The Atiyah–Singer index theorem

By Daniel S. Freed

In memory of Michael Atiyah

Abstract

The Atiyah–Singer index theorem, a landmark achievement of the early 1960s, brings together ideas in analysis, geometry, and topology. We recount some antecedents and motivations, various forms of the theorem, and some of its implications, which extend to the present.

1. Introduction

Consider the Riemann sphere . Let be a finite set, and to each suppose a nonzero integer  is given. A classical problem asks for a meromorphic function  with a zero or pole at each . If , then is a zero of multiplicity ; if , then is a pole of order . The solution is straightforward. Namely, is a rational function, unique up to the constant . In other words, now allowing , the solutions form a one-dimensional complex vector space. If we replace  by a closed Riemann surface of positive genus, then there is an obstruction to existence of a meromorphic function with specified zeros and poles. For example, an elliptic curve can be realized as a quotient of the complex line by the full lattice generated by  for some  with . A meromorphic function on  lifts to a doubly periodic function on , and the single constraint on the zeros and poles of a meromorphic function is . Proceeding from  to a general closed Riemann surface , we encounter more constraints. In fact, the constraints form a vector space whose dimension is the genus of , a topological invariant. Meromorphic functions are solutions to the Cauchy–Riemann equation, a linear elliptic partial differential equation. The solutions and obstructions to this elliptic PDE are “counted” via topology. In a more general form, this is the classical Riemann–Roch theorem.

The Atiyah–Singer index theorem, formulated and proved in 1962–63, is a vast generalization to arbitrary elliptic operators on compact manifolds of arbitrary dimension. The Fredholm index in question is the dimension of the kernel minus the dimension of the cokernel of a linear elliptic operator. The Atiyah–Singer theorem computes the index in terms of topological invariants of the operator and topological invariants of the underlying manifold. The theorem weaves together concepts and results in algebraic topology, algebraic geometry, differential geometry, and linear analysis; its ramifications go far beyond, in number theory, representation theory, operator algebras, nonlinear analysis, and theoretical physics. Furthermore, index theory itself is a sprawling enterprise. The basic Atiyah–Singer theorem spawned numerous generalizations and novel pathways. This paper—a tribute to Michael Atiyah—naturally focuses on aspects of his work and his influence. Even thus restricted, we can only skim the surface of this rich story.

There are antecedents of the index theorem from algebraic geometry and topology on the one hand, and from analysis on the other. We discuss these in turn in §2 and §3. The basic Atiyah–Singer theorem is the subject of §4. The first proof is based on cobordism and in broad outline follows Hirzebruch’s proofs of his signature and Riemann–Roch theorems. The second proof is based on -theory; it is inspired by Grothendieck’s Riemann–Roch theorem. In §5 we take up some of the extensions and variations of the basic theorem. These include an equivariant index theorem, the index theorem for parametrized families of operators, the index theorem for manifolds with boundary, and a few more. At this point our exposition makes a transition from global topological invariants of general linear elliptic operators to local geometric invariants of Dirac operators. Heat equation methods are the subject of §6, the first application being a local index theorem. New geometric invariants of Dirac operators appear in §7. In §8 we turn to physics, which was Atiyah’s focus after the mid-1980s and which provided an unanticipated playground for the circle of ideas surrounding the index theorem. We focus on anomalies in quantum theory, a subject to which Atiyah and Singer made an early contribution.

Each section of this paper has more introductory material, which we recommend even to the casual reader. Also, a lecture based on this paper may be viewed at Reference F1.

Michael had great mathematical and personal charisma. His writings capture his vibrancy, as did his lectures, some of which are available online. He wrote many wonderful expository articles about the index theorem, especially of the early period; you will enjoy perusing them.

I warmly thank Simon Donaldson, Charlie Reid, and Graeme Segal for their careful reading of and comments on an earlier version.

2. Antecedents and motivations from algebraic geometry and topology

Enumerative problems in algebraic geometry often lead to integers that have a topological interpretation. A classical example is the Riemann–Roch formula, which is our starting point in §2.1. The higher-dimensional generalization was taken up by Fritz Hirzebruch in the early 1950s, as we recount in §2.2. A few years later Alexander Grothendieck extended Hirzebruch’s theorem to a relative version, that is, to proper maps of complex manifolds. In the process he introduced -theory for sheaves. His ideas, briefly presented in §2.3, play a fundamental role in variations of the Atiyah–Singer index theorem a decade later. More immediately, as Graeme Segal writes in this volume Reference Seg4, Atiyah and Hirzebruch transported Grothendieck’s -theory over to algebraic topology. Raoul Bott’s computation of the stable homotopy groups of Lie groups, which took place during the same period as Hirzebruch’s and Grothendieck’s work on the Riemann–Roch theorem, is the cornerstone of their theory. Crucial for the index theorem are the resulting integrality theorems, of which we mention a few in §2.4. This led to a question—Why is the -genus an integer for a spin manifold?—which in early 1962 was the immediate catalyst for Atiyah and Singer’s collaboration.

2.1. The Riemann–Roch theorem

Let be a smooth projective curve over , i.e., a one-dimensional closed complex submanifold of a complex projective space. A divisor  is a finite set of points on  with an integer  attached to each point . A divisor determines a holomorphic line bundle on ; let denote the space of holomorphic sections of this bundle. We can describe  as the space of meromorphic functions on  which have a pole of order  at each . A basic problem in the theory of curves is: Compute the dimension of . While this is quite difficult in general, there is a topological formula for , where is a canonical divisor of . (The zero set of a holomorphic 1-form, weighted by the orders of the zeros, is a canonical divisor.)

Theorem 2.1 (Riemann–Roch).

Let be a smooth projective curve, and let be a divisor on . Then

Here is the genus of the curve , its fundamental topological invariant, which is defined to be . Also, is the sum of the integers which define the divisor . If , it can be shown that , so that in that case Equation 2.2 provides a complete solution to the problem of computing . Theorem 2.1 is the classical Riemann–Roch⁠Footnote1 formula. The Atiyah–Singer index theorem is a vast generalization of Equation 2.2, as we will see.

1

Riemann Reference Ri proved the inequality , and then Roch Reference Ro proved the more precise Equation 2.2. Sadly, Roch died of tuberculosis at the age of 26, just months after the 39-year-old Riemann succumbed to tuberculosis.

Backlinks: Reference 1, Reference 2, Reference 3.

Let us immediately note one consequence of the Riemann–Roch formula. Take to be the trivial divisor consisting of no points. Then is the space of constant functions and is the space of holomorphic differentials. We deduce from Equation 2.2 that the latter has dimension . It follows that is an integer, i.e., is even. Therefore, one-half the Euler number  is an integer, our first example of an integrality theorem. The proof is noteworthy: is an integer because it is the dimension of a vector space, namely .

In the last decade of the century, Noether, Enriques, and Castelnuovo generalized the Riemann–Roch inequality and equality to algebraic surfaces; see Equation 2.4 below.

2.2. Hirzebruch’s Riemann–Roch and signature theorems

We skip far ahead to the years 1945–1954 and the work of young Hirzebruch, based on two important developments in geometry. The first, initiated by Leray, is the theory of sheaves. The second are the results in Thom’s thesis, particularly those concerning bordism⁠Footnote2 groups of smooth manifolds. We state two of Hirzebruch’s main results, which are recounted in Reference H1.

2

Thom, Hirzebruch, and many others use “cobordism” in place of “bordism”; Atiyah Reference A10 clarified the relationship.

Backlinks: Reference 1, Reference 2, Reference 3.

Let be a nonsingular projective variety of complex dimension , and let be a holomorphic vector bundle. (In our discussion of curves we used divisors; recall that a divisor determines a holomorphic line bundle, which makes the link to our formulation here.) Then the cohomology groups  are defined via sheaf theory: is the vector space of holomorphic sections of , and for are derived from resolutions of the sheaf of holomorphic sections of . The cohomology groups are finite dimensional, which can be proved using the theory of elliptic differential operators and Dolbeault’s theorem. (See §§3.13.2.) The Euler characteristic is defined as the alternating sum

As for the case of Riemann surfaces, one often wants to compute , but in general depends on more than topological data. On the other hand, the Euler characteristic  does have a topological formula in terms of the Chern classes  and . The special case is the classical Riemann–Roch formula Equation 2.2. For a smooth projective algebraic surface () and the trivial bundle of rank , the result is commonly known as Noether’s formula:

In Equation 2.4 the Chern classes are evaluated on the fundamental class of  given by the natural orientation. The presence of 12 in the denominator gives an integrality theorem for the Chern numbers of a projective surface.

The solution to the Riemann–Roch problem for all —that is, the computation of Equation 2.3—is one of Hirzebruch’s signal achievements. Hirzebruch’s formula is expressed in terms of the Todd polynomials and the Chern character. Suppose that the tangent bundle splits as a sum of line bundles, and set . Then the Todd class is

This is a cohomology class of (mixed) even degree. Similarly, if is a sum of line bundles, with , then the Chern character is

The splitting principle in the theory of characteristic classes allows us to extend these definitions to  and  which are not sums of line bundles.

Theorem 2.7 (Hirzebruch’s Riemann–Roch theorem).

Let be a projective complex manifold, and let be a holomorphic vector bundle. Then

Hirzebruch’s second main theorem, which is a step in the proof of Theorem 2.7, is now called Hirzebruch’s signature theorem. Let  be a closed oriented real differentiable manifold of dimension  for some positive integer . Then there is a nondegenerate symmetric bilinear pairing on the middle cohomology  given by the cup product followed by evaluation on the fundamental class:

The signature  of this pairing is called the signature of . (The term “index” is used in place of “signature” in older literature.) Hirzebruch defines the -class as the polynomial in the Pontrjagin classes of  determined by the formal expression

where are the Chern roots of the complexified tangent bundle.⁠Footnote3 This is analogous to Equation 2.5: one first defines  in case splits as a sum of complex line bundles.

3

The total Pontrjagin class is defined by the expression .

Theorem 2.11 (Hirzebruch’s signature theorem).

The signature of a closed oriented smooth manifold  is

Hirzebruch’s proof uses Thom’s bordism theory Reference T1 in an essential way. Both sides of Equation 2.12 are invariant under oriented bordism and are multiplicative; for the signature, the former is a theorem of Thom Reference T2, §IV. Therefore, it suffices to verify Equation 2.12 on a set of generators of the (rational) oriented bordism ring, which had been computed by Thom. The even projective spaces  provide a convenient set of generators, and the proof concludes with the observation that the -class is characterized as evaluating to 1 on these generators. The Todd class enters the proof of Theorem 2.7 in a similar manner—its value on all projective spaces  is 1 and it is characterized by this property.

2.3. Grothendieck’s Riemann–Roch theorem

Hirzebruch’s Riemann–Roch theorem was extended in a new direction by Grothendieck Reference BS in 1957. A decisive step was Grothendieck’s introduction of -theory in algebraic geometry. Let be a smooth algebraic variety. Define as the free abelian group generated by coherent algebraic sheaves on , modulo the equivalence if there is a short exact sequence . One can replace “coherent algebraic sheaves” by “holomorphic vector bundles” in this definition, and one fundamental result is that the group  is unchanged. Thus Chern classes and the Chern character are defined for elements of . (Grothendieck refines these to take values in the Chow ring of .) If is a morphism of varieties, and a sheaf over , then is the sheaf on  associated to the presheaf . The assignment

extends to a homomorphism of abelian groups , as can be seen from the long exact sequence in sheaf cohomology.

Now let be a proper morphism between nonsingular irreducible quasiprojective varieties. There is a pushforward  in cohomology (or on the Chow rings).

Theorem 2.14 (Grothendieck’s Riemann–Roch theorem).

For we have

This reduces to Hirzebruch’s Riemann–Roch Theorem 2.7 upon taking to be a point and the -theory class of a holomorphic vector bundle.

One route to the Todd class is the special case in which is the inclusion of a divisor and is the class of the structure sheaf . Then for  and is extended by zero to . Let be the line bundle defined by the divisor . Observe that is the normal bundle to  in . The exact sequence of sheaves

leads to the equality

in . Set . Then from Equation 2.17,

On the other hand

Thus up to the Todd class of . To check Theorem 2.14 in this case, rewrite Equation 2.15 using the exact sequence

of vector bundles on  and the multiplicativity of the Todd genus,

This is what we checked in Equation 2.18 and Equation 2.19 for .

It is instructive at this stage to consider the inclusion of the zero section in a rank  vector bundle . Then the sheaf  fits into the exact sequence

of sheaves over . (Compare Equation 2.16.) Here is the (sheaf of sections of the) dual bundle to , and the arrows in Equation 2.22 at  are contraction by . Thus in  we have

where in -theory. Note that is the normal bundle to  in .

2.4. Integrality theorems in topology

One consequence of Hirzebruch’s Riemann–Roch Theorem 2.7 is that the characteristic number on the right hand side of Equation 2.8, which a priori is a rational number, is actually an integer. This integer is identified as a sum and difference of dimensions of vector spaces by the left hand side. On the other hand, the right hand side is defined for any almost complex manifold. Hirzebruch was led to ask (as early as 1954) whether the Todd genus of an almost complex manifold (much less a nonalgebraic complex manifold) is an integer Reference H3. He also asked analogous questions for real manifolds. Define the -classFootnote4 of a real manifold  by the formal expression

4

Hirzebruch had previously defined an -class which differs from the -class by a power of 2, hence the notation .

where are the Chern roots. This is a polynomial in the Pontrjagin classes. Then the Todd class of an almost complex manifold can be expressed as

In particular, depends only on the Pontrjagin classes and the first Chern class. It is reasonable to speculate that it was Equation 2.25 which motivated Hirzebruch to introduce the -class. Furthermore, since the second Stiefel–Whitney class  is the mod 2 reduction of , Hirzebruch asked: If a real manifold  has , i.e., if is a spin manifold, then is an integer?⁠Footnote5 This was proved true (initially up to a power of 2 in Reference BH2) by Borel and Hirzebruch Reference BH3 in the late 1950s using results of Milnor on cobordism Reference Mi1.

5

In Reference H3 Hirzebruch only asks a less sharp divisibility question (Problem 7 of that paper). The more precise form came later, along with the more general question: If a closed real manifold  admits an element  whose reduction mod 2 is , and is defined by Equation 2.25 (with replacing ), then is an integer?

The integrality proved, the obvious question presented itself:

A first answer to this question came from within algebraic topology, though not from traditional Eilenberg–MacLane cohomology theory. When Atiyah and Hirzebruch learned about Grothendieck’s work, they immediately set out to investigate possible ramifications in topology. The first step was to define -theory for arbitrary CW complexes  Reference AH1. The definition is as for algebraic varieties, but with “topological vector bundles” replacing “coherent algebraic sheaves”. The basic building blocks of topology are the spheres, and the calculation of  quickly reduces to that of the stable homotopy groups of the unitary group. By a fortunate coincidence Bott had just computed (in 1957) these homotopy groups Reference B1Reference B2. His periodicity theorem became the cornerstone of the new topological -theory. What results is a cohomology theory which satisfies all of the Eilenberg–MacLane axioms save one, the dimension axiom. Thus was born “extraordinary cohomology”. -theory is the subject of Graeme Segal’s paper in this volume Reference Seg4.

Returning to the Grothendieck program, Atiyah and Hirzebruch formulated a version of the Riemann–Roch theorem for smooth manifolds Reference AH2Reference H2. Let be a smooth map between differentiable manifolds, and suppose is “oriented” in the sense that there exists an element  with

Recall that Grothendieck’s theorem Equation 2.15 is stated in terms of a map . In the topological category we cannot push forward vector bundles, as we could sheaves in the algebraic category, so a new construction is needed.⁠Footnote6 Here we restrict our attention to embeddings of complex manifolds to simplify the presentation.⁠Footnote7 Then Equation 2.17 and Equation 2.23 motivate the definition of . Let be the normal bundle of  in . By the tubular neighborhood theorem, we can identify  with a neighborhood  of  in . The Thom complex is defined on the total space of  by contraction (compare Equation 2.22):

6

The definition of  was not given in the original paper Reference AH2; missing was the Thom class in -theory, which is closely related to the symbol of the Dirac operator. The Dirac operator enters the story in the collaboration of Atiyah and Singer (§4.1), and then the -theory Thom class and Thom isomorphism appear in Reference ABS, §12. See also the discussion in Reference Seg4, §1.

7

General case: Embed a closed manifold  in a sphere, and so factor an arbitrary map into an embedding followed by a projection: the composition . Bott Periodicity calculates the “shriek map” . For embeddings of real manifolds (with an orientation of the normal bundle) Clifford multiplication on spinors replaces Equation 2.28.

Notice that Equation 2.28 is exact for , so the resulting -theory element is supported on . By the tubular neighborhood theorem it is also defined on , and extension by zero yields the desired element . If is a vector bundle, then is defined by tensoring Equation 2.28 with .

The Atiyah-Hirzebruch Riemann–Roch theorem for smooth manifolds states

Once is defined, the proof is an exercise that compares Thom isomorphisms in -theory and cohomology. Specialize now to , and suppose . Choose the orientation class  to be zero. Then for in Equation 2.29 we deduce, in view of Equation 2.25, that is an integer. This argument by Atiyah and Hirzebruch provided a new proof of the integrality theorem for , and also a topological interpretation of the integer , so a first answer to Equation 2.26.

Still, that explanation was not considered satisfactory. As reported by Atiyah Reference A1, Hirzebruch realized that the signature is the difference in dimensions of spaces of harmonic differential forms, and he asked for a similar analytic interpretation of the -genus . Thus when Singer arrived for a sabbatical stay in Oxford in January 1962, the first question Atiyah asked him was, “Why is the -roof genus an integer for a spin manifold?” Singer Reference S1 responded, “Michael, why are you asking me that question? You know the answer to that.” But Atiyah was looking for something deeper, and he immediately had Singer hooked. By March the duo was in possession of the Dirac operator and the index formula. Then, nine months after that initial conversation, Atiyah and Singer completed the first proof of their eponymous index theorem.

3. Antecedents in analysis

The Atiyah–Singer index theorem brings the worlds of algebraic geometry and algebraic topology together with the worlds of differential geometry and global analysis. Our introduction to the latter in §3.1 begins with foundational theorems about harmonic differential forms and their relationship to cohomology. Geometric elliptic differential operators on Riemannian manifolds play a central role. We take up more general elliptic operators in §3.2, where we also recall basic facts about Fredholm operators. The Fredholm index, an integer-valued deformation invariant of a Fredholm operator, is the eponymous character of index theory. In §3.3 we give the reader an inkling of the activity around indices of elliptic operators during the years 1920–1963.

3.1. De Rham, Hodge, and Dolbeault

We begin with the de Rham and Hodge theorems, which exemplify the relationship between elliptic linear differential equations and topology. Let be a smooth -dimensional manifold, and consider the complex of differential forms