Real secondary index theory

Ulrich Bunke Göttingen, Jan 2003 Deligne cohomology Definition : B smooth manifold, A_{B}^{∗} sheaf of smooth forms

R_{B}^{k} : 0 → ℤ_{B} → A_{B}^{0} →^{d} A_{B}^{1} →^{d} ⋯ →^{d} A_{B}^{k} → 0

Definition : smooth Deligne cohomology H_{Del}^{k}(B) := H^{k−1}(B, R_{B}^{k−1}) exact sequences

0 → H^{k−1}(B, ℝ/ℤ) → H_{Del}^{k}(B) →^{R} A_{B}^{k}(B)

R - curvature

0 → H^{k−1}(B, ℝ)/H^{k−1}(B, ℤ) → H_{Del}^{k}(B) →^{v} H^{k}(B, ℤ) → 0

v - characteristic class

H^{k−1}(B, ℝ/ℤ) → H_{Del}^{k}(B) →^{v} H^{k}(B, ℤ)

equals Bockstein for ℤ ⊂ ℝ Lift of Chern classes V → B complex vector bundle c_{2m}(V) ∈ H^{2m}(B, ℤ) - Chern class consider connection ∇^{V} c_{2m}(∇^{V}) ∈ A_{B}^{2m}(B) - Chern-Weyl representative of c_{2m}(V)

(c_{2m}(V), c_{2m}(∇^{V})) ∈ H^{2m}(B, ℤ) ×_{H_{dR}^{2m}(B)} A_{B}^{2m}(B, d = 0)

Cheeger-Simons : There is a natural class ĉ_{2m}(V, ∇^{V}) ∈ H_{Del}^{2m}(B) such that

R(ĉ_{2m}(V, ∇^{V})) = c_{2m}(∇^{V})

and

v(ĉ_{2m}(∇^{V})) = c_{2m}(V)

Note : ĉ_{2} : \{ \begin{matrix}hermitean line bundles \\ with connections \end{matrix} \} /iso →^{∼} H_{Del}^{2}(B) geometric families A geometric family is the following collection of data :

1. E → B smooth locally trivial fibre bundle with closed n-dimensional fibres

2. fibrewise orientation

3. fibrewise Riemannian metric, horizontal subbundle (connection)

4. family of Dirac bundles V abbreviation E D - associated family of Dirac operators local index theory ind(E) ∈ K^{−n}(B) - index of D have local index forms

Ω^{k}(E) := [∫_{E/B} Â(∇^{T^{v}π})ch(∇^{T wist})] _{k} ∈ A_{B}^{k}(B)

Bismut : Ω^{k}(E) represents ch_{k}(ind(E)) in H_{dR}^{k}(B) filtration of K^{∗}(B) define Atiyah-Hirzebruch filtration

⊂ K_{k+1}^{∗} (B) ⊂ K_{k}^{∗}(B) ⊂ ⋯ ⊂ K_{0}^{∗}(B) = K^{∗}(B) by

K_{k}^{∗}(B) := _{f:Y →B,dim (Y)<k}^{⋂} ker(f^{∗})

observation : if x ∈ K_{2m}^{0}(B), then ch_{2m}(x) is the image of an integral obstruction class o_{2m}(x) ∈ H^{2m}(B, ℤ) (−1)^{m}(m − 1)!ch_{2m}(x) is the image of c_{2m}(x)

Deligne cohomology valued local index theory assume that ind(E) ∈ K_{k}^{−n}(B) define m by 2m = k or 2m − 1 = k There is a natural class ind_{Del}^{k}(E) ∈ H_{Del}^{k}(B) such that

R(ind_{Del}^{k}(E)) = Ω^{k}(E)

and

v(ind_{Del}^{k}(E)) = (−1)^{m}c_{k}(ind(E))

Characterization of ind_{Del}^{k}(E) Note that a Deligne cohomology class

u ∈ H_{Del}^{k}(B)

is characterized by its holonomy hol(u) : Let Z be a k − 1-cycle in a smooth manifold B′ and f : B′ → B a smooth map with image f(Z). we have R(f^{∗}u) = 0 f^{∗}u ∈ H^{k−1}(B′, ℝ/ℤ) define hol(u)(f_{∗}Z) :=< f^{∗}u, Z >∈ ℝ/ℤ what is hol(ind_{Del}^{k}(E))(f_{∗}Z)? by assumption : f^{∗}ind(E) = 0 choose family of smoothing operators Q such that f^{∗}D + Q is invertible (taming (f^{∗}E)_{t}) η((f^{∗}E)_{t}) - eta-form, dη((f^{∗}E)_{t}) = Ω(f^{∗}E)

hol(ind_{Del}^{k}(E))(f_{∗}Z) = [(m−1)! ∫_{Z} η^{k−1}((f^{∗}E)_{t})]_{mod ℤ} real structures and flat classes assume that V is the vertical spinor bundle twisted by a real vector bundle observe : k + n ≢ 0(4) implies Ω^{k}(E) = 0 thus R(ind_{Del}^{4k+2−n} (E)) = 0 hence ind_{Del}^{4k+2−n} (E) ∈ H^{k−1−n}(B, ℝ/ℤ) Problem : Find a topological expression for this cohomology class Lemma : ind_{Del}^{4k+2−n} (E) ∈ H^{k−1−n}(B, ℝ/ℤ) only depends on ind_{ℝ}(E) ∈ KO^{−n}(B). It is annihilated by 2. consider kernel of complexification

0 → U^{−n}(B) → KO^{−n}(B) → K^{−n}(B) there is a natural transformation d_{4k+1−n}^{−n} : U^{−n}(B) → H^{k−1−n}(B, ℝ/ℤ) such that ind_{Del}^{4k+2−n} (E) = d_{4k+1−n}^{−n} (ind_{ℝ}(E)) consider bundle of classifying spaces U/O → BO → BU note that U/O ≅ Ω^{7}KO ≅ KO^{1} n-fold loop : KO^{1−n} → KO^{−n} → K^{−n} a class in x ∈ U^{−n}(B) is a homotopy class of maps X : B → KO^{−n} such that (up to homotopy) it admits a lift X̂ : B → KO^{1−n} we will define classes

d̄_{4k+1−n} ∈ H^{4k+1−n} (KO^{1−n}, ℝ/ℤ) such that

d_{4k+1−n}^{−n} (x) = X̂^{∗}(d̄_{4k+1−n} )

(independent of the choice of the lift) let

T_{4k+1} ∈ H^{4k+1} (U/O, ℝ)

be the transgression of c_{4k+2} ∈ H^{4k+2} (BU, ℤ) then

(with 2m = 4k + 2 − n or 2m − 1 = 4k + 2 − n)

d̄_{4k+1−n} := [\frac{(m − 1)!}{(2k)!} Ω^{n}T_{4k+1} ] _{mod ℤ}

explicit computation

n H^{∗}(KO^{1−n}, ℤ)/tors  d̄_{4k+1−n}  other description
1  L(c_{4}, c_{8}, …)  \begin{matrix}[\frac{1}{2}ch_{0}] & k = 0 \\ 0 & k > 0 \end{matrix}
2  E(v_{3}, v_{7}, …)  0
3  L(u_{2}, u_{6}, …)  0
4  E(a_{1}, a_{5}, …)  0
5  L(y_{4}, y_{8}, …)  0
6  E(y_{3}, y_{7}, …)  [\frac{1}{2}y_{4k−5} ]
7  L(c_{2}, c_{6}, …)  [\frac{1}{2}c_{4k−6} ]
0  E(a_{1}, a_{5}, …)  [\frac{1}{2}a_{4k+1} ]  w_{2k}w_{2k+1}