Higher analytic indices and symbolic index pairing
Joint work with H. Moscovici.
Alexander Gorokhovsky
University of Colorado, Boulder
May 1, 2014
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
1 / 26
Connes-Moscovici construction
Connes-Moscovici localized analytic indices
Let D be an elliptic operator on a closed manifold M. Then D is Fredholm.
ind D = dim Ker D − dim Coker D ∈ Z.
Atiyah-Singer index theorem computes ind D in topological terms.
Connes-Moscovici: for [φ] ∈ H • (M) define
ind[φ] D ∈ C.
This contains the Fredholm index: ind1 D = ind D.
Connes-Moscovici also prove the corresponding index theorem.
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
2 / 26
Connes-Moscovici construction
Alexander-Spanier complex
C k (M) = C ∞ (M k+1 )
δ : C k (M) → C k+1 (M) is defined by
X
δφ(x0 , x1 , . . . , xk+1 ) =
(−1)i φ(x0 , . . . , xbi , . . . , xk+1 ).
The complex (C • (M), δ) is acyclic in positive degrees.
The locally zero subcomplex:
C0k (M) = {φ ∈ C k (M) | φ = 0 near diagonal of M k+1 }.
•
CAS
(M) = (C • (M)/C0• (M), δ)
•
H • (CAS
(M)) = H • (M).
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
3 / 26
Connes-Moscovici construction
Variants of the definition
Consider only antisymmetric cochains:
φ(xλ(0) , xλ(1) , . . . , xλ(k) ) = sgn(λ)φ(x0 , x1 , . . . , xk )
for all λ ∈ Sk+1 .
Consider (locally finite sums of) decomposable cochains.
For fi ∈ C ∞ (M)
(f0 ⊗ f1 ⊗ . . . fk )(x0 , x1 , . . . , xk ) = f0 (x0 )f1 (x1 ) . . . fk (xk ).
Finite sum of such cochains form a subcomplex of (C • , δ). Note:
X
δ(f0 ⊗ f1 ⊗ . . . fk ) =
(−1)i f0 ⊗ f1 . . . ⊗ fi−1 ⊗ 1 ⊗ fi ⊗ . . . fk
From now on C • (M) denotes the subcomplex of the antisymmetric
decomposable cochains. C0• (M) its locally zero subcomplex,
• (M) = C • (M)/C • (M).
CAS
0
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
4 / 26
Connes-Moscovici construction
Cochains on smoothing opertors
For φ = f0 ⊗ f1 ⊗ . . . fk and Ai ∈ Ψ−∞ (M) define
τφ (A0 , A1 , . . . , Ak ) = Tr A0 f0 A1 f1 . . . Ak fk ,
extend by linearity to antisymmetric decomposable φ. Viewed as a cyclic
cochain on Ψ−∞ (M), τφ satisfies
bτφ = τδφ , Bτφ = 0
and hence φ 7→ τφ defines a morphism of complexes
C • (M) → CC • (Ψ−∞ (M)).
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
5 / 26
Connes-Moscovici construction
Index and K -theory
Let Q be a parametrix for D, and define
S0 = 1 − QD, and S1 = 1 − DQ ∈ Ψ−∞ (M)
Then
S0 −(1 + S0 )Q
UD =
∈ M2 (Ψ(M))
D
S1
is an invertible operator with the inverse given by
S0 (1 + S0 )Q
−1
UD =
.
−D
S1
Using UD form the idempotent
2
1 0 −1
S0 S0 (1 + S0 )Q
PD = UD
U =
0 0 D
DS0
1 − S12
0 0
The index class PD −
∈ K0 (Ψ−∞ (M)) is well defined.
0 1
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
(1)
May 1, 2014
6 / 26
Connes-Moscovici construction
Definition of localized analytic indices
0 0
Notice that PD −
is the matrix of smoothing operators, and
0 1
0 0
ind D = Tr PD −
.
0 1
To define higher indices:
Choose a representative φ ∈ C • (M) of [φ] ∈ H k (M). δφ = 0 in a
neighborhood U of the diagonal in M k+2 .
Chose a parametrix Q for D with Schwartz kernel supported in a
sufficiently small (depending on U) neighborhood of the diagonal in
M 2.
Form PD and define
ind[φ] D = ck τφ (PD , PD , . . . , PD )
The result is well-defined.
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
7 / 26
Connes-Moscovici construction
bφ
Definition of Tr
Let R ∈ Ψ1 (M), R ≥ 0, invertible. For A ∈ Ψ(M) form
ζ(s) = Tr AR −s .
ζ(s) has a meromorphic continuation and near 0
ζ(s) =
1
b A + O(s).
Res A + Tr
s
bA
Res A is the noncommutative residue, does not depend on R. Tr
b
depends on R, for A trace class Tr(A)
= Tr A.
For φ = f0 ⊗ f1 ⊗ . . . fk and Ai ∈ Ψ(M) define
b φ (A0 , A1 , . . . , Ak ) = Tr(A
b 0 f0 A1 f1 . . . Ak fk )
Tr
b φ is a (discontinuous) cyclic cochain on Ψ(M).
Tr
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
8 / 26
Cyclic pairing interpretation
Definition of χφ
In the notations above set:
b φ − Tr
b δφ
χφ = (b + B) Tr
Lemma
If φ ∈ C0• (M) then χφ = 0
If Ai ∈ Ψ−∞ (M) for some i, then χφ (A0 , A1 , . . . , Ak+1 ) = 0.
Corollary
The map φ 7→ χφ defines a morphism of complexes
•
CAS
(M) → CC •+1 (S(M)),
where S(M) = Ψ(M)/Ψ−∞ (M) is the algebra of the complete symbols.
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
9 / 26
Cyclic pairing interpretation
Index pairing
To simplify notations, assume that D is an elliptic system (i.e. operates on
a sections of a trivial bundle). σ(D) ∈ S(M) is an invertible element,
therefore defines an element in K1 (S(M)), and Ch(σ(D)) ∈ HC• (S(M)).
Theorem
Let [φ] ∈ H • (M).
ind[φ] D = hχφ , Ch(σ(D))i
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
10 / 26
Cyclic pairing interpretation
Proof of the Theorem
Recall that
PD = UD
1 0 −1
U
0 0 D
From the (proof of) invariance of cyclic homology under inner
automorphisms,
Ch PD = Ch 1 + (b + B)TD ∈ CC• (Ψ(M))
for some explicit chain TD ∈ CC• (Ψ(M)). Complete symbol map induces
a morphism
σ : CC• (Ψ(M)) → CC• (S(M)),
and (b + B)σ(TD ) = 0.
Lemma
[σ(TD )] = Ch(σ(D)) ∈ HC• (S(M)).
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
11 / 26
Cyclic pairing interpretation
Proof of the Theorem (contd.)
Proof of the Theorem.
Assume k > 0.
b φ ((b + B)TD ) − Tr
b δφ (TD ) =
hχφ , Ch(σ(D))i = Tr
b φ (Ch PD − Ch 1) − Tr
b δφ (TD ) = ck τφ (PD , PD , . . . , PD ) − Tr
b δφ (TD )
Tr
If the parametrix Q used in construction of PD is supported in sufficiently
small neighborhood of the diagonal, depending on φ, then
b δφ (TD ) = 0.
Tr
Corollary
For any choice of the parametrix Q
ind[φ] D = ck τφ (PD , PD , . . . , PD ) − Trδφ (TD )
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
12 / 26
Cyclic pairing interpretation
The odd case
Assume that D is self-adjoint elliptic. One can define its index class in
K1 (Ψ−∞ (M)) as follows. Let ρ ∈ C ∞ (R) be such that ρ(λ) = 1 for large
λ, ρ(−λ) = −ρ(λ).
Then
U = e πiρ(D)
is invertible operator, representing ind D. U is homotopic to 1 in Ψ0 (M)
via the path t 7→ e πitρ(D) . Therefore
Ch U = (b + B)T
for certain chain T ∈ CC• (Ψ0 (M)).
Let φ be Alexander-Spanier cocycle. Define then
ind[φ] D = Trφ Ch U − Trδφ T
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
13 / 26
Cyclic pairing interpretation
The odd case (contd.)
Let P be the spectral projection on the positive spectrum for D.
ind[φ] D = hχφ , Ch(σ(P))i
Toeplitz picture: for f ∈ C ∞ (M) set
Tf = P f P
⊗ f1i . . . ⊗ fki


X X
indφ D = ck Tr 
sgn λTfλ(0) Tfλ(1) . . . Tfλ(k) 
Then if φ =
i
i f0
P
i
Alexander Gorokhovsky (CU Boulder)
λ∈Sk+1
Higher analytic indices
May 1, 2014
14 / 26
Computations
Cyclic cohomology of symbols
S 0 (M) = Ψ0 (M)/Ψ−∞ (M).
Theorem ( Wodzicki)
The principal symbol map
σpr : S 0 (M) → C ∞ (S ∗ M)
induces an isomorphism in periodic cyclic cohomology.
Theorem ( Wodzicki, Brylinski-Getzler )
There exists an isomorphism
HC • (S(M)) ∼
= H • (S ∗ M × S 1 )
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
15 / 26
Computations
Zakharevich map
There exists a canonical isomorphism
Z : H • (S ∗ M × S 1 ) → HC • (S(M))
(induced by a morphism of complexes). It has the following properties:
If φ = f0 ⊗ f1 . . . ⊗ fk is Alexander-Spanier cochain on M then
Z (π ∗ φ)(A0 , A1 , . . . Ak ) = Res A0 f0 A1 f1 . . . Ak fk =: Resφ (A0 , . . . , Ak ),
where π : S ∗ M × S 1 → M is the natural projection.
If φ is a cocycle
[χφ ] = Z (π ∗ φ ∪ [S 1 ])
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
16 / 26
Computations
The diagram
Theorem
The following diagram commutes:
i∗
• (S(M)) −
• (S 0 (M))
HCper
−−−→ HCper
x
x
σ∗

Z
 pr
I
H • (S ∗ M × S 1 ) −−−−→
H• (S ∗ M)
Here i : S 0 (M) ,→ S(M) is the inclusion map and I is given by
Z
I(α) =
α ∩ [Td]
S1
where [Td] ∈ H• (S ∗ M) is the dual of the pullback of the Todd class.
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
17 / 26
Computations
Index calculations
Applying this result to [S 1 ] ∈ H • (S ∗ M × S 1 ) we obtain
∗
•
χ1 = Z ([S 1 ]) = σpr
([Td]) ∈ HCper
(S 0 (M)).
Hence
ind D = hχ1 , Ch(σ(D))i = hCh(σpr (D)), [Td]i
More generally, in the even case
ind[φ] D = hχφ , Ch(σ(D))i = hπ ∗ φ ∪ Ch(σpr (D)), [Td]i.
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
18 / 26
Computations
The odd case
For φ =
i
i f0
P
⊗ f1i . . . ⊗ fki let DR(φ) =
P
i
f0i df1i . . . dfki . DR is a
morphism from Alexander-Spanier complex to de Rham complex.
Corollary
Assume φ =
i
i f0
P
⊗ f1i . . . ⊗ fki , δφ locally 0. Then

Tr 

X X
sgn λTf i Tf i
λ(0)
i
λ(1)
. . . Tf i  =
λ(k)
λ∈Sk+1
Z
DR(φ) ∧ Td ∧ π∗ (Ch(σpr (D)))
M
If k = dim M then δφ locally 0 for any φ (after modifying the notion of
locally 0). This extends result of Helton-Howe .
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
19 / 26
Computations
Vanishing results
Apply the diagram to 1 ∈ H • (S ∗ M × S 1 ). We obtain
•
[Res] = Z (1) = 0 ∈ HCper
(S 0 (M)).
Corollary (Wodzicki)
Let P ∈ Ψ0 be an idempotent. Then
Res P = 0
More generally, let φ =
cocycle. Then
P
i
i f0
⊗ f1i . . . ⊗ fki be an Alexander-Spanier
•
[Resφ ] = Z (π ∗ φ) = 0 ∈ HCper
(S 0 (M)).
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
20 / 26
Computations
Vanishing results (contd.)
Corollary
Let kPbe even and let P ∈ S 0 be an idempotent,
φ = i f0i ⊗ f1i . . . ⊗ fki – an Alexander-Spanier cocycle. Then
Resφ Ch(P) = const. ·
X
Res Pf0i Pf1i . . . Pfki = 0
i
Let k be odd, U ∈ S be a complete symbol of an elliptic operator
(i.e. invertible). Then
X
Resφ Ch(U) = const. ·
Res Uf0i U −1 f1i . . . U −1 fki = 0
i
Here const. is a nonzero constant.
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
21 / 26
The case of manifolds with boundary
Manifolds with boundary
(M, ∂M) – compact connected manifold with connected boundary.
Relative Alexander-Spanier complex:
•−1
•
•
CAS
(M, ∂M) = {(φ, ψ) | φ ∈ CAS
(M), ψ ∈ CAS
(∂M)}
δ(φ, ψ) = (δφ, φ|∂M − δψ).
Algebra of operators:
bΨ
operators of R. Melrose. b-trace:
b
Tr : b Ψ(M) → C.
It is not a trace, even on b Ψ−∞ . But coincides with the operator trace on
operators in b Ψ−∞ vanishing near the boundary.
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
22 / 26
The case of manifolds with boundary
Suspended operators
Let Y be a compact manifold without boundary. Ψsus(Y ) is the space of
order pseudodifferential operators on Y × R which are translation invariant
and have Schwartz kernel rapidly decreasing off-diagonal.
Let t be the variable on R, and τ the dual variable Given such an operator
b ) on Y :
A we can construct a family of operators A(τ
b ) = e −itτ ◦ A ◦ e itτ
A(τ
The linear functional Tr: for operators of sufficiently small order
Z∞
Tr(A) =
b )dτ.
A(τ
−∞
then extended.
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
23 / 26
The case of manifolds with boundary
Indicial family: homomorphism
I : b Ψ(M) → Ψsus (∂M).
If A ∈ b Ψ−∞ (M), B ∈ b Ψ(M)
b
Tr [A, B] = Tr I (A)[t, I (B)].
Define for φ = f0 ⊗ f1 . . . ⊗ fk – Alexander-Spanier cochain on M – then
b
Trφ (A0 , A1 , . . . Ak ) = bTr A0 f0 A1 f1 . . . Ak fk , Ai ∈ b Ψ
Similarly for ψ = g0 ⊗ g1 . . . ⊗ gk – Alexander-Spanier cochain on ∂M
Trψ (B0 , B1 , . . . Bk , Bk+1 ) = TrB0 g0 B1 g1 . . . Bk gk [Bk+1 , t], Bi ∈ Ψsus (∂M)
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
24 / 26
The case of manifolds with boundary
• (M)
For φ ∈ CAS
ρφ = (b + B) bTrφ − bTrδφ −Trφ|∂M .
• (∂M)
For ψ ∈ CAS
λψ = (b + B)Trψ − Trδψ
and
χ(φ,ψ) = ρφ + I ∗ λψ .
Then (φ, ψ) 7→ χ(φ,ψ) defines a morphism of complexes
•
CAS
(M, ∂M) → CC •+1 ( b Ψ(M)/ b Ψ−∞ (M)).
Alexander Gorokhovsky (CU Boulder)
Higher analytic indices
May 1, 2014
25 / 26
The case of manifolds with boundary
Index Theorem
Let D be an elliptic first order b operator, σ(D) ∈ b Ψ(M)/ b Ψ−∞ (M) its
d
+ D ∂ , xcomplete symbol. Assume that near the boundary D = dx
normal coordinate, D ∂ – first-order invertible self-adjoint operator on the
boundary.
Theorem
For (φ, ψ) – relative Alexander-Spanier cocycle
Z
hχ(φ,ψ) , Ch(σ(D))i =
M
DR(φ) ∧ Td(M) ∧ π∗ Ch(σpr (D))
Z
−
DR(ψ) ∧ Td(∂M) ∧ π∗ Ch(σpr (D ∂ )).
∂M
Alexander Gorokhovsky (CU Boulder)
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