nLab index

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

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Idea

The notion of index was originally defined

• For elliptic differential operators

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

• For Dirac operators

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)

• in KK-theory.

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.

There is also a quite general microlocal formulation of index theory due to Kashiwara and Schapira.

For elliptic differential and Fredholm operators

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.

For Dirac operators

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).

Proposition

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

$$D \colon \Gamma(X,\mathcal{E}) \to \Gamma(X, \mathcal{E})$$

with components (with respect to the $\mathbb{Z}_2$-grading) to be denoted

$$D = \left[ \array{ 0 & D^- \\ D^+ & 0 } \right] \,,$$

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 $D^2$:

$$\begin{aligned} ind(D^+) & \coloneqq dim(ker(D^+)) - dim(coker(D^+)) \\ & = dim(ker(D^+)) - dim(ker(D^-)) \\ & = sTr(ker(D)) \\ & = sTr( \exp(-t \, D^2) ) \;\;\; \forall t \gt 0 \end{aligned} \,.$$

This appears as (Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50), based on (MacKean-Singer 67).

Proof

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

1. $D |\psi\rangle \neq 0$ (since $D D |\psi\rangle = H |\psi\rangle = E |\psi\rangle \neq 0$);

2. 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.

Remark

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

• index = partition function .

This kind of argument appears throughout supersymmetric quantum field theory. In dimension 2 it controls the nature of the Witten genus.

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 $KK$ 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

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

  in the category $KK$ (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

$$ind \coloneqq \circ_{\mathbb{C}, A, B} \;\colon\; KK(\mathbb{C},A) \times KK(A,B) \to KK(\mathbb{C}, B)$$

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 $KK$ 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.

Related pages

• topological index

• analytical index

• fiber integration in K-theory

• Fredholm index, index of a Dirac operator

• index theorem

  • Atiyah-Singer index theorem

  • Mishchenko-Fomenko index theorem

  • Baum-Connes conjecture

• zeta function of an elliptic differential operator, eta function

• Poincaré–Hopf index theorem

• Gauss-Bonnet theorem

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	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	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

Original articles include

• Michael Atiyah, Isadore Singer, Index theory for skew-adjoint Fredholm operators (pdf)

Introduction with an eye towards application in mathematical physics:

• Mikio Nakahara, Chapter 12 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

• David D. Bleecker, Bernhelm Booß-Bavnbek: Index Theory with Applications to Mathematics and Physics, International Press of Boston (2013) [ISBN:9781571462640, pdf]

Lecture notes:

• Jean-Michel Bismut, Introduction to index theory, lecture notes (pdf)

A general introduction with an emphasis of indices as Gysin maps/fiber integration/Umkehr maps is in

• Chris Kottke, Talbot Workshop 2010 Talk 2: K-Theory and Index Theory (arXiv:1010.5002)

Textbook accounts include chapter III of

• H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989)

A standard textbook account of the description of indices of Dirac operators as partition functions in supersymmetric quantum mechanics is

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

based on original articles including

• H. MacKean, Isadore Singer, Curvature and eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967)

• Michael Atiyah, Raoul Bott, Vijay Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330

• Luis Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (Euclid)

• Ezra Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys. 92 (1983), 163-178. (pdf)

• 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)

• D. Quillen, Superconnection character forms and the Cayley transform. Topology 27

  (1988), no. 2, 211–238

For the more general discussion of indices of elliptic complexes see

• Peter Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem (pdf)

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)

A standard textbook account in the context of KK-theory is in section 24.1 of

• Bruce Blackadar, K-Theory for Operator Algebras