Index formulas

Relations between analytic and topological invariants of operators of a certain class. More precisely, index formulas establish a relation between the analytic index of a linear operator

( are topological vector spaces), defined by the formula

and measuring in this way the "difference" between the defective subspaces of (namely, the kernel and its cokernel ), and a topological index, namely some topological characteristic of the operator and the spaces , . For a general elliptic differential operator on a closed manifold, the problem of finding index formulas was posed towards the end of the 1950's [1] and solved in 1963 (see [2]), although special forms of index formulas were known even earlier, for example, the Gauss–Bonnet theorem and its multi-dimensional variants. Subsequently a number of generalizations of index formulas were obtained for objects of a more complex nature; in these cases, instead of the index, which is an integer, arbitrary complex numbers and more general objects (e.g. functions) may feature.

Elementary index formulas.

1) Let be the differentiable boundary of a bounded region and let be an elliptic pseudo-differential operator mapping the space of differentiable complex-valued vector functions on with values in into itself. Let be the manifold of tangent vectors to of length , oriented by means of the -form

where are local coordinates on , are the corresponding coordinates in the tangent space, and let be the oriented boundary of formed by the unit tangent vectors. Since is elliptic, its symbol is a non-singular -matrix function on . It turns out that the following Dynin–Fedosov formula holds for the index of [7]:

(1)

where is the exterior power of the matrix exterior form and denotes the trace of the -matrix form. In particular, if or if is a differential operator on an odd-dimensional manifold, then (this is not true, in general, for a pseudo-differential operator).

2) Let be an elliptic differential operator of the form

(where is a multi-index) in the space , and let be boundary differential operators from into of the form

The family of operators defines an elliptic boundary value problem if the function is non-singular on . Here are the coefficients of the polynomials

that are the remainders after division of the polynomials (in ) by the polynomial (in ), where

and is defined from the factorization , where

; , are, respectively, a unit tangent vector and the inward normal to ; (or ) is a polynomial (in ) without zeros in the upper (respectively, lower) -half-plane. By the index of the above-described boundary value problem one means the index of the corresponding linear operator from into taking into the set . It turns out that the index of the elliptic boundary value problem is the same as that of the elliptic pseudo-differential operator on whose symbol is given by the matrix . In particular, the index of the Dirichlet problem is zero. There are general index formulas for boundary value problems [16], , [27].

The Atiyah–Singer index formulas.

Let and be the spaces of infinitely-differentiable sections of the vector bundles and over a closed -dimensional differentiable manifold , and let be a (pseudo-differential) elliptic operator acting from into . The topological index of is defined as follows. Because of the ellipticity of the symbol of determines an isomorphism of the lifted vector bundles on :

where is the bundle of unit spheres of the cotangent bundle of . Let be the bundle of unit balls in ; this is a -dimensional manifold with boundary . By glueing the copies and of along their common boundary, one obtains a closed -dimensional manifold over which the vector bundle

is constructed, where and is used to identify and along . This vector bundle carries all the topological information required for the definition of the topological index. Namely:

(2)

Here is the cohomological Chern character of the bundle ; is the cohomological Todd class of the complexified cotangent bundle ; ; . The right-hand side represents the value of the -dimensional component of the element on the fundamental cycle of the manifold . Thus, the mapping determines a homomorphism that is trivial on the image of ; here is the Grothendieck group generated by complex vector bundles over .

The Atiyah–Singer index theorem states:

(3)

Formula (2) admits a number of modifications. The rational cohomology class , depending on the symbol , is introduced as follows. With the triple one can associate a difference element (cf. Difference element in -theory), which can be regarded as the first obstruction to extending the isomorphism to the whole of ,

where is the tangent bundle, which (by means of the Riemannian metric on ) can be identified with ; is the relative Grothendieck group of vector bundles over , and hence for the Chern character of : . The formula for the topological index of now takes the form:

(4)

where , .

The Thom isomorphism

then enables one to write (4) in the form

(5)

(As before, on the right-hand side of (4) and (5) are the values of the corresponding elements on the fundamental cycles, as in (2).)

The topological index is expressed in terms of -theory as follows. Let be a differentiable imbedding of in a Euclidean space, a tubular neighbourhood of in , which can be regarded as a real vector bundle over , so that is isomorphic (over ) to , the complexification of lifted to by the projection . Composition of the Thom isomorphism with the natural homomorphism induced by the imbedding induces a homomorphism . Let be the Bott periodicity isomorphism. Then the homomorphism does not depend on the imbedding and

Examples.

3) Let be a closed oriented Riemannian manifold, let be the bundle of complex exterior -forms over and let

be the exterior differentiation operator and its adjoint, respectively. The operator

where , , is elliptic and the index formula (3) holds for it; furthermore the topological index is equal to the Euler characteristic (the Hodge–de Rham theorem). For the Gauss–Bonnet theorem follows.

4) Let be the eigen -spaces of the involution , , where is the duality operator determined by the metric on , . The restriction of the operator to an operator from into , called the signature operator , is an elliptic operator for which the index formula (3) holds; furthermore, the analytic index is equal to the signature of the manifold , while the topological index is equal to the -genus (Hirzebruch's theorem).

5) Let be a holomorphic vector bundle over the complex compact manifold , let be the bundle of differential forms of type , let be the bundle of forms of type with coefficients in , and let be the -module of smooth sections of this bundle. Let be the Cauchy–Riemann–Dolbeault operator, its adjoint, and let , . Then the operator is an elliptic operator for which (3) holds; furthermore, the analytic index is equal to the Euler characteristic of with coefficients in the sheaf of germs of holomorphic sections of , while the topological index is , where is the Chern character of and is the Todd class of the tangent bundle to (the Riemann–Roch–Hirzebruch theorem).

Elliptic complexes.

In the more general situation which arises naturally, for example, in differential geometry, instead of a single operator one considers a complex of (pseudo-differential) operators

where the are differentiable vector bundles over the closed manifold and . By the symbol of the complex one means the corresponding sequence of principal symbols

where is the lifting of to by the projection . The complex is called elliptic if its symbol is an acyclic complex, that is, if it is exact everywhere outside the zero section. By the analytic index of the complex one means its Euler characteristic:

where is the -th cohomology group of . Two important examples of elliptic complexes are the de Rham complex and its complex analogue, the Dolbeault complex. The problem of computing in terms of the class of the complex in can be reduced to computing the index for a single operator [3].

If a compact group acts on (and commutes with the action of , that is, is a -complex), then is a -module, and is defined as an element of the ring of characters of the group . This is a function in . Here it turns out that the index theorem can be regarded as a generalization of the Lefschetz theorem on fixed points, since the topological index at a point can be expressed in terms of the index of the restriction of the symbol to the subset of fixed points of the mapping defined by .

Let be a topological cyclic group, that is, there exists an element in whose powers are dense in , let be the normal bundle to in and let be the class of the symbol of . Let be its restriction and let be the class generated by the standard complex of exterior powers of the bundle to (here , ). Then the Lefschetz number , which is equal to , is given by the formula

where is the natural extension of the topological index . The cohomological version of this formula is given by:

(6)

Without the compactness condition on , but under the hypothesis that is a zero-dimensional submanifold and that the action of is non-degenerate (that is, the graph of is transversal to the diagonal in ), there is an analogous formula, which can be expressed as follows. If , then leaves fixed while induces a linear mapping on the fibres , and

Finally, it is possible to weaken the condition of ellipticity of the -complex by considering so-called transversally-elliptic complexes; in this case, the index turns out to be a generalized function on the group (see [8]). In particular, if is finite, then transversal ellipticity is to equivalent to ellipticity, so that the previous formulas are applicable. If is a homogeneous space, then all the complexes of operators are transversally elliptic and in this case the index formula is in essence the same as the Frobenius reciprocity formula for the induced representations of the group .

Non-Fredholm operators.

In this case it is also sometimes possible to give another definition of the analytic index and to obtain corresponding index formulas.

Examples.

6) Let be a uniformly-elliptic operator on with almost-periodic coefficients. The analytic index is introduced by means of the relative dimension in the -factor (see von Neumann algebra) and is a real number (see [11]). There is a formula analogous to (1), but instead of the integral over the average value of the almost-periodic function is used.

7) Suppose that a discrete group acts freely on a manifold and that the quotient space is compact; let , be vector bundles over and let act on them in accordance with its action on . The analytic index of an elliptic operator on commuting with the action of is defined by the formula

(7)

where , are the orthogonal projections on and in , is any -invariant smooth density on and is defined, for any operator commuting with and having smooth kernel , by the formula

(here is any fundamental domain of the group on and is the trace of the matrix). It turns out that , where is the operator on whose symbol induces under the lifting to by the canonical projection [12]. Thus, the index formula for the operator can be obtained from the index formula for the operator on the compact manifold . This result enables one to reveal the non-triviality of spaces in which representations of discrete series are realized [13].

A formula of the same type can be obtained for invariant elliptic operators on homogeneous spaces of Lie groups, even without being discrete, with a natural generalization of the analytic index [20].

Another generalization of this situation can be obtained if one considers invariant operators on a manifold with an action of a locally compact group such that is compact [24].

8) If the coefficients of a uniformly-elliptic operator on form a homogeneous measurable random field, then it is possible to introduce the analytic index , which is a random variable (in the ergodic case, a real number) defined by formula (7) with replaced by . Here is constructed from the kernel of the operator by averaging over : . This example is a generalization of Example 6) and an analogous index formula holds for it .

9) Let be a compact manifold with a foliation and a longitudinal elliptic differential operator on , i.e. a differential operator containing only differentiations along the leaves and elliptic on every leaf. Suppose that there is a transverse measure on . Then a real-valued analytic index of can be defined and a formula of Atiyah–Singer type can be proved. Considering measured foliations, in this context one comes to a formula which generalizes that of Example 8), [18], [19].

Index formulas with values in -groups.

10) If a family of elliptic operators is given, parametrized by the points of a compact space , then its analytic index has been defined (see [15]). The topological index is constructed by analogy with formula (6) (all the constructions are carried out "fibrewise" over ) and the index theorem holds.

11) A more general theorem is obtained if one considers elliptic operators over a compact manifold acting in sections of vector bundles with fibres which are finitely-generated projective modules over a fixed -algebra . The analytic index here takes values in the group . If one takes with a compact , then one obtains the formula of Example 10). Also the equivalent situation (with a compact Lie group ) can be considered in this context [26], [29].

The case when is a -factor is of particular interest [28], implying the formula of Example 7).

12) There is a number of generalizations of the Atiyah–Singer formulas with the analytic index taking values in homology -groups or bivariant Kasparov -groups. Taking the Chern character and applying some kind of intersection index usually allows one to pass to the usual number-valued index formulas [23], [25]. Also, the longitudinal index theorem of Example 9) can be generalized in this manner [21].

13) Consider two generalized Dirac operators , which coincide near infinity (in particular, they are defined on Riemannian manifolds , which coincide near infinity, i.e. and are isometric for some compact subsets , ). Let , be positive near infinity and let there be the natural splittings

Then can be expressed by a formula of Atiyah–Singer type having important geometrical applications [22].

New analytic tools.

The Atiyah–Bott formula

provides a local expression of the index if one uses the asymptotic expansion of the traces on the right-hand side as . But this expression contains lower-order terms of the symbol of , so it seems difficult to see how the corresponding integrals cancel. It occurred that cancellation is obtained by using some symmetry and supersymmetry arguments. Also a probabilistic approach is effective to work with the traces of heat kernels. Families of elliptic operators can be considered in this way too [30]–[42].

References

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Comments

Several new proofs of the Atiyah–Singer index theorem have been given in recent years.

In his paper [a5], E. Witten suggested that supersymmetric quantum theory might provide the framework for a simple proof of the index theorem. Such a proof was realized by L. Alvarez-Gaumé [30] and subsequently by Friedan and Windey [a4]. These theoretical physicists relied on formal manipulations inside path integrals (including fermionic path integrals). So their proofs were certainly not rigorous. E. Getzler [38] found a rigorous version of their arguments which relied on pseudo-differential operator theory and the theory of supermanifolds. More recently, Getzler [39] found a proof whose geometric and algebraic parts are elementary and transparent. Independent of this work J.-M. Bismut [34] found a related proof using probabilistic methods.

For further material cf. also [a2], Chapt. XIX, [a1], Chapt. 12 and [a3], Chapt. 9.

References

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[a3]	M. Kaku, "Introduction to superstrings" , Springer (1988) MR0954610 Zbl 0655.58001
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