Integrable systemsand moduli spaces

Boris DUBROVINSISSA, Trieste

The Hebrew University of JerusalemEinstein Institute of Mathematics

April 30, 2015

Sunday 10 May 15

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thank you!

Sunday 10 May 15