Noncommutative Geometry and ConformalGeometry

(joint work with Hang Wang)

Raphael Ponge

Seoul National University

May 1, 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Superceded by the 3 papers below.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.

RP+HW: Noncommutative geometry and conformalgeometry. I. Local index formula and comformal invariants.To be 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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Group Actions on Manifolds

Fact

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

Solution Provided by NCG

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

C∞c (M) o Γ =∑

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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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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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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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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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).

Then(

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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Diffeomorphism-Invariant Geometry

Setup

M smooth manifold.

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

Fact

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

P =

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

(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).

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

√k)

(i.e., the spectral

triples are intertwined by a unitary operator).

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Twisted Spectral Triples

Definition (Connes-Moscovici)

A twisted spectral triple (A,H,D) (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][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:

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

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.

Examples

1 Connes-Tretkoff’s twisted spectral triples over NC toriassociated to conformal weights.

2 Invertible doubles of conformal perturbations of spectraltriples.

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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).

Twistings of ordinary spectral triples by scalingautomorphisms (Moscovici).

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

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Connections over a Spectral Triple

Setup

(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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σ-Connections over a Twisted Spectral Triple

Setup/Notation

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

E finitely generated projective (right) module over A.

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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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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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 ). Then

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

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

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Index Map

Definition

The index of D∇E is

ind D∇E :=1

2

(ind D+

∇E − ind D−∇E).

where ind D±∇E is the usual Fredholm index of D±∇E .

Remark

ind D∇E is actually an integer when σ = id, and in all mainexamples when σ 6= id.

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 such

thatindD [E ] = ind D∇E ∀(E ,∇E).

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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

2p,

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.20 / 40

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 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.22 / 40

Conformal Connes-Chern Character

Theorem (RP+HW)

1 The spectral triple(

C∞(M) o G , L2g (M, /S), /Dg

)σg

depends

on the choice of g ∈ C only up to unitary equivalence andconformal deformations in the realm of twisted spectral triples.

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

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.

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Computation of Ch(C)

Proposition (Ferrand-Obata)

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

Fact

If g ∈ C is G -invariant, then(

C∞(M) o G , L2g (M, /S), /Dg

)σg

is an

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

Consequence

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

Connes-Chern character of(

C∞(M) o G , L2g (M, /S), /Dg

), where G

is a group of isometries.

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Computation of Ch(C)

Remark

1 When G is a group of isometries, the Connes-Chern character

of(

C∞(M) o G , L2g (M, /S), /Dg

)is computed by using JLO or

CM representatives and a differentiable version of the localequivariant index theorem (Azmi, Chern-Hu).

2 We produce a new approach to equivariant heat kernelasymptotics that proves the local equivariant index theoremand computes the JLO cocycle in the same shot.

3 This approach was subsequently used by Yong Wang inseveral papers (e.g., equivariant eta cochain).

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Local Index Formula in Conformal Geometry

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.

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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Local Index Formula in Conformal Geometry

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

(2m)!

∑a

(2π)−a2

∫Mφ

a

A(RTMφ)∧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].

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Local Index Formula in Conformal Geometry

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 .

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Cyclic Homology of C∞(M) o G

Theorem (Brylinski-Nistor, Crainic)

Along the conjugation classes of G,

HP0(C∞(M) o G ) '⊕〈φ〉

⊕a

HevGφ(Mφ

a ),

where Gφ is the centralizer of φ and HevGφ(Mφ

a ) is the Gφ-invariant

even de Rham cohomology of Mφa .

Lemma

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

φ,

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

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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

A(RTMφ) ∧ νφ

(RN

φ)∧ ω.

Remark

Branson-Ørsted proved that for ω = 1 the above integral wasindependent of the choice of any metric g ∈ C preserved by φ.

Remark

The above invariants are not the type of conformal invariantsappearing in the Deser-Swimmer conjecture solved by Alexakis.

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Vaffa-Witten Inequality

Theorem (Vafa-Witten ‘84)

Let (Mn, g) be a compact spin Riemannian manifold. Then, thereexists a constant C > 0 such that, for any Hermitian vector bundleE over M equipped with a Hermitian connection ∇E , we have

|λ1(/D∇E )| ≤ C ,

where λ1(/D∇E ) is the smallest eigenvalue of the coupled Diracoperator /D∇E .

Remark

The proof combines the max-min principle with some version ofPoincare duality in K -theory.

Remark

Moscovici extended Vafa-Witten inequality to ordinary spectraltriples satisfying a suitable form of Poincare duality.

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Poincare Duality

Definition (Connes, Moscovici)

Two ordinary spectral triples (A1,H,D) and (A2,H,D) are inPoincare duality when

[a1, a2] = [[D, a1], a2] = 0 for all aj ∈ Aj .

The following bilinear form (·, ·)D : K0(A1)× K0(A2)→ Z isnondegenerate,

(E1, E2)D := ind D∇E1⊗E2 , Ej ∈ K0(Aj).

Definition (RP+HW)

Two twisted spectral triples (A1,H,D)σ1 and (A2,H,D)σ2 are inPoincare duality when

[a1, a2] = [[D, a1]σ1 , a2]σ2 = 0 for all aj ∈ Aj .

The following bilinear form (·, ·)D,σ : K0(A1)×K0(A2)→ Z isnondegenerate,

(E1, E2)D,σ := ind D∇E1⊗E2 , Ej ∈ K0(Aj).

Example

A Dirac spectral triple(

C∞(M), L2(M, /S), /Dg

)is in Poincare

duality with itself.

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Poincare Duality. Conformal Deformations

Example (RP+HW)

Consider the following data:

Ordinary spectral triples (A1,H,D) and (A2,H,D) inPoincare duality.

Inner automorphisms σj(a) = k2j ak−2j , aj ∈ A, with kj ∈ A+

j .

k = k1k2 ∈ A1 ⊗A2.

Then (A1,H, kDk)σ1 and (A2,H, kDk)σ2 are in Poincare duality.

Remark

When k1 = 1, the ordinary spectral triple (A1,H, kDk) is inPoincare duality with the twisted spectral triple (A2,H, kDk)σ2 .

Remark

The above results extend to pseudo-inner twistings of Poincaredual pairs of ordinary spectral triples.

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Poincare Duality. Further Examples

Further Examples

Duals of duals for cocompact discrete subgroups of semisimpleLie groups satisfying the Baum-Connes conjecture (Connes).

Ordinary and twisted spectral triples over noncommutativetori (Connes, RP+HW).

Spectral triples describing the Standard Model of particlephysics (Chamseddine, Connes, Marcolli).

Quantum projective line (D’Andrea-Landi).

Quantum Podles spheres (Dabrowski-Sitarz, Wagner).

Conformal deformations of all the above (RP+HW).

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σ-Hermitian Structures

Setup

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

E is a finitely generated projective module with σ-translate Eσ.

Definition (RP+HW)

A σ-Hermitian structure on E is given by

1 A Hermitian metric (·, ·) : E × E → A.

2 A right A-module isomorphism s : Eσ → E such that

(s σE(ξ), η) = σ[(ξ, s σE(η))

]∀ξ, η ∈ E .

Example

If σ(a) = kak−1, then any Hermitian structure on E gives rise to aσ-Hermitian structure with

s(ξ) =(σE)−1

(ξ)k−1 ∀ξ ∈ Eσ.35 / 40

σ-Hermitian σ-Connections

Definition (RP+HW)

A σ-connection ∇E on E is σ-Hermitian when

(ξ, s∇Eη)− (s∇Eξ, η) = dσ(s σE(ξ), η) ∀ξ, η ∈ E .

Proposition (RP+HW)

Assume ∇E is σ-Hermitian σ-connection. Then

1 The operator sD∇E is selfadjoint (and Fredholm).

2 |sD∇E | = |D∇E |.3 ind D+

∇E = dim ker D+∇E − dim ker D−∇E .

Definition

1 The eigenvalues of sD∇E are called the s-eigenvalues of D∇E .

2 We order them into a sequence λj(D∇E ), j = 1, 2, . . ., so that

|λ1(D∇E )| ≤ |λ2(D∇E )| ≤ · · · .36 / 40

Vafa-Witten Inequality for Twisted Spectral Triples

Theorem (RP+HW)

Let (A1,H,D)σ1 be a twisted spectral triple such that

1 (A1,H,D)σ1 has a Poincare dual (A2,H,D)σ2 .

2 σ1(a) = kak−1 for some positive element k ∈ A1.

3 dim K0(A)⊗Q <∞.

Then there is a constant C > 0 such that, for any Hermitianfinitely generated projective module E over A1 and anyσ1-Hermitian connection ∇E on E , we have

|λ1(D∇E )| ≤ C‖k−1‖.

Remark

For ordinary spectral triples with ordinary Poincare duality theinequality was obtained by Moscovici (‘97).

For k = 1 this extends Moscovici’s inequality to ordinaryspectral triples with twisted Poincare duals.

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Conformal Deformations of Spectral Triples

Theorem (RP+HW)

Consider the following data:

Ordinary spectral triples (A1,H,D) and (A2,H,D) inPoincare duality.

Inner automorphisms σj(a) = k2j ak−2j , aj ∈ A, with kj ∈ A+

j .

k = k1k2 ∈ A1 ⊗A2.

Then, there is a constant C > 0 independent of k1 and k2, suchthat, for any Hermitian module E over A1 equipped with aσ1-Hermitian connection ∇E , we have

|λ1 ((k1Dk1)∇E )| ≤ C‖k−11 ‖‖k1k2‖2.

Remark

There is also a version of the above results for pseudo-innertwistings of Poincare dual pairs of ordinary spectral triples.

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Vafa-Witten Inequality in Conformal Geometry

Theorem (RP+HW)

Let (M, g) be an even dimensional compact Riemannian spinmanifold. Then there is a constant C > 0 such that, for anyconformal factor k ∈ C∞(M), k > 0, and any Hermitian vectorbundle E equipped with a Hermitian connection ∇E , we have∣∣λ1(Dg ,∇E )

∣∣ ≤ C‖k‖∞, g := k−2g ,

where ‖k‖∞ is the maximum value of k.

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Further Results

Remark (RP+HW)

There are also versions of Vafa-Witten inequality for:

1 Twisted spectral triples over noncommutative tori associatedto conformal weights (with control of the dependence on theconformal weight).

2 Conformal deformations by group elements of spectral triplesassociated to duals of cocompact discrete subgroups ofsemisimple Lie groups satisfying the Baum-Connes conjecture(with control of the dependence on the conformal factor).

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