Index and secondary index theory for flat bundles with duality structure

U. Bunke and X. Ma

Index bundle of twisted signature operators

X ⟶ B

locally trivial fibre bundle with closed oriented even-dimensional fibres

analytic index π_{!}^{sign} : K(X) → K(B)

[E] ↦ index(D_{E}^{sign})

index bundle of fibrewise signature operator twisted by E

finer structure

If E is flat with parallel ϵ-symmetric form Q, then the index bundle carries a natural flat structure and parallel ϵ′-symmetric form.

explanation

- index(D_{E}^{sign}) is represented by H^{∗}(X/B, \underline{E}), the fibrewise cohomology of the sheaf \underline{E} of parallel sections of E.

- Fibrewise Poincaré duality and Q give ϵ′-symmetric form.

Witt group of flat duality bundles

X topological space F locally constant sheaf of finite-dimensional ℝ-vector spaces F^{∗} := \underline{Hom}(F, \underline{ℝ}) ϵ ∈ ℤ_{2} q : F → F^{∗} - ϵ-symmetric duality structure, i.e. q is isomorphism and q^{∗} = (−1)^{ϵ}q

L̂_{ϵ}(X) - semigroup generated by pairs (F, q), operation - ⊕ ∼ - relation ”lagrangian reduction”

- (F, q) ∼ 0 if there exists locally constant lagrangian subsheaf L ⊂ F, i.e. L = L^{⊥_{q}}

Definition 0.1 L_{ϵ}(X) := L̂_{ϵ}(X)/ ∼ Properties:

- L_{ϵ}(X) is an abelian group

- L(X) := L_{1}(X) ⊕ L_{−1}(X) is a ℤ_{2}-graded ring, product - ⊗ - X ↦ L(X) is contravariant and homotopy invariant

L(X) is a modification of a group L^{Lott}(X) introduced by J.Lott, where relation ”lagrangian reduction” is replaced by ”hyperbolic is trivial”.

Computation of L(X)

If X is connected, then L(X) only depends on π_{1}(X).

Rep(π_{1}(X)) - set of isomorphism classes of irreducible finite-dimensional real representations of π_{1}(X)

For (V, ρ) ∈ Rep(π_{1}(X)) we define the group A_{ϵ}(V, ρ) as follows:

• A_{ϵ}(V, ρ) := 0 iff (V, ρ) does not admit any ϵ-symmetric invariant form.

• A_{ϵ}(V, ρ) := ℤ_{2} iff (V, ρ) does admit an ϵ-symmetric invariant form q and (V, ρ, q) ∼ (V, ρ, −q).

• A_{ϵ}(V, ρ) := ℤ iff (V, ρ) does admit an ϵ-symmetric invariant form q and (V, ρ, q) ≁ (V, ρ, −q).

Proposition 0.2 If X is connected, then

L_{ϵ}(X) ≅ ⨁_{(V,ρ)∈Rep(π_{1}(X))} A_{ϵ}(V, ρ)

Primary analytic index map

X ⟶ B

locally trivial fibre bundle with closed oriented even-dimensional fibres

analytic index π_{∗}^{L} : L(X) → L(B)

(F, q) ↦ (H^{∗}Rπ_{∗}(F), π_{∗}(q))

where

- H^{∗}Rπ_{∗}(F) = ⊕_{i}R^{i}π_{∗}(F)

- orientation of T X/B gives

R^{dim(X/B)}π_{∗}\underline{ℝ} →^{∼} \underline{ℝ}

- π_{∗}(q) defined by composition

Rπ_{∗}F →^{Rπ_{∗}(q)} Rπ_{∗}F^{∗}
≅  Rπ_{∗}\underline{Hom}(F, \underline{ℝ})
→  \underline{Hom}(Rπ_{∗}F, Rπ_{∗}\underline{ℝ})
→^{[dim(X/B)]} \underline{Hom}(Rπ_{∗}F, \underline{ℝ})
≅  Rπ_{∗}(F)^{∗}

Properties:

- π_{∗}^{L} is natural w.r.t. pull-back of fibre bundles

- π_{∗}^{L} is L(B)-linear

- π_{∗}^{L} is functorial w.r.t. iterated fibre bundles

The analytic index π_{∗}^{L^{Lott}} : L^{Lott}(X) → L^{Lott}(B) defined by J. Lott is not functorial.

Relation with K-theoretic analytic index

L(X) ⟶^{b} K(X)
π_{!}^{L} ↓  π_{!}^{sign} ↓
L(B) ⟶^{b} K(B)

Definition of b : L(X) → K(X) : (F, ∇^{F}, Q) - flat real vector bundle with parallel form induced by (F, q), choose metric structure J ∈ End(F), i.e. J^{∗} ∘ Q : F → F^{∗} is a scalar product, J^{2} = (−1)^{ϵ}id_{F}, and J^{∗} ∘ Q = (−1)^{ϵ}Q ∘ J.

b([F, q]) := [ℂ ⊗_{ℝ} F, \frac{1}{\sqrt{(−1)^{ϵ}}} ⊗ J]

Atiyah-Singer Index theorem for families

L(X) ⟶^{b} K(X) ⟶^{ch} H^{ev}(X, ℝ)
π_{∗}^{L} ↓ π_{!}^{sign} ↓ ↓ ∫_{X/B} L(X/B) ∩ …
L(B) ⟶^{b} K(B) ⟶^{ch} H^{ev}(B, ℝ)

Characteristic forms

Let X be a manifold.

p := ch ∘ b : L(X) ⟶ H^{ev}(X, ℝ)

Given (F, q, J) we represent p([F, q]) by the explicit form

p(F, q, J) := ch(∇^{F_{ℂ},J}) := tr_{s} exp(−R^{∇^{F_{ℂ},J}}/2πi) . where

- \frac{1}{\sqrt{(−1)^{ϵ}}}J - ℤ_{2}-grading of F_{ℂ} := ℂ ⊗_{ℝ} F,

- ∇^{F_{ℂ},J} - even part of ∇^{F_{ℂ}}

- R^{∇} - curvature of the connection ∇

Secondary L-groups

We extend L(X) →^{p} H^{ev}(X, ℝ) to exact sequence of homotopy invariant functors

0 → H^{odd}(X, ℝ) → L̄(X) → L(X) →^{p} H^{ev}(X, ℝ)

We define L̄(X) modifying a corresponding definition of L̄^{Lott}(X) introduced by J. Lott

Definition of secondary L-groups

ˆL̄(X) is the semigroup generated by 4-tuples (F, q, J, ρ), where ρ ∈ Ω^{odd}(X)/im(d) is such that dρ = p(F, q, J).

∼ - relation ”lagrangian reduction” (F, q, J, ρ) ∼ (0, 0, 0, ρ + c̃h(∇^{⊕}, ∇^{F_{ℂ},J})) if there is a locally constant lagrangian subsheaf L ⊂ F.

notation:

- L ⊂ F subbundle corresponding to L,

- F_{ℂ} = L_{ℂ} ⊕ J(L_{ℂ}), ∇^{⊕} - part of ∇^{F_{C},J} preserving the decomposition, -c̃h - transgression Chern form

Definition 0.3 L̄(X) := ˆ L̄(X)/ ∼

Properties:

- L̄(X) is a Z_{2}-graded abelian group

- L̄(X) is a graded L(X)-module

- X ↦ L̄(X) is homotopy invariant contravariant functor

Secondary analytic index

X ⟶ B Secondary analytic index πL̄ ; ! :

H^{odd}(X, ℝ) → L̄(X) → L(X)
∫_{X/B}(L(X/B) ∩ …) ↓ πL̄ ; ∗ ↓ π_{∗}^{L} ↓
H^{odd}(B, ℝ) → L̄(B) → L(B)

Construction of πL̄ ; ∗:

Same definition as πL̄^{Lott} ; ∗

choose Riemannian metric on T X/B and horizontal subbundle T^{H}X

πL̄ ; ∗([F, q, J, ρ]) := [H^{∗}Rπ_{∗}F, π_{∗}(q), J′, ρ′]

where

- J′ induced by L^{2}-metric on H^{∗}Rπ_{∗}F given by fibrewise Hodge theory

ρ′ := ∫_{X/B} L(∇^{T X/B}) ∩ ρ − η

- ∇^{X/B} induced connection on T X/B

- η - eta form of the family of signature operators twisted by F and associated Bismut super connection

Properties of the secondary analytic index

- πL̄ ; ∗ is well-defined

Proof much more complicated than for π_{∗}^{L^{Lott}}

- πL̄ ; ∗ is natural with respect to pull-back

-πL̄ ; ∗ is functorial with respect to iterated fibre bundles

Not true for π_{∗}^{Lott}. Proof complicated

- πL̄ ; ∗ is L(B)-linear

Relation with K-theory

J. Lott constructed

\begin{aligned}H^{odd}(X, ℝ) & → & K^{ℝ/ℤ}(X) & → & K(X) & → & H^{ev}(X) \\ ↓ ∫_{X/Z}(L(X/Z) ∩ …) & & π_{!}^{sign,ℝ/ℤ} ↓ & & π_{!}^{sign} ↓ & & ↓ ∫_{X/Z}(L(X/Z) ∩ …) \\ H^{odd}(B, ℝ) & → & K^{ℝ/ℤ}(B) & → & K(B) & → & H^{ev}(B) \end{aligned} .

There is a natural map

L̄(X) → K^{ℝ/ℤ}(X)

such that the following is a refinement of the diagram above

\begin{aligned}H^{odd}(X, ℝ) & → & L̄(X) & → & L(X) & → & H^{ev}(X) \\ ↓ ∫_{X/Z}(L(X/Z) ∩ …) & & πL̄ ; ∗ ↓ & & π_{∗}^{L} ↓ & & ↓ ∫_{X/Z}(L(X/Z) ∩ …) \\ H^{odd}(B, ℝ) & → & L̄(B) & → & L(B) & → & H^{ev}(B) \end{aligned} .

The η-invariant

Let X be closed, oriented, and odd-dimensional. The eta-invariant induces a homomorphism η̄ : L̄(X) → ℝ/Z. Definition:

Same definition as J. Lott’s η̄^{Lott} : L̄^{Lott}(X) → ℝ/ℤ

choose Riemannian metric on X

η([F, q, J, ρ]) := η(D_{F}) − ∫_{X} L(∇^{T X}) ∩ ρ

η(D_{F}) - eta invariant of signature operator twisted by F

η̄ is well-defined complicated proof

There exists a natural extension

0 → ℤ → L̄^{ex}(X) → L̄(X) → 0

such that there is a lift

η : L̄^{ex}(X) → ℝ
Properties of L̄^{ex}(X)

- X ↦ L̄^{ex}(X) is functorial under degree-one homotopy equivalences

- we have a natural extension πL̄^{ex} ; ∗ of the index map such that

L̄^{ex}(X) → L̄(X)
πL̄^{ex} ; ∗ ↓  πL̄ ; ∗ ↓
L̄^{ex}(B) → L̄(B)