Noncommutative Geometry and Conformal Geometry: Local ...frederic/wcoas2014/RaphaelPonge.pdf ·...

Noncommutative Geometry and ConformalGeometry:

Local Index Formula and Conformal Invariants(joint work with Hang Wang)

Raphael Ponge

Seoul National University & UC Berkeley

November 1, 2014

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Main Results

Main Results (RP+HW)

Local index formula in conformal-diffeomorphism invariantgeometry.

Construction of a new class of conformal invariants.

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Main Results

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Main Results

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References

Main References

RP+HW: Index map, σ-connections, and Connes-Cherncharacter in the setting of twisted spectral triples.arXiv:1310.6131.

RP+HW: Noncommutative geometry and conformalgeometry. I. Local index formula and conformal invariants. Tobe posted on arXiv by next week.

RP+HW: Noncommutative geometry and conformalgeometry. II. Connes-Chern character and the local equivariantindex theorem. To be posted on arXiv by next week.

RP+HW: Noncommutative geometry and conformalgeometry. III. Poincare duality and Vafa-Witten inequality.arXiv:1310.6138.

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References

Main References

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References

Main References

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References

Main References

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References

Main References

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References

Main References

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Conformal Geometry

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Conformal Geometry

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Group Actions on Manifolds

If G is an arbitrary group of diffeomorphisms of a manifold M, thenM/G need not be Hausdorff (unless G acts freely and properly).

Solution Provided by NCG

Trade the space M/G for the crossed product algebra,

C∞c (M) o G =∑

fφuφ; fφ ∈ C∞c (M),

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

Proposition (Green)

If G acts freely and properly, then C∞c (M/G ) is Morita equivalentto C∞c (M) o G .

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If G is an arbitrary group of diffeomorphisms of a manifold M, thenM/G need not be Hausdorff

(unless G acts freely and properly).

C∞c (M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

Proposition (Green)

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C∞c (M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

Proposition (Green)

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C∞c (M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

Proposition (Green)

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C∞c (M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

Proposition (Green)

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The Noncommutative Torus

Example

Given θ ∈ R, let Z act on S1 by

k .z := e2ikπθz ∀z ∈ S1 ∀k ∈ Z.

Remark

If θ 6∈ Q, then the orbits of the action are dense in S1.

The crossed-product algebra Aθ := C∞(S1) oθ Z is generated bytwo operators U and V such that

U∗ = U−1, V ∗ = V−1, VU = e2iπθUV .

Remark

The algebra Aθ is called the noncommutative torus.

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Example

k .z := e2ikπθz ∀z ∈ S1 ∀k ∈ Z.

Remark

U∗ = U−1, V ∗ = V−1, VU = e2iπθUV .

Remark

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Example

k .z := e2ikπθz ∀z ∈ S1 ∀k ∈ Z.

Remark

U∗ = U−1, V ∗ = V−1, VU = e2iπθUV .

Remark

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Example

k .z := e2ikπθz ∀z ∈ S1 ∀k ∈ Z.

Remark

U∗ = U−1, V ∗ = V−1, VU = e2iπθUV .

Remark

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Example

k .z := e2ikπθz ∀z ∈ S1 ∀k ∈ Z.

Remark

U∗ = U−1, V ∗ = V−1, VU = e2iπθUV .

Remark

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Overview of Noncommutative Geometry

Classical NCG

Manifold M Spectral Triple (A,H,D)

Vector Bundle E over M Projective Module E over AE = eAq, e ∈ Mq(A), e2 = e

ind /D∇E ind D∇E

de Rham Homology/Cohomology Cyclic Cohomology/Homology

Atiyah-Singer Index Formula Connes-Chern Character Ch(D)

ind /D∇E =∫

A(RM) ∧ Ch(FE ) ind D∇E = 〈Ch(D),Ch(E)〉

Characteristic Classes Cyclic Cohomology for Hopf Algebras

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Classical NCG

Manifold M

Spectral Triple (A,H,D)

ind /D∇E =∫

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Classical NCG

ind /D∇E =∫

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Classical NCG

ind /D∇E =∫

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Classical NCG

ind /D∇E =∫

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Classical NCG

ind /D∇E =∫

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Classical NCG

ind /D∇E =∫

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Classical NCG

ind /D∇E =∫

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Spectral Triples

Definition (Connes-Moscovici)

A spectral triple (A,H,D) consists of

1 A Z2-graded Hilbert space H = H+ ⊕H−.

2 An involutive algebra A represented in H.3 A selfadjoint unbounded operator D on H such that

1 D maps H± to H∓.2 (D ± i)−1 is compact.3 [D, a] is bounded for all a ∈ A.

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Spectral Triples

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Spectral Triples

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Spectral Triples

2 An involutive algebra A represented in H.

3 A selfadjoint unbounded operator D on H such that1 D maps H± to H∓.2 (D ± i)−1 is compact.3 [D, a] is bounded for all a ∈ A.

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Spectral Triples

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Spectral Triples

1 D maps H± to H∓.

2 (D ± i)−1 is compact.3 [D, a] is bounded for all a ∈ A.

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Spectral Triples

1 D maps H± to H∓.2 (D ± i)−1 is compact.

3 [D, a] is bounded for all a ∈ A.

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Spectral Triples

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Dirac Spectral Triple

Example

(Mn, g) compact Riemannian spin manifold (n even) withspinor bundle /S = /S+ ⊕ /S−.

/Dg : C∞(M, /S)→ C∞(M, /S) is the Dirac operator of (M, g).

C∞(M) acts by multiplication on L2g (M, /S).

C∞(M), L2 (M, /S) , /Dg

)is a spectral triple.

Remark

We also get spectral triples by taking

H = L2(M,Λ•T ∗M) and D = d + d∗.

H = L2(M,Λ0,•T ∗CM) and D = ∂ + ∂∗

(when M is a complexmanifold).

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Example

C∞(M), L2 (M, /S) , /Dg

Remark

H = L2(M,Λ•T ∗M) and D = d + d∗.

H = L2(M,Λ0,•T ∗CM) and D = ∂ + ∂∗

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Example

C∞(M), L2 (M, /S) , /Dg

Remark

H = L2(M,Λ•T ∗M) and D = d + d∗.

H = L2(M,Λ0,•T ∗CM) and D = ∂ + ∂∗

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Example

C∞(M), L2 (M, /S) , /Dg

Remark

H = L2(M,Λ•T ∗M) and D = d + d∗.

H = L2(M,Λ0,•T ∗CM) and D = ∂ + ∂∗

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Example

C∞(M), L2 (M, /S) , /Dg

Remark

H = L2(M,Λ•T ∗M) and D = d + d∗.

H = L2(M,Λ0,•T ∗CM) and D = ∂ + ∂∗

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Example

C∞(M), L2 (M, /S) , /Dg

Remark

H = L2(M,Λ•T ∗M) and D = d + d∗.

H = L2(M,Λ0,•T ∗CM) and D = ∂ + ∂∗

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Diffeomorphism-Invariant Geometry

M smooth manifold.

G = Diff(M) full group diffeomorphism group of M.

The only G-invariant geometric structure of M is its manifoldstructure.

Theorem (Connes-Moscovici ‘95)

There is a spectral triple(C∞c (P) o G , L2 (P,Λ•T ∗P) ,D

), where

gijdx i ⊗ dx j ; (gij) > 0

is the metric bundle of M.

D is a “mixed-degree” signature operator, so that

D|D| = dH + d∗H + dV d∗V − d∗V dV .

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Diffeomorphism-Invariant Geometry

M smooth manifold.

G = Diff(M) full group diffeomorphism group of M.

The only G-invariant geometric structure of M is its manifoldstructure.

Theorem (Connes-Moscovici ‘95)

There is a spectral triple(C∞c (P) o G , L2 (P,Λ•T ∗P) ,D

), where

gijdx i ⊗ dx j ; (gij) > 0

is the metric bundle of M.

D is a “mixed-degree” signature operator, so that

D|D| = dH + d∗H + dV d∗V − d∗V dV .

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Conformal Change of Metric

Example

Consider a conformal change of metric,

g = k−2g , k ∈ C∞(M), k > 0.

Then the Dirac spectral triple(

C∞(M), L2g (M, /S), /D g

)is unitarily

equivalent to(

C∞(M), L2g (M, /S),

√k /Dg

(i.e., the spectral

triples are intertwined by a unitary operator).

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Example

g = k−2g , k ∈ C∞(M), k > 0.

C∞(M), L2g (M, /S), /D g

)is unitarily

equivalent to(

C∞(M), L2g (M, /S),

√k /Dg

(i.e., the spectral

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Example

g = k−2g , k ∈ C∞(M), k > 0.

C∞(M), L2g (M, /S), /D g

)is unitarily

equivalent to(

C∞(M), L2g (M, /S),

√k /Dg

(i.e., the spectral

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Example

g = k−2g , k ∈ C∞(M), k > 0.

C∞(M), L2g (M, /S), /D g

)is unitarily

equivalent to(

C∞(M), L2g (M, /S),

√k /Dg

(i.e., the spectral

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Example

g = k−2g , k ∈ C∞(M), k > 0.

C∞(M), L2g (M, /S), /D g

)is unitarily

equivalent to(

C∞(M), L2g (M, /S),

√k /Dg

(i.e., the spectral

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Example

g = k−2g , k ∈ C∞(M), k > 0.

C∞(M), L2g (M, /S), /D g

)is unitarily

equivalent to(

C∞(M), L2g (M, /S),

√k /Dg

(i.e., the spectral

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Example

g = k−2g , k ∈ C∞(M), k > 0.

C∞(M), L2g (M, /S), /D g

)is unitarily

equivalent to(

C∞(M), L2g (M, /S),

√k /Dg

(i.e., the spectral

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Twisted Spectral Triples

A spectral triple consists of

1 D maps H± to H∓.2 (D ± i)−1 is compact.3 is bounded for all a ∈ A.

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1 D maps H± to H∓.2 (D ± i)−1 is compact.3 [D,a] is bounded for all a ∈ A.

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2 An involutive algebra A represented in H.

3 A selfadjoint unbounded operator D on H such that1 D maps H± to H∓.2 (D ± i)−1 is compact.3 [D,a] is bounded for all a ∈ A.

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1 D maps H± to H∓.

2 (D ± i)−1 is compact.3 [D,a] is bounded for all a ∈ A.

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1 D maps H± to H∓.2 (D ± i)−1 is compact.

3 [D,a] is bounded for all a ∈ A.

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A twisted spectral triple (A,H,D)σ consists of

1 D maps H± to H∓.2 (D ± i)−1 is compact.3 is bounded for all a ∈ A.

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2 An involutive algebra A represented in H together with anautomorphism σ : A → A such that σ(a)∗ = σ−1(a∗) for alla ∈ A.

3 A selfadjoint unbounded operator D on H such that1 D maps H± to H∓.2 (D ± i)−1 is compact.3 is bounded for all a ∈ A.

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2 An involutive algebra A represented in H together with anautomorphism σ : A → A such that σ(a)∗ = σ−1(a∗) for alla ∈ A.

3 A selfadjoint unbounded operator D on H such that1 D maps H± to H∓.2 (D ± i)−1 is compact.3 [D, a]σ := Da− σ(a)D is bounded for all a ∈ A.

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Conformal Deformations of Spectral Triples

Proposition (Connes-Moscovici)

Consider the following:

An ordinary spectral triple (A,H,D).

A positive element k ∈ A with associated inner automorphismσ(a) = k2ak−2, a ∈ A.

Then (A,H, kDk)σ is a twisted spectral triple.

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Pseudo-Inner Twistings

Proposition (RP+HW)

Consider the following data:

A positive even operator ω =

(ω+ 00 ω−

)∈ L(H) so that

there are inner automorphisms σ±(a) = k±a(k±)−1

associated positive elements k± ∈ A in such way that

k+k− = k−k+ and ω±a = σ±(a)ω± ∀a ∈ A.

Set k = k+k− and σ(a) = kak−1. Then (A,H, ωDω)σ is atwisted spectral triple.

Example (RP+HW)

Connes-Tretkoff’s twisted spectral triples over NC tori associatedto conformal weights.

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Proposition (RP+HW)

(ω+ 00 ω−

)∈ L(H) so that

k+k− = k−k+ and ω±a = σ±(a)ω± ∀a ∈ A.

Example (RP+HW)

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Proposition (RP+HW)

(ω+ 00 ω−

)∈ L(H) so that

k+k− = k−k+ and ω±a = σ±(a)ω± ∀a ∈ A.

Example (RP+HW)

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Proposition (RP+HW)

(ω+ 00 ω−

)∈ L(H) so that

k+k− = k−k+ and ω±a = σ±(a)ω± ∀a ∈ A.

Set k = k+k− and σ(a) = kak−1.

Then (A,H, ωDω)σ is atwisted spectral triple.

Example (RP+HW)

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Proposition (RP+HW)

(ω+ 00 ω−

)∈ L(H) so that

k+k− = k−k+ and ω±a = σ±(a)ω± ∀a ∈ A.

Example (RP+HW)

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Proposition (RP+HW)

(ω+ 00 ω−

)∈ L(H) so that

k+k− = k−k+ and ω±a = σ±(a)ω± ∀a ∈ A.

Example (RP+HW)

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Further Examples

Conformal Dirac spectral triple (Connes-Moscovici).

Twisted spectral triples over NC tori associated to conformalweights (Connes-Tretkoff).

Twisted spectral triples associated to some quantumstatistical systems (e.g., Connes-Bost systems,supersymmetric Riemann gas) (Greenfield-Marcolli-Teh ‘13).

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Further Examples

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Further Examples

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Further Examples

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Connections over a Spectral Triple

(A,H,D) is an ordinary spectral triple.

E finitely generated projective (right) module over A.

Space of differential 1-forms:

Ω1D(A) := Spanadb; a, b ∈ A ⊂ L(H),

where db := [D, b].

Definition

A connection on a E is a linear map ∇E : E → E ⊗A Ω1D(A) such

that∇E(ξa) = ξ ⊗ da +

(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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where db := [D, b].

Definition

(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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where db := [D, b].

Definition

(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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where db := [D, b].

Definition

(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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where db := [D, b].

Definition

(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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where db := [D, b].

Definition

(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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σ-Connections over a Twisted Spectral Triple

Setup/Notation

(A,H,D)σ is a twisted spectral triple.

Space of twisted differential 1-forms:

Ω1D,σ(A) = Spanadσb; a, b ∈ A ⊂ L(H),

where dσb := [D, b]σ = Db − σ(b)D.

Definition

A σ-translate of E is a finitely generated projective module Eσtogether with a linear isomorphism σE : E → Eσ such that

σE(ξa) = σE(ξ)σ(a) ∀ξ ∈ E ∀a ∈ A.

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Setup/Notation

Definition

σE(ξa) = σE(ξ)σ(a) ∀ξ ∈ E ∀a ∈ A.

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Setup/Notation

Definition

σE(ξa) = σE(ξ)σ(a) ∀ξ ∈ E ∀a ∈ A.

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Setup/Notation

Definition

σE(ξa) = σE(ξ)σ(a) ∀ξ ∈ E ∀a ∈ A.

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Setup/Notation

Definition

σE(ξa) = σE(ξ)σ(a) ∀ξ ∈ E ∀a ∈ A.

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σ-Connections

Definition (RP+HW)

A σ-connection on a finitely generated projective module E is alinear map ∇E : E → Eσ ⊗A Ω1

D,σ(A) such that

∇E(ξa) = σE(ξ)⊗ dσa +(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

Example

If E = eAq with e = e2 ∈ Mq(A), then

1 Eσ = σ(e)Aq is a σ-translate.

2 It is equipped with the Grassmanian σ-connection,

∇E0 = (σ(e)⊗ 1)dσ.

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σ-Connections

Definition (RP+HW)

D,σ(A) such that

)a ∀a ∈ A ∀ξ ∈ E .

Example

∇E0 = (σ(e)⊗ 1)dσ.

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σ-Connections

Definition (RP+HW)

D,σ(A) such that

)a ∀a ∈ A ∀ξ ∈ E .

Example

∇E0 = (σ(e)⊗ 1)dσ.

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σ-Connections

Definition (RP+HW)

D,σ(A) such that

)a ∀a ∈ A ∀ξ ∈ E .

Example

∇E0 = (σ(e)⊗ 1)dσ.

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σ-Connections

Definition (RP+HW)

D,σ(A) such that

)a ∀a ∈ A ∀ξ ∈ E .

Example

∇E0 = (σ(e)⊗ 1)dσ.

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Coupling with σ-connections

Proposition (RP+HW)

The datum of σ-connection on E defines a coupled operator,

D∇E : E ⊗A dom D → Eσ ⊗A H,

of the form,

D∇E =

(0 D−∇E

D+∇E 0

where D±∇E : E ⊗ dom D± → Eσ ⊗H∓ are Fredholm operators.

Example

For a Dirac spectral triple (C∞(M), L2g (M), /S , /Dg ) and

E = C∞(M,E ), it can be shown that

1 Any connection ∇E on E defines a connection on E .

2 The corresponding coupled operator agrees with /D∇E .

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Proposition (RP+HW)

of the form,

D∇E =

(0 D−∇E

D+∇E 0

Example

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Proposition (RP+HW)

of the form,

D∇E =

(0 D−∇E

D+∇E 0

Example

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Proposition (RP+HW)

of the form,

D∇E =

(0 D−∇E

D+∇E 0

Example

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Proposition (RP+HW)

of the form,

D∇E =

(0 D−∇E

D+∇E 0

Example

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Proposition (RP+HW)

of the form,

D∇E =

(0 D−∇E

D+∇E 0

Example

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Index Map

Definition

The index of D∇E is

ind D∇E :=1

(ind D+

∇E − ind D−∇E).

where ind D±∇E = dim ker D±∇E − dim coker D±∇E .

Remark

When σ = id, and in all the main examples with σ 6= id, we have

ind D∇E = ind D+∇E .

Proposition (Connes-Moscovici, RP+HW)

1 ind D∇E depends only on the K -theory class of E .

2 There is a unique additive map indD,σ : K0(A)→ 12Z so that

indD [E ] = ind D∇E ∀(E ,∇E).

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Index Map

Definition

ind D∇E :=1

(ind D+

Remark

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Index Map

Definition

ind D∇E :=1

(ind D+

Remark

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Index Map

Definition

ind D∇E :=1

(ind D+

Remark

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Index Map

Definition

ind D∇E :=1

(ind D+

Remark

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Index Map

Definition

ind D∇E :=1

(ind D+

Remark

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Index Map

Definition

ind D∇E :=1

(ind D+

Remark

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Connes-Chern Character

Lemma (RP+HW)

Suppose that (A,H,D)σ is p-summable, i.e., Tr |D|−p <∞ forsome p ≥ 1. Assume further that D is invertible and E = eAq,e2 = e ∈ Mq(A). Then, for all k ≥ 1

ind D∇E =1

2Trγ(D−1[D, e]σ)2k

where γ = idH+ − idH− .

Theorem (Connes-Moscovici, RP+HW)

Assume (A,H,D)σ is p-summable. Then there is an even periodiccyclic class Ch(D)σ ∈ HP0(A), called Connes-Chern character,such that

ind D∇E = 〈Ch(D)σ,Ch(E)〉 ∀(E ,∇E),

where Ch(E) is the Chern character in periodic cyclic homology.

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Lemma (RP+HW)

Suppose that (A,H,D)σ is p-summable, i.e., Tr |D|−p <∞ forsome p ≥ 1.

Assume further that D is invertible and E = eAq,e2 = e ∈ Mq(A). Then, for all k ≥ 1

ind D∇E =1

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Lemma (RP+HW)

Suppose that (A,H,D)σ is p-summable, i.e., Tr |D|−p <∞ forsome p ≥ 1. Assume further that D is invertible and E = eAq,e2 = e ∈ Mq(A).

Then, for all k ≥ 12p,

ind D∇E =1

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Lemma (RP+HW)

ind D∇E =1

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Lemma (RP+HW)

ind D∇E =1

Assume (A,H,D)σ is p-summable.

Then there is an even periodiccyclic class Ch(D)σ ∈ HP0(A), called Connes-Chern character,such that

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Lemma (RP+HW)

ind D∇E =1

Assume (A,H,D)σ is p-summable. Then there is an even periodiccyclic class Ch(D)σ ∈ HP0(A)

, called Connes-Chern character,such that

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Lemma (RP+HW)

ind D∇E =1

Assume (A,H,D)σ is p-summable. Then there is an even periodiccyclic class Ch(D)σ ∈ HP0(A), called Connes-Chern character,

such that

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Lemma (RP+HW)

ind D∇E =1

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Lemma (RP+HW)

ind D∇E =1

where Ch(E) is the Chern character in periodic cyclic homology.21 / 31

Conformal Dirac Spectral Triple

1 Mn is a compact spin oriented manifold (n even).

2 C is a conformal structure on M.

3 G is a group of conformal diffeomorphisms preserving C.Thus, given any metric g ∈ C and φ ∈ G ,

φ∗g = k−2φ g with kφ ∈ C∞(M), kφ > 0.

4 C∞(M) o G is the crossed-product algebra, i.e.,

C∞(M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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C∞(M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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C∞(M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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3 G is a group of conformal diffeomorphisms preserving C.

Thus, given any metric g ∈ C and φ ∈ G ,

C∞(M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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C∞(M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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C∞(M) o G =∑

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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Lemma (Connes-Moscovici)

For φ ∈ G define Uφ : L2g (M, /S)→ L2

g (M, /S) by

Uφξ = k− n

2φ φ∗ξ ∀ξ ∈ L2

g (M, /S).

Then Uφ is a unitary operator, and

Uφ /DgU∗φ =√

kφ /Dg

√kφ.

The datum of any metric g ∈ C defines a twisted spectral triple(C∞(M) o G , L2

g (M, /S), /Dg

given by

1 The Dirac operator /Dg associated to g.

2 The representation fuφ → fUφ of C∞(M) o G in L2g (M, /S).

3 The automorphism σg (fuφ) := k−1φ fuφ.

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g (M, /S) by

Uφξ = k− n

2φ φ∗ξ ∀ξ ∈ L2

g (M, /S).

Uφ /DgU∗φ =√

kφ /Dg

√kφ.

g (M, /S), /Dg

given by

23 / 31

g (M, /S) by

Uφξ = k− n

2φ φ∗ξ ∀ξ ∈ L2

g (M, /S).

Uφ /DgU∗φ =√

kφ /Dg

√kφ.

g (M, /S), /Dg

given by

23 / 31

g (M, /S) by

Uφξ = k− n

2φ φ∗ξ ∀ξ ∈ L2

g (M, /S).

Uφ /DgU∗φ =√

kφ /Dg

√kφ.

g (M, /S), /Dg

given by

23 / 31

g (M, /S) by

Uφξ = k− n

2φ φ∗ξ ∀ξ ∈ L2

g (M, /S).

Uφ /DgU∗φ =√

kφ /Dg

√kφ.

g (M, /S), /Dg

given by

23 / 31

g (M, /S) by

Uφξ = k− n

2φ φ∗ξ ∀ξ ∈ L2

g (M, /S).

Uφ /DgU∗φ =√

kφ /Dg

√kφ.

g (M, /S), /Dg

given by

23 / 31

g (M, /S) by

Uφξ = k− n

2φ φ∗ξ ∀ξ ∈ L2

g (M, /S).

Uφ /DgU∗φ =√

kφ /Dg

√kφ.

g (M, /S), /Dg

given by

3 The automorphism σg (fuφ) := k−1φ fuφ.23 / 31

Conformal Connes-Chern Character

Theorem (RP+HW)

1 The Connes-Chern character Ch(/Dg )σg ∈ HP0(C∞(M) o G )is an invariant of the conformal class C.

2 For any even cyclic homology class η ∈ HP0(C∞(M) o G ),the pairing,

〈Ch(/Dg )σg , η〉,

is a scalar conformal invariant.

Definition

The conformal Connes-Chern character Ch(C) ∈ HP0(C∞(M)oG )is the Connes-Chern character Ch(/Dg )σg for any metric g ∈ C.

24 / 31

Theorem (RP+HW)

Definition

24 / 31

Theorem (RP+HW)

Definition

24 / 31

Theorem (RP+HW)

Definition

24 / 31

Computation of Ch(C)

Proposition (Ferrand-Obata)

If the conformal structure C is non-flat, then C contains aG -invariant metric.

If g ∈ C is G -invariant, then(

C∞(M) o G , L2g (M, /S), /Dg

ordinary spectral triple (i.e., σg = 1).

Consequence

When C is non-flat, we are reduced to the computation of the

Connes-Chern character of(

C∞(M) o G , L2g (M, /S), /Dg

), where G

is a group of isometries.

25 / 31

C∞(M) o G , L2g (M, /S), /Dg

Consequence

C∞(M) o G , L2g (M, /S), /Dg

), where G

25 / 31

C∞(M) o G , L2g (M, /S), /Dg

Consequence

C∞(M) o G , L2g (M, /S), /Dg

), where G

25 / 31

C∞(M) o G , L2g (M, /S), /Dg

Consequence

C∞(M) o G , L2g (M, /S), /Dg

), where G

25 / 31

Remark

1 When G is a group of isometries, the Connes-Chern characterof (C∞(M) o G , L2

g (M, /S), /Dg ) is represented by the CMcocycle.

2 The computation of the CM cocycle amounts to get a“differentiable version” of the local equivariant index theorem(LEIT) of Donnelly-Patodi, Gilkey and Kawasaki.

3 We produce a new proof of LEIT that allows us to computethe CM cocyle in the same shot.

4 This approach was various other applications (equivariant JLOcocycle, equivariant eta cochain, Yong Wang’s papers).

26 / 31

Remark

26 / 31

Remark

26 / 31

Remark

26 / 31

Remark

26 / 31

Local Index Formula in Conformal Geometry

C is a nonflat conformal structure on M.

g is a G -invariant metric in C.

Notation

Let φ ∈ G . Then

Mφ is the fixed-point set of φ; this is a disconnected sums ofsubmanifolds,Mφ =

⊔Mφ

a , dim Mφa = a.

N φ = (TMφ)⊥ is the normal bundle (vector bundle over Mφ).

Over Mφ, with respect to TM|Mφ = TMφ ⊕N φ, there aredecompositions,

φ′ =

(1 00 φ′|Nφ

), ∇TM = ∇TMφ ⊕∇Nφ

27 / 31

Notation

Let φ ∈ G . Then

⊔Mφ

a , dim Mφa = a.

φ′ =

(1 00 φ′|Nφ

), ∇TM = ∇TMφ ⊕∇Nφ

27 / 31

Notation

Let φ ∈ G . Then

⊔Mφ

a , dim Mφa = a.

φ′ =

(1 00 φ′|Nφ

), ∇TM = ∇TMφ ⊕∇Nφ

27 / 31

Notation

Let φ ∈ G . Then

⊔Mφ

a , dim Mφa = a.

φ′ =

(1 00 φ′|Nφ

), ∇TM = ∇TMφ ⊕∇Nφ

27 / 31

Notation

Let φ ∈ G . Then

Mφ is the fixed-point set of φ;

this is a disconnected sums ofsubmanifolds,Mφ =

⊔Mφ

a , dim Mφa = a.

φ′ =

(1 00 φ′|Nφ

), ∇TM = ∇TMφ ⊕∇Nφ

27 / 31

Notation

Let φ ∈ G . Then

⊔Mφ

a , dim Mφa = a.

φ′ =

(1 00 φ′|Nφ

), ∇TM = ∇TMφ ⊕∇Nφ

27 / 31

Notation

Let φ ∈ G . Then

⊔Mφ

a , dim Mφa = a.

φ′ =

(1 00 φ′|Nφ

), ∇TM = ∇TMφ ⊕∇Nφ

27 / 31

Notation

Let φ ∈ G . Then

⊔Mφ

a , dim Mφa = a.

φ′ =

(1 00 φ′|Nφ

), ∇TM = ∇TMφ ⊕∇Nφ

27 / 31

Theorem (RP + HW)

For any G -invariant metric g ∈ C, the conformal Connes-Cherncharacter Ch(/Dg )σg is represented by the periodic cyclic cocycleϕ = (ϕ2m) given by

ϕ2m(f 0Uφ0 , · · · , f2mUφ2m) =

(−i)n2

(2π)−a2

∫Mφ

A(RTMφ)∧νφ

φ)∧f 0df 1∧· · ·∧df 2m,

where φ := φ0 · · · φ2m, and f j := f j φ−10 · · · φ−1j−1, and

RTMφ)

:= det12

[RTMφ

sinh(RTMφ/2

)] ,νφ

:= det−12

[1− φ′|Nφe−R

28 / 31

Theorem (RP + HW)

For any G -invariant metric g ∈ C,

the conformal Connes-Cherncharacter Ch(/Dg )σg is represented by the periodic cyclic cocycleϕ = (ϕ2m) given by

ϕ2m(f 0Uφ0 , · · · , f2mUφ2m) =

(−i)n2

(2π)−a2

∫Mφ

A(RTMφ)∧νφ

φ)∧f 0df 1∧· · ·∧df 2m,

RTMφ)

:= det12

[RTMφ

sinh(RTMφ/2

)] ,νφ

:= det−12

28 / 31

Theorem (RP + HW)

ϕ2m(f 0Uφ0 , · · · , f2mUφ2m) =

(−i)n2

(2π)−a2

∫Mφ

A(RTMφ)∧νφ

φ)∧f 0df 1∧· · ·∧df 2m,

RTMφ)

:= det12

[RTMφ

sinh(RTMφ/2

)] ,νφ

:= det−12

28 / 31

Theorem (RP + HW)

ϕ2m(f 0Uφ0 , · · · , f2mUφ2m) =

(−i)n2

(2π)−a2

∫Mφ

A(RTMφ)∧νφ

φ)∧f 0df 1∧· · ·∧df 2m,

RTMφ)

:= det12

[RTMφ

sinh(RTMφ/2

)] ,νφ

:= det−12

28 / 31

Theorem (RP + HW)

ϕ2m(f 0Uφ0 , · · · , f2mUφ2m) =

(−i)n2

(2π)−a2

∫Mφ

A(RTMφ)∧νφ

φ)∧f 0df 1∧· · ·∧df 2m,

where φ := φ0 · · · φ2m,

and f j := f j φ−10 · · · φ−1j−1, and

RTMφ)

:= det12

[RTMφ

sinh(RTMφ/2

)] ,νφ

:= det−12

28 / 31

Theorem (RP + HW)

ϕ2m(f 0Uφ0 , · · · , f2mUφ2m) =

(−i)n2

(2π)−a2

∫Mφ

A(RTMφ)∧νφ

φ)∧f 0df 1∧· · ·∧df 2m,

where φ := φ0 · · · φ2m, and f j := f j φ−10 · · · φ−1j−1,

RTMφ)

:= det12

[RTMφ

sinh(RTMφ/2

)] ,νφ

:= det−12

28 / 31

Theorem (RP + HW)

ϕ2m(f 0Uφ0 , · · · , f2mUφ2m) =

(−i)n2

(2π)−a2

∫Mφ

A(RTMφ)∧νφ

φ)∧f 0df 1∧· · ·∧df 2m,

RTMφ)

:= det12

[RTMφ

sinh(RTMφ/2

)] ,νφ

:= det−12

28 / 31

Remark

The n-th degree component is given by

ϕn(f 0Uφ0 , · · · , fnUφn) =

∫M f 0df 1 ∧ · · · ∧ df n if φ0 · · · φn = 1,

0 if φ0 · · · φn 6= 1.

This represents Connes’ transverse fundamental class of M/G .

29 / 31

Remark

ϕn(f 0Uφ0 , · · · , fnUφn) =

∫M f 0df 1 ∧ · · · ∧ df n if φ0 · · · φn = 1,

0 if φ0 · · · φn 6= 1.

29 / 31

Remark

ϕn(f 0Uφ0 , · · · , fnUφn) =

∫M f 0df 1 ∧ · · · ∧ df n if φ0 · · · φn = 1,

0 if φ0 · · · φn 6= 1.

29 / 31

Cyclic Homology of C∞(M) o G

Theorem (Brylinski-Nistor, Crainic)

Along the conjugation classes of G,

HP0(C∞(M) o G ) '⊕〈φ〉

HevGφ(Mφ

where Gφ is the centralizer of φ and HevGφ(Mφ

a ) is the Gφ-invariant

even de Rham cohomology of Mφa .

Any closed form ω ∈ Ω•Gφ(Mφ

a ) defines a cyclic cycle ηω onC∞(M) o G via the transformation,

f 0df 1 ∧ · · · ∧ df k −→ Uφf 0 ⊗ f 1 ⊗ · · · ⊗ f k , f j ∈ C∞(Mφa )G

where f j is a Gφ-invariant smooth extension of f j to M.

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HP0(C∞(M) o G ) '⊕〈φ〉

HevGφ(Mφ

30 / 31

HP0(C∞(M) o G ) '⊕〈φ〉

HevGφ(Mφ

30 / 31

HP0(C∞(M) o G ) '⊕〈φ〉

HevGφ(Mφ

a ) defines a cyclic cycle ηω onC∞(M) o G

via the transformation,

30 / 31

HP0(C∞(M) o G ) '⊕〈φ〉

HevGφ(Mφ

30 / 31

HP0(C∞(M) o G ) '⊕〈φ〉

HevGφ(Mφ

30 / 31

Conformal Invariants

Theorem (RP+HW)

Assume that the conformal structure C is nonflat. Then

1 For any closed even form ω ∈ ΩevGφ(Mφ

a ), the pairing〈Ch(C), ηω〉 is a conformal invariant.

2 For any G -invariant metric g ∈ C, we have

〈Ch(C), ηω〉 =

∫Mφ

A(RTMφ) ∧ νφ

φ)∧ ω.

Remark

The above invariants are not the type of conformal invariantsappearing in the Deser-Schwimmer conjecture solved by SpyrosAlexakis in 2007 (about 600 pages).

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Theorem (RP+HW)

Assume that the conformal structure C is nonflat.

〈Ch(C), ηω〉 =

∫Mφ

A(RTMφ) ∧ νφ

φ)∧ ω.

Remark

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Theorem (RP+HW)

〈Ch(C), ηω〉 =

∫Mφ

A(RTMφ) ∧ νφ

φ)∧ ω.

Remark

31 / 31

Theorem (RP+HW)

〈Ch(C), ηω〉 =

∫Mφ

A(RTMφ) ∧ νφ

φ)∧ ω.

Remark

31 / 31

Theorem (RP+HW)

〈Ch(C), ηω〉 =

∫Mφ

A(RTMφ) ∧ νφ

φ)∧ ω.

Remark

31 / 31

Theorem (RP+HW)

〈Ch(C), ηω〉 =

∫Mφ

A(RTMφ) ∧ νφ

φ)∧ ω.

Remark

31 / 31

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