Noncommutative Geometry and Conformal
Geometry
(joint work with Hang Wang)
Raphaël 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Superceded by the 3 papers below.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-Chern
character in the setting of twisted spectral triples.
arXiv:1310.6131.
RP+HW: Noncommutative geometry and conformal
geometry. I. Local index formula and comformal invariants.
To be posted on arXiv soon.
RP+HW: Noncommutative geometry and conformal
geometry. II. Connes-Chern character and the local
equivariant index theorem. To be posted on arXiv soon.
RP+HW: Noncommutative geometry and conformal
geometry. III. Poincaré 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, then
M/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,
nX
o
Cc∞ (M) o Γ =
fφ uφ ; fφ ∈ Cc∞ (M) ,
uφ∗ = uφ−1 = uφ−1 ,
uφ f = (f ◦ φ−1 )uφ .
Proposition (Green)
If G acts freely and properly, then Cc∞ (M/G ) is Morita equivalent
to Cc∞ (M) o G .
4 / 40
The Noncommutative Torus
Example
Given θ ∈ R, let Z act on S 1 by
k.z := e 2ikπθ z
∀z ∈ S 1 ∀k ∈ Z.
Remark
If θ 6∈ Q, then the orbits of the action are dense in S 1 .
The crossed-product algebra Aθ := C ∞ (S 1 ) oθ Z is generated by
two operators U and V such that
U ∗ = U −1 ,
V ∗ = V −1 ,
VU = e 2iπθ UV .
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 A
E = eAq , e ∈ Mq (A), e 2 = e
indD
/ ∇E
ind D∇E
de Rham Homology/Cohomology
Cyclic Cohomology/Homology
Atiyah-Singer Index Formula
R
indD
/ ∇E = Â(R M ) ∧ Ch(F E )
Connes-Chern Character Ch(D)
ind D∇E = hCh(D), Ch(E)i
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
2
3
A Z2 -graded Hilbert space H = H+ ⊕ H− .
An involutive algebra A represented in H.
A selfadjoint unbounded operator D on H such that
1
2
3
D maps H± to H∓ .
(D ± i)−1 is compact.
[D, a] is bounded for all a ∈ A.
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Dirac Spectral Triple
Example
(M n , g ) compact Riemannian spin manifold (n even) with
spinor bundle S
/ =S
/+ ⊕ S
/ −.
D
/ g : 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/ ) ,D
/ g 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,• TC∗ M) and D = ∂ + ∂ (when M is a complex
manifold).
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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 manifold
structure.
Theorem (Connes-Moscovici ‘95)
There is a spectral triple Cc∞ (P) o G , L2 (P, Λ• T ∗ P) , D , where
P = gij dx i ⊗ dx j ; (gij ) > 0 is the metric bundle of M.
D is a “mixed-degree” signature operator, so that
D|D| = dH + dH∗ + dV dV∗ − dV∗ dV .
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Conformal Change of Metric
Example
(M n , g ) compact Riemannian spin manifold (n even) with
spinor bundle S
/ =S
/+ ⊕ S
/ −.
D
/ g : 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,
ĝ = k −2 g ,
k ∈ C ∞ (M), k > 0.
Then the Dirac spectral triple C ∞ (M), L2ĝ (M, S/ ) ,D
/ ĝ is unitarily
√
√ equivalent to C ∞ (M), L2g (M, S/ ) , k D
/ g 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
2
3
A Z2 -graded Hilbert space H = H+ ⊕ H− .
An involutive algebra A represented in H together with an
automorphism σ : A → A such that σ(a)∗ = σ −1 (a∗ ) for all
a ∈ A.
A selfadjoint unbounded operator D on H such that
1
2
3
D maps H± to H∓ .
(D ± i)−1 is compact.
[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 automorphism
σ(a) = k 2 ak −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).
+
ω
0
A positive even operator ω =
∈ L(H) so that
0 ω−
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 a
twisted spectral triple.
Examples
1
Connes-Tretkoff’s twisted spectral triples over NC tori
associated to conformal weights.
2
Invertible doubles of conformal perturbations of spectral
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Further Examples
Further Examples
Conformal Dirac spectral triple (Connes-Moscovici).
Twisted spectral triples over NC tori associated to conformal
weights (Connes-Tretkoff).
Twistings of ordinary spectral triples by scaling
automorphisms (Moscovici).
Twisted spectral triples associated to quantum statistical
systems (e.g., Connes-Bost systems, supersymmetric Riemann
gas) (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) := Span{adb; 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 + ∇E ξ 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) = Span{adσ b; a, b ∈ A} ⊂ L(H),
where dσ b := [D, b]σ = Db − σ(b)D.
Definition
A σ-translate of E is a finitely generated projective module E σ
together with a linear isomorphism σ E : E → E σ such that
σ E (ξa) = σ E (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 a
linear map ∇E : E → E σ ⊗A Ω1D,σ (A) such that
∇E (ξa) = σ E (ξ) ⊗ dσ a + ∇E ξ a
∀a ∈ A ∀ξ ∈ E.
Example
If E = eAq with e = e 2 ∈ Mq (A), then
1
E σ = 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 → E σ ⊗A H,
of the form,
D∇E =
0
+
D∇
E
−
D∇
E
0
,
±
± → E σ ⊗ H∓ are Fredholm operators.
where D∇
E : E ⊗ dom D
Example
For a Dirac spectral triple (C ∞ (M), L2g (M), S/ ,D
/ g ) 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
+
−
ind D∇
E − ind D∇E .
2
±
±
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 main
examples when σ 6= id.
Proposition (Connes-Moscovici, RP+HW)
1
2
ind D∇E depends only on the K -theory class of E.
There is a unique additive map indD,σ : K0 (A) → 12 Z such
that
indD [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 < ∞ for
some p ≥ 1. Assume further that D is invertible and E = eAq ,
e 2 = e ∈ Mq (A). Then, for all k ≥ 21 p,
ind D∇E =
o
1 n
Tr γ(D −1 [D, e]σ )2k ,
2
where γ = idH+ − idH− .
Theorem (Connes-Moscovici, RP+HW)
Assume (A, H, D)σ is p-summable. Then there is an even periodic
cyclic class Ch(D)σ ∈ HP0 (A) , called Connes-Chern character,
such that
ind D∇E = hCh(D)σ , Ch(E)i
∀(E, ∇E ),
where Ch(E) is the Chern character in periodic cyclic homology.
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Conformal Dirac Spectral Triple
Setup
1
M n 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 = kφ−2 g with kφ ∈ C ∞ (M), kφ > 0.
4
C ∞ (M) o G is the crossed-product algebra, i.e.,
nX
o
C ∞ (M) o G =
fφ uφ ; fφ ∈ Cc∞ (M) ,
uφ∗ = uφ−1 = uφ−1 ,
uφ f = (f ◦ φ−1 )uφ .
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Conformal Dirac Spectral Triple
Lemma (Connes-Moscovici)
For φ ∈ G define Uφ : L2g (M, S/ ) → L2g (M, S/ ) by
−n
Uφ ξ = kφ 2 φ∗ ξ
∀ξ ∈ L2g (M, S/ ).
Then Uφ is a unitary operator, and
p
p
Uφ D
/ g Uφ∗ = kφ D
/ g kφ .
Proposition (Connes-Moscovici)
The datum of any metric g ∈
C defines a twisted spectral triple
∞
2
given by
C (M) o G , Lg (M, S/ ),D
/g
σg
1
The Dirac operator D
/ g associated to g .
2
The representation fuφ → fUφ of C ∞ (M) o G in L2g (M, S/ ).
3
The automorphism σg (fuφ ) := kφ−1 fuφ .
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Conformal Connes-Chern Character
Theorem (RP+HW)
1
The spectral triple C ∞ (M) o G , L2g (M, S/ ),D
/g
σg
depends
on the choice of g ∈ C only up to unitary equivalence and
conformal deformations in the realm of twisted spectral triples.
2
The Connes-Chern character Ch(/
D g )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) o G )
is the Connes-Chern character Ch(/
D g )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 a
G -invariant metric.
Fact
If g ∈ C is G -invariant, then C ∞ (M) o G , L2g (M, S/ ),D
/g
σg
is an
ordinary spectral triple (i.e., σg = 1).
Consequence
When C is non-flat, we are reduced to the compuation of
the
∞
2
Connes-Chern character of C (M) o G , Lg (M, S/ ),D
/ g , 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/ ),D
/ g is computed by using JLO or
CM representatives and a differentiable version of the local
equivariant index theorem (Azmi, Chern-Hu).
2
We produce a new approach to equivariant heat kernel
asymptotics that proves the local equivariant index theorem
and computes the JLO cocycle in the same shot.
3
This approach was subsequently used by Yong Wang in
several papers (e.g., equivariant eta cochain).
25 / 40
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
M φ is the fixed-point set of φ; this is a disconnected sums of
submanifolds,
F
M φ = Maφ , dim Maφ = a.
N φ = (TM φ )⊥ is the normal bundle (vector bundle over M φ ).
Over M φ , with respect to TM|M φ = TM φ ⊕ N φ , there are
decompositions,
!
1
0
φ
φ
φ0 =
,
∇TM = ∇TM ⊕ ∇N .
0
0 φ|N φ
26 / 40
Local Index Formula in Conformal Geometry
Theorem (RP + HW)
For any G -invariant metric g ∈ C, the conformal Connes-Chern
character Ch(/
D g )σg is represented by the periodic cyclic cocycle
ϕ = (ϕ2m ) given by
ϕ2m (f 0 Uφ0 , · · · , f 2m Uφ2m ) =
n
Z
φ
a
(−i) 2 X
φ
(2π)− 2
Â(R TM )∧νφ R N ∧f 0 d f˜1 ∧· · ·∧d f˜2m ,
(2m)! a
Maφ
−1
where φ := φ0 ◦ · · · ◦ φ2m , and f˜j := f j ◦ φ−1
0 ◦ · · · ◦ φj−1 , and
#
φ
R TM /2
,
 R
:= det
sinh R TM φ /2
φ
φ
N
− 12
0
−R N
νφ R
:= det
1 − φ|N φ e
.
TM φ
"
1
2
27 / 40
Local Index Formula in Conformal Geometry
Remark
The n-th degree component is given by
R 0 1
˜
˜n if φ0 ◦ · · · ◦ φn = 1,
n
0
M f df ∧ · · · ∧ df
ϕn (f Uφ0 , · · · , f Uφn ) =
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 ,
MM
HP0 (C ∞ (M) o G ) '
HGevφ (Maφ ),
hφi
a
where G φ is the centralizer of φ and HGevφ (Maφ ) is the G φ -invariant
even de Rham cohomology of Maφ .
Lemma
Any closed form ω ∈ Ω•G φ (Maφ ) defines a cyclic cycle ηω on
C ∞ (M) o G via the transformation,
f 0 df 1 ∧ · · · ∧ df k −→ Uφ f˜0 ⊗ f˜1 ⊗ · · · ⊗ f˜k ,
φ
f j ∈ C ∞ (Maφ )G ,
where f˜j is a G φ -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
2
For any closed even form ω ∈ Ωev
(Maφ ), the pairing
Gφ
hCh(C), ηω i is a conformal invariant.
For any G -invariant metric g ∈ C, we have
Z
φ
φ
hCh(C), ηω i =
Â(R TM ) ∧ νφ R N ∧ ω.
Maφ
Remark
Branson-Ørsted proved that for ω = 1 the above integral was
independent of the choice of any metric g ∈ C preserved by φ.
Remark
The above invariants are not the type of conformal invariants
appearing in the Deser-Swimmer conjecture solved by Alexakis.
30 / 40
Vaffa-Witten Inequality
Theorem (Vafa-Witten ‘84)
Let (M n , g ) be a compact spin Riemannian manifold. Then, there
exists a constant C > 0 such that, for any Hermitian vector bundle
E 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 Dirac
operator D
/ ∇E .
Remark
The proof combines the max-min principle with some version of
Poincaré duality in K -theory.
Remark
Moscovici extended Vafa-Witten inequality to ordinary spectral
triples satisfying a suitable form of Poincaré duality.
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Poincaré Duality
Definition (Connes, Moscovici)
Two ordinary spectral triples (A1 , H, D) and (A2 , H, D) are in
Poincaré duality when
[a1 , a2 ] = [[D, a1 ], a2 ] = 0 for all aj ∈ Aj .
The following bilinear form (·, ·)D : K0 (A1 ) × K0 (A2 ) → Z is
nondegenerate,
(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 in
Poincaré 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 is
nondegenerate,
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Poincaré Duality. Conformal Deformations
Example (RP+HW)
Consider the following data:
Ordinary spectral triples (A1 , H, D) and (A2 , H, D) in
Poincaré duality.
Inner automorphisms σj (a) = kj2 akj−2 , aj ∈ A, with kj ∈ A+
j .
k = k1 k2 ∈ A1 ⊗ A2 .
Then (A1 , H, kDk)σ1 and (A2 , H, kDk)σ2 are in Poincaré duality.
Remark
When k1 = 1, the ordinary spectral triple (A1 , H, kDk) is in
Poincaré duality with the twisted spectral triple (A2 , H, kDk)σ2 .
Remark
The above results extend to pseudo-inner twistings of Poincaré
dual pairs of ordinary spectral triples.
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Poincaré Duality. Further Examples
Further Examples
Duals of duals for cocompact discrete subgroups of semisimple
Lie groups satisfying the Baum-Connes conjecture (Connes).
Ordinary and twisted spectral triples over noncommutative
tori (Connes, RP+HW).
Spectral triples describing the Standard Model of particle
physics (Chamseddine, Connes, Marcolli).
Quantum projective line (D’Andrea-Landi).
Quantum Podleś spheres (Da̧browski-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 E σ .
Definition (RP+HW)
A σ-Hermitian structure on E is given by
1
2
A Hermitian metric (·, ·) : E × E → A.
A right A-module isomorphism s : E σ → E such that
(s ◦ σ E (ξ), η) = σ (ξ, s ◦ σ E (η))
∀ξ, η ∈ E.
Example
If σ(a) = kak −1 , then any Hermitian structure on E gives rise to a
σ-Hermitian structure with
−1
s(ξ) = σ E (ξ)k −1
∀ξ ∈ E σ .
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σ-Hermitian σ-Connections
Definition (RP+HW)
A σ-connection ∇E on E is σ-Hermitian when
(ξ, s∇E η) − (s∇E ξ, η) = dσ (s ◦ σ E (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 Poincaré 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 Hermitian
finitely generated projective module E over A1 and any
σ1 -Hermitian connection ∇E on E, we have
|λ1 (D∇E )| ≤ C kk −1 k.
Remark
For ordinary spectral triples with ordinary Poincaré duality the
inequality was obtained by Moscovici (‘97).
For k = 1 this extends Moscovici’s inequality to ordinary
spectral triples with twisted Poincaré 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) in
Poincaré duality.
Inner automorphisms σj (a) = kj2 akj−2 , aj ∈ A, with kj ∈ A+
j .
k = k1 k2 ∈ A1 ⊗ A2 .
Then, there is a constant C > 0 independent of k1 and k2 , such
that, for any Hermitian module E over A1 equipped with a
σ1 -Hermitian connection ∇E , we have
|λ1 ((k1 Dk1 )∇E )| ≤ C kk1−1 kkk1 k2 k2 .
Remark
There is also a version of the above results for pseudo-inner
twistings of Poincaré 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 spin
manifold. Then there is a constant C > 0 such that, for any
conformal factor k ∈ C ∞ (M), k > 0, and any Hermitian vector
bundle E equipped with a Hermitian connection ∇E , we have
λ1 (Dĝ ,∇E ) ≤ C kkk∞ ,
ĝ := k −2 g ,
where kkk∞ 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 associated
to conformal weights (with control of the dependence on the
conformal weight).
2
Conformal deformations by group elements of spectral triples
associated to duals of cocompact discrete subgroups of
semisimple Lie groups satisfying the Baum-Connes conjecture
(with control of the dependence on the conformal factor).
40 / 40