nLab elliptic chain complex

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

$$\cdots \to \Gamma(E_{k+1}) \stackrel{P_k}{\to} \Gamma(E_k) \to \cdots$$

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

$$\cdots \to \pi^* E_{k+1} \stackrel{\sigma(P_k)}{\to} \pi^* E_k \to \cdots$$

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

For instance (Pati, def. 9.4.1).

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

Properties

Atiyah-Bott lemma

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

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

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

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

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 (…)

Related concepts

• index of an elliptic complex

References

• V. Pati, Elliptic complexes and index theory (pdf)

• Peter Gilkey, The Atiyah-Singer Index Theorem – Chapter 5 (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)