nLab
index

This page is about the notion of index in operator algebra. For other notions see elsewhere.

Idea

The notion of index was originally defined

• For elliptic differential operators

as an invariant correction of the kernel of such an operator. More generally the resulting notion is abstractly characterized as being the pairing operation (composition)

• in KK-theory.

For elliptic differential and Fredholm operators

The analytical index of an elliptic differential operator $D:\Gamma (E_1)\to \Gamma (E_2)$ 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.

Another useful way to think of this index is to interpret $D$ as being the Dirac operator of a spinning particle or more generally the supercharge of a system in supersymmetric quantum mechanics. Then the index is 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).

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.

General abstract definition in KK-theory

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 $\mathrm{KK}$ which is the additive and split exact localization of the category C*Alg of C*-algebras at the compact operators. For $ℂ$ the base C*-algebra of complex numbers the morphisms in this category have the following equivalent meaning:

• morphisms $ℂ\to A$ 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 $A\to 𝒞$ are K-homology classes which are represented by Fredholm operators $D$;

• the composition

  $$\mathrm{ind}(D_E):ℂ\stackrel{E}{\to }A\stackrel{D}{\to }ℂ\in \mathrm{KK}(ℂ,ℂ)\simeq \mathbb{Z}$$ind(D_E) \;\colon\; \mathbb{C} \stackrel{E}{\to} A \stackrel{D}{\to} \mathbb{C} \;\;\;\; \in KK(\mathbb{C}, \mathbb{C}) \simeq \mathbb{Z}

  in the category $\mathrm{KK}$ (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 $\mathrm{KK}(A,B)$ is the group of Fredholm operators $D$ on Hilbert module bundles over the C*-algebra $B$, and one takes the pairing

$$\mathrm{ind}≔{\circ }_{ℂ,A,B}:\mathrm{KK}(ℂ,A)\times \mathrm{KK}(A,B)\to \mathrm{KK}(ℂ,B)$$ind \coloneqq \circ_{\mathbb{C}, A, B} \;\colon\; KK(\mathbb{C},A) \times KK(A,B) \to KK(\mathbb{C}, B)

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 $\mathrm{KK}$ as as a generalized index map. Via the universal characterizatin of $\mathrm{KK}$ 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.

Related pages

• topological index

• analytical index

• index of a Dirac operator

• index theorem

  • Atiyah-Singer index theorem

  • Mishchenko-Fomenko index theorem

References

A standard textbook account of the description of indidces as partition functions in supersymmetric quantum mechanics is

• Nicole Berline, Ezra Getzler, Michèle Vergne, Heat Kernels and Dirac Operators, Springer Verlag Berlin (2004)

An explicit formula in Chern-Weil theory for indices of differential operators on Hilbert modules-bundles is discussed in detail in

• Thomas Schick, $L^2$-index, KK-theory, and connections, New York J. Math. 11 (2005) (arXiv:math/0306171)