Spectral Triples and Index Theory
1. Quantized Calculus
Raphaël Ponge
(University of Tokyo)
!! Noncommutative Geometry !!
Philosophy and Goal of NCG:
• Do differential geometry by trading spaces
for algebras.
• Translate the classical tools of differential
geometry in the language of operator algebras.
Noncommutative Manifolds:
• In NCG the analogue of a manifold is a spectral triple,
(A, H, D),
where:
- A is an algebra represented by bounded operators in the Hilbert space H.
- D is an unbounded selfadjoint operator on
H such that
D−1 is a compact operator,
[D, a] is a bounded for all a ∈ A.
Example (Dirac Spectral Triple):
• Let M be a compact spin Riemannian manifold.
• The following is a spectral triple,
!
"
∞
2
C (M ), L (M, S
/ ),D
/ ,
where D
/ is the Dirac operator acting the sections of the spinor bundle S
/.
!! Quantized Calculus (Connes) !!
Aim of the Quantized Calculus:
• Define an infinitesimal calculus for spectral
triples.
• Let (A, H, D) be a spectral triple.
Complex and Real Variables:
• From Quantum Mechanics:
Complex variable ←→ Operator on H
Real variable ←→ Selfadjoint operator
Infinitesimal Operators:
• Intituitely, an infinitesimal operator should be
such that
!T ! ≤ !
∀! > 0.
This gives T = 0!
• We relax the above condition to
For all ! > 0 there is a finite dimensional
subspace E ⊂ H such that "T|E ⊥ " ≤ !.
This is equivalent to T being a compact operator.
Characteristic Values:
• Let T ∈ L(H). The (k + 1)-th characteristic
value of T is
µk (T ) := inf{"T|E ⊥ "; dim E = k},
= inf{!T − R!; rk R ≤ k}.
• If T is compact, then
µk (T ) := (k + 1)-th eigenvalue of |T | :=
√
T ∗T .
Proposition. TFAE:
(i) T is a compact operator.
(ii) T is the norm-limit of finite-rank operators.
(iii) µk (T ) → 0 as k → ∞.
(iv) For all ! > 0 there is a finite dimensional
subspace E ⊂ H such that "T|E ⊥ " ≤ !.
Infinitesimals of order α > 0:
• T is an infinitesimal of order α when
µk (T ) = O(k−α).
• If T1 and T2 are infinitesimals of respective
orders α1 and α2, then
T1T2 is an infinitesimal of order α1 + α2,
T1 + T2 is an infinitesimal of order min(α1, α2).
Quantized Differential:
• Let (M n, g) be a Riemannian manifold.
• The volume element,
vg (x) :=
!
det g(x)dx1 ∧ · · · dxn
is an infinitesimal of order 1.
1.
• dx1, ..., dxn are infinitesimals of order n
• Any differential df =
1.
mal of order n
! ∂f
i is an infinitesidx
i
∂x
Definition:
The spectral triple (A, H, D) is p+-summable
when
!
µk (D−1) = O k
− 1p
"
.
That is, D−1 is an infinitesimal of order 1p .
Example:
• Let M be a compact spin Riemannian manifold of dimension n.
• The Dirac spectral triple
!
C ∞(M ), L2(M, S
/ ),D
/
is n+-summable.
"
• Set F := Sign D = D|D|−1.
Proposition:
If (A, H, D) is p+-summable, then
!
µk ([F, a]) = O k
− 1p
"
∀a ∈ A,
i.e.,[F, a] is an infinitesimal operator of order 1p .
• The quantized differential is
da := [F, a]
∀a ∈ A.
Noncommutative Integral:
• We seek for a linear functional such that:
(i) It is defined on infinitesimals of order 1.
(ii) It vanishes on infinitesimals of orders > 1.
(iii) It takes on non-negative values on selfadjoint operators with non-negative spectra.
(iv) It vanishes on differentials,
d(a0da1 · · · dap) = [F, a0da1 · · · dap].
• In order to have (iv) we require:
(iv)’ The NC integral is a trace.
• The usual trace,
Trace(T ) :=
!
!T ξk , ξk ",
(ξk ) orth. basis,
is a positive linear trace defined on trace-class
operators,
T is trace-class ⇐⇒ Trace |T | =
!
µk (T ) < ∞.
• An infinitesimal of order 1 need not be trace-class.
• The usual trace need not vanish on finiterank operators.
• The right NC analogue of the classical integral is provided by the Dixmier trace.
!
The Dixmier Trace −:
• If T is a (positive) infinitesimal operator of
order 1, then
µk (T ) = O(k−1),
σN (T ) :=
!
µk (T ) = O(log N ).
k<N
• The Dixmier trace is a clever way to extract
a"limit-point from the bounded sequence
!
σN (T )
log N
in such a way to get a linear trace.
• In particular,
"
!
1
µk (T ) → L ⇒ − T = L.
log N k<N
Example:
• Take H = L2(M ), w/ (M n, g) compact Riem. mfld.
• By the Weyl’s asymptotics,
! "2
k
c
λk (∆g ) ∼
n
c :=
,
(4π)
#
−n
2
Γ n
2+1
$ . Volg M.
where λk (∆g ) is the (k+1)-th (non-zero) eigenvalue of ∆g counted with multiplicity.
• Therefore,
µk
!
−n
∆g 2
!
k<N
!
−
−n
∆ 2
g
"
µk
= λk (∆g )
"
−n
∆g 2
=c=
#
c
∼ ,
k
∼ c. log N,
(4π)
"
−n
2
−n
2
Γ n
2+1
# . Volg M.
Summary:
Classical
Quantum
Complex variable
Operator on H
Real variable
Selfadjoint operator
Infinitesimal variable
Compact operator
Infinitesimal of order
α>0
Compact operator T s.t.
µk (T ) = O(k−α)
Differential
! ∂f
df = ∂xi dxi
Quantized differential
da = [F, a], F := Sign D
Integral
"
Dixmier trace −
!! Pseudodifferential Operators !!
• Let (M n, g) be a compact Riemann. manifold.
• The pseudodifferential operators (aka ΨDOs)
are continuous linear operators
P : C ∞(M ) −→ C ∞(M ).
• The ΨDOs form a filtrated algebra,
Ψ∗(M ) =
!
Ψm(M ),
m∈Z
P1P2 ∈ Ψm1+m2 (M )
∀Pj ∈ Ψmj (M ).
• Any P ∈ Ψm(M ) with m ≤ 0 extends to a
bounded operator,
P : L2(M ) −→ L2(M ).
This operator is compact if m < 0.
Examples:
• If P is a differential operator of order m, then
P ∈ Ψm(M ).
• If in addition P is elliptic, then
P −1 ∈ Ψ−m(M ).
• For all k ∈ Z,
k
2
(∆g )
∈ Ψk (M ).
!! The Noncommutative Residue !!
Theorem (Guillemin, Wodzicki):
1. The following formula defines a linear functional on ΨDOs:
Res P := Resz=0 Trace[P (∆g )
− 2z
]
∀P ∈ Ψ∗(M ).
2. We have
Res P =
!
M
cP (x)vg (x),
where cP (x) can be explicitly computed locally
in terms of the symbol of P .
Proposition (Guillemin, Wodzicki):
1. The noncommutative residue vanishes on
Ψ−(n+1)(M ) and on all differential operators.
2. The noncommutative residue is a trace,
Res[P1P2] = Res[P2P1]
∀Pj ∈ Ψ∗(M ).
Theorem (Wodzicki):
If M is connected, then any trace on Ψ∗(M )
is a constant multiple of the noncommutative
residue.
Example:
We have
−n
Res(∆g ) 2
#
−n
(4π) 2
=2 ! n "
vg (x),
M
Γ 2
n
=2
(4π)− 2
! " . Volg M.
Γ n
2
!! Quantized Calculus on a Manifold !!
Quantized Calculus:
• Let H be a Hilbert space (possibly coming
from a spectral triple (A, H, D)).
Classical
Quantum
Complex variable
Operator on H
Real variable
Selfadjoint operator
Infinitesimal variable
Compact operator
Infinitesimal of order
α>0
Compact operator T s.t.
µk (T ) = O(k−α)
Differential
! ∂f
i
df = ∂x
dx
i
Quantized differential
da = [F, a], F := Sign D
Integral
"
Dixmier trace −
!! Quantized Calculus on a Manifold !!
• Let (M n, g) be a compact Riemannian manifold.
• Take H = L2(M ).
Theorem (Connes):
Let P be a ΨDO of order −m, m > 0.
1. P is an infinitesimal of order m
n.
2. If P has order −n, then
!
1
− P = Res P.
n
Consequence:
• We extend the Dixmier trace to all ΨDOs by
letting
!
1
− P := Res P
n
∀P ∈ Ψ∗(M ).
• That is, we can integrate any ΨDO even if
it is not an infinitesimal of order ≥ 1.