The notion of index was originally defined
More generally the resulting notion is abstractly characterized as being the pairing operation (composition)
Even more generally, in generalized cohomology theory indices are given by genera and universal orientation in generalized cohomology, such as for instance the elliptic genus for elliptic cohomology and the Witten genus for tmf. See at genus for more on this generalized notion of indices.
One reason why this is an interesting invariant of an elliptic differential operator is that when deforming the operator by a compact operator then the dimension of both the kernel and the cokernel may change, but their difference remains the same. Hence one may think of the analytic index as a “corrected” version of its kernel, such as to make it be more invariant.
On the other hand, the topological index of an elliptic differential operator , is defined to be the pairing of the cup product of its Chern character and the Todd class of the base manifold with its fundamental class.
More generally such analytic and topological indices are defined for Fredholm operators.
The Atiyah-Singer index theorem assert that these two notios of index are in fact equal.
If the Fredholm operator in question happens to be a Dirac operator (such as that encoding the dynamics of a spinning particle or more generally the supercharge of a system in supersymmetric quantum mechanics) then the index of coincides with the partition function of this quantum mechanical system, namely the super-trace of the heat kernel of the corresponding Hamiltonian Laplace operator (Berline-Getzler-Vergne 04).
with components (with respect to the -grading) to be denoted
The last step here follows from an argument which is as simple as it is paramount whenever anything involves supersymmetry:
the point is that if a (hermitean) operator has a supercharge , in that , then all its non-vanishing eigenstates appear in “supermultiplet” pairs of the same eigenvalue: if has eigenvalue under , then
also has eignevalue (since ).
Therefore all eigenstates for non-vanishing eigenvalues appear in pairs whose members have opposite sign under the supertrace. So only states with contribute to the supertrace. But if and are hermitean operators for a non-degenerate inner product, then it follows that and hence these are precisely the states which are also annihilated by the supercharge (are in the kernel of ), hence are precisely only the supersymmetric states.
On these now the weight and hence the supertrace over this “Euclidean propagator” simply counts the number of supersymmetric states, signed by their fermion number.
If one thinks of as the time-evolution Hamiltonian of a system of supersymmetric quantum mechanics with the supercharge on the worldline, then is the space of supersymmetric quantum states, is the Euclidean time evolution operator and its supertrace is the partition function of the system. Hence we have the translation
This kind of argument appears throughout supersymmetric quantum field theory. In dimension 2 it controls the nature of the Witten genus.
More in detail, by the discussion there KK-theory (E-theory) is the category which is the additive and split exact localization of the category C*Alg of C*-algebras at the compact operators. For the base C*-algebra of complex numbers the morphisms in this category have the following equivalent meaning:
in the category (hence the Kasparov product) is the index of the Fredholm operator twisted by .
And hence even more generally one may regard any composition in as as a generalized index map. Via the universal characterizatin of itself, this then gives a fundamental and general abstract characterization of the notion of index:
|partition function in -dimensional QFT||supercharge||index in cohomology theory||genus||logarithmic coefficients of Hirzebruch series|
|0||push-forward in ordinary cohomology: integration of differential forms|
|1||spinning particle||Dirac operator||KO-theory index||A-hat genus||Bernoulli numbers|
|endpoint of 2d Poisson-Chern-Simons theory string||Spin^c Dirac operator twisted by prequantum line bundle||space of quantum states of boundary phase space/Poisson manifold||Todd genus||Bernoulli numbers|
|endpoint of type II superstring||Spin^c Dirac operator twisted by Chan-Paton gauge field||D-brane charge||Todd genus||Bernoulli numbers|
|2||superstring||Dirac-Ramond operator||superstring partition function||elliptic genus/Witten genus||Eisenstein series|
|self-dual string||M5-brane charge|
Original articles include
Lecture notes include
Textbook accounts include chapter III of
based on original articles including
D. Quillen, Superconnections and the Chern character Topology 24 (1985), no. 1, 89–95;
For the more general discussion of indices of elliptic complexes see
A standard textbook account in the context of KK-theory is in section 24.1 of