Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry...

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Noncommutative Geometry and Conformal Geometry (joint work with Hang Wang) Rapha¨ el Ponge Seoul National University & U.C. Berkeley June 27, 2014 1 / 21

Transcript of Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry...

Page 1: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Noncommutative Geometry and ConformalGeometry

(joint work with Hang Wang)

Raphael Ponge

Seoul National University & U.C. Berkeley

June 27, 2014

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References

Main References

RP+HW: Noncommutative geometry, conformal geometry,and the local equivariant index theorem. arXiv:1210.2032.Superceded by the 3 papers below.

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

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

RP+HW: Noncommutative geometry and conformalgeometry. II. Connes-Chern character and the localequivariant index theorem. To be posted on arXiv soon.

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

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Conformal Geometry

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Conformal Geometry

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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

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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Spectral Triples

Definition

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.

Example (Dirac Spectral Triple)

(C∞(M), L2g (M, /S), /Dg ),

where (Mn, g) is a compact Riemanian spin manifold (n even),/S = /S+ ⊕ /S− is the spinor bundle, and /D g is the Dirac operator.

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Spectral Triples

Definition

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.

Example (Dirac Spectral Triple)

(C∞(M), L2g (M, /S), /Dg ),

where (Mn, g) is a compact Riemanian spin manifold (n even),/S = /S+ ⊕ /S− is the spinor bundle, and /D g is the Dirac operator.

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Twisted Spectral Triples

Definition (Connes-Moscovici)

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

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

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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Examples

Example (Conformal Deformation of Spectral Triples)

Given an ordinary spectral triple (A,H,D), let k ∈ A, k > 0. Then

(A,H, kDk)σ, σ(a) = k2ak−2, a ∈ A,

is a twisted spectral triple.

Example (Conformal Change of Metric)

Let (C∞(M), L2g (M, /S), /Dg ) be a Dirac spectral triple. Consider

the conformal change of metric,

g = k−2g , k ∈ C∞(M), k > 0.

Then (C∞(M), L2g (M, /S), /D g ) is unitarily equivalent to

(C∞(M), L2g (M, /S),

√k /Dg

√k).

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Examples

Example (Conformal Deformation of Spectral Triples)

Given an ordinary spectral triple (A,H,D), let k ∈ A, k > 0. Then

(A,H, kDk)σ, σ(a) = k2ak−2, a ∈ A,

is a twisted spectral triple.

Example (Conformal Change of Metric)

Let (C∞(M), L2g (M, /S), /Dg ) be a Dirac spectral triple. Consider

the conformal change of metric,

g = k−2g , k ∈ C∞(M), k > 0.

Then (C∞(M), L2g (M, /S), /D g ) is unitarily equivalent to

(C∞(M), L2g (M, /S),

√k /Dg

√k).

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Further Examples

Further Examples

Conformal Dirac spectral triple (Connes-Moscovici).

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

Poincare duals of some ordinary spectral triples (RP+HW,Part 3).

Twisted spectral triples associated to quantum statisticalsystems (e.g., Connes-Bost systems, supersymmetric Riemanngas) (Greenfield-Marcolli-Teh ‘13).

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σ-Connections

Definition (Bimodule of σ-Differential Forms)

Ω1D,σ(A) = Spanadσb; a, b ∈ A ⊂ L(H),

where dσb := [D, b]σ = Db − σ(b)D.

Definition (σ-Connection)

Let E be a finitely generated projective module over A.

1 A σ-translate 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.

2 A σ-connection is a given by a σ-translate Eσ and a linearmap ∇E : E → Eσ ⊗ Ω1

D,σ(A) such that

∇E(ξa) = σE(ξ)⊗ dσa +(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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σ-Connections

Definition (Bimodule of σ-Differential Forms)

Ω1D,σ(A) = Spanadσb; a, b ∈ A ⊂ L(H),

where dσb := [D, b]σ = Db − σ(b)D.

Definition (σ-Connection)

Let E be a finitely generated projective module over A.

1 A σ-translate 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.

2 A σ-connection is a given by a σ-translate Eσ and a linearmap ∇E : E → Eσ ⊗ Ω1

D,σ(A) such that

∇E(ξa) = σE(ξ)⊗ dσa +(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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σ-Connections

Definition (Bimodule of σ-Differential Forms)

Ω1D,σ(A) = Spanadσb; a, b ∈ A ⊂ L(H),

where dσb := [D, b]σ = Db − σ(b)D.

Definition (σ-Connection)

Let E be a finitely generated projective module over A.

1 A σ-translate 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.

2 A σ-connection is a given by a σ-translate Eσ and a linearmap ∇E : E → Eσ ⊗ Ω1

D,σ(A) such that

∇E(ξa) = σE(ξ)⊗ dσa +(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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σ-Connections

Definition (Bimodule of σ-Differential Forms)

Ω1D,σ(A) = Spanadσb; a, b ∈ A ⊂ L(H),

where dσb := [D, b]σ = Db − σ(b)D.

Definition (σ-Connection)

Let E be a finitely generated projective module over A.

1 A σ-translate 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.

2 A σ-connection is a given by a σ-translate Eσ and a linearmap ∇E : E → Eσ ⊗ Ω1

D,σ(A) such that

∇E(ξa) = σE(ξ)⊗ dσa +(∇Eξ

)a ∀a ∈ A ∀ξ ∈ E .

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Operator D∇E

Proposition (RP+HW)

1 The data of a σ-connection ∇E defines a closed unboundedoperator,

D∇E =

(0 D−∇E

D+∇E 0

), D±∇E : E ⊗H± → Eσ ⊗H∓.

2 The operators D±∇E are Fredholm.

Definition (Index of D∇E )

ind D∇E =1

2

(ind D+

∇E − ind D−∇E),

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

Remark

In all the main examples ind D∇E is actually an integer.

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Operator D∇E

Proposition (RP+HW)

1 The data of a σ-connection ∇E defines a closed unboundedoperator,

D∇E =

(0 D−∇E

D+∇E 0

), D±∇E : E ⊗H± → Eσ ⊗H∓.

2 The operators D±∇E are Fredholm.

Definition (Index of D∇E )

ind D∇E =1

2

(ind D+

∇E − ind D−∇E),

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

Remark

In all the main examples ind D∇E is actually an integer.

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Operator D∇E

Proposition (RP+HW)

1 The data of a σ-connection ∇E defines a closed unboundedoperator,

D∇E =

(0 D−∇E

D+∇E 0

), D±∇E : E ⊗H± → Eσ ⊗H∓.

2 The operators D±∇E are Fredholm.

Definition (Index of D∇E )

ind D∇E =1

2

(ind D+

∇E − ind D−∇E),

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

Remark

In all the main examples ind D∇E is actually an integer.

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Operator D∇E

Proposition (RP+HW)

1 The data of a σ-connection ∇E defines a closed unboundedoperator,

D∇E =

(0 D−∇E

D+∇E 0

), D±∇E : E ⊗H± → Eσ ⊗H∓.

2 The operators D±∇E are Fredholm.

Definition (Index of D∇E )

ind D∇E =1

2

(ind D+

∇E − ind D−∇E),

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

Remark

In all the main examples ind D∇E is actually an integer.10 / 21

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Index Map and Connes-Chern Character

Proposition (Connes-Moscovici, RP+HW)

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

indD [E ] = ind D∇E ∀(E ,∇E).

Theorem (Connes-Moscovici, RP+HW)

Assume that Tr |D|−p <∞ for some p ≥ 1. Then there is a(periodic) cyclic cohomology class Ch(D)σ ∈ HP0(A) , calledConnes-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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Index Map and Connes-Chern Character

Proposition (Connes-Moscovici, RP+HW)

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

indD [E ] = ind D∇E ∀(E ,∇E).

Theorem (Connes-Moscovici, RP+HW)

Assume that Tr |D|−p <∞ for some p ≥ 1. Then there is a(periodic) cyclic cohomology class Ch(D)σ ∈ HP0(A) , calledConnes-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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Conformal Dirac Spectral Triple

Setup

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 (discrete) crossed-product algebra, i.e.,

C∞(M) o G =∑

fφuφ; fφ ∈ C∞c (M),

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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Page 23: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Conformal Dirac Spectral Triple

Setup

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 (discrete) crossed-product algebra, i.e.,

C∞(M) o G =∑

fφuφ; fφ ∈ C∞c (M),

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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Conformal Dirac Spectral Triple

Setup

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 (discrete) crossed-product algebra, i.e.,

C∞(M) o G =∑

fφuφ; fφ ∈ C∞c (M),

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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Conformal Dirac Spectral Triple

Setup

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 (discrete) crossed-product algebra, i.e.,

C∞(M) o G =∑

fφuφ; fφ ∈ C∞c (M),

u∗φ = u−1φ = uφ−1 , uφf = (f φ−1)uφ.

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Conformal Dirac Spectral Triple

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.

Proposition (Connes-Moscovici)

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

g (M, /S), /Dg

)σg

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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Conformal Dirac Spectral Triple

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.

Proposition (Connes-Moscovici)

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

g (M, /S), /Dg

)σg

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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Conformal Dirac Spectral Triple

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.

Proposition (Connes-Moscovici)

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

g (M, /S), /Dg

)σg

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.13 / 21

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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 structure C.

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

〈Ch(/Dg )σg , η〉,

is a conformal invariant.

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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 structure C.

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

〈Ch(/Dg )σg , η〉,

is a conformal invariant.

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Computation of Ch(/Dg)σg

Strategy

1 Thanks to the conformal invariance we can choose any metricg ∈ C to compute Ch(/Dg )σg .

2 If the conformal structure C is nonflat, then it contains a Ginvariant metric C.

3 If g ∈ C is G -invariant, then σg = 1, and so the conformal

Dirac spectral triple(

C∞(M) o G , L2g (M, /S), /Dg

)σg

is an

ordinary spectral triple.

4 In this case, the Connes-Chern character is computed as aconsequence of a new heat kernel proof of the localequivariant index theorem of Atiyah-Segal, Donelly-Patodi,Gilkey.

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Page 32: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Computation of Ch(/Dg)σg

Strategy

1 Thanks to the conformal invariance we can choose any metricg ∈ C to compute Ch(/Dg )σg .

2 If the conformal structure C is nonflat, then it contains a Ginvariant metric C.

3 If g ∈ C is G -invariant, then σg = 1, and so the conformal

Dirac spectral triple(

C∞(M) o G , L2g (M, /S), /Dg

)σg

is an

ordinary spectral triple.

4 In this case, the Connes-Chern character is computed as aconsequence of a new heat kernel proof of the localequivariant index theorem of Atiyah-Segal, Donelly-Patodi,Gilkey.

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Page 33: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Computation of Ch(/Dg)σg

Strategy

1 Thanks to the conformal invariance we can choose any metricg ∈ C to compute Ch(/Dg )σg .

2 If the conformal structure C is nonflat, then it contains a Ginvariant metric C.

3 If g ∈ C is G -invariant, then σg = 1, and so the conformal

Dirac spectral triple(

C∞(M) o G , L2g (M, /S), /Dg

)σg

is an

ordinary spectral triple.

4 In this case, the Connes-Chern character is computed as aconsequence of a new heat kernel proof of the localequivariant index theorem of Atiyah-Segal, Donelly-Patodi,Gilkey.

15 / 21

Page 34: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Computation of Ch(/Dg)σg

Strategy

1 Thanks to the conformal invariance we can choose any metricg ∈ C to compute Ch(/Dg )σg .

2 If the conformal structure C is nonflat, then it contains a Ginvariant metric C.

3 If g ∈ C is G -invariant, then σg = 1, and so the conformal

Dirac spectral triple(

C∞(M) o G , L2g (M, /S), /Dg

)σg

is an

ordinary spectral triple.

4 In this case, the Connes-Chern character is computed as aconsequence of a new heat kernel proof of the localequivariant index theorem of Atiyah-Segal, Donelly-Patodi,Gilkey.

15 / 21

Page 35: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Computation of Ch(/Dg)σg

Strategy

1 Thanks to the conformal invariance we can choose any metricg ∈ C to compute Ch(/Dg )σg .

2 If the conformal structure C is nonflat, then it contains a Ginvariant metric C.

3 If g ∈ C is G -invariant, then σg = 1, and so the conformal

Dirac spectral triple(

C∞(M) o G , L2g (M, /S), /Dg

)σg

is an

ordinary spectral triple.

4 In this case, the Connes-Chern character is computed as aconsequence of a new heat kernel proof of the localequivariant index theorem of Atiyah-Segal, Donelly-Patodi,Gilkey.

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Page 36: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Notation

Setup

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 (a even).

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

.

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Notation

Setup

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 (a even).

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

.

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Page 38: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Notation

Setup

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 (a even).

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

.

16 / 21

Page 39: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Local Index Formula in Conformal Geometry

Theorem (RP + HW)

For any G -invariant metric g ∈ C, the Connes-Chern characterCh(/Dg )σg is represented by the periodic cyclic cocycle ϕ = (ϕ2m)given by

ϕ2m(f 0uφ0 , · · · , f2muφ2m) =

(−i)n2

(2m)!

∑0≤a≤n

(2π)−a2

∫Mφ

a

A(RTMφa )∧νφ

(RN

φ)∧f 0df 1∧· · ·∧df 2m,

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

A(

RTMφ)

:= det12

[RTMφ

/2

sinh(RTMφ/2

)] ,νφ

(RN

φ)

:= det−12

[1− φ′|Nφe−R

Nφ].

17 / 21

Page 40: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Local Index Formula in Conformal Geometry

Theorem (RP + HW)

For any G -invariant metric g ∈ C,

the Connes-Chern characterCh(/Dg )σg is represented by the periodic cyclic cocycle ϕ = (ϕ2m)given by

ϕ2m(f 0uφ0 , · · · , f2muφ2m) =

(−i)n2

(2m)!

∑0≤a≤n

(2π)−a2

∫Mφ

a

A(RTMφa )∧νφ

(RN

φ)∧f 0df 1∧· · ·∧df 2m,

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

A(

RTMφ)

:= det12

[RTMφ

/2

sinh(RTMφ/2

)] ,νφ

(RN

φ)

:= det−12

[1− φ′|Nφe−R

Nφ].

17 / 21

Page 41: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Local Index Formula in Conformal Geometry

Theorem (RP + HW)

For any G -invariant metric g ∈ C, the Connes-Chern characterCh(/Dg )σg is represented by the periodic cyclic cocycle ϕ = (ϕ2m)given by

ϕ2m(f 0uφ0 , · · · , f2muφ2m) =

(−i)n2

(2m)!

∑0≤a≤n

(2π)−a2

∫Mφ

a

A(RTMφa )∧νφ

(RN

φ)∧f 0df 1∧· · ·∧df 2m,

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

A(

RTMφ)

:= det12

[RTMφ

/2

sinh(RTMφ/2

)] ,νφ

(RN

φ)

:= det−12

[1− φ′|Nφe−R

Nφ].

17 / 21

Page 42: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Local Index Formula in Conformal Geometry

Theorem (RP + HW)

For any G -invariant metric g ∈ C, the Connes-Chern characterCh(/Dg )σg is represented by the periodic cyclic cocycle ϕ = (ϕ2m)given by

ϕ2m(f 0uφ0 , · · · , f2muφ2m) =

(−i)n2

(2m)!

∑0≤a≤n

(2π)−a2

∫Mφ

a

A(RTMφa )∧νφ

(RN

φ)∧f 0df 1∧· · ·∧df 2m,

where φ := φ0 · · · φ2m,

and f j := f j φ−10 · · · φ−1j−1, and

A(

RTMφ)

:= det12

[RTMφ

/2

sinh(RTMφ/2

)] ,νφ

(RN

φ)

:= det−12

[1− φ′|Nφe−R

Nφ].

17 / 21

Page 43: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Local Index Formula in Conformal Geometry

Theorem (RP + HW)

For any G -invariant metric g ∈ C, the Connes-Chern characterCh(/Dg )σg is represented by the periodic cyclic cocycle ϕ = (ϕ2m)given by

ϕ2m(f 0uφ0 , · · · , f2muφ2m) =

(−i)n2

(2m)!

∑0≤a≤n

(2π)−a2

∫Mφ

a

A(RTMφa )∧νφ

(RN

φ)∧f 0df 1∧· · ·∧df 2m,

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

and

A(

RTMφ)

:= det12

[RTMφ

/2

sinh(RTMφ/2

)] ,νφ

(RN

φ)

:= det−12

[1− φ′|Nφe−R

Nφ].

17 / 21

Page 44: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Local Index Formula in Conformal Geometry

Theorem (RP + HW)

For any G -invariant metric g ∈ C, the Connes-Chern characterCh(/Dg )σg is represented by the periodic cyclic cocycle ϕ = (ϕ2m)given by

ϕ2m(f 0uφ0 , · · · , f2muφ2m) =

(−i)n2

(2m)!

∑0≤a≤n

(2π)−a2

∫Mφ

a

A(RTMφa )∧νφ

(RN

φ)∧f 0df 1∧· · ·∧df 2m,

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

A(

RTMφ)

:= det12

[RTMφ

/2

sinh(RTMφ/2

)] ,νφ

(RN

φ)

:= det−12

[1− φ′|Nφe−R

Nφ].

17 / 21

Page 45: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Local Index Formula in Conformal Geometry

Remark

The n-th degree component of ϕ 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 .

18 / 21

Page 46: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Cyclic Homology of C∞(M)o G

Notation

Let φ ∈ G . Then

〈φ〉 is the conjugation class of φ.

Gφ = ψ ∈ G ; ψ φ = φ ψ is the stabilizer of φ.

H•(Mφa ) is the Gφ-invariant cohomology of Mφ

a

Theorem (Brylinski-Nistor, Crainic)

Along the conjugation classes of G,

HP•(C∞(M) o G ) '⊕〈φ〉∈〈G〉

⊕0≤a≤n

H•(Mφa )G

φ.

19 / 21

Page 47: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Cyclic Homology of C∞(M)o G

Notation

Let φ ∈ G . Then

〈φ〉 is the conjugation class of φ.

Gφ = ψ ∈ G ; ψ φ = φ ψ is the stabilizer of φ.

H•(Mφa ) is the Gφ-invariant cohomology of Mφ

a

Theorem (Brylinski-Nistor, Crainic)

Along the conjugation classes of G,

HP•(C∞(M) o G ) '⊕〈φ〉∈〈G〉

⊕0≤a≤n

H•(Mφa )G

φ.

19 / 21

Page 48: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Construction of Geometric Cycles

Proposition (Brylinski-Getzler, Crainic, RP+HW)

1 To any Gφ-invariant closed diff. form ω on Mφa is naturally

associated an even cyclic cycle ηω on C∞(M) o G .

2 If ω = f 0df 1 ∧ · · · ∧ df m, then

ηω =∑σ∈Sm

ε(σ)f 0 ⊗ f σ(1) ⊗ · · · ⊗ f σ(m−1) ⊗ f σ(m)uφ,

where f j is a suitable smooth extension of f j to M.

20 / 21

Page 49: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Construction of Geometric Cycles

Proposition (Brylinski-Getzler, Crainic, RP+HW)

1 To any Gφ-invariant closed diff. form ω on Mφa is naturally

associated an even cyclic cycle ηω on C∞(M) o G .

2 If ω = f 0df 1 ∧ · · · ∧ df m, then

ηω =∑σ∈Sm

ε(σ)f 0 ⊗ f σ(1) ⊗ · · · ⊗ f σ(m−1) ⊗ f σ(m)uφ,

where f j is a suitable smooth extension of f j to M.

20 / 21

Page 50: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Conformal Invariants

Theorem (RP+HW)

Let ω be as in the previous slide. For any metric g ∈ C define

Ig (ω) = 〈Ch(/Dg )σg , ηω〉.

Then

1 Ig (ω) is an invariant of the conformal structure C depending

only on the class of ω in H•(Mφa )Gφ .

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

Ig (ω) =

∫Mφ

a

A(RTMφ) ∧ νφ

(RN

φ)∧ ω.

Remark

The above invariants are not of the same type as those consideredby S. Alexakis in his solution of the Deser-Swimmer conjecture.

21 / 21

Page 51: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Conformal Invariants

Theorem (RP+HW)

Let ω be as in the previous slide. For any metric g ∈ C define

Ig (ω) = 〈Ch(/Dg )σg , ηω〉.

Then

1 Ig (ω) is an invariant of the conformal structure C depending

only on the class of ω in H•(Mφa )Gφ .

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

Ig (ω) =

∫Mφ

a

A(RTMφ) ∧ νφ

(RN

φ)∧ ω.

Remark

The above invariants are not of the same type as those consideredby S. Alexakis in his solution of the Deser-Swimmer conjecture.

21 / 21

Page 52: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Conformal Invariants

Theorem (RP+HW)

Let ω be as in the previous slide. For any metric g ∈ C define

Ig (ω) = 〈Ch(/Dg )σg , ηω〉.

Then

1 Ig (ω) is an invariant of the conformal structure C depending

only on the class of ω in H•(Mφa )Gφ .

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

Ig (ω) =

∫Mφ

a

A(RTMφ) ∧ νφ

(RN

φ)∧ ω.

Remark

The above invariants are not of the same type as those consideredby S. Alexakis in his solution of the Deser-Swimmer conjecture.

21 / 21

Page 53: Noncommutative Geometry and Conformal Geometry (joint work … · RP+HW: Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index

Conformal Invariants

Theorem (RP+HW)

Let ω be as in the previous slide. For any metric g ∈ C define

Ig (ω) = 〈Ch(/Dg )σg , ηω〉.

Then

1 Ig (ω) is an invariant of the conformal structure C depending

only on the class of ω in H•(Mφa )Gφ .

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

Ig (ω) =

∫Mφ

a

A(RTMφ) ∧ νφ

(RN

φ)∧ ω.

Remark

The above invariants are not of the same type as those consideredby S. Alexakis in his solution of the Deser-Swimmer conjecture.

21 / 21