Lagrangian Floer theory
of arbitrary genus
and
Gromov-Witten invariant
Kenji Fukaya
(Kyoto University)
at
University Miami (US)
1
L = {(Lk , bk )}
A finite set of pairs
Lk
(relatively spin) Lagrangian submanifolds
bk ÎH odd (L;L 0 ) weak bounding cochains
A cyclic unital filtered A inifinity category
L
set of objects
set of morphisms
F, Oh, Ohta, Ono (FOOO) (+ Abouzaid FOOO (AFOOO))
2
Hochshild cohomology
Hochshild homology
Cyclic cohomology
Cyclic homology
q
Open closed maps
FOOO , AFOOO
p
3
Gromov-Witten invariant
GWg, : H(X;L)Ä ® L
Counting genus g pseudo-holomorphic maps intersecting with cycles in X
4
Problem to study
or
Compute
GWg, ( p ( x1 ) ,… , p ( x
))
in terms of the structures of
5
structure.
relation
6
Inner product and cyclicity
is (up to sign) a Poincare duality on
H(L Ç L';L0 )
mk (x1,… , xk ), x0 = ± mk (x0, x1,… , xk-1 ), xk
cyclicity
8
Problem to study
or
Compute
GWg, ( p ( x1 ) ,… , p ( x
))
in terms of the structures of
9
Answer
NO we can't !
GWg, ( p ( x1 ) ,… , p ( x
))
is determined by the structures of
in case
g = 0,
=3
In general we need extra information.
I will explain those extra information below.
It is Lagrangian Floer theory of higher genus (loop).
10
dIBL structure (differential involutive bi-Lie structure) on B
3 kinds of operations
differential
dd = 0
Lie bracket
Jacobi
co Lie Bracket
{ }
}{
co Jacobi
is a derivation
is a coderivation
with respect to d
11
}a{= å a1,c a2,c
{ } is compatible with }{
å ±a {a ,b} ± a {a ,b}
+ å ±b {b ,a} ± b {b ,a}
1,c
1,c
2,c
2,c
2,c
2,c
1,c
1,c
+}{a,b}{= 0
Involutive
a
å {a
a }=0
1,c 2,c
12
IBL infinity structure = its homotopy everything analogue
operations :
P n,m : EnB ® EmB
EnB = B Ä
Ä B / Sn
n
Homotopy theory of IBL infinity structure is built (Cielibak-Fukaya-Latschev)
13
(C,
,d)
(
B= B C
cyc
chain complex with inner product
) (dual cyclic bar complex)
*
j Î( B C ) , j
cyc
*
g st = es ,et
i1 ik
has a structure of dIBL algebra
(cf. Cielibak-F-Latschev)
= j (e , , e )
i1
ik
ei
basis of C
(gst ) = (g st )-1
14
{j ,y }l1
}j {i1
lk+k ' -2
ik ; j1
jl
= å ±gstj l1
= å ±gstj ia
la-1 sla+b+1 ic
ia-1 sjb
yl
a
la+b t lc
lk+k ' -2
jb-1t
15
There is a category version.
Dual cyclic bar complex
has dIBL structure
Remark: This structure does NOT (yet) use
operations mk
except the classical part of m1, that is the usual boundary operator.
16
Cyclic
stucture (operations mk ) on
satisfying Maurer-Cartan equation
1
dM 1,0 + {M 1,0 ,M 1,0 } = 0
2
This is induced by a holomorphic DISK
17
is given by
M 1,0 (x1,… , xk ) = mk-1 (x1,… , xk-1 ), xk
1
dM 1,0 + {M 1,0 ,M 1,0 } = 0
2
relation among
mk
18
Theorem (Lagrangian Floer theory of arbitrary genus) (to be written up)
There exists
1
dM ,g +
2
M
å å
1+ 2 =
,g
ÎE B
{M
+1 g1 +g2 =g
1,g1
such that BV master equation
,M
2 ,g2
}out +}M
-1,g
{+{M
} =0
+1,g-1 int
is satisfied.
The gauge equivalence class of
{M
,g
}
is well-defined.
19
Note:
{ } :B ÄB ® B
induces
and
{ }int : EnB ® En-1B
{x1
xn, y1
ym }out = å ±{xi , y j }x1
xˆi
xn y1
yˆ j
ym
i, j
{x1
xn }int = å ±{xi , x j }x1
xˆi
xˆ j
yn
i, j
20
}{: B ® B Ä B
induces
}{: En-1B ® EnB
}x1
xn {= å ±x1 }xi { xˆ j
yn
i
21
M
Î( E B)
*
,g
is obtained from moduli space of genus g
bordered Riemann surface with
boundary components
L
L
g = 2,
=3
L
22
1
dM ,g +
2
{M
,M
1,g1
å å
1+ 2 =
+1 g1 +g2 =g
}
2 ,g2 out
{M
1,g1
,M
}M
2 ,g2
-1,g
}out +}M
{
-1,g
} =0
{+{M
+1,g-1 int
{M
+1,g-1 int
}
23
Remark：
(1): In case the target space M is a point, a kind of this theorem appeared
in papers by various people including Baranikov, Costello Voronov, etc. (In
Physics there is much older work by Zwieback.)
(2): Theorem itself is also expected to hold by various people including F for
a long time.
(3): The most difficult part of the proof is transversality. It becomes
possible by recent progress on the understanding of transversality
issues. It works so far only over
. It also requires machinery from
homological algebra of IBL infinity structure to work out the
problem related to take projective limit, in the same way as A infinity case
of [FOOO]. This homological algebra is provided by Cielibak-F-Latshev.
(4): Because of all these, the novel part of the proof of this theorem is
extremely technical. So I understand that it should be written up carefully
before being really established.
(5): In that sense the novel point of this talk is the next theorem (in slide 33)
which contains novel point in the statement also.
24
Relation to `A model Hodge structure'
We need a digression first.
Let
Put
Z(x1,x 2 ) = p ( x1 ) , p ( x 2 )
We (AFOOO) have an explicit formula to calculate it based on Cardy relation.
25
Formula for
Z(x1,x 2 ) = p ( x1 ) , p ( x 2 )
26
Theorem (AFOOO, FOOO ....)
Z(x1,x 2 ) = p ( x1 ) , p ( x 2 )
a basis of
27
Hochshild complex
Let
be the operator obtained by `circle' action.
Hochshild homology
HH* (W(L), W(L))
is homology of the free loop space of
L.
B;H* (L(L)) ® H*+1 (L(L))
is obtained from the S1 action on the free loop space.
28
Hochshild complex
Let
be the operator obtained by `circle' action.
Proposition
Z(Bx1,x2 ) = (}M 1,0 {) (x1,x 2 )
it implies that there exists
such that
because
d K + Kd = B
BK = 0
if Z is non-degenerate
±}M 1,0 { = dM 2,0 + {M 2,0, M 1,0 }out
2nd of BV master equation
29
Corollary (Hodge – de Rham degeneration) (Conjectured by Kontsevitch-Soibelman)
If
Z
is non-degenerate then
Remark: This uses only
To recover
M 2,0
: moduli space of annulus.
GWg, ( p ( x1 ) ,… , p ( x
we must to use all the informations
M
))
,g
30
Remark: Why this is called `Hodge – de Rham degeneration' ?
Hodge structure uses
¶, ¶
with
¶¶ + ¶¶ = ¶¶ = ¶¶ = 0
Ker¶ Ker¶ Kerd
d = ¶+ ¶
@
@
Im ¶ Im ¶ Im d
Kerdu is independent of u
we may rewrite this to d = ¶ + u¶
u
H (du ) @
Im du
One main result of Hodge theory is
We have
Put
d B + Bd = BB = dd = 0
du = d + uB
d K + Kd = B
BK = 0
H (du ) @
Kerdu
is independent of u
Im du
31
Floer's boundary operator
f = PO Landau-Gizburg potential
Table from
Saito-Takahashi's
paper
FROM PRIMITIVE FORMS
TO FROBENIUS MANIFOLDS
(Similar table is also in a paper
by Katzarkov-Kontsevich-Pantev)
Z
plus paring
between HH* and HH*
32
MainTheorem (work in progress)
The gauge equivalence class of
{M
GWg, ( p ( x1 ) ,… , p ( x
,g
}
determines Gromov-Witten invariants
))
for
if
Z
is non-degenerate.
33
Idea of the proof of Main theorem
Metric Ribbon tree
Bordered Riemann surface
Example: 2 loop
35
1
1
1
36
37
Moduli of metric ribbon graph
=
Moduli of genus g Riemann surface
with marked points
etc.
This isomorphism was used in Kontsevich's proof of Witten conjecture
38
is identified with moduli space of bordered Riemann surface.
Fix
C1, ,C
and consider the family
R(c) = (cC1, , cC )
parametrised by
39
Consider one parameter family of moduli spaces
L
L
40
Study the limit when
?
?
41
etc.
(various combinatorial types
depending on
.)
S
42
Counting
gives
GW1,2 ( p ( x1 ), p ( x 2 ))
More precisely integrating the forms
x1, x 2,…
on the moduli space
by the evaluation map using the boundary marked points
43
Counting
We obtain numbers that can be calculated from
{M
,g
}
44
Actually we need to work it out more carefully.
Let
and try to compute
GWg,1 ( p ( x))
Need to use actually
45
Forgetting
u
defines
is identified with the total space of complex line bundle over
(the fiber of
is identified to the tangent space of the unique
interior marked point.)
(The absolute value corresponds to c
the phase S1 corresponds to the extra freedom to glue.)
46
is not a trivial bundle. (Its chern class is Mumford-Morita class.)
So
is not homlogous to a class in
the boundary.
47
is homologous to a class on the boundary
-1
and
(D)
p
here D is Poincare dual to the c1
a class on the boundary
p -1 (D)
48
òp
-1 -1
P (D)
Because
ev* (x) =
something coming from boundary.
p -1P -1 (D) is a union of S1 orbits, p -1P -1 (D) = S1W
then
ò
S1W
ev* (x) =
ò
W
ev* (Bx) = ò ev* (d Kx) = ò ev* (Kx)
¶W
W
Hodge to de Rham degeneration
¶W = ¶1W + S W '
1
and
ò
S1W ¢
ev* (Kx) = ò ev* (BKx) = 0
W¢
49
A a class on the boundary
ò
*
A
ev (x)
and
ò
¶W1
are determined by
ev* (Kx)
{M
,g
}
and
QED
50