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Noncommutative Geometry and ConformalGeometry
(joint work with Hang Wang)
Raphael Ponge
Seoul National University
May 1, 2014
1 / 40
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Superceded by the 3 papers below.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
3 / 40
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 .
4 / 40
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.
5 / 40
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
6 / 40
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.
21 / 40
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.
23 / 40
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.
24 / 40
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
.
26 / 40
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].
27 / 40
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 .
28 / 40
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.
29 / 40
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.
30 / 40
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.
31 / 40
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.
33 / 40
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.
38 / 40
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.
39 / 40
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).
40 / 40