
+
K∑
k=1
βkmβkn = 0. (4.2)
The first part (4.1) of these equations are called “Darboux equations” in the literature. Thus the
Darboux equations are partial-flatness conditions, and have to be supplemented by the second
equations (4.2) to ensure flatness.
If the rotation coefficients are symmetric, i.e., βmn = βnm, the metric components satisfy
the conditions (Egorov conditions)
∂
∂λm
(
h2n
)
=
∂
∂λn
(
h2m
)
that ensure the existence of a potential φ (Egorov potential) such that
h2n =
∂φ
∂λn
.
If the Darboux equations and the Egorov condition are satisfied, the flatness condition (4.2)
reduces to the equations
K∑
k=1
∂βmn
∂λk
= 0. (4.3)
A diagonal metric that satisfies the Darboux equations and the Egorov conditions is called an
“Egorov metric”. Thus an Egorov metric is associated with a symmetric (βmn = βnm) solution
of the coupled system of the Darboux equations and (4.3). This system is closely related to the
K-wave system [12, 14].
If one starts from a solution of the Darboux equations (4.1), the Lame´ coefficients are reco-
vered as a solution of the equations
∂hn
∂λm
= hmβmn.
The Darboux equations are integrability conditions of these linear equations. These equations
leaves some arbitrariness in the Lame´ coefficients. Two sets hn, h˜n of Lame´ coefficients have
the same rotation coefficients if and only if their ratios wn = h˜n/hn satisfy the equations
1
wm − wn
∂wn
∂λm
=
∂ log hn
∂λm
. (4.4)
6We do not use Einstein’s convention in this paper.
Old and New Reductions of Dispersionless Toda Hierarchy 13
Any solution of these equations thus gives a transformation on the set of Lame´ coefficients with
the same rotational coefficients. This transformation is called “Combescure transformation” in
the theory of orthogonal curvilinear coordinate systems [10, 35].
4.2 Implications of Gibbons–Tsarev equations
Let us consider the Gibbons–Tsarev equations (3.4) in the language of Darboux equations and
Egorov metrics. There are three metrics that are of particular interest:
K∑
n=1
(hndλn)
2, hn =
√
αn/γn, (4.5)
K∑
n=1
(h˜ndλn)
2, h˜n =
√
αnγn, (4.6)
K∑
n=1
(hˆnd log λn)
2, hˆn =
√
αnλn/γn. (4.7)
(4.5) and (4.6) underlie the hodograph solutions of the hydrodynamic equations (3.3). (4.5)
and (4.7) are the metrics that we shall consider in the next section in the context of Frobenius
structures.
We show below that the rotation coefficients of these metrics are symmetric and satisfy the
Darboux equations (with respect to λn’s in the first and second cases and log λn’s in the third
case). Egorov potentials themselves can be readily identified as one can see from (3.5):
hn
2 =
∂φ
∂λn
, h˜n
2 =
∂u2
∂λn
, hˆn
2 =
∂φ
∂ log λn
.
The first two cases (4.5) and (4.6) are closely related.
Theorem 5. The Lame´ coefficients of (4.5) and (4.6) have the same rotation coefficients
βmn = βnm =
√
αmαnγmγn
(γm − γn)2
. (4.8)
These rotation coefficients satisfy the Darboux equations (4.1).
Proof. Do straightforward calculations using the Gibbons–Tsarev equations (3.4). The rotation
coefficients of hn’s can be calculated as
1
hm
∂hn
∂λm
=
hn
hm
∂ log hn
∂λm
=
√
αnγm
αmγn
(
1
2αn
∂αn
∂λm
−
1
2γn
∂γn
∂λm
)
=
√
αnγm
αmγn
(
1
2αn
αmαn(γm + γn)
(γm − γn)2
−
1
2γn
αmγn
γm − γn
)
=
√
αmαnγmγn
(γm − γn)2
.
In much the same way, the rotation coefficients of h˜n’s can be calculated as
1
h˜n
∂h˜n
∂λm
=
h˜n
h˜m
∂ log h˜n
∂λm
=
√
αnγn
αmγm
(
1
2αn
∂αn
∂λm
+
1
2γn
∂γn
∂λm
)
=
√
αnγn
αmγm
(
1
2αn
αmαn(γm + γn)
(γm − γn)2
+
1
2γn
αmγn
γm − γn
)
=
√
αmαnγmγn
(γm − γn)2
.
Thus the rotation coefficients of hn’s and h˜n’s turn out to coincide. Differentiating them with
respect to λk and doing some algebra, one can derive the Darboux equations. 
14 K. Takasaki
Corollary 1. h˜n’s are a Combescure transformation of hn’s, and the ratios γn = h˜n/hn satis-
fy (4.4).
The right hand side of (4.4) can be calculated explicitly as
∂ log hn
∂λm
=
αmγn
(γm − γn)2
,
thus (4.4) coincides with (3.9). Since (3.9) is a special case of (3.8), the characteristic speeds Vkn
and V¯kn, too, generate Combescure transformations of hn. An analogous fact is known for the
dispersionless KP hierarchy [27] and universal Whitham hierarchy [23] as well (see the remark
below). Thus, as stressed by Tsarev [35], the notion of Combescure transformations lies in the
heart of integrability of hydrodynamic systems such as (3.3).
The third case (4.7) is of somewhat different nature. It is log λn’s rather than λn’s that are
used to formulate the Darboux equations.
Theorem 6. The rotation coefficients
βˆmn =
1
hˆm
∂hˆn
∂ log λm
, m 6= n,
of the Lame´ coefficients of (4.7) are related to (4.8) as
βˆmn =
√
λmλnβmn,
and satisfy the Darboux equations
∂βˆmn
∂ log λk
+ βˆmkβˆkn = 0 for k 6= m,n. (4.9)
Proof. Do straightforward calculations. Note, in particular, that the left hand side of (4.9)
and (4.1) are related as
∂βˆmn
∂ log λk
+ βˆmkβˆkn =
√
λmλnλk
(
∂βmn
∂λk
+ βmkβkn
)
. 
Remark 5. If the rotation coefficients βˆmn are homogeneous functions of λk’s of degree zero,
the Darboux equations (4.9) and the Euler equations
N∑
k=1
∂βˆmn
∂ log λk
=
N∑
k=1
λk
∂βˆmn
∂λk
= 0
imply flatness of (4.7). This is indeed the case for the metrics ( , ) considered in the next
section.
Remark 6. In the chordal case (3.7), the two sets hn =
√
αn, h˜n =
√
αnγn of Lame´ coefficients
amount to (4.5) and (4.6). They have the same rotation coefficients
βmn = βnm =
√
αmαn
(γm − γn)2
that satisfy the Darboux equations (4.1). Consequently, h˜n’s are a Combescure transformation
of hn’s, and the ratios h˜n/hn = γn satisfy the equations
1
γm − γn
∂γn
∂λm
=
∂ log hn
∂λm
=
αm
(γm − γn)2
.
The last equations are fundamental equations in the hodograph solutions of the Benney equa-
tions [21, 22], the dispersionless KP hierarchy [27] and the universal Whitham hierarchy [23, 32].
Old and New Reductions of Dispersionless Toda Hierarchy 15
5 Flat coordinates
5.1 Flat coordinates in Case I
Let us recall Dubrovin and Zhang’s construction [16] of two Frobenius structures on the param-
eter space of the Landau–Ginzburg potential
λ(p) = p−N
N∏
i=1
(p− bi). (5.1)
The Frobenius structures are realized by the following inner products (or metrics) 〈 , 〉, ( , ) and
cubic forms 〈 , , 〉, ( , , ) for vector fields on the parameter space of the Laurent polynomial:
〈∂, ∂′〉 =
M∑
n=1
res
p=γn
[
∂λ(p) · ∂′λ(p)
dλ(p)
(d log p)2
]
, (5.2)
〈∂, ∂′, ∂′′〉 =
M∑
n=1
res
p=γn
[
∂λ(p) · ∂′λ(p) · ∂′′λ(p)
dλ(p)
(d log p)2
]
, (5.3)
(∂, ∂′) =
M∑
n=1
res
p=γn
[
∂ log λ(p) · ∂′ log λ(p)
d log λ(p)
(d log p)2
]
, (5.4)
(∂, ∂′, ∂′′) =
M∑
n=1
res
p=γn
[
∂ log λ(p) · ∂′ log λ(p) · ∂′′ log λ(p)
d log λ(p)
(d log p)2
]
. (5.5)
The cubic forms are used to define two commutative and associative product structures ◦, ? of
vector fields
〈∂ ◦ ∂′, ∂′′〉 = 〈∂, ∂′ ◦ ∂′′〉 = 〈∂, ∂′, ∂′′〉, (∂ ? ∂′, ∂′′) = (∂, ∂′ ? ∂′′) = (∂, ∂′, ∂′′).
These two Frobenius structures are a prototype of Dubrovin’s duality [13, 29].
When ∂, ∂′, ∂′′ are derivatives in λn’s, one can use the Lo¨wner equations (3.11) to evaluate
these inner products and cubic forms as follows
〈
∂
∂λm
,
∂
∂λn
〉
= δmn
αn
γn
,
〈
∂
∂λk
,
∂
∂λm
,
∂
∂λn
〉
= δkmn
αn
γn
, (5.6)
(
∂
∂λm
,
∂
∂λn
)
= δmn
αn
γnλn
,
(
∂
∂λk
,
∂
∂λm
,
∂
∂λn
)
= δkmn
αn
γnλn2
, (5.7)
where δkmn = δkmδmn (i.e., δkmn is equal to 1 if k = m = n and 0 otherwise). Thus (5.2)
and (5.4) correspond to the Egorov metrics (4.5) and (4.7) considered in the last section. Note
that (5.6) and (5.7) hold as far as the Lo¨wner equations are satisfied. In particular, they are
valid for the general case of (1.1) as well.
As shown by Dubrovin and Zhang [16], the first metric (5.2) has a system of flat coordinates
q1, . . . , qM−1, q¯0, q¯1, . . . , q¯N defined by the residue formula
qn = res
p=∞
[
z(p)n
n
d log p
]
, q¯0 = φ, q¯n = res
p=0
[
z¯(p)n
n
d log p
]
.
The inner products of ∂/∂qn’s and ∂/∂q¯n’s are calculated explicitly as
〈
∂
∂qm
,
∂
∂qn
〉
= M˜δm+n,M˜ ,
〈
∂
∂q¯m
,
∂
∂q¯n
〉
= Nδm+n,N ,
〈
∂
∂qm
,
∂
∂q¯n
〉
= 0.
Unfortunately, this construction of flat coordinates does not work for the more general
Landau–Ginzburg potential (1.1). Actually, it seems likely that the metric (5.2) is no longer
flat in other cases7. In this respect, it is very significant that Brini et al. [4] extended Dubrovin
7We thank one of the referees for pointing out this possibility.
16 K. Takasaki
and Zhang’s dual pair of Frobenius structures to the Lax function (2.4) of the dispersionless
Ablowitz–Ladik hierarchy.
On the other hand, the construction of the second Frobenius structure is valid for the general
case of (1.1) as well. This fact is shown by Chang [9] in the case where N = −1 and κi’s are
arbitrary. log bi’s are flat coordinates of the second metric (5.4). One can confirm, with slightest
modification of Chang’s calculations, that this is also the case for an arbitrary value of N as
the following result of calculations of the inner product shows
(
∂
∂bi
,
∂
∂bj
)
= (1− δij)
κiκj
Nbibj
+ δij
(κi −N)κi
Nbi2
.
We omit details of these calculations, which are parallel to the proof of Lemma 3 below. Let us
note that flatness of (5.8) is also a consequence of homogeneity of the rotation coefficients (cf.
Remark 5).
Remark 7. Speaking more precisely, the definition of a Frobenius manifold requires some more
data, in particular, an Euler vector field E and associated scaling properties [12]. In the present
setting, the Landau–Ginzburg potential λ(p) has natural homogeneity, and one can use the
vector field
E =
M∑
n=1
λn
∂
∂λn
=
1
M˜
M∑
i=1
bi
∂
∂bi
as an Euler vector field. Furthermore, one can introduce the prepotential F as a function of the
flat coordinates tn and express the cubic form as
(
∂
∂tl
,
∂
∂tm
,
∂
∂tn
)
=
∂3F
∂tl∂tm∂tn
.
In the following, we refer to the notion of Frobenius manifolds in a loose sense, and focus our
consideration on flatness of metrics.
5.2 Flat coordinates in Case II
We now turn to the Landau–Ginzburg potential (1.2). The goal is to present a set of flat
coordinates for the inner product
(∂, ∂′) =
M+N∑
n=1
res
p=γn
[
∂ log λ(p) · ∂′ log λ(p)
d log λ(p)
(d log p)2
]
. (5.8)
This inner product corresponds to the Egorov metric (4.7). Its flatness is ensured by homogeneity
of the rotation coefficients (cf. Remarks 5 and 10). The associated cubic form is defined by
(∂, ∂′, ∂′′) =
M+N∑
n=1
res
p=γn
[
∂ log λ(p) · ∂′ log λ(p) · ∂′′ log λ(p)
d log λ(p)
(d log p)2
]
,
though we shall not study its implications. Let us mention that the technical details and the
final result of the following consideration are remarkably similar to the case of Ferguson and
Strachan [20].
Let us start from the natural coordinates b1, . . . , bM , c1, . . . , cN of the parameter space. One
can calculate part of the inner product explicitly as follows.
Old and New Reductions of Dispersionless Toda Hierarchy 17
Lemma 3.
(
∂
∂bi
,
∂
∂bj
)
= −δij
κi
bi2
,
(
∂
∂bi
,
∂
∂ck
)
= δkN
κi
NbicN
.
Proof. The derivative of log λ(p) with respect to p is a function as shown in (3.17). The
derivatives with respect to bi and ck take the simple form
∂ log λ(p)
∂bi
= −
κi
p− bi
,
∂ log λ(p)
∂ck
= p−k.
Consequently, the inner products in question can be expressed as
(
∂
∂bi
,
∂
∂bj
)
=
M+N∑
n=1
res
p=γn
[
κiκj
(p− bi)(p− bj)
(
∂ log λ(p)
∂p
)−1 dp
p2
]
,
(
∂
∂bi
,
∂
∂ck
)
=
M+N∑
n=1
res
p=γn
[
−
κip−k
p− bi
(
∂ log λ(p)
∂p
)−1 dp
p2
]
.
Since
(
∂ log λ(p)
∂p
)−1
=
pN+1
M∏
k=1
(p− bk)
Q(p)
, Q(p) = M˜
M+N∏
n=1
(p− γn),
the 1-forms in the residues are rational and have poles of the first order at p = γ1, . . . , γM+N .
Other possible poles are located at p = bi, bj , 0. The latter poles, however, can disappear because
of zeros of the numerator in this expression of (∂ log λ(p)/∂p)−1. For example, if i 6= j, the first
1-form is non-singular at p = bi, bj as well as at p = 0. Since the residue theorem says that the
sum of all residues is equal to 0, one can conclude that
(
∂
∂bi
,
∂
∂bj
)
= 0 for i 6= j.
By the same reasoning, one can confirm that
(
∂
∂bi
,
∂
∂ck
)
= 0 for k < N.
As regards the remaining cases, one can use the residue theorem to rewrite the inner products
as
(
∂
∂bi
,
∂
∂bi
)
= − res
p=bi
[
κi2
(p− bi)2
(
∂ log λ(p)
∂p
)−1 dp
p2
]
,
(
∂
∂bi
,
∂
∂cN
)
= − res
p=0
[
−
κip−N
p− bi
(
∂ log λ(p)
∂p
)−1 dp
p2
]
.
In view of the local expression
(
∂ log λ(p)
∂p
)−1
=
p− bi
κi
+O((p− bi)
2 as p→ bi
and
(
∂ log λ(p)
∂p
)−1
= −
pN+1
NcN
+O
(
pN+2
)
as p→ 0,
one can readily calculate the residues and confirm the statement for the remaining cases. 
18 K. Takasaki
This lemma shows that log bi’s are part of flat coordinates. If one can further find a partial
change of coordinates (c1, . . . , cN )→ (q¯0, . . . , q¯N−1) that is independent of bi’s and flat in them-
selves, the new coordinate system log b1, . . . , log bM , q¯0, . . . , q¯N−1 are totally flat. We shall show
that
q¯0 = φ, q¯n = res
p=0
[
z¯(p)n
n
d log p
]
, n = 1, . . . , N − 1, (5.9)
give such coordinates.
To this end, we need to know some properties of q¯n’s (which are defined for n ≥ N as well
by the same residue formula). Let p¯(ζ) denote the inverse function of ζ = z¯(p). It has a Laurent
expansion of the form
p¯(ζ) = eφζ−1
(
1 + u¯1ζ
−1 + · · ·
)
.
Therefore log p¯(ζ) is also well-defined as a series of the form
log p¯(ζ) = − log ζ + φ+ u¯1ζ
−1 + · · · .
Lemma 4. q¯n’s coincide with the coefficients of the expansion of log p¯(ζ):
log p¯(ζ) = − log ζ + q¯0 + q¯1ζ
−1 + · · ·+ q¯nζ
−n + · · · . (5.10)
Proof. One can rewrite the definition (5.9) of q¯n’s as
q¯n = res
p=0
[
z¯(p)n
n
d log p
]
= res
ζ=∞
[
ζn
n
d log p¯(ζ)
]
= − res
ζ=∞
[
log p¯(ζ)d
(
ζn
n
)]
= − res
ζ=∞
[
log p¯(ζ)ζn−1dζ
]
.
This implies that log p¯(ζ) has a Laurent expansion as (5.10) shows. 
Lemma 5. For n = 1, . . . , N − 1, q¯n is a polynomial of cN−n, . . . , cN−1 and e−φ of the form
q¯n =
1
N
e(n−N)φcN−n + higher orders in cN−n+1, . . . , cN−1 (5.11)
and qN is a function of bi’s only
q¯N =
1
N
log
M∏
i=1
(−bi)
κi . (5.12)
Proof. Since z¯(p) = (log λ(p))1/N and cN = eNφ, one can calculate its n-th power as a Laurent
series of the form
z¯(p)n =
(
eNφp−N + cN−1p
1−N + · · ·+ c1p
−1 + log
M∏
i=1
(−bi) +O(p)
)n/N
.
Extracting the p0 term yields (5.11) and (5.12). 
(5.11) implies that the map (c1, . . . , cN ) 7→ (q¯0, . . . , q¯N−1) is invertible. We now choose bi’s
and q¯0, . . . , q¯N−1 as a new coordinate system on the parameter space of λ(p), and consider λ(p)
to be a function of p and these coordinates.
Old and New Reductions of Dispersionless Toda Hierarchy 19
Lemma 6. The derivatives of log λ(p) with respect to q¯n’s can be expressed as
∂ log λ(p)
∂q¯n
=
(
−ζ−n +O
(
ζ−N−1
))∣
∣
ζ=z¯(p)
d log λ(p)
d log p
(5.13)
in a neighborhood of p = 0.
Proof. We can use the so called “thermodynamic identity” [12]
∂ log λ(p)
∂q¯n
d log p = −
∂ log p¯(ζ)
∂q¯n
∣
∣
∣
∣
ζ=z¯(p)
d log λ(p)
to rewrite the derivatives of log λ(p) with respect to q¯n’s as
∂ log λ(p)
∂q¯n
= −
∂ log p¯(ζ)
∂q¯n
∣
∣
∣
∣
ζ=z¯(p)
d log λ(p)
d log p
.
Let us recall the expansion (5.10) of log p¯(ζ). In this expansion, the coefficients of ζ0, . . . , ζN−1
are q¯n’s themselves and, as (5.12) shows, the next leading coefficient qN is a function of bi’s
only. Consequently,
∂ log p¯(ζ)
∂q¯n
= ζ−n +O
(
ζ−N−1
)
. 
Remark 8. This is a place where the technical details slightly deviate from the case of Ferguson
and Strachan [20]. As in their case, the flat coordinates under construction are a mixture of the
two types of coordinates, bi’s and q¯n’s, presented in the Case I. There is, however, a delicate
difference in the derivation of the vital equalities (5.13). In their case, they could derive these
equalities in a rather straightforward manner. In our case, we need a small piece of extra
consideration on the special structure of qN as explained above.
Theorem 7. The inner product of the derivatives in bi’s and q¯n’s can be expressed as
(
∂
∂bi
,
∂
∂bj
)
= −δij
κi
bi2
,
(
∂
∂q¯m
,
∂
∂q¯n
)
= Nδm+n,N ,
(
∂
∂bi
,
∂
∂q¯n
)
= −δn0
κi
bi
.
In particular, log bi’s and q¯n’s are flat coordinates.
Proof. Since the expression
∂ log λ(p)
∂bi
= −
κi
p− bi
of derivatives with respect to bi’s persists to be true, the foregoing calculations of the inner
products of ∂/∂bi’s are also valid. To consider the inner products containing ∂/∂q¯n’s, we note
the equality
∂ log λ(p)
∂q¯n
=
N∑
k=1
∂ck
∂q¯n
p−k
as well. Thus the 1-forms in the expression
(
∂
∂q¯m
,
∂
∂q¯n
)
=
M+N∑
n=1
res
p=γn
[
∂ log λ(p)
∂q¯m
∂ log λ(p)
∂q¯n
(
∂ log λ(p)
∂p
)−1 dp
p2
]
,
20 K. Takasaki
(
∂
∂bi
,
∂
∂q¯n
)
=
M+N∑
n=1
res
p=γn
[
∂ log λ(p)
∂bi
∂ log λ(p)
∂q¯n
(
∂ log λ(p)
∂p
)−1 dp
p2
]
of the inner products are rational, and can have extra poles at p = 0 in addition to the first order
poles at p = qn’s. By the residue theorem, the sum over the residues at qn’s can be converted
to the residues at p = 0:
(
∂
∂q¯m
,
∂
∂q¯n
)
= − res
p=0
[
∂ log λ(p)
∂q¯m
∂ log λ(p)
∂q¯n
(
∂ log λ(p)
∂p
)−1 dp
p2
]
,
(
∂
∂bi
,
∂
∂q¯n
)
= − res
p=0
[
−
κi
p− bi
∂ log λ(p)
∂q¯n
(
∂ log λ(p)
∂p
)−1 dp
p2
]
.
To evaluate the residues of p = 0, let us recall (5.13). The residues in the last equalities can be
thereby evaluated as
res
p=0
[
∂ log λ(p)
∂q¯m
∂ log λ(p)
∂q¯n
(
∂ log λ(p)
∂p
)−1 dp
p2
]
= res
p=0
[(
−ζ−m +O
(
ζ−N−1
))(
−ζ−n +O
(
ζ−N−1
))∣
∣
ζ=z¯(p)d log λ(p)
]
= res
ζ=∞
[(
−ζ−m +O
(
ζ−N−1
))(
−ζ−n +O
(
ζ−N−1
))
NζN−1dζ
]
= −Nδm+n,N
and
res
p=0
[
∂ log λ(p)
∂bi
∂ log λ(p)
∂q¯n
(
∂ log λ(p)
∂p
)−1 dp
p2
]
= res
p=0
[(
−
κi
p− bi
)
(
−ζ−n +O
(
ζ−N−1
))∣
∣
ζ=z¯(p)d log p
]
= δn0
κi
bi
.
This completes the proof. 
Remark 9. As in the case of Ferguson and Strachan [20], we have been unable to find a dual
Frobenius structure. A naive idea will be to consider the same inner product (5.2) and the cubic
form (5.3) as used for Dubrovin and Zhang’s Landau–Ginzburg potential. This, however does
not work because the 1-forms in the definition of the inner product and the cubic form have
essential singularities at p = 0, and one cannot use the residue theorem.
Remark 10. Unlike the Landau–Ginzburg potential of Ferguson and Strachan [20], our Landau–
Ginzburg potential (1.2) has natural quasi-homogeneity. We can use the vector field
E =
M+N∑
n=1
λn
∂
∂λn
=
1
M˜
M∑
i=1
bi
∂
∂bi
+
1
M˜
N∑
k=1
kck
∂
∂ck
as an Euler vector field.
6 Conclusion
We have examined the two Landau–Ginzburg potentials (1.1) and (1.2) and the associated
reductions of the dispersionless Toda hierarchy. (1.1) is a generalization of Dubrovin and
Zhang’s Landau–Ginzburg potential [16], and contains distinct examples such as the dispersion-
less Ablowitz–Ladik hierarchy [4]. (1.2) is a transcendental function, and its logarithm resembles
Ferguson and Strachan’s generalized waterbag model [20] of the dispersionless KP hierarchy.
We have observed that these quite different Landau–Ginzburg potentials have very similar
features:
Old and New Reductions of Dispersionless Toda Hierarchy 21
• Consistency of the reduction can by confirmed by a direct method as shown in the proof
of Theorem 1.
• The radial Lo¨wner equations can be derived in much the same way as shown in the proof of
Theorem 2. Once these equations are obtained, the generalized hodograph method works
automatically and in a unified way.
• They have natural homogeneity. This ensures flatness of the third Egorov metric (4.7),
hence of the inner products (5.4) and (5.8). Even the construction of flat coordinates of
these inner products are partially similar as shown in the proof of Theorem 7.
Meanwhile, we have encountered a rather complicated situation in the problem of flatness
of the first Egorov metric (4.5) (equivalently, the inner product (5.2)) in the cases other than
Dubrovin and Zhang’s Landau–Ginzburg potential (5.1). It is well known that non-flatness of
the Egorov metric implies non-locality of an underlying Hamiltonian structure [6]. Presumably,
this issue should be considered case-by-case. Of particular interest is the reduced Lax func-
tion (2.4) of the dispersionless Ablowitz–Ladik hierarchy, which is shown to have a dual pair of
Frobenius structures [4]. Extending their result to other specializations of (1.1) is an important
open problem.
Our second Landau–Ginzburg potential (1.2), too, raises many open problems. Questions
posed by Ferguson and Strachan [20] to the waterbag models of the dispersionless KP hierarchy
can be restated for our model. Moreover, the degenerate case (2.5) of this Landau–Ginzburg
potential will have its own special properties and applications.
Acknowledgements
We thank the referees for many valuable comments. This work is partly supported by JSPS
Grants-in-Aid for Scientific Research No. 21540218 and No. 22540186 from the Japan Society
for the Promotion of Science.
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