Principal hierarchies of infinite-dimensional Frobenius manifolds: the extended 2D Toda lattice

0.38 MB·English

by  Guido Carlet

PRINCIPAL HIERARCHIES OF
INFINITE-DIMENSIONAL FROBENIUS MANIFOLDS:
THE EXTENDED 2D TODA LATTICE
2
1
GUIDO CARLET AND LUCA PHILIPPE MERTENS
0
2

n Abstract. We define a dispersionless tau-symmetric bihamilto-
a nianintegrablehierarchyonthespaceofpairsoffunctionsanalytic
J
inside/outsidetheunitcirclewithsimplepolesat0/∞respectively,
4
which extends the dispersionless 2D Toda hierarchy of Takasaki
2
and Takebe. Then we construct the deformed flat connection of

] the infinite-dimensional Frobenius manifold M0 introduced in [4]
h
and,byexplicitlysolvingthedeformedflatnessequations,weprove
p
that the extended 2D Toda hierarchy coincides with principal hi-
-
h erarchy of M0.
t
a
m
[

Contents
2
v
Introduction 2
3
1. The extended dispersionless 2D Toda hierarchy 7
4
3 1.1. The dispersionless 2D Toda hierarchy 7
5 1.2. Analytic setting 8
.
9 1.3. The extended hierarchy: Lax formulation 11
0
1.4. The extended hierarchy: bi-Hamiltonian formulation 15
1
1.5. Tau symmetry 17
1
: 2. The principal hierarchy of M 18
v 0
i 2.1. The manifold M0 as a bundle on the space of parametrized
X
simple curves 18
r 2.2. The metric 19
a
2.3. The Levi-Civita connection 20
2.4. Flat coordinates 22
2.5. The associative product and the deformed flat connection 23
2.6. Deformed flat coordinates 26
2.7. Levelt basis, monodromy and orthogonality. 35
2.8. The principal hierarchy 40
Concluding remarks 43
References 46
Date: 24 January 2012.
2000 Mathematics Subject Classification. 53D45, 35Q58.
Key words and phrases. 2D Toda, Frobenius manifold, principal hierarchy.
1
2 G. CARLET AND L.PH. MERTENS
Introduction
The theory of Frobeniusmanifolds [6], originating as a geometric for-
mulation of the associativity equations of two-dimensional topological
field theory [16, 5], has proven to be an important tool in the study and
classification of bi-Hamiltonian tau-symmetric integrable hierarchies of
PDEs with one spatial variable [9].
The possibility of extending the Frobenius manifolds techniques to
the realm of integrable PDEs with two spatial variables has been re-
cently proposed in [3] where, in collaboration with B. Dubrovin, we
have constructed an infinite-dimensional Frobenius manifold naturally
associated with the bi-Hamiltonian structure of the dispersionless 2D
Toda hierarchy.
In this work we further develop the program of [3]. First we show
that the insights coming from the theory of Frobenius manifolds allow
us to solve the problem of finding an extension of the dispersionless 2D
Toda hierarchy. Such problem has been open since the introduction
of the extended [4] and extended bigraded [2] Toda hierarchies, which
are characterized by extra “logarithmic” flows, which are essential in
the framework of [9] and for the description of Gromov-Witten poten-
tials [10, 12], but cannot be obtained by reduction of the 2D Toda flows
of [15, 14].
The usual Lax formulation of the dispersionless 2D Toda hierarchy
is given in terms of two formal Laurent series, the Lax symbols
λ(z) = z +u +..., λ¯(z) = u¯ z−1 +u¯ +...,
0 −1 0
and Lax equations (4) which define an infinite set of commuting vector
fields on the loop space of formal Lax symbols. In order to define a
larger set of flows on this loop space we need to impose some analyt-
icity conditions. First we require that, instead of formal power series,
¯ ¯
λ(z), λ(z) are “holomorphic Lax symbols”, i.e. λ(z), resp. λ(z), is
a holomorphic function on a neighborhood of closure of the exterior,
resp. interior, part of the unit circle in Riemann sphere C∪{∞} ad-
mitting a simple pole at ∞, resp. 0, with the normalization condition
λ(z) = z + O(1) for |z| → ∞. Second we impose a “winding num-
¯
bers condition”, namely we require that the functions λ(z), λ(z) and
¯
w(z) := λ(z) +λ(z), when restricted to the unit circle |z| = 1, define
analyticcurves inC× := C\{0}whichhavewindingnumber aroundthe
origin respectively equal to 1, −1 and 1. We denote by M the space
1
of pairs of holomorphic Lax symbols satisfying the winding numbers
condition.
INFINITE-DIMENSIONAL PRINCIPAL HIERARCHIES 3
The dispersionless 2D Toda equations define evolutionary flows on
pairs of holomorphic Lax symbols depending smoothly on the variable
x, hence, in particular, on the loop space of M ,
1
LM = C∞(S1,M ).
1 1
We define the extended dispersionless 2D Toda hierarchy as a set of
commuting flows on LM which include the usual 2D Toda flows. We
1
summarize the first part of our results in the following theorem.
Theorem. Let Q , for αˆ ∈ Zˆ = Z ∪{u,v} and p > 0, be functions
αˆ,p
¯
of λ, λ defined by
(λ+λ¯)α+1(λ¯ −λ)p
Q = − for α 6= −1,
α,p
(α+1)(2p)!!
(−λ)p λ¯ (λ¯ −λ)p
Q = − log 1+ +c −1 − ,
−1,p p
p! λ (2p)!!
(cid:18) (cid:19)
(cid:16) (cid:17)
(−λ)p λ¯ λ¯p
¯ ¯
Q = − log 1+ +c −1 + log λ(λ+λ) −c −1 ,
v,p p p
p! λ p!
(cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
λ¯p+1 (cid:0) (cid:1)
Q =
u,p
(p+1)!
where c = 1+···+ 1 are the harmonic numbers (with c = c = 0).
p p 0 −1
The Lax equations
¯
∂λ ∂λ
¯
= {−(Q ) ,λ}, = {(Q ) ,λ},
∂tαˆ,p αˆ,p − ∂tαˆ,p αˆ,p +
define a tau-symmetric hierarchy of commuting flows on the loop space
of M .
1
These flows admit a bi-Hamiltonian formulation with respect to the
dispersionless 2D Toda Poisson brackets {,} and {,}
1 2
∂
· = {·,H }
∂tαˆ,p αˆ,p 1
with recursion relations
{·,H } = (α+p+2){·,H } ,
α,p 2 α,p+1 1
{·,H } = (p+1){·,H } +2{·,H } ,
v,p 2 v,p+1 1 u,p 1
{·,H } = (p+2){·,H }
u,p 2 u,p+1 1
for p > −1, and Hamiltonians
1 dz
¯
H = Q (λ(z),λ(z)) dx.
αˆ,p αˆ,p+1
2πi z
IS1 I|z|=1
4 G. CARLET AND L.PH. MERTENS
The dispersionless 2D Toda flows are finite combinations of the flows
defined above.
The main motivation for the definition of this hierarchy comes from
the construction of an infinite-dimensional semisimple Frobenius mani-
foldM associatedwiththePoisson pencil ofthe2DTodahierarchy [3].
0
The Frobenius manifold M is defined on the space of pairs of holo-
0
morphic Lax symbols with certain additional conditions (ensuring the
invertibility of the metric η and the well-posedness of the Riemann-
Hilbert problem defining the flat coordinates). In this article we will
¯
further assume that λ(z), λ(z) satisfy the winding numbers condition
mentioned above, i.e. we assume that M is an open subset of M .
0 1
The main geometrical object associated with a Frobenius manifold is
the deformed flat connection ∇˜, which is, roughly speaking, defined by
deforming the Levi-Civita connection of the metric η using the product
of the Frobenius manifold. The flatness of ∇˜ is a basic result of the
theory of (finite-dimensional) Frobenius manifolds, which essentially
encodes all the axioms of the Frobenius manifold itself. The deformed
flat connection allows to associate with a Frobenius manifold a quasi-
linear integrable hierarchy, called Principal hierarchy. This procedure
is based on the construction of a system of deformed flat coordinates
depending on the deformation parameter ζ. The choice of a certain
normal form of the solution of the deformed flatness equations pro-
vides the so-called Levelt system of deformed flat coordinates. The
Hamiltonian densities of the Principal hierarchy are defined as the co-
efficients in the ζ -expansion of the analytic part of the Levelt system
of deformed flat coordinates. Here we will not review in detail these
Frobenius manifold constructions but rather refer the reader to [7, 9].
In the second part of this article we construct the deformed flat
connection ∇˜ of the Frobenius manifold M and we explicitly solve the
0
associateddeformedflatnessequations. Weshowthatthedeformedflat
coordinates are indeed a Levelt system, thus we obtain the Principal
hierarchy of M , which coincides with the extended dispersionless 2D
0
Toda hierarchy, when restricted to LM .
0
INFINITE-DIMENSIONAL PRINCIPAL HIERARCHIES 5
More precisely let us consider the following functions on M ×C×,
0
which are holomorphic in the parameter ζ in a neighborhood of ζ = 0
θα(ζ) = − 1 (λ+λ¯)α+1eλ¯−2λζ dz for α 6= −1, (2a)
2πi α+1 z
I|z|=1
¯
θ−1(ζ) = − 1 e−λζ log 1+ λ +Ein(−λζ)−1 +eλ¯−2λζ dz,
2πi λ z
I|z|=1(cid:20) (cid:18) (cid:18) (cid:19) (cid:19) (cid:21)
(2b)
¯
1 λ
θ (ζ) = −e−λζ log 1+ +Ein(−λζ)−1 + (2c)
v
2πi λ
I|z|=1h (cid:18) (cid:18) (cid:19) (cid:19)
dz
+eλ¯ζ logλ¯(λ+λ¯)−Ein(λ¯ζ)−1 , (2d)
z
1 eλ¯ζ(cid:0)−1 dz (cid:1)i
θ (ζ) = (2e)
u
2πi ζ z
I|z|=1
and the functions
yα(ζ) = ζα+21θα(ζ), (3a)
yv(ζ) = ζ21 ζ−1θv(ζ)+2log(ζ)θu(ζ) , (3b)
1
yu(ζ) = ζ2θ(cid:0)u(ζ), (cid:1) (3c)
which are multivalued in ζ on C× := C\{0}.
Theorem. The sequence of functions {y (ζ)} on M ×C× forms a
αˆ αˆ∈Z 0
Levelt basis of deformed flat coordinates for M .
0
The Principal hierarchy is given by the set of vector fields on LM
0
defined by the Poisson structure associated with the flat metric η on
M and the Hamiltonians
0
H = θ dx,
αˆ,p αˆ,p+1
IS1
where the Hamiltonian densities θ are obtained by expanding at
αˆ,p
ζ = 0 the analytic part of the Levelt basis of deformed flat coordinates
θ (ζ) = θ ζp.
αˆ αˆ,p
p>0
X
Since
1 dz
θ = Q ,
αˆ,p αˆ,p
2πi z
I|z|=1
the Hamiltonians of the Principal hierarchy are equal to those of the
extended 2D Toda hierarchy defined before.
6 G. CARLET AND L.PH. MERTENS
Theorem. The Principal hierarchy of the Frobenius manifold M co-
0
incides with the extended dispersionless 2D Toda hierarchy restricted
on LM .
0
Recently other examples of infinite-dimensional Frobenius manifolds
have appeared in the literature. In [13] Raimondo has constructed
a Frobenius manifold structure on a vector subspace of the space of
Schwartz functions S(R) on the real line which is associated with
the dispersionless Kadomtsev-Petviashvili hierarchy (dKP). Wu and
Xu [17] have defined a family of Frobenius manifolds on the space of
pairs of certain even functions meromorphic in the interior/exterior
of the unit disk in C, which are related to the dispersionless two-
component BKPhierarchy. Inbothcasestheauthorsdefine, essentially
by bihamiltonian recursion, dispersionless hierarchies which extend the
original 2 + 1 systems. Note that the definition of such hierarchies is
somehow simpler since it does not require the construction in terms of
the Levelt normal form of the deformed flat connection, as presented
here.
The article is organized as follows.
In the first section we define the extended dispersionless 2D Toda
hierarchy on the loop space of holomorphic Lax symbols with certain
conditions on the winding numbers. We first recall some basic facts on
thedispersionless2DTodahierarchyandintroducetheanalyticsetting.
Next we give the Lax and bi-Hamiltonian formulation of the extended
flows and show that they indeed contain the usual dispersionless flows
of the 2D Toda hierarchy.
In the second section we study the deformed flat connection associ-
atedwiththeinfinite-dimensional FrobeniusmanifoldM definedin[3].
0
We obtain simple expressions for the metric and its Levi-Civita con-
nection in a new set of “mixed” coordinates. Necessary background
from the theory of the Frobenius manifold M is recalled when nec-
0
essary. We solve explicitly the deformed flatness equations and prove
that our solution provides a Levelt system of deformed flat coordinates.
Expanding in the deformation parameter we obtain the Hamiltonian
densities of the Principal hierarchy of M which coincide with those of
0
theextended2DTodahierarchy. Finallyanalternative, thoughslightly
morecomplicated, system of deformed flat coordinates satisfying anor-
thogonality condition (cf. [9, Theorem 3.6.4]), and the corresponding
Principal hierarchy are presented.
INFINITE-DIMENSIONAL PRINCIPAL HIERARCHIES 7
1. The extended dispersionless 2D Toda hierarchy
In this section we define an extension of the dispersionless 2D Toda
hierarchy introduced by Takasaki and Takebe [14] as the small disper-
sion limit of the 2D Toda hierarchy of Ueno and Takasaki [15]. To
¯
perform such extension we assume that Lax functions λ, λ are non-
vanishing holomorphic functions on neighborhoods of z = ∞, 0 re-
spectively, which contain the unit circle, and that they satisfy certain
analytic assumptions. We begin by recalling the standard formulation
of the dispersionless 2D Toda hierarchy.
1.1. The dispersionless 2D Toda hierarchy. The dispersionless
2D Toda hierarchy is an infinite set of commuting quasi-linear PDEs
for two sets of variables u , u¯ depending on a “space” variable x and
k l
two series of independent “time” variables t = (t ) , t¯= (t¯ ) . Let
k k>0 k k>0
the Lax symbols
λ(z,x) = z + u (x)zk, λ¯(z,x) = u¯ (x)zl.
k l
k60 l>−1
X X
betwoformalLaurentseriesinz. Thedispersionless2DTodahierarchy
is defined by the Lax equations
¯
∂λ ∂λ
= {(λn) ,λ}, = {(λn) ,λ¯}, (4a)
+ +
∂t ∂t
n n
¯
∂λ ∂λ
= {(λ¯n) ,λ}, = {(λ¯n) ,λ¯}. (4b)
∂t¯ − ∂t¯ −
n n
The bracket of two functions of the variables z,x is defined by
∂f ∂g ∂g ∂f
{f(z,x),g(z,x)} = z −z ,
∂z ∂x ∂z ∂x
while the notations ( ) , ( ) represent projections taken with respect
+ −
to the variable z:
f zk = f zk, f zk = f zk.
k k k k
! !
Xk + Xk>0 Xk − Xk<0
The equations (4) are formal Laurent series in z: each coefficient
defines an evolutionary quasi-linear equation involving a finite number
of dependent variables u , u¯ . Such equations have the remarkable
k l
property of defining commutative flows
∂ ∂
, = 0
∂s ∂s
(cid:20) n m(cid:21)
for s equal to either t or t¯ .
n n n
8 G. CARLET AND L.PH. MERTENS
The flows (4) admit a bi-Hamiltonian formulation [1]
∂
· = {·,H } = −{·,H } , (5a)
n 1 n−1 2
∂t
n
∂
· = {·,H¯ } = {·,H¯ } (5b)
∂t¯ n 1 n−1 2
n
with Hamiltonians given by
λn+1 dz λ¯n+1 dz
¯
H = − Res dx, H = − Res dx,
n n
n+1 z n+1 z
Z Z
where the residue of a formal series is Res f zkdz = f . The hy-
k k z 0
drodynamic type Poisson brackets {,} and {,} are compatible, i.e.
1 2
P
any their linear combination is still a Poisson bracket. These Poisson
brackets have been defined in [1]. Their definition is recalled below in
Proposition 1.
1.2. Analytic setting. DenoteD theclosedunitdiscintheRiemann
0
sphere C ∪ {∞}, D the closure of the complement of D and S1 =
∞ 0
D ∩ D the unit circle. For a compact subset K of the Riemann
0 ∞
sphere, denote by H(K) the space of holomorphic functions on K, i.e.
functions which extend holomorphically to a neighbourhood of K.
Foreachp ∈ Zthespaceofholomorphicfunctionsonaneighborhood
of S1 splits in a direct sum
H(S1) = zpH(D )⊕zp−1H(D ).
0 ∞
The projections
( ) : H(S1) → zpH(D ), ( ) : H(S1) → zp−1H(D )
>p 0 6p−1 ∞
are given by
zp ζ−pf(ζ)
(f) (z) = f zk = dζ,
>p k
2πi ζ −z
k>p I|z|<|ζ|
X
zp ζ−pf(ζ)
(f) (z) = f zk = − dζ
6p−1 k
2πi ζ −z
k6p−1 I|z|>|ζ|
X
for f(z) = f zk ∈ H(S1). As usual ( ) = ( ) and ( ) = ( ) .
k∈Z k + >0 − 6−1
The symbol ( ) : H(S1) → C denotes the coefficient of zk in the
k
P
Laurent series expansion, i.e. (f) = 1 f(z)z−kdz.
k 2πi |z|=1 z
Define the infinite-dimensional manifold M as the affine subspace
H
1
¯
M = (λ(z),λ(z)) ∈ zH(D )⊕ H(D ) λ(z) = z+O(1) for z → ∞
∞ 0
z
in the(cid:8)direct sum of the vector spaces zH(cid:12)(D ) and 1H(D ). We wi(cid:9)ll
(cid:12) ∞ z 0
sometimes refer to M asthe spaceof pairsofholomorphic Laxsymbols.
INFINITE-DIMENSIONAL PRINCIPAL HIERARCHIES 9
¯ ¯
For (λ,λ) ∈ M, the functions λ(z), λ(z) have the following Laurent
series expansions
λ(z) = z + u zk, λ¯(z) = u¯ zk
k k
k60 k>−1
X X
at ∞ and 0 respectively.
ˆ ¯
The tangent space at a point λ = (λ(z),λ(z)) ∈ M will be identified
with a space of pairs of functions
1
∼
T M = H(D )⊕ H(D )
λˆ ∞ z 0
where a vector ∂ is associated with the pair X = (X(z),X¯(z)) given
X
by X(z) = ∂ λ(z) and X¯(z) = ∂ λ¯(z).
X X
ˆ
The cotangent bundle at λ ∈ M will be given also by a space of pairs
of functions
T∗M ∼= H(D )⊕zH(D )
λˆ 0 ∞
by representing a covector α as the pair (α(z),α¯(z)) by the residue
pairing
1 dz
< α,X >= α(z)X(z)+α¯(z)X¯(z) . (6)
2πi z
I|z|=1
(cid:2) (cid:3)
Note that this residue pairing is slightly different from the pairing used
in [3].
A point in the loop space LM of smooth maps from S1 to the M is
given by a pair of functions (λ(z,x),λ¯(z,x)). Here S1 = R mod 2π,
while the symbol S1 will denote the unit circle in C.
¯
A tangent vector at a point (λ(z,x),λ(z,x)) ∈ LM is clearly iden-
tified with a map from S1 to H(D ) ⊕ 1H(D ) and a 1-form with
∞ z 0
a map from S1 to H(D ) ⊕ zH(D ). The pairing of a vector X =
0 ∞
(X(z,x),X¯(z,x)) and a 1-form α = (α(z,x),α¯(z,x)) is
1 dz
< α,X >= α(z,x)X(z,x)+α¯(z,x)X¯(z,x) dx,
2πi z
IS1I|z|=1
(cid:2) (cid:3)
which is the natural extension of the pairing (6).
The equations (4) defining the 2D Toda flows specify, for each n > 0,
a vector field over LM. Indeed, note that equations (4) are of the form
¯ ¯
(∂ λ,∂ λ) = ({−Q ,λ},{Q ,λ})
t t − +
where Q = λn or λ¯n. For (λ,λ¯) ∈ LM, at fixed x ∈ S1, we have Q(z) ∈
H(S1) and we can easily check that this implies that {Q ,λ} ∈ H(D )
− ∞
and {Q ,λ¯} ∈ 1H(D ). Hence ({−Q ,λ},{Q ,λ¯}) ∈ T LM.
+ z 0 − + λˆ
10 G. CARLET AND L.PH. MERTENS
Recall that a Poisson bracket {,} of two local functionals F, G on
i
LM is written in terms of a Poisson operator P from the cotangent to
i
the tangent space of LM as follows
{F,G} =< dF,P (dG) > .
i i
The bi-Hamiltonian structure of the 2D Toda hierarchy was defined
in [1]. We recall here the formulas for the dispersionless limit of the
Poisson brackets, in the analytic setting.
Proposition 1. The maps P : T∗LM → TLM define compatible
i
Poisson brackets on LM. Such maps, given a 1-form ωˆ = (ω,ω¯) ∈
T∗LM at λˆ = (λ,λ¯) ∈ LM, are defined by
λˆ
¯
P (ωˆ) = −{λ,(ω −ω¯) }+({λ,ω}+{λ,ω¯}) ,
1 − 60
¯ ¯
{λ,(ω −ω¯) }+({λ,ω}+{λ,ω¯})
(cid:0) + >0
P (ωˆ) = {λ,(λω +λ¯ω¯) }−λ({λ,ω}+{λ¯,ω¯}) +zλ′φ ,
2 − (cid:1)60 x
−{λ¯,(λω,+λ¯ω¯) }+λ¯({λ,ω}+{λ¯,ω¯}) +zλ¯′φ
(cid:0) + >0 x
where
(cid:1)
1 dz
¯
φ = {λ,ω}+{λ,ω¯} .
x
2πi z
I|z|=1
(cid:0) (cid:1)
The Hamiltonians
1 λn+1 dz 1 λ¯n+1 dz
¯
H = − dx, H = − dx
n n
2πi n+1 z 2πi n+1 z
ZS1I|z|=1 ZS1I|z|=1
define local functionals on LM and generate the Hamiltonian vector
fields (4) according to (5). Summarizing well known facts in this ana-
lytic setting:
Proposition 2. The dispersionless 2D Toda hierarchy equations (4)
define a set of bi-Hamiltonian commuting vector fields on LM, with
respect to the Poisson brackets {,} and with recursion relations (5).
i
Note that we have chosen to represent 1-forms on M by elements
of H(D ) ⊕ zH(D ) using the pairing (6). One can more generally
0 ∞
represent a 1-form by a pair of functions in H(S1) ⊕ H(S1): the 1-
form does not change by adding to the representative an element in
1H(D )⊕z2H(D ) (recall that zpH(D ) and zpH(D ) are seen here
z ∞ 0 ∞ 0
as subspaces of H(S1), for any p ∈ Z). The freedom of choosing the
representative extends ofcoursetothe1-formsontheloopspace. Later
we will need the following easy to check observation: