Atiyah–Singer index theorem


In differential geometry, the Atiyah–Singer index theorem, proved by, states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.

History

The index problem for elliptic differential operators was posed by. 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. Friedrich Hirzebruch and Armand 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.
The Atiyah–Singer theorem was announced by. The proof sketched in this announcement was never published by them, though it appears in the book. It appears also in the "Séminaire Cartan-Schwartz 1963/64" that was held in Paris simultaneously with the seminar led by Richard Palais at Princeton University. The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof replaced the cobordism theory of the first proof with K-theory, and they used this to give proofs of various generalizations in the papers.
If D is a differential operator on a Euclidean space of order n in k variables, then its symbol is the function of 2k variables
, given by dropping all terms of order less than n and replacing by. So the symbol is homogeneous in the variables y, of degree n. The symbol is well defined even though does not commute with because we keep only 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, and so is elliptic as this is nonzero whenever any of the 's are nonzero. The wave operator has symbol, which is not elliptic if, 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. 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 to the cotangent space of X. The differential operator is called elliptic if the element of Hom is invertible for all non-zero cotangent vectors at any point x of X.
A key property 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 parametrix 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, other than the kernel, are finite-dimensional.

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 dimension of the kernel of D, and the dimension of the cokernel of D. In other words,
This is sometimes called the analytical index of D.
Example: Suppose that the manifold is the circle, and D is the operator d/dx − λ for some complex constant λ. Then the kernel is the space of multiples of exp 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 indeed vary continuously, and can be given in terms of topological data by the index theorem.

Topological index

The topological index of an elliptic differential operator between smooth vector bundles and on an -dimensional compact manifold is given by
in other words the value of the top dimensional component of the mixed cohomology class on the fundamental homology class of the manifold.
Here,
One can also define the topological index using only K-theory. If X is a compact submanifold of a manifold Y then there is a pushforward map from K to K. The topological index of an element of
K is defined to be the image of this operation with Y some Euclidean space, for which K can be naturally identified with the integers Z. This map is independent of the embedding of X in Euclidean space. Now a differential operator as above naturally defines an element of K, and the image in Z under this map "is" the topological index.
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 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:
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. Many important invariants of a manifold 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.

Extensions of the Atiyah–Singer index theorem

Teleman index theorem

Due to , :
The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and Lipschitz manifolds,, the extension of Atiyah–Singer's signature operator to Lipschitz manifolds, Kasparov's K-homology and topological cobordism.
This result shows that the index theorem is not merely a differentiable statement, but rather a topological statement.

Connes–Donaldson–Sullivan–Teleman index theorem

Due to, :
This theory is based on a signature operator S, defined on middle degree differential forms on even-dimensional quasiconformal manifolds.
Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds. The work "provides local constructions for characteristic classes based on higher dimensional relatives of the measurable Riemann mapping in dimension two and the Yang–Mills theory in dimension four."
These results constitute significant advances along the lines of Singer's program Prospects in Mathematics. At the same time, they provide, also, an effective construction of the rational Pontrjagin classes on topological manifolds. The paper provides a link between Thom's original construction of the rational Pontrjagin classes and index theory.
It is important to mention that the index formula is a topological statement. The obstruction theories due to Milnor, Kervaire, Kirby, Siebenmann, Sullivan, Donaldson show that only a minority of topological manifolds possess differentiable structures and these are not necessarily unique. Sullivan's result on Lipschitz and quasiconformal structures shows that any topological manifold in dimension different from 4 possesses such a structure which is unique.
The quasiconformal structures and more generally the Lp-structures,
p > n/2,
introduced by M. Hilsum, are the weakest analytical structures on topological manifolds of dimension n for which the
index theorem is known to hold.

Other extensions

Euler characteristic

Suppose that M is a compact oriented manifold. If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d*, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold. The index formula for this operator yields the Chern-Gauss-Bonnet theorem.

Hirzebruch–Riemann–Roch theorem

Take X to be a complex manifold with a holomorphic 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 with i even or odd, and we let the differential operator D be the sum
restricted to E. Then the analytical index of D is the holomorphic Euler characteristic of V:
The topological index of D is given by
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
with the differential given by. Then the i'th cohomology group is just the coherent cohomology group Hi, so the analytical index of this complex is the holomorphic Euler characteristic Σ i dim. As before, the topological index is chTd.

Hirzebruch signature theorem

The Hirzebruch signature theorem states that the signature of a compact oriented 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
times the Hodge * operator. The operator D is the Hodge Laplacian
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.

 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 an even integer 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.

Proof techniques

Pseudodifferential operators

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 proofs 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 of positive order is not a differential operator, but is a pseudodifferential operator.
Also, there is a direct correspondence between data representing elements of K, S) and symbols of elliptic pseudodifferential operators.
Pseudodifferential operators have an order, which can be any real number or even −∞, and have symbols, 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.

Cobordism

The initial proof was based on that of the Hirzebruch–Riemann–Roch theorem, and involved cobordism theory and pseudodifferential operators.
The idea of this first proof is roughly as follows. Consider the ring generated by pairs 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, 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.

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 to 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, where it is trivial.

Heat equation

gave a new proof of the index theorem using the heat equation, see e.g..
The proof is also published in and.
If D is a differential operator with adjoint D*, then D*D and DD* are self adjoint operators whose non-zero eigenvalues have the same multiplicities. However their zero eigenspaces may have different multiplicities, as these multiplicities are the dimensions of the kernels of D and D*. Therefore, the index of D is given by
for any positive t. The right hand side is given by the trace of the difference of the kernels of two heat operators. These have an asymptotic expansion for small positive t, which can be used to evaluate the limit as t tends to 0, giving a proof of the Atiyah–Singer index theorem. The asymptotic expansions for small t appear very complicated, but invariant theory shows that there are huge cancellations between the terms, which makes it possible to find the leading terms explicitly. These cancellations were later explained using supersymmetry.

Theoretical references

The papers by Atiyah are reprinted in volumes 3 and 4 of his collected works,