| Atiyah–Singer index theorem | |
|---|---|
| Name | Atiyah–Singer index theorem |
| Field | Differential geometry; Mathematical physics |
| Introduced | 1960s |
| Authors | Sir Michael F. Atiyah; Isadore M. Singer |
| Related | Dirac operator, K-theory, Elliptic operator, index |
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a foundational result in differential geometry and mathematical physics that equates the analytical index of an elliptic differential operator on a compact manifold with a topological index computed from characteristic classes. Its bridge between analysis, topology and geometry has deep consequences for Quantum Physics and Quantum field theory because it links spectral properties of operators like the Dirac operator to global topological invariants that control anomalies, quantization and spectral flow in gauge theories.
The theorem concerns elliptic differential operators D acting between sections of vector bundles over a compact manifold M. The analytic index, ind_a(D), is defined as dim ker D − dim coker D, an integer describing imbalance of solutions. The topological index, ind_t(D), is computed via K-theory and characteristic classes such as the Chern character and the Todd class or Â-genus depending on the geometric structure. The Atiyah–Singer theorem asserts ind_a(D) = ind_t(D) for any elliptic operator D on a compact manifold, providing a computable topological formula for an analytic quantity. This equality generalizes classical results like the Gauss–Bonnet theorem and the Riemann–Roch theorem.
Analytic index: For an elliptic operator D (e.g., the Dolbeault operator on a complex manifold or the Dirac operator on a spin manifold), the analytic index is a Fredholm index of a linear operator between Hilbert spaces of square-integrable sections. The notion uses functional analysis and spectral theory developed in context of Hilbert space methods and self-adjoint operator theory.
Topological index: Using topological K-theory and the symbol class σ(D) in K^0(T^*M), one constructs ind_t(D) typically via the pushforward (Gysin) map and characteristic classes like the Chern character and Todd class or Â-genus. Examples: the de Rham operator recovers the Euler characteristic (Gauss–Bonnet), while the Dolbeault operator yields the Hirzebruch–Riemann–Roch index for complex varieties and coherent sheaves studied by Hirzebruch and others.
Multiple proofs exist, reflecting deep connections between topology and analysis. Atiyah and Singer's original proof used topological K-theory and cobordism arguments; later proofs invoke heat kernel methods and local index theory developed by Patodi, Getzler, and Gilkey. Heat kernel/probabilistic proofs use short-time asymptotics of the heat operator exp(−t D^2) and the McKean–Singer formula to relate local curvature forms to global indices. Other approaches employ K-homology and Kasparov theory from operator algebras and noncommutative geometry as developed by Connes, connecting the index theorem to cyclic cohomology and spectral triples.
Key ideas: symbol calculus and the construction of parametrix; reduction to K-theoretic computations on the cotangent bundle; localization of index density via characteristic classes; and the use of heat kernel asymptotics to obtain local index formulas.
In Quantum field theory, the theorem underpins understanding of chiral and gauge anomalies by linking the non-conservation of currents to topological invariants computed from background gauge and gravitational fields. The Atiyah–Singer index computes differences in zero modes of fermionic operators, controlling the fermion determinant phases and anomaly inflow in models studied by Witten, Hawking and others. In condensed matter, index-theoretic ideas classify topological phases such as quantum Hall states and topological insulators via K-theory and the bulk-boundary correspondence. In string theory and M-theory, indices count BPS states and contribute to topological terms in effective actions; computations often invoke the Dirac index on moduli spaces and index localization techniques used by localization theorems in supersymmetric field theories.
Dirac operator: On a compact spin manifold, the Dirac operator D has index given by the Â-genus: ind D = ∫_M Â(M). In physics, the kernel of D counts massless fermion zero modes in background gauge fields; this counting enters instanton calculus in Yang–Mills theory and the study of the ADHM construction of instantons.
Anomalies: The index theorem relates the chiral anomaly to characteristic classes like tr(F∧F) (the second Chern character) producing the Adler–Bell–Jackiw anomaly formula. The index counts net chiral zero modes responsible for non-conservation of axial current in gauge backgrounds.
Spectral flow: The spectral flow of a family of self-adjoint operators is computed by index-theoretic data; results of Atiyah–Patodi–Singer link spectral asymmetry (the η-invariant) to boundary corrections in index formulas important for fermions on manifolds with boundary and for global anomalies.
The original theorem admits numerous extensions: the Atiyah–Patodi–Singer index theorem for manifolds with boundary introduces η-invariants; equivariant index theorems incorporate group actions and lead to fixed-point formulas (Atiyah–Bott, Berline–Vergne); families index theorem treats continuous families of operators and yields characteristic classes in parameter space; and extensions in noncommutative geometry generalize index theory to C*-algebra settings (Kasparov theory, Connes–Moscovici). Other generalizations include heat kernel local index formulas, index theory on singular spaces, and relations to mirror symmetry and moduli space computations in algebraic geometry and string theory.
Category:Differential geometry Category:Mathematical physics