Atiyah–Singer index theorem

The Grothendieck–Riemann–Roch theorem was one of the main motivations behind the index theorem because the index theorem is the counterpart of this theorem in the setting of real manifolds. Now, if there's a map ${\displaystyle f:X\to Y}$ of compact stably almost complex manifolds, then there is a commutative diagram[28]

if ${\displaystyle Y=*}$ is a point, then we recover the statement above. Here ${\displaystyle K(X)}$ is the Grothendieck group of complex vector bundles. This commutative diagram is formally very similar to the GRR theorem because the cohomology groups on the right are replaced by the Chow ring of a smooth variety, and the Grothendieck group on the left is given by the Grothendieck group of algebraic vector bundles.

Due to (Teleman 1983), (Teleman 1984):

For any abstract elliptic operator (Atiyah 1970) on a closed, oriented, topological manifold, the analytical index equals the topological index.

The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and Lipschitz manifolds (Teleman 1980), (Teleman 1983), the extension of Atiyah–Singer's signature operator to Lipschitz manifolds (Teleman 1983), Kasparov's K-homology (Kasparov 1972) and topological cobordism (Kirby & Siebenmann 1977).

This result shows that the index theorem is not merely a differentiability statement, but rather a topological statement.

Due to (Donaldson & Sullivan 1989), (Connes, Sullivan & Teleman 1994):

For any quasiconformal manifold there exists a local construction of the Hirzebruch–Thom characteristic classes.

This theory is based on a signature operator S, defined on middle degree differential forms on even-dimensional quasiconformal manifolds (compare (Donaldson & Sullivan 1989)).

Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds (see page 678 of (Connes, Sullivan & Teleman 1994)). The work (Connes, Sullivan & Teleman 1994) "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 (Singer 1971). At the same time, they provide, also, an effective construction of the rational Pontrjagin classes on topological manifolds. The paper (Teleman 1985) provides a link between Thom's original construction of the rational Pontrjagin classes (Thom 1956) 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 (Sullivan 1979) shows that any topological manifold in dimension different from 4 possesses such a structure which is unique (up to isotopy close to identity).

The quasiconformal structures (Connes, Sullivan & Teleman 1994) and more generally the L^p-structures, p > n(n+1)/2, introduced by M. Hilsum (Hilsum 1999), are the weakest analytical structures on topological manifolds of dimension n for which the index theorem is known to hold.

• 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
  $${\displaystyle 0\rightarrow E_{0}\rightarrow E_{1}\rightarrow E_{2}\rightarrow \dotsm \rightarrow E_{m}\rightarrow 0}$$

  
  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 (1993) of the Atiyah–Patodi–Singer index theorem.
• 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. See also: equivariant index theorem.
• Atiyah (1976) 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 & Schmid (1977) to rederive properties of the discrete series representations of semisimple Lie groups.
• The Callias index theorem is an index theorem for a Dirac operator on a noncompact odd-dimensional space. The Atiyah–Singer index is only defined on compact spaces, and vanishes when their dimension is odd. In 1978 Constantine Callias, at the suggestion of his Ph.D. advisor Roman Jackiw, used the axial anomaly to derive this index theorem on spaces equipped with a Hermitian matrix called the Higgs field.[29] The index of the Dirac operator is a topological invariant which measures the winding of the Higgs field on a sphere at infinity. If U is the unit matrix in the direction of the Higgs field, then the index is proportional to the integral of U(dU)ⁿ⁻¹ over the (n−1)-sphere at infinity. If n is even, it is always zero.
  • The topological interpretation of this invariant and its relation to the Hörmander index proposed by Boris Fedosov, as generalized by Lars Hörmander, was published by Raoul Bott and Robert Thomas Seeley.[30]

Suppose that ${\displaystyle M}$ is a compact oriented manifold of dimension ${\displaystyle n=2r}$. If we take ${\displaystyle \Lambda ^{\text{even}}}$ to be the sum of the even exterior powers of the cotangent bundle, and ${\displaystyle \Lambda ^{\text{odd}}}$ to be the sum of the odd powers, define ${\displaystyle D=d+d^{*}}$, considered as a map from ${\displaystyle \Lambda ^{\text{even}}}$ to ${\displaystyle \Lambda ^{\text{odd}}}$. Then the analytical index of ${\displaystyle D}$ is the Euler characteristic ${\displaystyle \chi (M)}$ of the Hodge cohomology of ${\displaystyle M}$, and the topological index is the integral of the Euler class over the manifold. The index formula for this operator yields the Chern–Gauss–Bonnet theorem.

The concrete computation goes as follows: according to one variation of the splitting principle, if ${\displaystyle E}$ is a real vector bundle of dimension ${\displaystyle n=2r}$, in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles ${\displaystyle l_{1},\,\ldots ,\,l_{r}}$ such that ${\displaystyle E\otimes \mathbb {C} =l_{1}\oplus {\overline {l_{1}}}\oplus \dotsm l_{r}\oplus {\overline {l_{r}}}}$. Therefore, we can consider the Chern roots ${\displaystyle x_{i}(E\otimes \mathbb {C} )=c_{1}(l_{i})}$, ${\displaystyle x_{r+i}(E\otimes \mathbb {C} )=c_{1}{\mathord {\left({\overline {l_{i}}}\right)}}=-x_{i}(E\otimes \mathbb {C} )}$, ${\displaystyle i=1,\,\ldots ,\,r}$.

Using Chern roots as above and the standard properties of the Euler class, we have that ${\textstyle e(TM)=\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )}$. As for the Chern character and the Todd class,[31]

${\displaystyle {\begin{aligned}\operatorname {ch} {\mathord {\left(\Lambda ^{\text{even}}-\Lambda ^{\text{odd}}\right)}}&=1-\operatorname {ch} (T^{*}M\otimes \mathbb {C} )+\operatorname {ch} {\mathord {\left(\Lambda ^{2}T^{*}M\otimes \mathbb {C} \right)}}-\ldots +(-1)^{n}\operatorname {ch} {\mathord {\left(\Lambda ^{n}T^{*}M\otimes \mathbb {C} \right)}}\\&=1-\sum _{i}^{n}e^{-x_{i}}(TM\otimes \mathbb {C} )+\sum _{i<j}e^{-x_{i}}e^{-x_{j}}(TM\otimes \mathbb {C} )+\ldots +(-1)^{n}e^{-x_{1}}\dotsm e^{-x_{n}}(TM\otimes \mathbb {C} )\\&=\prod _{i}^{n}\left(1-e^{-x_{i}}\right)(TM\otimes \mathbb {C} )\\[3pt]\operatorname {Td} (TM\otimes \mathbb {C} )&=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )\end{aligned}}}$

Applying the index theorem,

${\displaystyle \chi (M)=(-1)^{r}\int _{M}{\frac {\prod _{i}^{n}\left(1-e^{-x_{i}}\right)}{\prod _{i}^{r}x_{i}}}\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )=(-1)^{r}\int _{M}(-1)^{r}\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )=\int _{M}e(TM)}$

which is the "topological" version of the Chern-Gauss-Bonnet theorem (the geometric one being obtained by applying the Chern-Weil homomorphism).

Take X to be a complex manifold of (complex) dimension n 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 (0, i) with i even or odd, and we let the differential operator D be the sum

${\displaystyle {\overline {\partial }}+{\overline {\partial }}^{*}}$

restricted to E.

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

${\displaystyle 0\rightarrow V\rightarrow V\otimes \Lambda ^{0,1}T^{*}(X)\rightarrow V\otimes \Lambda ^{0,2}T^{*}(X)\rightarrow \dotsm }$

with the differential given by ${\displaystyle {\overline {\partial }}}$. 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 of V:

${\displaystyle \operatorname {index} (D)=\sum _{p}(-1)^{p}\dim H^{p}(X,V)=\chi (X,V)}$

Since we are dealing with complex bundles, the computation of the topological index is simpler. Using Chern roots and doing similar computations as in the previous example, the Euler class is given by ${\textstyle e(TX)=\prod _{i}^{n}x_{i}(TX)}$ and

${\displaystyle {\begin{aligned}\operatorname {ch} \left(\sum _{j}^{n}(-1)^{j}V\otimes \Lambda ^{j}{\overline {T^{*}X}}\right)&=\operatorname {ch} (V)\prod _{j}^{n}\left(1-e^{x_{j}}\right)(TX)\\\operatorname {Td} (TX\otimes \mathbb {C} )=\operatorname {Td} (TX)\operatorname {Td} \left({\overline {TX}}\right)&=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}\prod _{j}^{n}{\frac {-x_{j}}{1-e^{x_{j}}}}(TX)\end{aligned}}}$

Applying the index theorem, we obtain the Hirzebruch-Riemann-Roch theorem:

${\displaystyle \chi (X,V)=\int _{X}\operatorname {ch} (V)\operatorname {Td} (TX)}$

In fact we get a generalization of it to all complex manifolds: Hirzebruch's proof only worked for projective complex manifolds X.

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 ${\displaystyle i^{k(k-1)}}$ times the Hodge star operator. The operator D is the Hodge Laplacian

${\displaystyle D\equiv \Delta \mathrel {:=} \left(\mathbf {d} +\mathbf {d^{*}} \right)^{2}}$

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.

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.

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(B(X), S(X)) (clutching functions) and symbols of elliptic 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 versions of the index theorem can be extended from elliptic differential operators to elliptic pseudodifferential operators.

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.

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.

Atiyah, Bott, and Patodi (1973) gave a new proof of the index theorem using the heat equation, see e.g. Berline, Getzler & Vergne (1992). The proof is also published in (Melrose 1993) and (Gilkey 1994).

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

${\displaystyle \operatorname {index} (D)=\dim \operatorname {Ker} (D^{*})=\operatorname {Tr} \left(e^{-tD^{*}D}\right)-\operatorname {Tr} \left(e^{-tDD^{*}}\right)}$

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.

• Mazzeo, Rafe. "The Atiyah–Singer Index Theorem: What it is and why you should care" (PDF). Archived from the original on June 24, 2006. Retrieved January 3, 2006.{{cite web}}: CS1 maint: bot: original URL status unknown (link) Pdf presentation.
• Voitsekhovskii, M.I.; Shubin, M.A. (2001) [1994], "Index formulas", Encyclopedia of Mathematics, EMS Press
• Wassermann, Antony. "Lecture notes on the Atiyah–Singer Index Theorem". Archived from the original on March 29, 2017.

• Raussen, Martin; Skau, Christian (2005), "Interview with Michael Atiyah and Isadore Singer" (PDF), Notices of AMS, pp. 223–231
• R. R. Seeley and other (1999) Recollections from the early days of index theory and pseudo-differential operators - A partial transcript of informal post–dinner conversation during a symposium held in Roskilde, Denmark, in September 1998.