Atiyah–Singer index theorem

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In the mathematics of manifolds and differential operators, the Atiyah–Singer index theorem states that for an elliptic differential operator on a compact manifold, the analytical index (closely related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other important theorems (such as the Riemann-Roch theorem) as special cases, and has applications in theoretical physics.

It was proved by Michael Atiyah and Isadore Singer in 1963.

Contents

[edit] Notation

  • X is a compact smooth manifold M (without boundary).
  • E and F are smooth vector bundles over X.
  • D is an elliptic differential operator from E to F. So in local coordinates it acts as a differential operator, taking smooth sections of E to smooth sections of F.

[edit] The symbol of a differential operator

If D is a differential operator of order n in k variables

x1, ..., xk,

then its symbol is the function of 2k variables

x1, ... , xk, y1, ..., yk,

given by dropping all terms of order less than n and replacing ∂/∂xi by yi. So the symbol is homogeneous in the variables y, of degree n. The symbol is well defined even though ∂/∂xi does not commute with xi because we only keep the highest order terms and differential operators commute "up to lower-order terms". The operator is called elliptic if the symbol is nonzero whenever at least one y is nonzero.

Example: The Laplace operator in k variables has symbol y12 + ... + yk2, and so is elliptic as this is nonzero whenever all the y's are nonzero. The wave operator has symbol −y12 + ... + yk2, which is not elliptic if k ≥ 2, as the symbol vanishes for some non-zero values of the ys.

The symbol of a differential operator of order n on a smooth manifold X is defined in much the same way using local coordinate charts, and is a function on the cotangent bundle of X, homogeneous of degree n on each cotangent space. (In general, differential operators transform in a rather complicated way under coordinate transforms; however, the highest order terms transform like tensors so we get well defined homogeneous functions on the cotangent spaces that are independent of the choice of local charts.) More generally, the symbol of a differential operator between two vector bundles E and F is a section of the pullback of the bundle

Hom(E, F)

to the cotangent space of X. The differential operator is called elliptic if the element of Hom(Ex, Fx) is invertible for all non-zero cotangent vectors at any point x of X.

A key propety of elliptic operators is that they are almost invertible; this is closely related to the fact that their symbols are almost invertible. More precisely, an elliptic operator D on a compact manifold has a (non-unique) pseudoinverse D′ such that DD′ − 1 and D′D − 1 are both compact operators. An important consequence is that the kernel of D is finite dimensional, because all eigenspaces of compact operators are finite-dimensional. (The pseudoinverse of an elliptic differential operator is almost never a differential operator. However, it is an elliptic pseudodifferential operator.)

[edit] The analytical index

As the elliptic differential operator D has a pseudoinverse, it is a Fredholm operator. Any Fredholm operator has an index, defined as the difference between the (finite) dimension of the kernel of D (solutions of Df = 0), and the (finite) dimension of the cokernel of D (the constraints on the right-hand-side of an inhomogeneous equation like Df = g, or equivalently the kernel of the adjoint operator). In other words,

Index(D) = Dim Ker(D) − Dim Coker(D) = Dim Ker(D) − Dim Ker(D*).

This is sometimes called the analytical index of D.

Example. Suppose that the manifold is the circle (thought of as R/Z), and D is the operator d/dx − λ for some complex constant λ. (This is the simplest example of an elliptic operator.) Then the kernel is the space of multiples of exp(λx) if λ is an integral multiple of 2πi and is 0 otherwise, and the kernel of the adjoint is a similar space with λ replaced by its complex conjugate. So D has index 0. This example shows that the kernel and cokernel of elliptic operators can jump discontinuously as the elliptic operator varies, so there is no nice formula for their dimensions in terms of continuous topological data. However the jumps in the dimensions of the kernel and cokernel are the same, so the index, given by the difference of their dimensions, does vary continuously, and can be given in terms of topological data by the index theorem.

[edit] The topological index

The topological index of an elliptic differential operator D between smooth vector bundles E and F on an n-dimensional compact manifold X is given by

ch(D)Td(X)[X],

in other words the value of the top dimensional component of the mixed cohomology class ch(D)Td(X) on the fundamental homology class of the manifold X. Here,

  • Td(X) is the Todd class of the manifold X.
  • ch(D) is equal to φ−1(ch(d(E,F, σ(D)), where
  • φ is the Thom isomorphism from Hk(X,Q) to Hn+k(B(X)/S(X),Q)
  • B(X) is the unit ball bundle of the cotangent bundle of X, and S(X) is its boundary, and p is the projection to X.
  • ch is the Chern character from K-theory K(X) to the rational cohomology ring H(X,Q).
  • d(p*E,p*F, σ(D)) is the "difference element" of K(B(X)/S(X)) associated to two vector bundles p*E and p*F on B(X) and an isomorphism σ(D) between them on the subspace S(X).
  • σ(D) is the symbol of D

Another method of defining the topological index uses K theory. If X is a compact submanifold of a manifold Y then there is a pushforward operation from K(TX) to K(TY). The topological index of an element of K(TX) is defined to be the image of this operation with Y some Euclidean space, for which K(TY) can be naturally identified with the integers Z. This index is independent of the embedding of X in Euclidean space.

[edit] The Atiyah-Singer index theorem

As usual, D is an elliptic differential operator between vector bundles E and F over a compact manifold X.

The index problem is the following: compute the (analytical) index of D using only the symbol s and topological data derived from the manifold and the vector bundle. The Atiyah-Singer index theorem solves this problem, and states:

The analytical index of D is equal to its topological index.

In spite of its formidable definition, the topological index is usually straightforward to evaluate explicitly. So this makes it possible to evaluate the analytical index. (The cokernel and kernel of an elliptic operator are in general extremely hard to evaluate individually; the index theorem shows that we can usually at least evaluate their difference.) Many important invariants of a manifold (such as the signature) can be given as the index of suitable differential operators, so the index theorem allows us to evaluate these invariants in terms of topological data.

Although the analytical index is usually hard to evaluate directly, it is at least obviously an integer. The topological index is by definition a rational number, but it is usually not at all obvious from the definition that it is also integral. So the Atiyah-Singer index theorem implies some deep integrality properties, as it implies that the topological index is integral.

The index of an elliptic differential operator obviously vanishes if the operator is self adjoint. It also vanishes if the manifold X has odd dimension, though there are pseudodifferential elliptic operators whose index does not vanish in odd dimensions.

[edit] Examples

[edit] The Euler characteristic

Suppose that X is a compact oriented manifold. If we take E and F to be the sums of the even and odd exterior powers of the tangent bundle, and D to be d + d* restricted to E, then the topological index of D is the Euler characteristic of M, while the analytical index is the alternating sum of the dimensions of the de Rham cohomology groups.

[edit] Hirzebruch-Riemann-Roch theorem

Take X to be a complex manifold with a complex vector bundle V. We let the vector bundles E and F be the sums of the bundles of differential forms with coefficients in V of type (0,i) with i even or odd, and we let the differential operator D be the sum

\overline\partial + \overline\partial^*

restricted to E. Then the analytical index of D is the holomorphic Euler characteristic of V:

index(D) = Σ(−1)p dim Hp(X,V)

The topological index of D is given by

index(D) = ch(V)Td(X)[X],

the product of the Chern character of V and the Todd class of X evaluated on the fundamental class of X. By equating the topological and analytical indices we get the Hirzebruch-Riemann-Roch theorem. In fact we get a generalization of it to all complex manifolds: Hirzebruch's proof only worked for projective complex manifolds X.

This derivation of the Hirzebruch-Riemann-Roch theorem is more natural if we use the index theorem for elliptic complexes rather than elliptic operators. We can take the complex to be

0→V→V⊗Λ0,1T*(X)→V⊗Λ0,2T*(X)...

with the differential given by \overline\partial. Then the i'th cohomology group is just the coherent cohomology group Hi(X, V), so the analytical index of this complex is the holomorphic Euler characteristic Σ (−1)i dim(Hi(X, V)). As before, the topological index is ch(W)Td(X)[X].

[edit] Hirzebruch signature theorem

The Hirzebruch signature theorem states that the signature of a compact smooth manifold X of dimension 4k is given by the L genus of the manifold. This follows from the Atiyah-Singer index theorem applied to the following signature operator.

The bundles E and F are given by the +1 and −1 eigenspaces of the operator on the bundle of differential forms of X, that acts on k-forms as

ik(k−1)

times the Hodge * operator. The operator D is

d + d*

restricted to E, where d is the Cartan exterior derivative and d* is its adjoint.

The analytic index of D is the signature of the manifold X, and its topological index is the L genus of X, so these are equal.

[edit] The  genus and Rochlin's theorem

The  genus is a rational number defined for any manifold, but is in general not an integer. Borel and Hirzebruch showed that it is integral for spin manifolds, and even if in addition the dimension is 4 mod 8. This can be deduced from the index theorem, which implies that the  genus for spin manifolds is the index of a Dirac operator. The extra factor of 2 in dimensions 4 mod 8 comes from the fact that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so the index is even.

In dimension 4 this result implies Rochlin's theorem that the signature of a 4 dimensional spin manifold is divisible by 16: this follows because in dimension 4 the  genus is minus one eighth of the signature.

[edit] History

The index problem for elliptic differential operators was posed in 1959 by Israel Gelfand (On Elliptic Equations, in Russian Mathematical Surveys, 1960). He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann-Roch theorem and its generalization the Hirzebruch-Riemann-Roch theorem, and the Hirzebruch signature theorem. Hirzebruch and Borel had proved the integrality of the  genus of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Isadore Singer in 1961).

The first research announcement was a 1963 paper (Atiyah-Singer). The proof sketched in this announcement was never published by them, though it appears in the book by Palais. Their first published proof replaced the coborism theory of the first proof with K-theory, and they used this to give proofs of various generalizations in a sequence of papers (listed in the references below).

In 1973 Atiyah, Bott, and Patodi gave a new proof of the index theorem using the heat equation.

In 1988 Getzler, motivated by ideas of Witten and Alvarez-Gaume, gave a short proof of the local index theorem for operators that are locally Dirac operators. (This covers most of the useful cases.)

[edit] Proof techniques

[edit] Pseudodifferential operators

Main article: pseudodifferential operator

Pseudodifferential operators can be explained easily in the case of constant coefficient operators on Euclidean space. In this case, constant coefficient differential operators are just the Fourier transforms of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions.

Many proof of the index theorem use pseudodifferential operators rather than differential operators. The reason for this is that for many purposes there are not enough differential operators. For example, a pseudoinverse of an elliptic differential operator is almost never a differential operator, but is a pseudodifferential operator. Also it is not true in general that all elements of K(B(X),S(X)) are represented by symbols of differential operators, and not all homotopies between symbols can be raised to homotopies between differential operators. However the analogous facts are true for pseudodifferential operators.

Pseudodifferential operators have an order, which can be any real number or even −∞, and have symbols (which are no longer polynomials on the cotangent space), and elliptic differential operators are those whose symbols are invertible for sufficiently large cotangent vectors. Most version of the index theorem can be extended from elliptic differential operators to elliptic pseudodifferential operators.

[edit] Cobordism

The initial proof was based on that of the Hirzebruch-Riemann-Roch theorem (1954), and involved cobordism theory and pseudodifferential operators.

The idea of this first proof is roughly as follows. Consider the ring generated by pairs (X,V) where V is a smooth vector bundle on the compact smooth oriented manifold X, with relations that the sum and product of the ring on these generators are given by disjoint union and product of manifolds (with the obvious operations on the vector bundles), and any boundary of a manifold with vector bundle is 0. This is similar to the cobordism ring of oriented manifolds, except that the manifolds also have a vector bundle. The topological and analytical indices are both reinterpreted as functions from this ring to the integers. Then one checks that these two functions are in fact both ring homomorphisms. In order to prove they are the same, it is then only necessary to check they are the same on a set of generators of this ring. Thom's cobordism theory gives a set of generators; for example, complex vector spaces with the trivial bundle together with certain bundles over even dimensional spheres. So the index theorem can be proved by checking it on these particularly simple cases.

[edit] K theory

Atiyah and Singer's first published proof used K theory rather than cobordism. If i is any inclusion of compact manifolds from X to Y, they defined a 'pushforward' operation i! on elliptic operators of X to elliptic operators of Y that preserves the index. By taking Y so be some sphere that X embeds in this reduces the index theorem to the case of spheres. If Y is a sphere and X is some point embedded in Y, then any elliptic operator on Y is the image under i! of some elliptic operator on the point. This reduces the index theorem to the case of a point, when it is trivial.

[edit] Heat equation

[edit] Generalizations

  • The Atiyah-Singer theorem applies to elliptic pseudodifferential operators in much the same way as for elliptic differential operators. In fact, for technical reasons most of the early proofs worked with pseudodifferential rather than differential operators: their extra flexibility made some steps of the proofs easier.
  • Instead of working with an elliptic operator between two vector bundles, it is sometimes more convenient to work with an elliptic complex
0 → E0 → E1 →E2 → ... → Em →0
of vector bundles. The difference is that the symbols now form an exact sequence (off the zero section). In the case when there are just two non-zero bundles in the complex this implies that the symbol is an isomorphism off the zero section, so an elliptic complex with 2 terms is essentially the same as an elliptic operator between two vector bundles. Conversely the index theorem for an elliptic complex can easily be reduced to the case of an elliptic operator: the two vector bundles are given by the sums of the even or odd terms of the complex, and the elliptic operator is the sum of the operators of the elliptic complex and their adjoints, restricted to the sum of the even bundles.
  • If the manifold is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index. These conditions can be local (like demanding that the sections in the domain vanish at the boundary) or more complicated global conditions (like requiring that the sections in the domain solve some differential equation). The local case was worked out by Atiyah and Bott, but they showed that many interesting operators (e.g., the signature operator) do not admit local boundary conditions. To handle these operators, Atiyah, Patodi and Singer introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder. This point of view is adopted in the proof of Melrose of the Atiyah-Patodi-Singer index theorem (see reference below).
  • Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space Y. In this case the index is an element of the K-theory of Y, rather than an integer. If the operators in the family are real, then the index lies in the real K-theory of Y. This gives a little extra information, as the map from the real K theory of Y to the complex K theory is not always injective.
  • If there is a group action of a group G on the compact manifold X, commuting with the elliptic operator, then one replaces ordinary K theory with equivariant K-theory. Moreover, one gets generalizations of the Lefschetz fixed point theorem, with terms coming from fixed point submanifolds of the group G.
  • Atiyah showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient. The kernel of the elliptic operator is in general infinite dimensional in this case, but it is possible to get a finite index using the dimension of a module over a von Neumann algebra; this index is in general real rather than integer valued. This version is called the L2 index theorem, and was used by Atiyah and Wilfried Schmid to rederive properties of the discrete series representations of semisimple Lie groups. See Astérisque 32-3 (9176) 43-72.

[edit] Abel Prize citation

When Michael Atiyah and Isadore Singer were awarded the Abel Prize by the Norwegian Academy of Science and Letters in 2004, the prize announcement explained the Atiyah–Singer index theorem in these words:

   
“
 Scientists describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas, called differential equations, involving their rates of change. Such formulas may have an "index," the number of solutions of the formulas minus the number of restrictions that they impose on the values of the quantities being computed. The Atiyah–Singer index theorem calculated this number in terms of the geometry of the surrounding space.

A simple case is illustrated by a famous paradoxical etching of M. C. Escher, "Ascending and Descending," where the people, going uphill all the time, still manage to circle the castle courtyard. The index theorem would have told them this was impossible.
   
”

[edit] References

[edit] Books

[edit] Papers

The papers by Atiyah are reprinted in volume 3 of his collected works, ISBN 0-19-853277-6.

  • Atiyah, Michael F. and Singer, Isadore M., The Index of Elliptic Operators on Compact Manifolds, Bull. Amer. Math. Soc. 69, 322-433, 1963. An announcement of the index theorem.
  • Atiyah, Michael F. and Singer, Isadore M., The Index of Elliptic Operators I Ann. Math. 87, 484-530, 1968. This gives a proof using K theory instead of cohomology.
  • M. F. Atiyah; G. B. Segal The Index of Elliptic Operators: II The Annals of Mathematics 2nd Ser., Vol. 87, No. 3 (May, 1968), pp. 531-545 This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K thoery.
  • Atiyah, Michael F. and Singer, Isadore M.,The Index of Elliptic Operators III.The Annals of Mathematics 2nd Ser., Vol. 87, No. 3 (May, 1968), pp. 546-604. This papers hows how to convert from the K-theory version to a version using cohomology.
  • Atiyah, Michael F. and Singer, Isadore M.,The Index of Elliptic Operators IV. The Annals of Mathematics 2nd Ser., Vol. 93, No. 1 (Jan., 1971), pp. 119-138. This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family.
  • Atiyah, Michael F. and Singer, Isadore The Index of Elliptic Operators V. The Annals of Mathematics 2nd Ser., Vol. 93, No. 1 (Jan., 1971), pp. 139-149. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information.
  • M. F. Atiyah; R. Bott A Lefschetz Fixed Point Formula for Elliptic Differential Operators. Bull. Am. Math. Soc. 72 (1966), 245-50. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
  • M. F. Atiyah; R. Bott A Lefschetz Fixed Point Formula for Elliptic Complexes: A Lefschetz Fixed Point Formula for Elliptic Complexes: I II. Applications The Annals of Mathematics 2nd Ser., Vol. 86, No. 2 (Sep., 1967), pp. 374-407 and Vol. 88, No. 3 (Nov., 1968), pp. 451-491. These gives the proofs and some applications of the results announced in the previous paper.
  • I. M. Gel'fand, On elliptic equations, Russ. Math.Surv. 15 (3) (1960) p 113-123, reprinted in volume 1 of his collected works, p. 65-75, ISBN 0-387-13619-3. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data.
  • E. Getzler, Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem Commun. Math. Phys., 92 (1983) pp. 163–178
  • E. Getzler, A short proof of the local Atiyah–Singer index theorem Topology, 25 (1988) pp. 111–117

[edit] External links

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