nLab
index

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

Context

Index theory

Functional analysis

Operator algebra

Contents

Idea

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.

For elliptic differential and Fredholm operators

The analytical index of an elliptic differential operator D:Γ(E 1)Γ(E 2)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 DD, 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 DD (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 DD coincides with the partition function of this quantum mechanical system, namely the super-trace of the heat kernel exp(tD 2)\exp(-t D^2) of the corresponding Hamiltonian Laplace operator D 2D^2 (Berline-Getzler-Vergne 04).

Proposition

Let (X,g)(X,g) be a compact Riemannian manifold and \mathcal{E} a smooth super vector bundle and indeed a Clifford module bundle over XX. Consider a Dirac operator

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

with components (with respect to the 2\mathbb{Z}_2-grading) to be denoted

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

where D =(D +) *D^- = (D^+)^\ast. Then D +D^+ is a Fredholm operator and its index is the supertrace of the kernel of DD, as well as of the heat kernel of D 2D^2:

ind(D +) dim(ker(D +))dim(coker(D +)) =dim(ker(D +))dim(ker(D )) =sTr(ker(D)) =sTr(exp(tD 2))t>0. \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 HH has a supercharge DD, in that H=D 2H = D^2, then all its non-vanishing eigenstates appear in “supermultiplet” pairs of the same eigenvalue: if |ψ|\psi\rangle has eigenvalue E>0E \gt 0 under HH, then

  1. D|ψ0D |\psi\rangle \neq 0 (since DD|ψ=H|ψ=E|ψ0D D |\psi\rangle = H |\psi\rangle = E |\psi\rangle \neq 0);

  2. also D|ψD |\psi\rangle has eignevalue EE (since [H,D]=0[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|ψ=0H |\psi\rangle = 0 contribute to the supertrace. But if HH and DD are hermitean operators for a non-degenerate inner product, then it follows that (D 2|ψ=0)(D|ψ=0)(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 DD), hence are precisely only the supersymmetric states.

On these now the weight exp(tD 2)=1\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 2D^2 as the time-evolution Hamiltonian of a system of supersymmetric quantum mechanics with DD the supercharge on the worldline, then ker(D)ker(D) is the space of supersymmetric quantum states, exp(tD 2)\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 KKKK 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 A\mathbb{C} \to A are operator K-cohomology classes which are represented by “vector bundles over the space represented by AA”, namely by Hilbert modules EE over AA;

  • morphisms A𝒞A \to \mathcal{C} are K-homology classes which are represented by Fredholm operators DD;

  • the composition

    ind(D E):EADKK(,) 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 KKKK (hence the Kasparov product) is the index of the Fredholm operator DD twisted by EE.

More generally, if BB is some other chosen base C*-algebra then KK(A,B)KK(A,B) is the group of Fredholm operators DD on Hilbert module bundles over the C*-algebra BB, and one takes the pairing

ind ,A,B:KK(,A)×KK(A,B)KK(,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 BB. (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 KKKK as as a generalized index map. Via the universal characterizatin of KKKK 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:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

References

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

  • H. MacKean, Isadore Singer, Curvature and eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967)
  • Michael Atiyah, Raoul Bott, V. K. 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

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

Revised on August 29, 2014 05:27:37 by Urs Schreiber (89.204.138.134)