the hebrew university of jerusalem einstein institute of ... · 1 g=1 (6g5)!! 24g1 ·(g1)! z 6g+4 2...
TRANSCRIPT
![Page 1: The Hebrew University of Jerusalem Einstein Institute of ... · 1 g=1 (6g5)!! 24g1 ·(g1)! z 6g+4 2 P 1 g=0 (6g1)!! 24g ·g! z 6g 2 P 1 g=0 6g+1 6g1 (6g1)!! 24g ·g! z 6g+2 P 1 g=1](https://reader035.vdocument.in/reader035/viewer/2022070909/5f92d2f5e52af67f3444f46e/html5/thumbnails/1.jpg)
Integrable systemsand moduli spaces
Boris DUBROVINSISSA, Trieste
The Hebrew University of JerusalemEinstein Institute of Mathematics
April 30, 2015
Sunday 10 May 15
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Plan
1. Integrable Hamiltonian systems
2. Hamiltonian PDEs
3. KdV
4. Deligne - Mumford moduli spaces and Witten - Kontsevich solution to KdV
Mg,n
5. Construction of KdV: a recipe using Mg,n
6. More general class of integrable hierarchie (Witten’s programme) and -classes�
Sunday 10 May 15
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Hamiltonian systems
x = J rH
x 2 M
a manifold (phase space), H = H(x) Hamiltonian
J operator of Poisson bracket
E.g., dimM = 2n, x = (q1, . . . , qn, p1, . . . , pn), J =
✓0 1
�1 0
◆
qi = @H@pi
pi = �@H@qi
9=
; , i = 1, . . . , n
{f, g} = hrf, J rgi
Integrability: commuting Hamiltonians{Hi, Hj} = 0, H = H1
+ completeness ) commuting flowsdx
dti= J rHi,
d
dti
✓dx
dtj
◆=
d
dtj
✓dx
dti
◆
Sunday 10 May 15
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Infinite-dimensional analogue: Hamiltonian PDEs
ut
= F (u, ux
, uxx
, . . . ) = J�H
�u(x)for u = u(x, t)
a dynamical system on the space of functions u(x)
Functional (Hamiltonian)
A skew-symmetric operator J of Poisson bracket
H = H[u] =1
2⇡
Z 2⇡
0h(u, u
x
, . . . ) dx
Fréchet derivative (Euler - Lagrange operator)
�H
�u(x)=
@h
@u� d
dx
@h
@ux
+d2
dx2
@h
@uxx
� . . .
Sunday 10 May 15
![Page 5: The Hebrew University of Jerusalem Einstein Institute of ... · 1 g=1 (6g5)!! 24g1 ·(g1)! z 6g+4 2 P 1 g=0 (6g1)!! 24g ·g! z 6g 2 P 1 g=0 6g+1 6g1 (6g1)!! 24g ·g! z 6g+2 P 1 g=1](https://reader035.vdocument.in/reader035/viewer/2022070909/5f92d2f5e52af67f3444f46e/html5/thumbnails/5.jpg)
Example. Korteweg - de Vries (KdV) equation
Integrability: an infinite system of commuting PDEs
(uti)tj =�utj
�ti
uti = F
i
(u, ux
, uxx
, . . . ) = J�H
i
�u(x)i = 0, 1, 2, . . .
) common solution u = u(x, t0, t1, t2, . . . )
for given Cauchy data u0(x) = u(x, 0, 0, 0, . . . )
J =@
@x
ut
= uux
+✏2
12uxxx
=@
@x
�H
�u(x), H =
Z ✓u3
6� ✏2
24u2x
◆dx
Sunday 10 May 15
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Example. KdV hierarchy
ut0 = ux = @x�H0�u(x)
ut1 = uux + ✏2
12uxxx = @x�H1�u(x)
ut2 = u2
2! ux + ✏2
12 (2uxuxx + uuxxx) +✏4
240uV = @x
�H2�u(x)
. . . . . . . . . . . . . . . . . . . . . . . .
Constructions:
• Lax representation, isospectral deformations
• Infinite-dimensional Grassmannians
• Baker - Akhiezer functions
Sunday 10 May 15
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Example. KdV hierarchy
ut0 = ux = @x�H0�u(x)
ut1 = uux + ✏2
12uxxx = @x�H1�u(x)
ut2 = u2
2! ux + ✏2
12 (2uxuxx + uuxxx) +✏4
240uV = @x
�H2�u(x)
. . . . . . . . . . . . . . . . . . . . . . . .
Constructions:
• Lax representation, isospectral deformations
• Infinite-dimensional Grassmannians
• Baker - Akhiezer functions
• Topology of moduli spacesSunday 10 May 15
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Deligne - Mumford moduli spaces of stable algebraic curves
Mg,n = {(Cg, x1, . . . , xn)} / ⇠
Tautological line bundles
Mg,n
Li
T ⇤xiC
g
i := c1 (Li) 2 H2�Mg,n
�, i = 1, . . . , n
· ··0
1
1
M0,4 = P1
M0,3 = pt
····
0
1
1z
M1,1 = {ellipticcurves}
Sunday 10 May 15
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Witten - Kontsevich solution to KdV
Then the tau-function of this solution
u(x, t0, t1, . . . ; ✏)|t=0 = x
⌧ = ⌧(t0, t1, . . . ; ✏)
such that ✏
2 @2log ⌧
@x
2= u(t; ✏) reads
where
nonzero only if p1 + · · ·+ pn = 3g � 3 + n
log ⌧(t; ✏) =X
g�0
✏2g�2Fg(t)
Fg(t) =X
n
1
n!
Xtp1 . . . tpn
Z
Mg,n
p11 . . . pn
n
Sunday 10 May 15
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In physics literature (Witten et al.):
the tau-function = partition function of 2D quantum gravity
log ⌧(t) =DeP
i�0 ti⌧iE
⌧0 = 1, ⌧1, ⌧2, . . . observables
time variables of KdV hierarchy = coupling constants
Correlators
h⌧p1 . . . ⌧pni =Z
Mg,n
p11 . . . pn
n , p1 + · · ·+ pn = 3g � 3 + n
Sunday 10 May 15
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Moreover, Hamiltonian densities Hp
[u] =
Zhp
(u, ux
, . . . ; ✏) dx
are two-point correlation functions
hp = hh⌧0⌧p+1ii = ✏
2 @2log ⌧(t)
@x @tp+1h�1 = u
h0 = u
2
2 + ✏
2
12uxx
h1 = u
3
3! +✏
2
24
�u2x
+ 2uuxx
�+ ✏
4
240uxxxx
h2 = u
4
4! +✏
2
24
�uu2
x
+ u2uxx
�+ ✏
4
480
�3u2
xx
+ 4ux
uxxx
+ 2uuxxxx
�+ ✏
6
6720u(6)
etc. Hence@hp�1
@tq=
@hq�1
@tptau-symmetry
NB: change of densities hp
7! hp
+ @x
(. . . )
does not change the PDEsSunday 10 May 15
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Construction of KdV: a recipe using Mg,n
Start from KdVε=0 vt0 = v
x
vt1 = v v
x
vt2 = v
2
2! vx. . . . . . . . .
vtk = v
k
k! vx
(change notations: u(x, t) 7! v(x, t) )
Verify commutativity:
Exercise. Prove
=@
@x
v
k+1
(k + 1)!
(vtk)tl =@
@x
@
@tl
v
k+1
(k + 1)!=
@
2
@x
2
v
k+l+1
k! l! (k + l + 1)= (k $ l)
Derive the WK solution at g=0: v(t) =X
n�1
1
n
X
k1+···+kn=n�1
tk1
k1!. . .
tkn
kn!
@
nv
@tk1 . . . @tkn
=@
n
@x
n
v
k1+···+kn+1
k1! . . . kn!(k1 + · · ·+ kn + 1)
Sunday 10 May 15
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Next, to recover full KdV do a substitution v ! u
u = v + ✏2@2x
2
4 1
24
log vx
+ ✏2✓
vxxxx
1152v2x
� 7 vxx
vxxx
1920v3x
+
v3xx
360v4x
◆+O �
✏4�
| {z }
3
5
�F
Then(“quasitriviality”, B.D., Y.Zhang).
• KdVε=0(v)=0 KdV(u)=0
• Operator of Poisson bracket unchangedJ =@
@x
• New Hamiltonian densities
hp
(u, ux
, . . . ) = h0p
(v) + ✏2@x
@tp+1�F
are differential polynomials in u satisfying tau-symmetry
Sunday 10 May 15
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E.g., start with vt
= v vx
, plug
u = v +✏2
24
(log vx
)
xx
+O �✏4�
ut
= vt
+✏2
24
✓vxt
vx
◆
xx
+ · · · = v vx
+✏2
24
✓v v
xx
+ v2x
vx
◆
xx
+ . . .
then
= v vx
+
✏2
24
[(v (log vx
)
x
)
xx
+ vxxx
] = v vx
+
✏2
24
[(v (log vx
)
xx
)
x
+ 2vxxx
]
= uux
+✏2
12uxxx
+O �✏4�
(cf. N.Ibragimov, V.Baikov, R.Gazizov, ’89)
Sunday 10 May 15
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Geometrical meaning of the substitution (Dijkgraaf, Witten 1990): expressing higher genera via genus zero:
�F =X
g�1
✏2g�2Fg
⇣v, v
x
, . . . , v(3g�2)⌘
plug the genus zero solution
vx
= vx
(t)v = v(t) =X
n�1
1
n
X
k1+···+kn=n�1
tk1
k1!. . .
tkn
kn!, etc. Then
Fg
⇣v(t), v
x
(t), . . . , v(3g�2)(t)⌘=
X 1
n!
Xtk1 . . . tkn
Z
Mg,n
k11 . . . kn
n
= Fg
(t), g � 1
Sunday 10 May 15
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How to find the substitution? Solve loop equation
L(�)e�F = 0 8�
L(�) =X
k�0
�Ak(�)� ✏2Bk(�)
� @
@v(k)� ✏2
2
X
k,l�0
Ckl(�)@2
@v(k)@v(l)+
1
16(v � �)2=
X
m��1
Lm
�m+2
Ak
(�) = @k
x
✓1
v � �
◆+
kX
j=1
✓kj
◆@j�1x
✓1pv � �
◆@k�j+1x
✓1pv � �
◆
Bk
(�) = � 1
16@k+2x
✓1
(v � �)2
◆
Ckl
(�) = @k+1x
✓1pv � �
◆@l+1x
✓1pv � �
◆
Commutation relations [Lm,Ln] = (m� n)Lm+n, m, n � �1
Sunday 10 May 15
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Digression: on computation of intersection numbers
h⌧k1 . . . ⌧kni =Z
Mg,n
k11 . . . kn
n
Let M(z) =1
2
0
B@�P1
g=1(6g�5)!!
24g�1·(g�1)!z�6g+4 �2
P1g=0
(6g�1)!!24g·g! z�6g
2P1
g=06g+16g�1
(6g�1)!!24g·g! z�6g+2
P1g=1
(6g�5)!!24g�1·(g�1)!z
�6g+4
1
CA
(cf. Faber - Zagier series). Then (M.Bertola, B.D., Di Yang, 2015)
Xh⌧k1 . . . ⌧kni
(2k1 + 1)!!
z2k1+21
. . .(2kn + 1)!!
z2kn+2n
= � 1
n
X
r2Sn
TrM(zr1) · · ·M(zrn)Qnj=1(z
2rj � z2rj+1
)� �n,2
z21 + z22(z21 � z22)
2, n � 2
Sunday 10 May 15
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Proof uses Lax representation for KdV hierarchyn -th equation , Ltn = [An, L]
L = @2x
+ 2u Lax operator, An
=
1
(2n+ 1)!!
⇣L
2n+12
⌘
+
(set ✏ = 1) Eigenfunctions L = z2 satisfy
tn = An , n � 0
At t = 0 u = u0(x) = x
000 + 2x 0 = z
2 0 ⇠ Airy equation
Use
Ai(z) ⇠ e�⇣
2p⇡z1/4
1X
k=0
(6k � 1)!!
(2k � 1)!!
(�216 ⇣)�k
k!, ⇣ =
2
3z3/2, z ! 1, | arg z| < ⇡
Sunday 10 May 15
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More general integrable Hamiltonian tau-symmetric hierarchies conjecturally depend on an infinite number of parameters s1, s2, . . . (a deformation of KdV)
(B.D., S.-Q.Liu, Y.Zhang, D.Yang, 2014)
Construction uses Hodge potential
ut
= uux
+ ✏2⇣u
xxx
12� s1ux
uxx
⌘
+ ✏4� s160
u(5) + s21
✓uxx
uxxx
+1
5ux
u4
◆� 4s31
5
�2u
x
u2xx
+ u2x
uxxx
�
�s26
�2u
x
u2xx
+ u2x
uxxx
�i+O(✏6)
Hg(t, s) =X
n�0
1
n!
Xtp1 . . . tpn
Z
Mg,n
eP
k�1 skchk(E) p11 . . . pn
n
Hodge bundle
Mg,n
E
H0 (T ⇤Cg)
(even components of Chern character vanish ch2i (E) = 0)
Sunday 10 May 15
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Clearly Hg(t, 0) = Fg(t) Redenote s2k�1 7! � B2k
(2k)!sk
Thm.1) H0 = F0, for g � 1 Hg = (Fg +�Hg)v=v(t),v
x
=vx
(t),...
where�H
g
2 Qhs1, . . . , sg; v, v
±1x
, vxx
, . . . , v(3g�3)i
�H1 = �1
2s1v
E.g.,
�H2 = s1
✓11v2
xx
480v2x
� vxxx
40vx
◆+
7
40s21vxx �
✓s3110
+s248
◆v2x
RecallF1 =
1
24
log vx
, F2 =
v(4)
1152v2x
� 7vxx
vxxx
1920v3x
+
v3xx
360v4x
Sunday 10 May 15
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2) The substitution
v 7! u = v + ✏2@2x
X
g�1
✏2g�2Hg
h0p
=vp+2
(p+ 2)!7! h
p
= h0p
+ ✏2@x
@tp+1
X
g�1
✏2g�2Hg
transforms KdVε=0 to a new integrable Hamiltonian tau-symmetric hierarchy
=X
g�0
✏2gh[g]p
, h[g]p
2 Qhs1, . . . , sg;u, ux
, . . . , u(2g+2)i
utp =@
@x
�H[u; s]
�u(x), H[u; s] =
Zhp dx, p = 0, 1, 2, . . .
@hp�1
@tq=
@hq�1
@tpSunday 10 May 15
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E.g., for t = t1 one obtains a deformation of KdV
depending on the parameters s1, s2, . . .
ut
= uux
+ ✏2⇣u
xxx
12� s1ux
uxx
⌘
+ ✏4� s160
u(5) + s21
✓uxx
uxxx
+1
5ux
u(4)
◆� 4s31
5
�2u
x
u2xx
+ u2x
uxxx
�
�s26
�2u
x
u2xx
+ u2x
uxxx
�i+O(✏6)
Sunday 10 May 15
![Page 23: The Hebrew University of Jerusalem Einstein Institute of ... · 1 g=1 (6g5)!! 24g1 ·(g1)! z 6g+4 2 P 1 g=0 (6g1)!! 24g ·g! z 6g 2 P 1 g=0 6g+1 6g1 (6g1)!! 24g ·g! z 6g+2 P 1 g=1](https://reader035.vdocument.in/reader035/viewer/2022070909/5f92d2f5e52af67f3444f46e/html5/thumbnails/23.jpg)
Conjecture. This is a universal deformation of KdVε=0 in the class of Hamiltonian tau-symmetric integrable hierarchies. For
Example 1. For sk = � B2k
2k(2k � 1)
s2k�1, for k � 1
one obtains intermediate long wave eq. (A.Buryak)
ut
= uux
+X
g�1
✏2gsg�1 |B2g|(2g)!
u2g+1
Example 2. For sk = (4k � 1)B2k
2k(2k � 1)s2k�1, k � 1
one obtains (?) Volterra equation (=discrete KdV)
ut
=1
✏
⇣eu(x+✏) � eu(x�✏)
⌘(checked up to ✏12)
s = 0 obtain KdV
Sunday 10 May 15
![Page 24: The Hebrew University of Jerusalem Einstein Institute of ... · 1 g=1 (6g5)!! 24g1 ·(g1)! z 6g+4 2 P 1 g=0 (6g1)!! 24g ·g! z 6g 2 P 1 g=0 6g+1 6g1 (6g1)!! 24g ·g! z 6g+2 P 1 g=1](https://reader035.vdocument.in/reader035/viewer/2022070909/5f92d2f5e52af67f3444f46e/html5/thumbnails/24.jpg)
Thank you!
Sunday 10 May 15