nLab
elliptic chain complex

Context

Index theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

The notion of elliptic chain complex is the generalization of the notion of elliptic operator from single linear maps to chain complexes of linear maps.

Definition

For XX a smooth manifold and {E k} k\{E_k\}_{k \in \mathbb{Z}} a collection of vector bundles over XX, a chain complex of differential operators between the spaces of sections of these bundles

Γ(E k+1)P kΓ(E k) \cdots \to \Gamma(E_{k+1}) \stackrel{P_k}{\to} \Gamma(E_k) \to \cdots

is called an elliptic chain complex if the corresponding sequence of symbols

π *E k+1σ(P k)π *E k \cdots \to \pi^* E_{k+1} \stackrel{\sigma(P_k)}{\to} \pi^* E_k \to \cdots

(where π:T *XX\pi \colon T^* X \to X is the cotangent bundle) is an exact sequence.

For instance (Pati, def. 9.4.1).

For a single differential operator PP this says that 0π *E 1σ(P)π *E 000 \to \pi^* E_1 \stackrel{\sigma(P)}{\to} \pi^* E_0 \to 0 is exact, which means that σ(P)\sigma(P) is an isomorphism, hence that PP is an elliptic operator.

Properties

Atiyah-Bott lemma

If (,d)(\mathcal{E}, d) is an elliptic complex of smooth sections =Γ X(E)\mathcal{E} = \Gamma_X(E) of a vector bundle EXE \to X overa compact closed manifold XX, then the inclusion

(,d)(¯,d) (\mathcal{E},d) \hookrightarrow (\overline{\mathcal{E}}, d)

into the complex of distributional sections is a quasi-isomorphism, in fact a homotopy equivalence.

This is due to (Atiyah-Bott). A localized refinement (suitable for factorization algebras of local observables) appears as Gwilliam, lemma 5.2.13.

Examples

The classical examples of elliptic complexes are discussed also in (Gilkey section 3).

de Rham complex

Let XX be a compact smooth manifold. Then the de Rham complex is an ellptic complex. The corresponding index of an elliptic complex is the Euler characteristic

Ind(Ω (X),d)=χ(X)= p=0 dimX(1) pdimH dR p(X,) Ind(\Omega^\bullet(X),d) = \chi(X) = \sum_{p = 0}^{dim X} (-1)^p dim H_{dR}^p(X, \mathbb{C})

The Yang-Mills complex

(…)

The Dolbeault complex

(…) Dolbeault complex

The index of an elliptic complex of the Dolbeault complex is the arithmetic genus

Spin complex

(…) index is A-hat genus (…)

References

  • V. Pati, Elliptic complexes and index theory (pdf)
  • Michael Atiyah, Raoul Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374–407. MR 0212836 (35 #3701)
  • Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)

Revised on February 19, 2013 15:14:58 by Urs Schreiber (80.81.16.253)