- Atiyah–Singer index theorem
mathematicsof 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 Atiyahand Isadore Singerin 1963.
*"X" is a
compactsmooth manifold(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".
The symbol of a differential operator
If "D" is a differential operator of order "n" in "k" variables
:"x"1, ..., "x""k",
then its symbol is the function of 2"k" variables
:"x"1, ... , "x""k", "y"1, ..., "y""k",
given by dropping all terms of order less than "n" and replacing ∂/∂"x""i" by "y""i". So the symbol is homogeneous in the variables "y", of degree "n". The symbol is well defined even though ∂/∂"x""i" does not commute with "x""i" 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 "y"12 + ... + "y""k"2, and so is elliptic as this is nonzero whenever any the "y"i""'s are nonzero. The wave operator has symbol −"y"12 + ... + "y""k"2, which is not elliptic if "k" ≥ 2, as the symbol vanishes for some non-zero values of the "y"s.
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 bundleof "X", homogeneous of degree "n" on each cotangent space. (In general, differential operators transform in a rather complicated way under coordinate transforms (see jet bundle); 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
to the cotangent space of "X". The differential operator is called "elliptic" if the element of Hom("E""x", "F""x") 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 (non-unique)
parametrix(or 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, other than the kernel, are finite-dimensional. (The pseudoinverse of an elliptic differential operator is almost never a differential operator. However, it is an elliptic pseudodifferential operator.)
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 cokernelof "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(λ"x") if λ is an integral multiple of 2π"i" 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.
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 classof the manifold "X".Here,
*Td("X") is the
Todd classof the manifold "X".
*ch("D") is equal to φ−1(ch(d("E","F", σ("D")), where
*φ is the
Thom isomorphismfrom H"k"("X",Q) to H"n+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 characterfrom 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.
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.
The Euler characteristic
Suppose that "X" 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" to be "d" + "d"*, considered as a map from "E" to F. Then the topological index of "D" is the
Euler characteristicof "M" and the analytical index is the alternating sum of the dimensions of the de Rham cohomology groups.
Take "X" to be a
complex manifoldwith 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
restricted to "E". Then the analytical index of "D" is the
holomorphic Euler characteristicof "V":
:index("D") = Σ(−1)"p" dim "H""p"("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
with the differential given by . Then the "i"'th cohomology group is just the coherent cohomology group H"i"("X", "V"), so the analytical index of this complex is the holomorphic Euler characteristic Σ (−1)"i" dim(H"i"("X", "V")). As before, the topological index is ch("V")Td("X") ["X"] .
Hirzebruch signature theorem
Hirzebruch signature theoremstates that the signature of a compact smooth manifold "X" of dimension 4"k" is given by the L genusof 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
:"d" + "d"*
restricted to "E", where "d" is the Cartan
exterior derivativeand "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 Â genus and Rochlin's theorem
Â genusis 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 Â 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 theoremthat the signature of a 4 dimensional spin manifold is divisible by 16: this follows because in dimension 4 the Â genus is minus one eighth of the signature.
The index problem for elliptic differential operators was posed in 1959 by harvs|txt|first=Israel |last=Gel'fand|authorlink=Israel Gelfand|year=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 theoremand its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Hirzebruch and Borel had proved the integrality of the Â genusof 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 Singer in 1961).
The first announcement of the Atiyah–Singer theorem was the paper harv|Atiyah|Singer|1963. The proof sketched in this announcement was never published by them, though it appears in the book harv|Palais|1965. Their first published proof harv|Atiyah|Singer|1968a replaced the
cobordismtheory of the first proof with K-theory, and they used this to give proofs of various generalizations in a sequence of papers from 1968 to 1971, listed in the references below.
harvs|txt=yes|last=Atiyah |author2-link=Raoul Bott|last2=Bott|author3-link=Vijay Kumar Patodi|last3=Patodi|year=1973 gave a new proof of the index theorem using the
heat equation, described in harv|Melrose|1993.
harvtxt|Getzler|1983 motivated by ideas of harvs|txt=yes|authorlink=Edward Witten|last=Witten|year=1982 and Alvarez-Gaume, gave a short proof of the local index theorem for operators that are locally
Dirac operators; this covers many of the useful cases.
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 anelliptic differential operator of positive order is not a differential operator, but is a pseudodifferential operator. Also, there is a direct correspondence between data representating 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 symbolsare invertible for sufficiently large cotangent vectors. Most version 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 theoryand 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, when it is trivial.
harvs|txt=yes|last=Atiyah |author2-link=Raoul Bott|last2=Bott|author3-link=Vijay Kumar Patodi|last3=Patodi|year=1973 gave a new proof of the index theorem using the
heat equation, described in harv|Melrose|1993 and harv|Gilkey|1994. harvtxt|Berline|Getzler|Vergne|2004 describe a simpler heat equation proof exploiting supersymmetry.
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 :Index("D") = dim Ker("D") − dim Ker("D"*) = Tr(e−"tD"*"D") − Tr(e−"tDD"*)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 arehuge cancellations between the terms, which makes it possible to find the leading terms explicitly. These cancellations were later explained using supersymmetry.
*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 "
::0 → "E"0 → "E"1 →"E"2 → ... → "E""m" →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.
manifoldis 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 harvtxt|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 actionof 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".
*harvtxt|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 harvtxt|Atiyah|Schmid|1977 to rederive properties of the discrete series representations of semisimple Lie groups.
Abel Prize citation
Michael Atiyahand Isadore Singerwere awarded the Abel Prizeby the Norwegian Academy of Science and Lettersin 2004, the prize announcement explained the Atiyah–Singer index theorem in these words:
cquote|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.
The papers by Atiyah are reprinted in volumes 3 and 4 of his collected works, harvs|last=Atiyah|year1=1988a|year2=1988b
*citation|id=MR|0420729|last=Atiyah|first= M. F.
chapter=Elliptic operators, discrete groups and von Neumann algebras|title= Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974)|pages= 43–72|series= Asterisque|volume= 32–33|publisher= Soc. Math. France, Paris|year= 1976
*citation|last1= Atiyah|first1= Michael F. |last2=Singer|first2= Isadore M. |title=The Index of Elliptic Operators on Compact Manifolds|journal= Bull. Amer. Math. Soc.|volume= 69|pages= 322–433|year= 1963 |url=http://www.ams.org/bull/1963-69-03/S0002-9904-1963-10957-X/home.html An announcement of the index theorem.
*citation|last1= Atiyah|first1= Michael F. |last2=Singer|first2= Isadore M. |url=http://links.jstor.org/sici?sici=0003-486X%28196805%292%3A87%3A3%3C484%3ATIOEOI%3E2.0.CO%3B2-M |title=The Index of Elliptic Operators I|journal= Ann. Math. |volume=87|pages= 484–530|year= 1968a This gives a proof using K theory instead of cohomology.
*citation |first=M. F. |last=Atiyah|first2= G. B.|last2= Segal |url=http://links.jstor.org/sici?sici=0003-486X%28196805%292%3A87%3A3%3C531%3ATIOEOI%3E2.0.CO%3B2-2 |title=The Index of Elliptic Operators: II|journal= The Annals of Mathematics 2nd Ser.|volume= 87 |issue=3|year= 1968|pages= 531–545 This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K theory.
*citation|last1= Atiyah|first1= Michael F. |last2=Singer|first2= Isadore M. |url=http://links.jstor.org/sici?sici=0003-486X%28196805%292%3A87%3A3%3C546%3ATIOEOI%3E2.0.CO%3B2-V |title=The Index of Elliptic Operators III.|journal=The Annals of Mathematics 2nd Ser.|volume= 87|issue=3|year= 1968b|pages= 546–604 This paper shows how to convert from the K-theory version to a version using cohomology.
*citation|last1= Atiyah|first1= Michael F. |last2=Singer|first2= Isadore M. |url=http://links.jstor.org/sici?sici=0003-486X%28197101%292%3A93%3A1%3C119%3ATIOEOI%3E2.0.CO%3B2-2 |title=The Index of Elliptic Operators IV.|journal= The Annals of Mathematics 2nd Ser.|volume= 93|issue=1|year= 1971|pages= 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.
*citation|last1= Atiyah|first1= Michael F. |last2=Singer|first2= Isadore M. |url=http://links.jstor.org/sici?sici=0003-486X%28197101%292%3A93%3A1%3C139%3ATIOEOV%3E2.0.CO%3B2-V |title=The Index of Elliptic Operators V.|journal=The Annals of Mathematics 2nd Ser.|volume= 93|issue= 1|year= 1971|pages= 139–149. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information.
*citation|first=M. F.|last= Atiyah|first2= R.|last2= Bott |title=A Lefschetz Fixed Point Formula for Elliptic Differential Operators.|journal= Bull. Am. Math. Soc. |volume=72 |year=1966|pages= 245–50
url=http://www.ams.org/bull/1966-72-02/S0002-9904-1966-11483-0/home.html|doi=10.1090/S0002-9904-1966-11483-0 . This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
*citation|first=M. F.|last= Atiyah|first2= R.|last2= Bott |title=A Lefschetz Fixed Point Formula for Elliptic Complexes: I |url=http://links.jstor.org/sici?sici=0003-486X%28196709%292%3A86%3A2%3C374%3AALFPFF%3E2.0.CO%3B2-N
journal=The Annals of Mathematics 2nd Ser.|volume= 86|issue=2 |year= 1967|pages= 374–407 and citation|first=M. F.|last= Atiyah|first2= R.|last2= Bott |title=A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications.
url=http://links.jstor.org/sici?sici=0003-486X%28196811%292%3A88%3A3%3C451%3AALFPFF%3E2.0.CO%3B2-B |journal=The Annals of Mathematics 2nd Ser.|volume=88|issue=3|year= 1968|pages=451–491 These give the proofs and some applications of the results announced in the previous paper.
*citation|id=MR|0650828 |last=Atiyah|first= M.|last2= Bott|first2= R.|last3= Patodi|first3= V. K.|title= On the heat equation and the index theorem|journal= Invent. Math.|volume= 19 |year=1973|pages= 279–330| doi=10.1007/BF01425417citation|id=MR|0650829|title= Errata |journal=Invent. Math.|volume= 28 |year=1975|pages= 277–280 |doi=10.1007/BF01425562|author= Atiyah, M.
*citation|id=MR|0463358|last= Atiyah|first= Michael|last2= Schmid|first2= Wilfried |title=A geometric construction of the discrete series for semisimple Lie groups|journal= Invent. Math.|volume= 42 |year=1977|pages= 1–62|doi=10.1007/BF01389783, citation|id=MR|0550183|title= Erratum:|journal= Invent. Math. |volume= 54 |year=1979|issue= 2|pages=189–192|doi=10.1007/BF01408936|author= Atiyah, Michael
*citation|id=MR|0951894|last= Atiyah|first= Michael|title= Collected works. Vol. 3. Index theory: 1. |series=Oxford Science Publications|publisher= The Clarendon Press, Oxford University Press|publication-place=New York|year= 1988a| isbn= 0-19-853277-6
*citation|id=MR|0951895|last= Atiyah|first= Michael|title= Collected works. Vol. 4. Index theory: 2. |series=Oxford Science Publications|publisher= The Clarendon Press, Oxford University Press|publication-place=New York|year= 1988b| isbn= 0-19-853278-4
*citation|first1=Nicole |last1=Berline|first2= Ezra |last2=Getzler|first3= Michèle|last3= Vergne|title= Heat Kernels and Dirac Operators|year=2004|isbn=3540200622 This gives an elementary proof of the index theorem for the Dirac operator, using the heat equation and supersymmetry.
*citation|authorlink=Jean-Michel Bismut|first=Jean-Michel|last=Bismut|title=The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem.|url=http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WJJ-4D8DXG0-8J-1&_cdi=6880&_user=10&_orig=search&_coverDate=06%2F01%2F1984&_sk=999429998&view=c&wchp=dGLbVzb-zSkWW&md5=0d679ec499c8595e59bbc7b047a752b8&ie=/sdarticle.pdf|journal=J. Funct. Analysis|year=1984|volume=57|pages=56–99|doi=10.1016/0022-1236(84)90101-0 Bismut proves the theorem for elliptic complexes using probabilistic methods, rather than heat equation methods.
*citation|first=I. M. |last=Gel'fand|title=On elliptic equations|journal=Russ. Math.Surv.|volume= 15 |issue=3|year=1960|pages= 113–123|doi=10.1070/rm1960v015n03ABEH004094 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.
*citation|first=E.|last= Getzler|url=http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103940796 |title=Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem |journal=Commun. Math. Phys.|volume= 92 |year=1983|pages= 163–178 |doi=10.1007/BF01210843
*citation|first=E. |last=Getzler|doi=10.1016/0040-9383(86)90008-X |title=A short proof of the local Atiyah–Singer index theorem|journal=Topology|volume= 25 |year=1988|pages= 111–117
* Free online textbook that proves the Atiyah–Singer theorem with a heat equation approach
* Free online textbook.
*citation|first=Richard S. |last=Palais |year=1965|title=Seminar on the Atiyah–Singer Index Theorem| series=Annals of Mathematics Studies|volume=57| isbn=0691080313 This describes the original proof of the theorem (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)
*citation|last=Shanahan|first= P. |url=http://thepiratebay.org/tor/3524522/Shanahan_P._The_Atiyah-Singer_index_theorem__an_introduction_(Le |title=The Atiyah–Singer index theorem: an introduction|isbn=0387086609 |series=Lecture Notes in Mathematics |volume=638|publisher= Springer|year= 1978
*citation|id=MR|0683171|first=E.|last= Witten|title=Supersymmetry and Morse theory|journal= J. Diff. Geom. |volume= 17 |year=1982|pages= 661–692
* Rafe Mazzeo: [http://math.stanford.edu/~mazzeo/Web/Talks/asit3.pdf "The Atiyah–Singer Index Theorem: What it is and why you should care"] . Pdf presentation.
*Raussen, Skau, [http://www.ams.org/notices/200502/comm-interview.pdf Interview with Atiyah, Singer] , Notices AMS 2005.
*R. R. Seeley and other, [http://mmf.ruc.dk/~Booss/recoll.pdf Recollections from the early days of index theory and pseudo-differential operators ]
*springer|id=I/i050650|title=Index formulas|first=M.I. |last=Voitsekhovskii|first2=M.A.|last2=Shubin
*A. J. Wassermann, [http://iml.univ-mrs.fr/~wasserm/ Lecture Notes on the Atiyah–Singer Index Theorem]
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