This page is about the notion of index in analysis/operator algebra. For other notions see elsewhere.
noncommutative topology, noncommutative geometry
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The notion of index was originally defined
as an invariant correction of the kernel of such an operator (namely corrected by the cokernel). The definition has particularly nice properties in the special case
where it coincides with the partition function of supersymmetric quantum mechanics.
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.
The analytical index of an elliptic differential operator $D \colon \Gamma(E_1) \to \Gamma(E_2)$ is defined to be the the difference between the dimension of its kernel and that of its cokernel.
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 $D$, 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 $D$ (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 $D$ coincides with the partition function of this quantum mechanical system, namely the super-trace of the heat kernel $\exp(-t D^2)$ of the corresponding Hamiltonian Laplace operator $D^2$ (Berline-Getzler-Vergne 04).
Let $(X,g)$ be a compact Riemannian manifold and $\mathcal{E}$ a smooth super vector bundle and indeed a Clifford module bundle over $X$. Consider a Dirac operator
with components (with respect to the $\mathbb{Z}_2$-grading) to be denoted
where $D^- = (D^+)^\ast$. Then $D^+$ is a Fredholm operator and its index is the supertrace of the kernel of $D$, as well as of the heat kernel of $D^2$:
This appears as (Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50), based on (MacKean-Singer 67).
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 $H$ has a supercharge $D$, in that $H = D^2$, then all its non-vanishing eigenstates appear in “supermultiplet” pairs of the same eigenvalue: if $|\psi\rangle$ has eigenvalue $E \gt 0$ under $H$, then
$D |\psi\rangle \neq 0$ (since $D D |\psi\rangle = H |\psi\rangle = E |\psi\rangle \neq 0$);
also $D |\psi\rangle$ has eignevalue $E$ (since $[H,D] = 0$).
Therefore all eigenstates for non-vanishing eigenvalues appear in pairs whose members have opposite sign under the supertrace. So only states with $H |\psi\rangle = 0$ contribute to the supertrace. But if $H$ and $D$ are hermitean operators for a non-degenerate inner product, then it follows that $(D^2 |\psi\rangle = 0) \Leftrightarrow (D|\psi\rangle = 0)$ and hence these are precisely the states which are also annihilated by the supercharge (are in the kernel of $D$), hence are precisely only the supersymmetric states.
On these now the weight $\exp(- t D^2) = 1$ and hence the supertrace over this “Euclidean propagator” simply counts the number of supersymmetric states, signed by their fermion number.
If one thinks of $D^2$ as the time-evolution Hamiltonian of a system of supersymmetric quantum mechanics with $D$ the supercharge on the worldline, then $ker(D)$ is the space of supersymmetric quantum states, $\exp(-t \, D^2)$ 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.
The abstract universal characterization of indices is: the index is the pairing in KK-theory/E-theory.
More in detail, by the discussion there KK-theory (E-theory) is the category $KK$ which is the additive and split exact localization of the category C*Alg of C*-algebras at the compact operators. For $\mathbb{C}$ the base C*-algebra of complex numbers the morphisms in this category have the following equivalent meaning:
morphisms $\mathbb{C} \to A$ are operator K-cohomology classes which are represented by “vector bundles over the space represented by $A$”, namely by Hilbert modules $E$ over $A$;
morphisms $A \to \mathcal{C}$ are K-homology classes which are represented by Fredholm operators $D$;
the composition
in the category $KK$ (hence the Kasparov product) is the index of the Fredholm operator $D$ twisted by $E$.
More generally, if $B$ is some other chosen base C*-algebra then $KK(A,B)$ is the group of Fredholm operators $D$ on Hilbert module bundles over the C*-algebra $B$, and one takes the pairing
to be the index map relative $B$. (See e.g. Schick 05, section 6.) This is the case that the Mishchenko-Fomenko index theorem applies to.
And hence even more generally one may regard any composition in $KK$ as as a generalized index map. Via the universal characterizatin of $KK$ itself, this then gives a fundamental and general abstract characterization of the notion of index:
The index pairing is the composition operation in the KK-localization of C*Alg, hence in noncommutative stable homotopy theory.
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
Original articles include
(…)
Lecture notes include
A general introduction with an emphasis of indices as Gysin maps/fiber integration/Umkehr maps is in
Textbook accounts include chapter III of
A standard textbook account of the description of indices of Dirac operators as partition functions in supersymmetric quantum mechanics is
based on original articles including
D. Quillen, Superconnections and the Chern character Topology 24 (1985), no. 1, 89–95;
Varghese Mathai, Daniel Quillen, Superconnections, Thom classes, and equivariant differential forms. Topology 25 (1986), no. 1, 85–110;
Ezra Getzler, A short proof of the Atiyah-Singer index theorem, Topology 25 (1986), 111-117 (pdf)
For the more general discussion of indices of elliptic complexes see
An explicit formula in Chern-Weil theory for indices of differential operators on Hilbert modules-bundles is discussed in detail in
A standard textbook account in the context of KK-theory is in section 24.1 of