Atiyah-Singer index rate

from Wikipedia, the free encyclopedia

The Atiyah-Singer index theorem is a central statement from global analysis , a mathematical branch of differential geometry . It says that for an elliptic differential operator on a compact manifold the analytic index ( Fredholm index , closely related to the dimension of the solution space ) is equal to the apparently more general, but easier to compute topological index . (This is defined using topological invariants .)

So you can do without calculating the analytical index, which is difficult to determine. The sentence is therefore particularly important for applications, although it emphasizes the abstract.

Many other important theorems, such as the Riemann-Roch theorem or the Gauss-Bonnet theorem, are special cases. The theorem was proven in 1963 by Michael Atiyah and Isadore M. Singer : They received the Abel Prize 2004 for it . The theorem also has applications in theoretical physics .

A quote

First of all, a quote from the official tribute to Atiyah and Singer for the 2004 Abel Prize : “Scientists describe the world in terms of sizes and forces that are changeable in space and time. The laws of nature are often expressed by formulas for their rate of change, called differential equations. Such formulas can have an 'index', the number of solutions to the formulas minus the number of constraints they place on the quantities to be calculated. The Atiyah-Singer index theorem calculates this index from the geometry of the underlying space. A simple example is provided by M. C. Escher's famous paradoxical picture 'ascending and descending', in which people climb up all the time but still move in a circle around the courtyard. The index rate would have told them that that was impossible. "

Mathematical preliminaries

  • is a smooth compact manifold (without a margin) .
  • and are smooth vector bundles across .
  • is an elliptic differential operator of on . This means that in local coordinates it acts as a differential operator that maps smooth sections of the vector bundle onto those of .

The symbol of a differential operator

If a differential operator of order in variables

its "symbol" is a function of the variable

,

which is given by the fact that all the terms of a lower order than omits and by replacing. The symbol is thus homogeneous in the variables of degree . It is well defined (although not commutated), since only the highest term was kept and differential operators commute down to lower terms . The operator is called elliptic if the symbol is not equal to 0, if at least one is not equal to 0.

Example for : The Laplace operator in variables has the symbol and is therefore elliptical, since it is not equal to 0 if one of the is not equal to 0. The wave equation, on the other hand, has the symbol which is for not elliptical. The symbol disappears here for some not equal to 0.

The symbol of a differential operator of order on a smooth manifold is similarly defined using local coordinate maps. It is a function of the cotangent bundle of and is homogeneous in degree on each cotangent space . More general is the symbol of a differential operator between two vector bundles and an intersection of the withdrawn bundle

to the cotangent space of . The differential operator is called elliptic if the element of is invertible for all cotangential vectors not equal to 0 at every point of .

An important property of elliptic operators is that they are almost invertible, which is closely related to the fact that their symbols are almost invertible. More precisely, this means that an elliptic operator on a compact manifold has a (ambiguous) parametrix , so that both and its adjoint, compact operators . The parametrix of an elliptical differential operator is usually not a differential operator, but an integral operator (more generally: an elliptical so-called pseudo differential operator ). An important consequence is that the kernel of is finite dimensional, since all eigenspaces of compact operators are finite dimensional.

The two indices

The analytical index

Since the elliptic differential operator has a pseudo inverse , it is a Fredholm operator . Every Fredholm operator has an index, defined as the difference between the (finite) dimensions of the kernel of ( i.e. the solutions of the homogeneous equation ) and the coke kernel of (the restricting conditions on the right-hand side ( inhomogeneous equations such as ), or equivalently: the kernel of the adjoint operator ), i.e.

Example for : Assume that the manifold is a circle (as thought) and the operator for a complex constant . (This is the simplest example of an elliptic operator). Then the kernel of , i.e. the part mapped to zero, is equal to the space spanned by all terms of the form , if , and equal to 0 in the other cases. The kernel of the adjoint operator is simply replaced by its complex-conjugate.

In this example it has the index 0, as with self-adjoint operators , although the operator is not self-adjoint. The example shows at the same time that the kernel and coke of an elliptic operator can jump discontinuously if the elliptic operator is varied in such a way that the “other cases” mentioned above are covered. So even with this simple example there is no nice formula for the index expressed by topological quantities . Since the jumps in the dimensions of core and coke core are the same, their difference, the index, is not zero, but changes continuously and can be expressed in terms of topological quantities.

The topological index

In the following the compact manifold is additionally -dimensional and orientable and denotes its tangential bundle . Furthermore, let and be two Hermitian vector bundles.

The topological index of the elliptic differential operator between the intersections of the smooth vector bundles and is through

given. In other words, it is the value of the component of the mixed höchstdimensionalen cohomology (mixed cohomology class) on the fundamental homology class of where:

  • the so-called Todd class of . The i-th Chern class of the bundle is denoted by. Further be
  • ,
where the Thom isomorphism , the unit ball bundle of the cotangent bundle , its edge, the Chern character of the topological K-theory on the rational cohomology ring , the difference element ("difference element") from to the two vector bundles and and the main symbol which is an isomorphism on the subspace . The object can also be understood equivalently as the Chern character of the index bundle .

Another method of defining the topological index systematically uses the K-theory. If there is a compact submanifold of , then there is a pushforward operation of to . The topological index of an element of is defined as the image of this operation, where is a Euclidean space that can be identified naturally with the integers . The index is independent of the embedding of in Euclidean space .

The Atiyah-Singer index theorem (equality of indices)

Let D again be an elliptic differential operator between two vector bundles E and F over a compact manifold X.

The index problem consists of the following task: The analytic index of D is to be calculated, using only the symbols and topological invariants of the manifold and the vector bundles. The Atiyah-Singer Index Theorem solves this problem and, in a nutshell:

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

The topological index can generally be calculated well, despite its complex formulation, and in contrast to the analytical index. Many important invariants of the manifold (like the signature) can be expressed as an index of certain differential operators and thus by topological quantities.

Although the analytical index is difficult to calculate, it is at least an integer, while the topological index could also be "rational" by definition and "wholeness" is by no means obvious. The index rate also makes deep-seated statements for the topology.

The index of an elliptic operator obviously vanishes if the operator is self-adjoint . Even with manifolds of odd dimensions the index disappears for elliptic differential operators, but there are elliptic pseudo differential operators whose index does not vanish for odd dimensions.

Examples

The Euler-Poincaré characteristic

The manifold is compact and orientable. Let the sum of the even outer products of the cotangential bundle, the sum of the odd outer products, be a mapping from to . Then the topological index of is the Euler-Poincaré characteristic of and the analytical index results from the index theorem as the alternating sum of the dimensions of the De Rham cohomology groups .

The Hirzebruch-Riemann-Roch theorem

Let be a complex manifold with a complex vector bundle . The vector bundles and are the sums of the bundles of the differential forms with coefficients in of type (0, i) , where is even or odd. Let the differential operator be the sum

restricted to , where is the Dolbeault operator and its adjoint operator. Then the analytic index of is the holomorphic Euler-Poincaré characteristic of :

The topological index of is through

given as the product of the Chern character of and the Todd class of , calculated on the fundamental class of

Equating topological and analytical indexes yields the Hirzebruch-Riemann-Roch theorem, which generalizes the Riemann-Roch theorem. In fact, Hirzebruch only proved the theorem for projective complex manifolds in the above form; it is generally valid for complex manifolds.

This derivation of the Hirzebruch-Riemann-Roch theorem can also be derived “more naturally” using the index theorem for elliptic complexes instead of elliptic operators. The complex is through

given with the differential . Then the -th cohomology group is precisely the coherent coherent group , so that the analytical index of this complex is the holomorphic Euler characteristic . As before is the topological index .

Signature set by Hirzebruch

Let be an oriented compact smooth manifold of dimension . Hirzebruch's signature theorem says that the signature of the manifold is given by the L gender of . This follows from the Atiyah-Singer index theorem applied to the signature operator . The Atiyah-Singer index rate says in this special case

This statement was proven in 1953 by Friedrich Hirzebruch using the cobordism theory .

The Â-gender and Rochlin's theorem

The gender is a rational number defined for any manifold, but in general not an integer. Armand Borel and Friedrich Hirzebruch showed that it is whole for spin manifolds and even if the dimension is also congruent 4 modulo 8. This can be deduced from the index theorem, which assigns the index of a Dirac operator to the gender for spin manifolds . The extra factor 2 in the dimensions that are congruent 4 modulo 8 comes from the quaternionic structure of the core and co-core of the Dirac operator in these cases. As complex vector spaces, they have even dimensions, so the index is also even.

In dimension 4, Rochlin's theorem follows from this that the signature of a 4-dimensional spin manifold is divisible by 16, since there the signature is equal to (−8) times the Â-gender.

history

The index problem for elliptic differential operators was posed by Israel Gelfand in 1959 ( On Elliptic Equations, in the Russian Mathematical Surveys, 1960). He noticed the homotopy invariance of the (analytic) index and asked for a formula for the index that contains only topological invariants. Further motivations for the index rate were the Riemann-Roch theorem and its generalization, the Hirzebruch-Riemann-Roch theorem, and Hirzebruch's signature theorem. As mentioned, Hirzebruch and Borel had proven the integer of the Â-gender of a spin manifold, and Atiyah suggested that this could be explained if it were the index of the Dirac operator (which is mainly treated in physics ) . (In mathematics, this operator was "rediscovered" in 1961 by Atiyah and Singer.)

The first announcement was published in 1963, but the proof outlined there was never published (but appeared in the anthology of Palais). The first published proof used the K-theory instead of the cobordism theory , which was also used for the following proofs of various generalizations.

In 1973 Atiyah, Raoul Bott and Patodi gave a new proof with the help of the heat conduction equation (diffusion equation).

In 1983 Ezra Getzler gave a “brief” proof of the local index theorem for Dirac operators (which includes most of the standard cases) using supersymmetric methods, based on ideas from Edward Witten and Luis Alvarez-Gaumé .

Michael Francis Atiyah and Isadore Manual Singer were awarded the Abel Prize in 2004 for proving the index set .

Evidence Techniques

Pseudo differential operators

For example, while differential operators with constant coefficients in Euclidean space are Fourier transforms of multiplication with polynomials, the corresponding pseudo differential operators are Fourier transforms of multiplication with more general functions. Many proofs of the index theorem use such pseudo differential operators, since there are “not enough” differential operators for many purposes. For example, the pseudo-inverse of an elliptic differential operator is almost never a differential operator, but it is a pseudo differential operator. For most versions of the index set there is such an extension to pseudo differential operators. The proofs become more flexible by using these generalized differential operators.

Cobordism

The original proof, like that of the Hirzebruch-Riemann-Roch theorem by Hirzebruch in 1954, was based on the use of the Kobordism theory and also used pseudo differential operators.

The idea of ​​the proof was roughly as follows: Consider the ring created by the pairs , where there is a smooth vector bundle on a smooth, compact, orientable manifold . The sum and the product in this ring are given by the disjoint union and the product of manifolds (with corresponding operations on the vector bundles). Every manifold that is the edge of a manifold disappears in this calculus. The procedure is the same as in the cobordism theory, only that here the manifolds also carry vector bundles. Analytical and topological index are interpreted as functions on this ring with values ​​in the integers. After checking whether the indices interpreted in this way are ring homomorphisms, one only has to prove their equality for the generators of the ring. These result from René Thom's cobordism theory, e.g. B. complex vector spaces with the trivial bundle and certain bundles over spheres of even dimension. The index rate only has to be considered on relatively simple manifolds.

K theory

The first published evidence by Atiyah and Singer used K-theory instead of cobordism. Let be an arbitrary inclusion of compact manifolds from to Then one can define a pushforward operation from elliptic operators to to elliptic operators to which receives the index. If one takes it as an embedded sphere, the index rate is reduced to the case of spheres. If is a sphere and a point is embedded in, then every elliptic operator on is under the image of an elliptic operator on the point. This reduces the index rate to the case of one point where it is trivial.

Thermal equation

Atiyah, Raoul Bott and Vijay Kumar Patodi gave a new proof of the index theorem using the conduction equation .

If a differential operator with adjungiertem operator is, are and self-adjoint operators whose non-vanishing eigenvalues have the same multiplicity. However, their eigenspaces with eigenvalue zero can have different multiples, since these are the dimensions of the kernels of and . The index of is therefore through

given for any positive . The right hand side of the equation is given by the trace of the difference in the kernels of two heat conduction operators. Their asymptotic expansion for small positives can be used to determine the limit value towards 0 and thus to provide a proof of the Atiyah-Singer index theorem. The limit value smaller is rather complicated at first glance, but since many of the terms cancel each other out, the leading terms can be specified explicitly. A deeper reason why many of the terms cancel each other was later provided by methods of theoretical physics, namely by so-called supersymmetry .

Generalizations

  • Instead of using elliptic operators between vector bundles, it is sometimes more advantageous to work with an elliptic complex of vector bundles:
The difference is that the symbols now form an exact sequence (outside of the zero section). In the case of exactly two bundles (not equal to zero) in the complex, it follows that the symbol outside the zero intersection is an isomorphism, so that an elliptic complex with two terms is essentially the same as an elliptic operator between two vector bundles. Conversely, the index set of an elliptic complex can easily be reduced to the index set of an elliptic operator. The two vector bundles are given by the sum of the even or odd terms in the complex, and the elliptic operator is the sum of the operators of the elliptic complex and its adjoints, restricted to the sum of the even bundles.
  • If the manifold has a boundary, the domain of the elliptic operator must be restricted to ensure a finite index. These additional conditions can be local and demand the disappearance of the sections (the vector bundles) on the edge, or more complex global conditions, e.g. B. require the fulfillment of certain differential equations by the cuts. Atiyah and Bott examined the local case, but they also showed that many interesting operators such as the signature operator do not allow local boundary conditions. Atiyah, Patodi and Singer introduced global boundary conditions for these cases, which correspond to the appending of a cylinder to the edge of the manifold and which restricted the domain of definition to such sections that are square-integrable along the cylinder. The view was used by Melrose in his proof of the Atiyah-Patodi-Singer index theorem.
  • Instead of an elliptic operator, one can consider whole families, parameterized by a space. In this case, the index is an element of the K-theory of instead of an integer. If the operators are real, the index in the real K-theory is of which, compared to the complex K-theory, sometimes provides some additional information about the manifold.
  • If there is an action of a group explained on the compact manifold that commutes with the elliptic operator, the ordinary K-theory is replaced by equivariant K-theory (Atiyah, Bott). Here one obtains generalizations of the Lefschetz fixed point theorem , whereby the fixed points refer to the submanifolds that are invariant under .
  • Atiyah also showed how one can extend the index theorem to some non-compact manifolds if discrete groups with compact quotients operate on them. The kernel of the elliptic operator in this case is mostly infinite dimensional, but one can get a finite index using the dimension of a module over a Von Neumann algebra . In general, this index is real rather than an integer. This L 2 index theorem was used by Atiyah and Wilfried Schmid to derive new sentences about the discrete series in the representation theory of semi-simple Lie groups (Astérisque Vol. 32, No. 3, 1976, pp. 43-72).

literature

  • Michael Francis Atiyah: Collected works. Volume 3: Index theory. 1. Clarendon Press, Oxford 1988, ISBN 0-19-853277-6 (reprint of papers cited below).
  • MF Atiyah, IM Singer: The index of elliptic operators on compact manifolds. In: Bulletin American Mathematical Society. Vol. 69, 1963, ISSN  0273-0979 , pp. 322-433 (announcement).
  • MF Atiyah, R. Bott: A Lefschetz Fixed Point Formula for Elliptic Complexes. I. In: Annals of Mathematics. 2. Series, Vol. 86, No. 2, Sept. 1967, pp. 374-407.
  • MF Atiyah, R. Bott: A Lefschetz Fixed Point Formula for Elliptic Complexes. II. In: Annals of Mathematics. 2. Series, Vol. 88, No. 3, Nov. 1968, pp. 451-491 (Evidence and Applications).
  • MF Atiyah, IM Singer: The index of elliptic operators I. In: Annals of Mathematics. Vol. 87, No. 3, May 1968, ISSN  0003-486X , pp. 484-530 (proof with K-theory), online (PDF; 3.7 MB) .
  • MF Atiyah, GB Segal : The index of elliptic operators II. In: Annals of Mathematics. Vol. 87, No. 3, May 1968, pp. 531-545 (with equivariant K-theory as a kind of Lefschetz fixed point theorem).
  • MF Atiyah, IM Singer: The index of elliptic operators III. In: Annals of Mathematics. Vol. 87, No. 3, May 1968, pp. 546-604 (cohomology instead of K-theory).
  • MF Atiyah, IM Singer: The index of elliptic operators IV. In: Annals of Mathematics. Vol. 93, No. 1, Jan. 1971, pp. 119-138 (with families of operators).
  • MF Atiyah, IM Singer: The index of elliptic operators V. In: Annals of Mathematics. Vol. 93, No. 1, Jan. 1971, pp. 139–149 (real instead of complex operators).
  • Bernhelm Booss : Topology and Analysis. Introduction to the Atiyah-Singer index formula. Springer, Berlin a. a. 1977, ISBN 3-540-08451-7 .
  • Israel Gelfand : On elliptic equations. In: Russian Mathematical Surveys. Vol. 15, No. 3, 1960, pp. 113-123, doi: 10.1070 / RM1960v015n03ABEH004094 (or Gelfand: Collected Papers. Volume 1. Springer, Berlin et al. 1987, ISBN 3-540-13619-3 ).
  • Ezra Getzler : Pseudodifferential operators on supermanifolds and the Atiyah-Singer Index Theorem. In: Communications in Mathematical Physics. Vol. 92, No. 2, 1983, pp. 163-178, online (PDF; 1.7 MB) .
  • Ezra Getzler: A short proof of the local Atiyah-Singer Index Theorem. In: Topology. Vol. 25, No. 1, 1988, ISSN  0040-9383 , pp. 111-117, online (PDF; 455 kB) .
  • Nicole Berline , Ezra Getzler, Michèle Vergne : Heat Kernels and Dirac Operators. Springer-Verlag, Berlin a. a. 2004, ISBN 3-540-20062-2 (proof with diffusion equation and supersymmetry).
  • Richard S. Palais : Seminar on the Atiyah-Singer-Index Theorem (= Annals of Mathematics Studies 57, ISSN  0066-2313 ). Princeton University Press, Princeton NJ 1965.
  • John Roe : Elliptic operators, topology and asymptotic methods (= Chapman & Hall / CRC Research Notes in Mathematics 395). 2nd edition, 1st reprint. Chapman and Hall / CRC, Boca Raton FL et al. a. 2001, ISBN 0-582-32502-1 .
  • MI Voitsekhovskii MA Shubin: Index formulas . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
  • S.-T. Yau (Ed.): The founders of index theory: reminiscences of Atiyah, Bott, Hirzebruch and Singer. International Press, Somerville 2003.

In the physics literature:

  • Tohru Eguchi , Peter B. Gilkey , Andrew J. Hanson: Gravitation, Gauge Theories and Differential Geometry. Physics Reports, Vol. 66, 1980, pp. 213-393.
  • Mikio Nakahara: Geometry, Topology and Physics. Institute of Physics , 2nd edition 2003, Chapter 12 (Index Theorems).

Web links

References and comments

  1. Core and co-core are generally very difficult to calculate, but using the index rate there is a relatively simple formula for the difference between the dimensions.
  2. ^ Nicole Berline, Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of mathematical sciences 298). Berlin u. a. Springer 1992, ISBN 0-387-53340-0 , p. 161.
  3. Hirzebruch: The Signature Theorem. Reminiscences and recreation. Prospects in Mathematics, Annals of Mathematical Studies, Volume 70, 1971, pp. 3-31.
  4. M. Atiyah, R. Bott, VK Patodi: On the Heat Equation and the Index Theorem. Inventiones Mathematicae, Volume 19, 1973, pp. 279-330.
  5. Alvarez-Gaumé: Supersymmetry and the Atiyah-Singer index theorem. In: Comm. Math. Phys. Volume 90, 1983, pp. 161–173, Online ( Memento of the original dated January 29, 2017 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. . Independently, Daniel Friedan gave evidence based on supersymmetry in 1984, Daniel Friedan, P. Windey: Supersymmetric derivation of the Atiyah-Singer Index and the Chiral Anomaly. In: Nuclear Physics. Volume 235, 1984, pp. 395-416, PDF. @1@ 2Template: Webachiv / IABot / projecteuclid.org