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Geometry of loop spaces
Yoshiaki Maeda
Tohoku Forum for Creativity,Tohoku University
Conference: XIX Geometry SeminarAugust 28-September 4, 2016
Joint work with S. Rosenberg and F. Torres-Ardila
1
Introduction
• M : compact, connected orientable C∞-manifold.
• Diff(M): the diffeomorphism group of M
Question: What can we say about the topology of Diff(M) ?
This is a difficult question!
2
Some known results
A few results are known:
• Diff(S1) ∼ O(2)
• Diff(S2) ∼ O(3) (Smale)
• Diff(S3) ∼ O(4) (Hatcher)
• Diff(S1 × S2) ∼ O(2)×O(3)×ΩSO(3) (Hatcher)
There are some results on the fundamental groups for diffeomorphism
groups of surfaces and 3-manifolds.
3
Fundamental groups of diffeomorphism groups
We will consider the following question:
Question: When M has an S1-action, is π1(Diff(M), Id) infinite?
Note that a circle action gives a loop of diffeomorphisms of M .
We consider the case dim(M) = 5
Let (M4, J, g, ω) be a compact Kaehler surface with an integral Kaehler
form ω ∈ H2(M4,Z), and let Mk be the circle bundle associated to
k[ω] ∈ H2(M4,Z) for k ∈ Z.
4
Results (1)
We set
|R|∞ = max|R(ei, ej, ek, eℓ)||e1, e2, e3, e4 is orthonormal basis
where R is the Riemannian curvature of M4.
Theorem A: Let (M4, J, g, ω) be as above. Assume that for
k ∈ Z− 0,
|R|∞ <6
7k2 +
3πσ(M4)
7vol(M4)k2
where σ(M4) is the signature of M4. Then π1(Diff(Mk)) is infinite.
More precisely, the loop of diffeomorphisms of Mk given by rotation
in the circle fiber give an element of infinite order in π1(Diff(Mk), id).
5
Results (2)
In particular, we have
Corollary B: Theorem A holds for the following cases:
• k >> 0
• M = T4 and k 6= 0
6
Tools for proving Theorem A
• Geometry of Loop Space
• Psuedo-differential algebra bundles for loop spaces
• Wodzicki-Chern-Simons classes
7
Geometry of Loop Space
Let
• (M, g): a compact oriented smooth Riemannian manifold.
• LM : loop space of M=γ : S1 → M,γ ∈ C∞
• For s, we consider the Sobolev space Hs(S1;TM) as a model for
TγLM ; X ∈ Hs(S1;TM) with X(θ) ∈ γ(θ) is a vector field along γ.
The inner product on Hs(S1;TM) is given by
〈X, Y 〉s =1
2π
∫ 2π
0〈(1 +∆)sX(θ), Y (θ)〉γ(θ)dθ
where
• 〈. , . 〉 is the inner product on (M, g)
• ∆ = D∗D, D = Ddθ, D∗ : formal adjoint of D.
8
Hs-connection of loop space (1)
We consider the Levi-Civita connenction defined by
2〈∇sXY,Z〉s
= X〈Y, Z〉s + Y 〈X,Z〉s − Z〈X, Y 〉s
+〈[X, Y ], Z〉s + 〈[Z,X], Y 〉s − 〈[Y, Z], X〉s
Proposition 1:
For s=0:
∇0XY = DXY = δXY +ΓX(Y )
where δXY is the variation of Y in X direction and Γ is the Christoffel
symbols of the Riemannian metric of (M, g).
9
Hs-connection of loop space (2)
Proposition 2:
• For s = 1:
∇1XY = DXY +
1
2(1 +∆)−1
[−∇γ(R(X, γ))Y −R(X, γ)∇γY −∇γ(R(Y, γ)X)
−R(Y, γ)∇γX +R(X,∇γY )γY )γ −R(∇γX,Y )γ]
where R is the curvature tensor of (M, g)
• We can give general formulas for any s > 12
• The connection 1-form ωsX is taken values in zeroth order ΨDOs
acting on Y
10
Remarks on ΨDOs (1)
A very brief explanation of ΨDOs.
A differential operator of order m,
Pf(x) =m∑
|α|=k=0
a(x)α(∂x)αf(x),
can be written using Fourier transform and Fourier inversion as
Pf(x) =
∫ ∫a(x, ξ)e<x−y|ξ>f(y)dydξ
where a(x, ξ) =∑m
k=|α|=0a(x)α(iξ)α.
11
Remarks on ΨDOs (2)
Definition
• A ΨDO P of order m is defined by the above integral transfor-
mation using a more general function a(x, ξ).
• a(x, ξ) must have the following asymptotic expansion:
a(x, ξ) ∼ am(x, ξ) + am−1(x, ξ) + · · ·
where ak(x, ξ) is homogeneous of degree k in ξ
• ak(x, ξ) is called the symbol of a(x, ξ) of order k and is denoted
by σk(a).
12
Hs-curvature of Loop space
We compute the curvature form Ωs of ∇s:
Ωs(X, Y ) = ∇sX∇s
Y −∇sY∇s
X −∇s[X,Y ]
Proposition 3:
• s=0: Ω0(X, Y ) = R(X, Y ) (the curvature on (M, g))
• s = 1: Ω1(X, Y ) is a ΨDO of order zero.
σ0(Ω1(X, Y )) = R(X, Y )
σ−1(Ωs(X, Y )) = P−1(R,∇R)
where P−1(R,∇R) is a polynomial in R and ∇R.
13
Chern Classes and Chern-Simons Classes
We review Chern-Weil theory and Chern-Simons theory in finite di-
mensions:
• G: finite dimensional Lie group with Lie algebra g
• gk = g
⊗k
• F → M : a principal G-bundle over M
• θ: connection on F
• ΩF : curvature of θ
• Ik(G) = P : gk → C|P : symmetric, multilinear, Ad-invariant
(degree k Ad-invariant polynomial)
14
Chern Classes
Chern-Weil Theory
(P ∈ Ik(G),ΩF ∈ Λ2(F, g)) −→ P ΩF ∈ Λ2k(F)
−→ [P(ΩM)] ∈ Λ2k(M)
15
Chern-Weil theory for U(n)-bundles
The special case of G=U(n):
• Consider the invariant polynomial defined by P(A) = trace(Ak),
where A ∈ u(n). The corresponding cohomology class is the kth
Chern class.
• For a U(n)-connection on a U(n)-bundle, the curvature form Ω
takes values in the Lie algebra u(n).
16
Chern-Simons Classes (for 2 connections)
• θ0, θ1: connections on F
• For P ∈ Ik(G),
P(Ω0)− P(Ω1) = dCSP (θ0, θ1)
where
CSP(θ0, θ1) =
∫ 1
0P(θ0 − θ1,Ωt, · · · ,Ωt)dt
θt = tθ0 + (1− t)θ1, Ωt = dθt + θt ∧ θt.
Remark: These theories extend to associated vector bundles.
17
Chern-Simons Classes for ΨDO bundles
We extend Chern-Simons classes to an infinite dimensional setting
(which includes the case of loop spaces).
• We consider Chern-Weil and Chern-Simons theory for the infinite
dimensional Lie group ΨDO∗0
• Main example: The Levi-Civita connection 1-forms for the metrics
on loop space take values in ΨDO≤0, the Lie algebra of ΨDO∗0
• We use the ”Wodzicki residue” to produce invariant polynomials
on ΨDO≤0
18
Wodzicki residue
• P : ΨDO of order zero acting on (sections of a bundle over) a
compact manifold N with symbol σP (x, ξ) and asymptotic expan-
sion
σP (x, ξ) ∼∑k=0
σP−k(x, ξ)
where σP−k(x, ξ) is homogeneous of order −k.
• The Wodzicki residue of P is defined by
resw(P) =1
(2π)n
∫S∗N
tr(σP−n(x, ξ)) dξdx
where S∗N is the unit cosphere bundle over N .
19
WCS class for loop space
Definition: The k-th WCS class for the Levi-Civita connections
∇0,∇1 on the loop space LM is defined by
CSW2k−1(∇1,∇0) =
1
k!
∫ 1
0
∫S∗S1
tr[σ−1((ω1 − ω0) ∧Ωk−1t )]dt
where Ωt = dωt + ωt ∧ ωt, ωt = tω0 + (1− t)ω1.
20
Application of WCS class to circle actions
Let M be a compact oriented manifold.
Let a0, a1 : S1 ×M −→ M be smooth actions.:
Definition:
• a0 and a1 are smoothly homotopic if there exists a C∞ map
F : [0,1] × S1 × M −→ M such that F(0, θ,m) = a0(θ,m) and
F(1, θ,m) = a1(θ,m).
• a0 and a1 are smoothly homotopic through actions if F(t, ·, ·) is
an action for all t.
Example: standard rotation actions on S2 are smoothly homo-
topic.
21
Remark on circle actions
We can rewrite an action a : S1×M −→ M in the following two ways:
• a determines aD : S1 −→ Diff(M) by
aD(θ)(m) = a(θ,m).
Since aD(0) = id, we have [aD] ∈ π1(Diff(M), id).
• a determines aL : M −→ LM given by
aL(m)(θ) = a(θ,m)
This determines a class [aL] ∈ Hn(LM,Z) (n = dim M) by setting
[aL] = aL∗ [M ].
22
Some properties
Proposition 4:
Let dim M = 2k−1. Let a0, a1 : S1×M → M be two smooth actions.
(1) If∫[aL0]
CSW2k−1 6=
∫[aL1]
CSW2k−1, then a0, a1 are not smoothly homo-
topic through actions. Moreover, [aD0 ] 6= [aD1 ] ∈ π1(Diff(M), id)
(2) If∫[aL1]
CSW2k−1 6= 0, then π1(Diff(M), id) is infinite.
23
Proof of Proposition 4, (2)
Let an be the n-th iterate of a1, i.e. an(θ,m) = a1(nθ,m).
Then,
CSW2k−1 =
∫S1
γ(θ)f(θ)dθ
where f(θ) is periodic. Each loop γ ∈ aL1(M) corresponds to the loop
γ(n·) ∈ aLn. Therefore,∫S1 γ(θ)f(θ)dθ is replaced by
∫d
dθγ(nθ)f(nθ) = n
∫ 2π
0γ(θ)f(θ)dθ
Thus, ∫[aLn]
CSw2k−1 = n
∫[aL1]
CSw2k−1
Therefore, each [aLn] ∈ π1(Diff(M), id) is distinct.
24
Proof of Theorem A:
We recall the situation for Theorem A: (with slightly different nota-
tion)
We consider the case dim M = 5
• (M,J, g, ω) is a compact Kaehler surface with integral Kaehler form
ω ∈ H2(M,Z).
• As in geometric quantization, we construct the S1-bundle with
connection η ∈ Λ1(Lk)
Lk → M
with curvature Ωk = kω.
• Let Mk be the total space of Lk.
25
Sasakian Structure
Mk has a Sasakian structure, i.e. there is a geometry on Mk with
many special features:
• The horizontal space of the connection is H = Ker(η)
• For a vertical vector ξ satisfying η(ξ) = 1, we have dη(ξ, ·) = 0
• ξ is the characteristic vector field of the fibration π : Mk → M
• Φ(Xp) = (J[π∗X]π(p))L has Φ2 = −id on horizontal vectors; Φ = 0
on vertical vectors.
• etc.
Define a metric g on Mk by
g(X, Y ) = g(π∗X, π∗Y ) + η(X)η(Y )
Then π : Mk → M is a Riemannian submersion.
26
Formulas
Let X = XH + XV be the decomposition of X ∈ TMk into horizontal
and vertical components for the Levi-Civita connection ∇ associated
to the metric g. For X ∈ TM, let XL be its horizontal lift.
Lemma:
∇XLYL = (∇XY )L + kg(JX, Y )ξ, ∇XLξ = −k(JX)L
Proposition:
g(R(XL, Y L)ZL,WL) = R(X, Y,Z,W ) + k2[−〈JY, Z〉〈JX,W 〉
+〈JX,Z〉〈JY,W 〉+2〈JX, Y 〉〈JZ,W 〉]
g(R(XL, Y L, ZL, ξ) = 0
g(R(ξ, XL, Y L, ξ) = k2〈X,Y 〉
where R and R are the curvature tensor for ∇ and ∇, respectively.
27
Computation of the WCS form
We now compute CSW5 on Mk, using the formula for R.
Let γ be a fiber. Set γ = ξ and let ei be an orthonormal frame of
TMk with ξ = e1. We can take ei = eLi for i = 2,3,4,5.
We can write down the WCS formula for the case dim M = 4: after
some calculations of σR(x, ξ),
CSW5,γ(e1, e2, e3, e4, e5)
=1
30
∑σ
sgn(σ)R(eσ(1), eℓ, ξ, en)R(eσ(2), eσ(3), er, eℓ)R(eσ(4), eσ(5), en, er)
28
WCS formula
Now we take the orthonormal basis e2, Je2, e3, Je3 for M .
Plugging the curvature formula for the submersion into the WCS for-
mula, we have
Proposition:
CSW5,γ(ξ, e2, Je2, e3, Je3)
=k2
3032π2p1(Ω)(e2, Je2, e3, Je3) + 32k2[3R(e2, Je2, e3, Je3)
−R(e2, e3, e2, e3)−R(e2, Je3, e2, Je3) +R(e2, Je2, e2, Je2)
+R(e3, Je3, e3, Je3)] + 192k4
where p1(Ω) is the first Pontrjagin form.
29
Proof of Theorem A (1)
Set
|R|∞ = max|R(ei, ej, ek, eℓ)|i, j, k, ℓ ∈ 2,3,4,5
Here, we use the ”old notation” for the orthonormal frame e2, e3, e4, e5.
Thus, we have
Proposition: Let [aLk ] ∈ H5(LMk,R) be the class associated to the
rotation action in the fiber of π : Mk → M. Then∫[aL
k]CSW
5 > 0 if
k2(32πp1(Ω)(e2, Je2, e3, Je3)− 224k2|R|∞ +192k4) > 0
pointwise on M .
30
Proof of Theorem A (2)
We first recall:
Proposition If∫[aL
k]CSW
5 > 0, then the loop of diffeomorphisms of Mk
given by the rotation in the circle fiber gives an element of infinite
order in π1(Diff(Mk))
By this Proposition, we have
Corollary A:
The loop of diffeomorphisms of Mk given by the rotation in the circle
fiber gives an element of infinite order in π1(Diff(Mk)) provided
|R|∞ <6
7k2 +
3πσ(M)
7vol(M)k2, k 6= 0
where σ(M) is the signature of M .
31
Results (3)
In particular, we have
Corollary B: Theorem A holds in the following cases:
• k >> 0
• M = T4 and k 6= 0
32
The case of complex projective space
These arguments can be refined for M = P2(C) by computing the
curvature explicitly for the Fubini-Study metric. We get:
Theorem:
Let |k| > 1. Then the loop of diffeomorphisms of P2(C)k given by
the rotation in the circle fiber gives an element of infinite order in
π1(Diff(P2(C)k)).
Proof: A calculation gives∫P2(C)
CSW5 > 0 provided k2(k2 − 1)2 > 0.
33
Remark 1
For k = 1, we note that P2(C)1 = S5. Then π1(Diff(P2(C)1)) =
π1(Diff(S5)) contains the image of π1(Isom(S5)) = π1(SO(6)) = Z2
as a subgroup. Rotating the fiber is an action by isometries, so it has
order at most 2. Thus the Theorem must fail for k = 1.
For k = 0, P2(C)0 = P2(C)× S1. This should be the easiest case, but
we get no information.
Another example
These calculations are also explicit for M = S2 × S2 with the product
metric:
Theorem:
The loop of diffeomorphisms of (S2 × S2)k given by the rotation of
the circle fiber gives an element of infinite order in π1(Diff(S2 × S2k))
for k 6= 0.
34
Remark 2
Why dim(M)=2k-1
Our construction of CSW is valid only for the ”odd dimentional” case
Why dim (M) = 5
For the case 2k − 1 = 3, we get
Theorem:
CSW3 = 0
So, our argument does not work for dim(M) = 3.
35
Thank you
36
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