Editing Atiyah–Singer index theorem (section)

===Other extensions===
*The Atiyah–Singer theorem applies to elliptic [[pseudodifferential operator]]s in much the same way as for elliptic differential operators. In fact, for technical reasons most of the early proofs worked with pseudodifferential rather than differential operators: their extra flexibility made some steps of the proofs easier.
*Instead of working with an elliptic operator between two vector bundles, it is sometimes more convenient to work with an ''[[elliptic complex]]'' <math display="block">0\rightarrow E_0 \rightarrow E_1 \rightarrow E_2 \rightarrow \dotsm \rightarrow E_m \rightarrow 0</math> of vector bundles. The difference is that the symbols now form an exact sequence (off the zero section). In the case when there are just two non-zero bundles in the complex this implies that the symbol is an isomorphism off the zero section, so an elliptic complex with 2 terms is essentially the same as an elliptic operator between two vector bundles. Conversely the index theorem for an elliptic complex can easily be reduced to the case of an elliptic operator: the two vector bundles are given by the  sums of the even or odd terms of the complex, and the elliptic operator is the sum of the operators of the elliptic complex and their adjoints, restricted to the sum of the even bundles.
*If the [[manifold]] is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index. These conditions can be local (like demanding that the sections in the domain vanish at the boundary) or more complicated global conditions (like requiring that the sections in the domain solve some differential equation). The local case was worked out by Atiyah and Bott, but they showed that many interesting operators (e.g., the [[signature operator]]) do not admit local boundary conditions. To handle these operators, [[Michael Atiyah|Atiyah]], [[Vijay Kumar Patodi|Patodi]] and [[Isadore Singer|Singer]] introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder. This point of view is adopted in the proof of {{harvtxt|Melrose|1993}} of the [[Atiyah–Patodi–Singer index theorem]].
*Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space ''Y''. In this case the index is an element of the K-theory of ''Y'', rather than an integer. If the operators in the family are real, then the index lies in the real K-theory of ''Y''. This gives a little extra information, as the map from the real K-theory of ''Y'' to the complex K-theory is not always injective.
*If there is a [[Group action (mathematics)|group action]] of a group ''G'' on the compact manifold ''X'', commuting with the elliptic operator, then one replaces ordinary K-theory with [[Equivariant algebraic K-theory|equivariant K-theory]].  Moreover, one gets generalizations of the [[Lefschetz fixed-point theorem]], with terms coming from fixed-point submanifolds of the group ''G''. See also: [[equivariant index theorem]].
*{{harvtxt|Atiyah|1976}} showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient. The kernel of the elliptic operator is in general infinite dimensional in this case, but it is possible to get a finite index using the dimension of a module over a [[von Neumann algebra]]; this index is in general real rather than integer valued. This version is called the '''''L''<sup>2</sup> index theorem''', and was used by {{harvtxt|Atiyah|Schmid|1977}} to rederive properties of the [[discrete series representation]]s of [[semisimple Lie group]]s.
*The [[Callias index theorem]] is an index theorem for a Dirac operator on a noncompact odd-dimensional space.  The Atiyah–Singer index is only defined on compact spaces, and vanishes when their dimension is odd. In 1978 [[Constantine Callias]], at the suggestion of his Ph.D. advisor [[Roman Jackiw]], used the [[chiral anomaly|axial anomaly]] to derive this index theorem on spaces equipped with a [[Hermitian matrix]] called the [[Higgs field]].<ref>[https://projecteuclid.org/download/pdf_1/euclid.cmp/1103904395  Index Theorems on Open Spaces]</ref> The index of the Dirac operator is a topological invariant which measures the winding of the Higgs field on a sphere at infinity.  If ''U'' is the unit matrix in the direction of the Higgs field, then the index is proportional to the integral of ''U''(''dU'')<sup>''n''−1</sup> over the (''n''−1)-sphere at infinity.  If ''n'' is even, it is always zero.
**The topological interpretation of this invariant and its relation to the [[Hörmander index]] proposed by [[Boris Fedosov]], as generalized by [[Lars Hörmander]], was published by [[Raoul Bott]] and [[Robert Thomas Seeley]].<ref>[https://projecteuclid.org/download/pdf_1/euclid.cmp/1103904396  Some Remarks on the Paper of Callias]</ref>