Kernel Formula Approach to the Universal Whitham Hierarchy

0.17 MB·English

by  Hsin-Fu Shen

Kernel Formula Approach to the Universal
Whitham Hierarchy
Hsin-Fu Shen1, Niann-Chern Lee2, and Ming-Hsien Tu3 ∗
1 Department of Mechanical Engineering, WuFeng Institute of Technology,
Chiayi 621, Taiwan,
2 General Education Center, National Chinyi University of Technology,
0
1
Taichung 411, Taiwan
0
2
3 Department of Physics, National Chung Cheng University,

n
a Chiayi 621, Taiwan
J

2
January 12, 2010
1

]
I
S
.
n
Abstract
i
l
n
We derivethe dispersionlessHirotaequationsofthe universalWhitham hierarchyfrom
[

the kernelformula approachproposedby Carrolland Kodama. Besides, we also verify the
1
v associativity equations in this hierarchy from the dispersionless Hirota equations and give
4
3 a realization of the associative algebra with structure constants expressed in terms of the
9
residue formulas.
1
1. Keywords: kernel formula, Lax formulation, dispersionless Hirota equations, Whitham
0
hierarchy, WDVV equations, residue formula.
0
1
:
v
i
X
r
a
∗[email protected]
1
1 Introduction
Dispersionless integrable hierarchies have been an active subject of research in recent years.
ThedispersionlessKP(dKP)hierarchy[12,13,23,25]anddispersionlessToda(dToda)hierarchy[22,
24] are now the prototype systems in several branches of physics and mathematics, such as
topological field theories, matrix models, string theory, Laplacian growth problem and confor-
mal maps (see, e.g., [1, 5, 4, 2, 15, 16, 21, 29, 31, 32], and references therein). The solution
of the dispersionless integrable hierarchies can be characterized by a τ-function whose loga-
rithm, F = logτ, called free energy satisfies a set of dispersionless Hirota (dHirota) equations
and gives the solutions to the associativity equations or Witten-Dijkgraaf-Verlinde-Verlinde
(WDVV) equations introduced in the context of topological field theories[10, 8, 30].
Itwasshown[3]thattheassociativity equationsareencodedinthedHirotaequationsforthe
dKP and the dToda hierarchies. There are several works devoted to derive the corresponding
dHirota equations. For the dKP hierarchy, Takasaki and Takebe[25] derived them by taking
the dispersionless limit of the differential Fay identity. Later, Carroll and Kodama[6] studied
it from the approach of kernel formulas. As for the dToda hierarchy, Wiegmann and Zabrodin
et al.[29] investigated the dHirota equations in the language of conformal mappings. Teo[28]
derived the dHirota equations for both the dKP and dToda hierarchies in the content of
complex analysis using the notion of Faber polynomials and Grunsky coefficients.
In this work we shall focus on the universal Whitham hierarchy which was introduced by
Krichever[16]asauniversalframeworkforbothdispersionlessintegrablesystemsandWhitham
modulation equations. Inparticular, thegenus-zero case isthemain target of recentresearches
(see e.g., [19, 20, 11, 17, 18, 26, 27]) since the universal Whitham hierarchy is the master
equation of many other dispersionless integrable systems. Based on the work of Carroll and
Kodama[6](seealso[7])weliketogeneralizethekernelformulaapproachforthedKPhierarchy
to derive the dHirota equations of the universal Whitham hierarchy. Moreover, motivated by
theworkofBoyarsky etal.[3], weintendtoprovideadirectverification ofassociative equations
in the universal Whitham hierarchy from the dHirota equations point of view.
Thispaperisorganizedasfollows. Inthenextsection,webrieflyrecalltheLaxformulation
for the universal Whitham hierarchy of zero genus case. In section 3, the kernel formula
approach is employed to derive the dHirota equations of the universal Whitham hierarchy. In
Section 4, following the work of [3] , we will prove the associativity equation of the universal
Whitham hierarchy from these dHirota equations and give a realization of the associative
algebra. The situations of the finite-dimensional reduction are also briefly discussed. Section
5 is devoted to the concluding remarks.
2
2 Lax Formulation of the Universal Whitham Hierarchy
Theuniversal Whitham hierarchy of genus zero with N+1 marked points at q (α = 1,··· ,N)
α
and ∞ on Riemann sphere is defined by the Lax functions[16]
λ (p) = p+ u p−j+1
0 j
j=2
X
r
λ (p) = α + u (p−q )j−1, α = 1,··· ,N
α αj α
p−q
α
j=1
X
whereu andu arefunctionsofthetimevariablest (n = 1,2,···)andt (α = 1,··· ,N;n =
j αj 0n αn
0,1,···). One can regard λ as a map defined in domain Ω containing p = ∞, while λ in
0 0 α
Ω containing p = q . λ and λ satisfy the Lax equations
α α 0 α
∂ λ (p) ={B (p),λ (p)}, ∂ = ∂/∂t (1)
αn β αn β αn αn
where the Poisson bracket is defined by {f,g} = ∂ f∂ g−∂ f∂ g and
p 01 01 p
B (p) = (λn)
0n 0 (0,≥0)
B (p) = (λn) , n = 1,2,···
αn α (α,<0)
B (p) = −log(p−q ).
α0 α
Following the notation used in [19, 26] we shall denote () and () as the projections
(0,≥0) (α,<0)
to nonnegative powers in p and negative powers in (p−q ), respectively. Therefore
α
λn(p) = B (p)+O(p−1), p → ∞,
0 0n
λn(p) = B (p)+O(1), p → q .
α αn α
In particular, B (p)= p and B (p) = r /(p−q ). The Lax equations can be written as the
01 α1 α α
Zakharov-Shabat equations
∂ B (p)−∂ B (p)+{B (p),B (p)} = 0
βn αm αm βn αm βn
which is equivalent to ω∧ω = 0 with
∞ N ∞
ω = dB ∧dt + dB ∧dt .
0n 0n αn αn
n=1 α=1n=0
X XX
Introducing Darboux coordinate (λ ,M ) in Ω and (λ ,M ) in Ω , we have
0 0 0 α α α
ω = dλ ∧dM in Ω
0 0 0
= dλ ∧dM in Ω
α α α
3
where the Orlov-Schulman functions M and M are defined by[19, 26]
0 α
N
t
M = nt λn−1+ 00 + λ−n−1v , t = − t
0 0n 0 λ 0 0n 00 α0
0
n=1 n=1 α=1
X X X
t
M = nt λn−1+ α0 + λ−n−1v ,
α αn α λ α αn
α
n=1 n=1
X X
satisfies
∂ M = {B ,M }
αn β αn β
and the canonical relation {λ ,M } = {λ ,M }= 1.
0 0 α α
From the expressions of the symplectic two-from ω we have, in Ω ,
0
∞ N ∞
d M dλ + B dt + B dt = 0.
0 0 0n 0n αn αn
!
n=1 α=1n=0
X XX
Therefore, there exists a S-function, S , such that
0
∞ N ∞
dS = M dλ + B dt + B dt .
0 0 0 0n 0n αn αn
n=1 α=1n=0
X XX
On the other hand, in Ω , we have
α
∞ N ∞
dS = M dλ + B dt + B dt .
α α α 0n 0n αn αn
n=1 α=1n=0
X XX
It turns out that the Orlov-Schulman functions M and M have the expressions
0 α
∂S (t) ∂S (t)
0 α
M = , M =
0 α
∂λ ∂λ
0 α
then the S-functions admit expansions of the form
λ−n
S = t λn+t logλ − 0 v ,
0 0n 0 00 0 n 0n
n=1 n=1
X X
λ−n
S = t λn +t logλ +φ − α v .
α αn α α0 α α n αn
n=1 n=1
X X
The Hamilton-Jacobi equations are defined by
∂ S = B (p ) = B (∂ S )
αn β αn β αn 01 β
4
hence
λ−m
B = ∂ S (t)= λn − 0 ∂ v
0n 0n 0 0 m 0n 0m
m=1
X
λ−m
= ∂ S (t)= ∂ φ − α ∂ v
0n α 0n α 0n αm
m
m=1
X
λ−m
B = ∂ S = − 0 ∂ v
αn αn 0 αn 0m
m
m=1
X
λ−m
= ∂ S = δ λn +∂ φ − β ∂ v
αn β αβ β αn β m αn βm
m=1
X
λ−m
B = ∂ S = −logλ − 0 ∂ v
α0 α0 0 0 α0 0m
m
m=1
X
λ−m
β
= ∂ S = δ logλ +∂ φ − ∂ v .
α0 β αβ β α0 β α0 βm
m
m=1
X
Theaboveequationsprovidesomerelationsbetweenthecoefficients(u ,u ,r ,q )and(φ ,v ).
j jα α α α αn
For instance, from B = p we have
01
λ−m
p = ∂ S (λ) = λ − 0 ∂ v , in Ω
01 0 0 01 0m 0
m
m=1
X
λ−m
p = ∂ S (λ) = ∂ φ − α ∂ v , in Ω
01 α 01 α 01 αm α
m
m=1
X
which implies u = ∂ v , q = ∂ φ and r = −∂ v . On the other hand, B (p ) =
2 01 01 α 01 α α 01 α1 α0 α
∂α0Sα = −log(pα−qα) which yields rα = e−∂α0φα. In fact, there exist a F function [16, 19, 26]
such that
α
v = F , v = F , φ = −F + t log(−1), (2)
0n 0n αn αn α α0 β0
β=1
X
where Fαn = ∂αnF. Then we see that u2 = ∂021F, the dKP potential. And from rα = e−∂α0φα,
∂01∂α1F −e∂α20F = 0, which is just the dToda field equation. We remark here that the Lax
equation for the t flow is ∂ λ = {B ,λ} = −1 ∂λ if setting q to be zero. The Poisson
α0 α0 α0 p ∂t01 α
bracket may be written as, not so strictly, {f,g} = −(p∂f ∂g −p ∂f ∂g). The time variable
∂p∂tα0 ∂tα0∂p
−t seems to play the role of the parameter s in the usual Poisson bracket of the dToda
α0
hierarchy[22,24,25]. ThereforetheuniversalWhithamhierarchy canbeviewed asanassembly
of one dKP hierarchy and N copies of dToda-like hierarchies.
5
3 Carroll-Kodama Approach
3.1 Kernel Formulas
In contrast to the dKP and dToda hierarchy, the Whitham hierarchy contains N + 1 Lax
functions λ and λ that are defined in domains Ω and Ω (α = 1,··· ,N), respectively. To
0 α 0 α
derive the kernel formulas of the universal Whitham hierarchy we shall generalize the method
of kernel formula to the domains Ω , Ω , Ω ∩Ω and Ω ∩Ω (α 6=β).
0 α 0 α α β
In Ω , from the expression of B = p = ∂ S , we have
0 01 01 0
λ−m
λ −p− 0 F = 0.
0 01,0m
m
m=1
X
Multiplying above by λn−1∂ λ and taking the projection () then we obtain
0 p 0 (0,≥0)
n−1
F
01,0m
∂ Q (λ)−p∂ Q (λ)− ∂ Q (λ) = 0, n≥ 1 (3)
p 0,n+1 p 0,n p 0,n−m
m
m=1
X
where Q (λ) = B (λ)/n. Taking the summation µ−n and noting that ∂ Q = 1 we
0,n 0n n=0 p 0,1
have P
∞
F
µ−p (λ)− 01,0mµ−m ∂ Q (λ)µ−n = 1
0 p 0,n
m
!
m=1 n=1
X X
or
1
= ∂ Q (λ)µ−n. (4)
p 0,n
p (µ)−p (λ)
0 0
n=1
X
Integrating above with respect to p (λ) by fixing the integration constant at µ → ∞, we find
0
µ B (λ)
log = 0,n µ−n (5)
p (µ)−p (λ) n
0 0
n=1
X
Rewriting the expressions of p = ∂ S and B = ∂ S in kernel formula (5) as
0 01 0 0n 0n 0
p (λ) = λ−D (λ)F ,
0 0 01
B (λ) = λn−D (λ)F
0n 0 0n
where we have used the symbols D (z) = ∞ z−n∂ . Then the kernel formula (5) becomes
α n=1 n αn
P ∂ (D (µ)−D (λ))F
eD0(µ)D0(λ)F = 1− 01 0 0 . (6)
µ−λ
In Ω , from B = ∂ S , we have
α αn αn α
r λ−m
B = α = λ +∂ φ − α F
α1 α α1 α α1,αm
p−q m
α
m=1
X
6
Multiplying above by λn−1∂ λ and taking the projection () then we obtain
α p α (0,≤−2)
n−1
r F
α α1,αm
∂ Q (λ)− −∂ φ ∂ Q (λ)− ∂ Q (λ) = 0, n ≥ 1 (7)
p α,n+1 α1 α p α,n p α,n−m
p−q m
α
(cid:18) (cid:19) m=1
X
where Q (λ) = B (λ)/n. Taking the summation µ−n and noting that ∂ Q =
α,n αn n=0 p α,1
− rα we have P
(p−qα)2
∞
r F r
µ− α −∂ φ − α1,αmµ−m ∂ Q (λ)µ−n = − α
p−q α1 α m p α,n (p−q )2
α ! α
(cid:18) (cid:19) m=1 n=1
X X
or
p (µ)−q
α α = ∂ Q (λ)µ−n. (8)
p α,n
(p (µ)−p (λ))(p (λ)−q )
α α α α
n=1
X
Integrating above with respect to p (λ) by fixing the integration constant at µ → ∞, we find
α
p (λ)−q B (λ)
log α α = α,n µ−n (9)
p (λ)−p (µ) n
α α
n=1
X
Rewriting the expressions of p −q , p and B in kernel formula (9) as
α α α αn
1
p −q = e−∂α0Sα(λ) = − e∂α0(∂α0+Dα(λ))F
α α
λ
p (λ) = ∂ S (λ) = ∂ φ −D (λ)F
α 01 α 01 α α 01
B = ∂ S (λ) = λn +∂ φ −D (λ)F
αn αn α αn α α αn
then the kernel formula (9) becomes
µλ
e(∂α0+Dα(µ))(∂α0+Dα(λ))F = − ∂ (D (µ)−D (λ))F. (10)
01 α α
µ−λ
In Ω ∩Ω , let us rewrite (5) as
0 α
µ = ePn=1 ∂0nSn0(λ)µ−n.
∂ S (µ)−∂ S (λ)
01 0 01 0
By replacing S (λ) by S (λ) and using the relations
0 α
∂ S (λ) = ∂ φ −D (λ)F
01 α 01 α α 01
∂ S (λ) = ∂ φ −D (λ)F
0n α 0n α α 0n
then we have
µeD0(µ)(∂α0+Dα(λ))F = µ−∂ (D (µ)−∂ −D (λ))F. (11)
01 0 α0 α
On the other hand, one can derive (11) from the kernel formula (9) by expressing it as
e−∂α0Sα(λ) = ePn=1 ∂αnSnα(λ)µ−n.
∂ S (λ)−∂ S (µ)
01 α 01 α
7
After replacing S (λ) by S (λ), then the above equation becomes
α 0
λeD0(λ)(∂α0+Dα(µ))F = λ−∂ (D (λ)−∂ −D (µ))F
01 0 α0 α
which, after making an exchange λ ↔ µ, coincides with (11).
In Ω ∩Ω with α 6= β, we rewrite (9) as
α β
e−∂α0Sα(λ) = ePn=1 ∂αnSnα(λ)µ−n.
∂ S (λ)−∂ S (µ)
01 α 01 α
By replacing S (λ) by S (λ), then above equation becomes
α β
e−∂α0Sβ(λ) = ePn=1 ∂αnSnβ(λ)µ−n. (12)
∂ S (λ)−∂ S (µ)
01 β 01 α
Since
e−∂α0Sβ(λ) = e−∂α0φβ+∂α0Dβ(λ)F = −ǫ e∂α0(∂β0+Dβ(λ))F,
αβ
∂ S (λ)−∂ S (µ) = ∂ (D (µ)+∂ −∂ −D (λ))F,
01 β 01 α 01 α α0 β0 β
F
∂ S (λ) = ∂ φ − αn,βmλ−m,
αn β αn β
m
m=1
X
where φ is defined by (2) and
β
+1, α≤ β
ǫ =
αβ

−1, α> β

Thus eq.(12) can be expressed as

ǫ e(∂α0+Dα(µ))(∂β0+Dβ(λ))F = −∂ (D (µ)+∂ −∂ −D (λ))F (13)
αβ 01 α α0 β0 β
Eqs. (6), (10), (11) and (13) constitute the dispersionless Hirota equations of the genus-zero
universal Whitham hierarchy. They describe algebraic relations between second derivatives of
the free energy F. Remarkably, our results coincide with that by Takasaki and Takebe [26]
whoderived these equations from the Hamilton-Jacobian equations in the context of the Faber
polynomials.
3.2 Dispersionless Hirota Equations
Using the parametrization[27]: tˆ = nt , tˆ = t and tˆ = nt . the corresponding time
0n 0n α0 α0 αn αn
derivative is written as ∂ˆ = ∂ . Then the four dispersionless Hirota equations can be
αn ∂tˆαn
8
rewritten as[26] :
p (µ)−p (λ)
Dˆ (µ)Dˆ (λ)F = log 0 0 ,
0 0
µ−λ
µλ(p (µ)−p (λ))
Dˆ (µ)Dˆ (λ)F = log α α ,
α α
µ−λ
(14)
p (µ)−p (λ)
Dˆ (µ)Dˆ (λ)F = log 0 α ,
0 α
µ
p (µ)−p (λ)
Dˆ (µ)Dˆ (λ)F = log α β ,
α β
ε
αβ
where we have use the new symbols : Dˆ (z) = ∞ z−n∂ = ∞ z−n∂ˆ , and Dˆ (z) =
0 n=1 n 0n n=1 0n α
∂α0 + ∞n=1 z−nn∂αn = ∞n=0z−n∂ˆαn. This set oPf equations will bPe our main interests in the
followiPngdiscussion. NoPtethattheyhavethesameformifapplyingthedifferentiation operator
Dˆ (ν) on them. The result can be summarized as a single equation:
δ
(Dˆ (µ)−Dˆ (λ))Dˆ (ν)Fˆ
Dˆ (µ)Dˆ (λ)Dˆ (ν)F = − α β δ 01, (15)
α β δ
p (µ)−p (λ)
α β
where α,β,δ = 0,1,2,··· ,N. In fact, we may rewrite (14) as a set of equations satisfied by
the second derivatives of F. Letting µ → λ in (14) we have
kFˆ = P (X), X = 0,X = Fˆ
0k,01 k+1 1 j≥2 0n,0m
n+m=j
X
kFˆ = eFˆ0α0,α0P (X), X = Fˆ
αk,01 k−1 j≥1 αn,αm
n+m=j
X
k+1
kFˆ = P (X)− Y P (X), X = Fˆ ,Y = nFˆ
0k,01 k+1 l k+1−l j≥1 0n,αm j 0n,αm
l=1 n+m=j n+m=j
X X X
k
kFˆαk,01 = ∓eFˆα0,β0 YlPk−l(X), Xj≥1 = Fˆαn,βm,Yj = nFˆαn,βm
l=1 n+m=j n+m=j,n≥1
X X X
where k ≥ 1 and Pk(X) are Schur polynomials defined by ePj=1Xjsj = k=0Pk(X)sk. The
first nontrivial one of each dHirota equation is given below P
Fˆ −Fˆ −Fˆ2 /2 = 0
03,01 02,02 01,01
Fˆ −eFˆα0,α0 = 0
α1,01
Fˆ +Fˆ +Fˆ2 /2 = 0
01,01 02,α0 01,α0
Fˆα1,01±eFˆα0,β0Fˆα1,β0 = 0
We remark that, in terms of t , the first and the second equations are just the dKP and
αn
dToda equations, respectively.
9
4 Associativity Equations in the Whitham Hierarchy
4.1 WDVV equations
Let F be a function of the time variables tˆ where the subscribes J = {(0,n),(α,m)} with
J
α = 1,··· ,N;n ≥1,m ≥ 0anddenotingFˆ = ∂ˆ F, Fˆ = ∂ˆ ∂ˆ F,etc. Letusdefinetˆ = tˆ
J J KL J K I 01
andintroducethenondegenerate metric ηˆ = Fˆ whichprovides atransformation between
JK JKI
{tˆ } and {Fˆ }. Thus
J KI
∂Fˆ ∂Fˆ
Fˆ = JK AI = CˆA ηˆ (16)
JKL A ∂FˆAI ∂tˆL A JK AL
X X
where the coefficients CˆA defined above connect Fˆ and ηˆ . Notice that setting J = I
JK JKL AL
one has ηˆ = CˆA ηˆ and hence CˆA = δA. If CˆL are structure constants of an
KL A IK AL IK K JK
associative algebrPa generated by {φˆJ}, i.e., φˆJ · φˆK = LCˆJLKφˆL, then the associativity
condition (φˆJ ·φˆK)·φˆL = φˆJ ·(φˆK ·φˆL) yields CˆJMKCˆMNL =PCˆJNMCˆKML, or, equivalently,
CˆM Fˆ = CˆM Fˆ .
JK MLN KL JMN
M M
X X
The above equation is known as the WDVV equations arising in the context of topological
field theory[10, 8, 30]. Define a four indices quantity Xˆ = CˆL Fˆ , then the
JKMN L JK LMN
WDVV equations can be replaced by the condition that XˆJKMN is sPymmetric with respect to
permutations of any indices. Using the operator Dˆ (z) introduced in the previous section, it
α
is convenient to define the generating functions for Fˆ , Fˆ , CˆL and Xˆ as follows:
JK JKL JK JKLM
Dˆ (u)Dˆ (v)F = u−nv−mFˆ , (17)
α β αn,βm
n,m=0
X
Dˆ (u)Dˆ (v)Dˆ (w)F = u−nv−mw−lFˆ , (18)
α β γ αn,βm,γl
n,m,l=0
X
CˆJ(u ,v ) = u−nv−mCˆJ , (19)
α β αn,βm
n,m=0
X
Xˆ(u ,v ,w ,z ) = u−nv−mw−kz−lXˆ . (20)
α β γ δ αn,βm,γk,δl
n,m,k,l=0
X
Hence the WDVV equations can be expressed in terms of the generating function as
Xˆ(u ,v ,w ,z )= Xˆ(u ,w ,v ,z ).
α β γ δ α γ β δ
4.2 From dHirota to WDVV
Applying ∂ˆ on the first line of dHirota equation(14), we get
αl
(Dˆ (u)−Dˆ (v))∂ˆ Fˆ
Dˆ (u)Dˆ (v)∂ˆ F = − 0 0 αl 01
0 0 αl
p (u)−p (v)
0 0
10