symplectic geometry of langlangian submanifoldresearch.ipmu.jp/seminars/pdf/fukaya.pdf ·...
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Symplectic Geometryof
Langlangian submanifold
Kenji FUKAYA
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Symplectic Geometry ?
Origin Hamiltonian Dynamics
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€
q1,L,qn position
€
p1,L, pn momentum
€
H (q1,L,qn; p1,L, pn;t) Hamiltonian
Hamiltonian’s equation
€
dqidt
=∂H∂pi
dp1dt
= −∂H∂qi
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Hamilton’s equation is invariant of the coordinate change
€
Qi =Qi (q1,L,qn , p1,L, pn )Pi = Pi (q1,L,qn , p1,L, pn )
€
dPi ∧∑ dQi = dpi ∧∑ dqi
canonical transformaiton = symplectic diffeomorphism
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Symplectic manifold
€
Ui has local coordinate
€
q1,L,qn, p1,L, pn
coordinate change is symplectic diffeomorphism
€
ω = dpi ∧∑ dqi is globally defined.
€
dω = 0
€
ω∧L∧ω = volume form.
€
X = UUi
symplectic form
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Two important sources of symplectic Geometry
(1) Hamiltonian dynamics
(2) Algebraic or Kahler geometry
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(1) Hamiltonian dynamics
€
X =T *M cotangent bundle.
€
q1,L,qn local coordinate of
€
p1,L, pn coordinate of the cotangent vector
€
M
€
ω = dpi ∧∑ dqisymplectic form
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(2) Algebraic or Kahler geometry
Solution set of polynomial equation has a symplectic structure
(Fubini-Study form)
€
X = (x, y, z,w) x5 + y5 + z5 +w5 =1{ }Example
(Take closure in projective space.)
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Clasical mechanics Hamiltonian mechanics
€
⊃
Quantum mechanics
Hamilton formalism play important role
Symplectic geometry?
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Global Symplectic Geometry ?
It is not clear whether Global Symplectic Geometry is related to the origin of symplectic geometry, that is Physics.
On the other hand, from Mathematical point of viewLocal symplectic geometry is trivial.
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Riemannian geometry
€
Rijkl curvature (how locally spacetime curves.)
The most important quantity of Riemannian geometry.
There is no curvature in symplectic geometry.(Dauboux’s theorem 19th century.)
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There is nontrivialGlobal Symplectic Geometry.
This is highly nontrivial fact and was established by using
“string theory’’
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Classical mechanics Hamiltonian mechanics
€
⊃
Quantum mechanics Symplectic geometry
?
QFT ? String ?
Why?
Global symplectic geometry
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I will discuss geometry of Lagrangian submanifold as an example of nontrivial Global Symplectic geometry.
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Lagrangian submanifold
Example
€
f (q1,L,qn ) a function of .
€
q1,L,qn
€
pi =∂f∂qi, i =1,L,n
defines an n dimensional submanifold, Lagrangian submanifold L.
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q graph of .
€
p =∂f∂q
€
f (q) = area
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no longer a graphfinally becomes a manifold without boundary
function
no longer a graph
Lagrangian submanifolds
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Definition:
€
L ⊂ X
€
ω = 0 on L.
€
dimL = 12 dimX
L is a Lagrangian submanifold.
a submanifold of X.
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Role of Lagrangian submanifold
Lagrangian submanifold of T*M is a generalization of a function on M
Symplectic diffeomorphism: X X is a Lagrangian submanifold of
€
→
€
X × X
is a Lagrangian submanifold
€
Rn ⊂ Cn
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Role of Lagrangian submanifold
Lagrangian submanifold of T*M is a generalization of a function on M
Symplectic diffeomorphism: X X is a Lagrangian submanifold of
is a Lagrangian submanifold
€
Rn ⊂ Cn
X X
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Role of Lagrangian submanifold
Lagrangian submanifold of T*M is a generalization of a function on M
Symplectic diffeomorphism: X X is a Lagrangian submanifold of
€
→
€
X × X
is a Lagrangian submanifoldRn Cn
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Role of Lagrangian submanifold
Lagrangian submanifold of T*M is a generalization of a function on M
Symplectic diffeomorphism: X X is a Lagrangian submanifold of
€
→
is a Lagrangian submanifold
€
Rn ⊂ Cn
Lagrangian submanifold is the correct boundary condition for open string.
D brane
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Symplectic Geometry - analogy from algebraic geometry
= Hamiltonian dynamics + Lagrangain submanifold + epsilon
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no longer a graph
Lagrangian submanifold of 2 dimensional Euclidean space
= Circle
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Classify Lagrangian submanifolds of ?
The first interesting case n = 3.
The case n = 2. Answer: 2 dimensional torus. (Easy (except the case of Klein bottle))
€
Cn
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Theorem (Gromov, 1980’)
3 sphere S3 is NOT a Lagrangian submanifold of C3 .
We need “open string theory” to prove this.
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If L is a Lagrangian submanifolds in R2n = Cn then there exists a disc which bounds it.
(1)
holomorphic.
L
€
∂D2 → L
€
ϕ : D2 →Cn
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(1I)
Such a disc can not exists if L is sphere S3
because
L is S3
€
ϕ*ωD2∫ = 0
holomorphic.
€
ϕ
€
ϕ*ωD2∫ > 0
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Classical mechanics Hamiltonian mechanics
€
⊃
Quantum mechanics Symplectic geometry
?QFT ? String ?
Why?
Global symplectic geometry
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To go further we need to be more systematic.
Approximate Geometry by Algebra.
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Poincare (begining of 20th century)
X : space
H(X) : Homology group
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Poincare (begining of 20th century)
X : space
H(X) : Homology group
Algebraic topology=
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Poincare (begining of 20th century)
X : space
H(X) : Homology group
Linear story
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Begining of 21th century
we are now working on non Linear story
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Classify the Lagrangian submanifolds of ?
€
C3
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Thurston-Perelman
3 manifolds are one of the 8 types of spaces
Which among those 8 types is a Lagrangian submanifold of ?
€
C3
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3 manifold Lagrangian submanifold?
€
S3 No(Gromov)
€
R3 Yes
€
H 3 No(Viterbo)
€
R× S2 Yes
€
R×H 2 Yes
€
SL(2,R) No (F)
€
Sol No (F)
€
Nil No (F)
Answer
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3 manifold Lagrangian submanifold?
S3 : Curvature =1 No(Gromov)
R3 : Curvature =0 Yes
H3 : Curvature=-1 No(Viterbo)
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3 manifoldLagrangian
submanifold?
€
SL(2,R) No (F)
€
Sol No (F)
€
Nil No (F)
€
a bc d
ad − bc =1
€
* *0 *
€
1 * *0 1 *0 0 1
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Method : Count the discs.
= Open string
holomorphic.
L
€
∂D2 → L
€
ϕ : D2 →C3
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Count the discs.
Obtain numbers. (Many numbers)
Those sysem of numbers has a structure.
Obtain algebraic system; (something like group)
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● In theoretical Physics,
Higher (>4) dimensional spaces are (at last) begining to be studied.
● Space of dimension > 4 will never directly observable from human.
● They will be seen to us only through some system of numbers which
can be checked by experiments.
● The role of higher dimensional geometry in physics here seems to be
to provide a way to understand some huge list of numbers.
● This is the same as what I said about algebraic topology.
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DifficultyCounting the discs is actually a difficult problem.
Counting the discs = Counting the number of solution of Non Linear differential equations
Keep going and understand Lagrangian submanifold by open string theory.
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If L is a Lagrangian submanifold in R2n = Cn then there exists a disc which bounds it.
First and essential step to show S3 is not a Lagrangian submanifold.
Counting the discs is difficult.
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Difficulty
Counting the discs is actually a difficult problem.
Counting the discs = Counting the number of solutions of Non Linear differential equations
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Physics helps
Mirror symmetry (discovered in 1990’)
provides (potentially) a powerful tool to compute the number of discs.
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Classical mechanics Hamiltonian mechanics
€
⊃
Quantum mechanics Symplectic geometry
?QFT ? String ?
Why?
Global symplectic geometry
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(Homological) Mirror symmetry (Konsevitch 1994)
€
X
€
X^
Symplectic manifold Complex manifold
€
LLagrangian submanifold(A brane)
€
E→ X^
Holomorphic vector bundle
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(Homological) Mirror symmetry
Difficult problem of counting discs
becomes
Attackable problem of complex geometry
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(Homological) Mirror symmetry
Difficult problem of counting discs
becomes
Attackable problem of complex geometry
Global, Non Linear
Local, Linear
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Difficult problem of counting discs
becomes
Attackable problem of complex geometry
Global, Non Linear
Local, Linear
Non perturbative
Perturbative
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Theorem (Seidel-Smith-F, Nadler)
Compact Lagrangian submanifold L of is the same as as D-brane
if
L, M are simply conneted and L is spin.
€
T *M
€
M ⊂ T *M
A special case of a version of a conjecture by Arnold.
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L is the same as M as D-brane
○ Homolog group of L is homology group of M.
○ [L] = [M] in H(T*M).
implies in particular
Arnold conjectured stronger conclusion in 1960’s.
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(Homological) Mirror symmetry (Kontsevitch 1992)
€
X
€
X^
Symplectic manifold Complex manifold
€
LLagrangian submanifold(A brane)
€
E→ X^
Holomorphic vector bundle
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In our case is noncompact and situaltion is slightly different.
Symplectic manifold
€
LLagrangian submanifold(A brane)
Flat vector bundle€
T *M
€
X^
=T *M (or M )€
X =T *M
€
E→M
● String theory of is gauge theory on M. (Witten 1990’)
€
T *M
● M is simply connected Flat bundle on M is trivial.
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● By proving a (small) part of (homological) Mirror symmetry conjecture we get new insight on Lagrangian submanifolds.
● Then we enhance conjecture and make it more precise and richer.
● Solving some more parts we get another insight.
● Conjecture now is becoming richer and richer contain many interesting and attackable open problems.
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I want to keep going and understand Global symplectic geometry
by using the ideas from
String theory.
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Hamiltonian Dynamics
€
H :T *M → R
Quantum mechanics
Global symplectic manifold X
?
€
−1 ∂ψ∂t
= H q, −1∂ ∂q( )ψ
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In Hamiltonian formalism (symplectic geometry)
€
q1,L,qn position and
€
p1,L, pn momentum
play the same role
€
Qi = piPi = −qi
is a canonical transformation
This transformation is NOT allowed in Lagrangian formalism
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Symmetry between
€
q and
€
pstill exists in quantum mechanics.
Fourier transformation
€
q×
€
∂∂q€
H (q, p)⇔ H q, ∂∂q
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But this symmetry (after quantization) does not
survive in global geometry.
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€
X =T *M
€
q1,L,qnCoordinate change between are nonlinear.
Coordinate change between are linear.
€
p1,L, pn
Coordinate change does NOT mix up
€
q1,L,qnand
€
p1,L, pn
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In algebraic geometry, there is no way to say which coordinate is
€
qand which coordinate is .
€
p
There is no way to associate an operator to a function (Hamiltonian) on .
€
X
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Definition:Lagrangian submanifold L of a symplectic manifold X
€
L ⊂ X
€
ω = 0 on L.
€
dim L =12dim X
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€
dpi =∂ 2 f∂qi∂qjj
∑ dqj
€
pi =∂f∂qi
€
ω = dpi ∧∑ dqi =∂ 2 f∂qi∂qji, j
∑ dqj ∧dqi = 0
On L.