Questions tagged [index-theory]

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107 votes
8 answers
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What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem. The ...
Qiaochu Yuan's user avatar
81 votes
3 answers
12k views

Intuitive explanation for the Atiyah-Singer index theorem

My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem. I'm trying to learn the ...
Daniel Moskovich's user avatar
61 votes
3 answers
5k views

Atiyah-Singer theorem-a big picture

So far I made several attempts to really learn Atiyah-Singer theorem. In order to really understand this result a rather broad background is required: you need to know analysis (pseudodifferential ...
truebaran's user avatar
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50 votes
0 answers
12k views

Atiyah's paper on complex structures on $S^6$

M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$. https://arxiv.org/abs/1610.09366 It relies on the topological $K$-theory $KR$ and in ...
David C's user avatar
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48 votes
0 answers
17k views

What is the current understanding regarding complex structures on the 6-sphere?

In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
jdc's user avatar
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45 votes
0 answers
2k views

What is the "real" meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...
Sebastian Goette's user avatar
27 votes
2 answers
2k views

Atiyah-Singer style index theorem for elliptic cohomology?

In 1994, Mike Hopkins wrote a paper called Topological Modular Forms, the Witten Genus, and the Theorem of the Cube. As usual, the introduction was fantastic, explaining the power of various cobordism ...
David White's user avatar
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24 votes
3 answers
3k views

How we do actually compute the topological index in Atiyah-Singer?

This is migrated by math.stackexchange as I did not receive an answer. I do not know if it is too naive for this site. I am taking a lectured class in Atiyah-Singer this semester. While the class is ...
Bombyx mori's user avatar
  • 6,141
21 votes
2 answers
2k views

Applications of Atiyah-Singer using pseudodifferential operators

Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...
AlexE's user avatar
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18 votes
3 answers
3k views

Atiyah-Patodi-Singer Eta invariant and Chern-Simons form

I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant: 1) Is eta a topological invariant (...
Gian's user avatar
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18 votes
2 answers
1k views

What is the role of equivariance in the Atiyah-Singer index theorem?

I'm trying to read through "The index of elliptic operators I," and here's what I understand of the structure of the proof: Define (using purely K-theoretic means) a homomorphism $K_G(TX) \to R(G)$...
Akhil Mathew's user avatar
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18 votes
2 answers
1k views

Karoubi versus Kasparov K-theory

I have the following, probably very elementary question: Let $Cl^{p,q}$ be the Clifford algebra on generators $e_i$, $i=1, \ldots, p+q$ with $e_i e_j = -e_j e_i$ and $e_{i}^{2}=-1$ for $i=1,\ldots,p$, ...
Johannes Ebert's user avatar
17 votes
3 answers
1k views

Geometric meaning of L-genus

Is there any reasonable geometric meaning of the L-genus for smooth manifolds? or perhaps easier, for complex algebraic surfaces? The question came up after a friend and I realized that we don't ...
Mauricio's user avatar
  • 1,415
15 votes
1 answer
999 views

Atiyah's proof of the moduli space of SD irreducible YM connections

In the paper "Self-duality in Four-dimensional Riemannian Geometry" (1978), Atiyah, Hitchin and Singer present a proof that the space of self-dual irreducible Yang-Mills connections is a ...
Quaere Verum's user avatar
15 votes
0 answers
331 views

Beyond smoothness-the clear picture about the notion of a differential form

In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $...
truebaran's user avatar
  • 9,140
14 votes
5 answers
3k views

Generalization of winding number to higher dimensions

Is there a natural geometric generalization of the winding number to higher dimensions? I know it primarily as an important and useful index for closed, plane curves (e.g., the Jordan Curve Theorem), ...
Joseph O'Rourke's user avatar
14 votes
1 answer
365 views

Is it possible to classify finite dimensional vector bundles in terms of Fredholm operators?

Let ${\mathcal K}$ be the space of Fredholm operators on a Hilbert space. It is well known that ${\mathcal K}$ represents $K$-theory. Let ${\mathcal K}_0$ be the path component of ${\mathcal K}$ of ...
Gregory Arone's user avatar
13 votes
1 answer
1k views

Atiyah-Singer for pseudodifferential operators via heat kernel?

The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...
AlexE's user avatar
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13 votes
1 answer
703 views

Atiyah-Singer index theorem, pairing between K-homology and K-theory and Chern character

There is a general (abstract) index theorem in noncommutative geometry: you take a K-theory class and K-homology class (which is represented by a triple $(A,H,F)$) and you pair them together. This ...
truebaran's user avatar
  • 9,140
13 votes
0 answers
1k views

Eta-Invariant and Atiyah-Patodi-Singer Index Theorem

In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-...
Valac's user avatar
  • 615
12 votes
3 answers
812 views

Index of a family of operators

In the usual setting of the Atiyah-Singer index theorem the situation is as follows: we have a closed smooth manifold $M$ without boundary and $D$ is some elliptic differential operator acting on ...
truebaran's user avatar
  • 9,140
12 votes
0 answers
537 views

Elementary-ish geometric proof of Hirzebruch signature theorem for Riemannian 4-manifolds?

The Hirzebruch signature theorem tells us that for a smooth compact oriented 4-manifold, the signature $\sigma(M)$ is proportional to the first Pontryagin number of $M$: $$ 3\sigma(M)= p_1(M) = k \...
David Roberts's user avatar
  • 33.8k
11 votes
2 answers
1k views

Elliptic operators corresponds to non vanishing vector fields

Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ...
Ali Taghavi's user avatar
11 votes
1 answer
586 views

supersymmetry and the de Rham complex

In Alvarez-Gaume's paper "Supersymmetry and the index theorem" there is given a certain supersymmetric Lagrangian whose quantization, apparently, leads to the de Rham Laplacian on the exterior ...
SUSY student's user avatar
10 votes
2 answers
2k views

Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ (...
truebaran's user avatar
  • 9,140
10 votes
1 answer
583 views

Index of Dirac operator and Chern character of symmetric product twisting bundle

I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...
Guest123412341234's user avatar
10 votes
1 answer
769 views

Baum Connes conjecture and Atiyah-Singer index theorem

Baum Connes conjecture is considered as a far generalisation of the Atiyah Singer index theorem (in K-theoretical formulation). I would like to understand how the latter follows from this conjecture. ...
truebaran's user avatar
  • 9,140
10 votes
0 answers
172 views

Baum Connes conjecture and abstract isomorphism

Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...
truebaran's user avatar
  • 9,140
10 votes
0 answers
732 views

What is Quillen's contribution to index theorem?

In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...
Zhaoting Wei's user avatar
  • 8,657
10 votes
0 answers
399 views

Between Being a Connection and Being an Elliptic Operator

Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect ...
S.Z.'s user avatar
  • 463
9 votes
2 answers
1k views

Understanding the analytic index map of the Atiyah-Singer index theorem

Hi, I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn. I do not understand why the analytic index map ...
AlexE's user avatar
  • 2,926
9 votes
1 answer
706 views

Is there a version of the Poincaré–Hopf theorem for manifold with corners?

As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...
Ya He's user avatar
  • 93
9 votes
2 answers
420 views

Matrix of cosecants appearing in equivariant index computations

In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8....
Rudolf Zeidler's user avatar
9 votes
1 answer
463 views

Functoriality for wrong way maps

In the K-theory formulation of the index theorem one defines the topological index in terms of the so called wrong way maps. Those maps are defined for embeddings of compact manifolds $i:X \to Y$: see ...
truebaran's user avatar
  • 9,140
9 votes
0 answers
512 views

Why is the symbol map in Atiyah–Singer paper continuous?

I am reading "Index of elliptic operators: I", by Atiyah and Singer these days and I am trying to understand all the paper. I find it difficult to verify the following statement on page 512:...
Mahmoud Abdelrazek's user avatar
9 votes
0 answers
218 views

Torsion in Atiyah Singer index formula

In the papers of Atiyah and Singer, they first show their index theorem and then derive some index formulas. For the Fredholm index living in the integers, they use the fact that on spheres the Chern ...
InfiniteLooper's user avatar
9 votes
0 answers
256 views

Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2. I was wondering if there exists any reference concerning the ...
Gian's user avatar
  • 405
9 votes
0 answers
404 views

Why does the index of the Dirac operator on a manifold with boundary live inside the Pfaffian line of the boundary Dirac operator?

I am trying to understand a statement from page 72 of What is an elliptic object? by Stolz and Teichner. They have a spin Riemannian 2-manifold $\Sigma$, with boundary 1-manifold $Y$. The Dirac ...
Sam Gunningham's user avatar
8 votes
1 answer
367 views

Do we have a "topological assembly map" in the Baum-Connes conjecture?

In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index $$ \text{a-ind}: K^*_G(TM)\...
Zhaoting Wei's user avatar
  • 8,657
8 votes
1 answer
525 views

Further directions of index theory

The Atiyah-Singer theorem is a major achievement of twentieth century mathematics. It has inspired a lot of work and people started to develop generalizations of this theorem. I would like to know the ...
truebaran's user avatar
  • 9,140
8 votes
1 answer
517 views

KK-theoretical proof of Atiyah-Singer index theorem

Does anyone know of any detailed proof of the Atiyah-Singer Index Theorem using KK-theory/ Kasparov products? References to any papers and textbooks are greatly appreciated. Thanks!
user119667's user avatar
8 votes
1 answer
270 views

Noncommutative Fredholm operators

Let $A$ be a unital $C^*$-algebra and $F:H_A\rightarrow H_A$ a Fredholm operator on the standard Hilbert $A$-module $H_A:=l^2(A)$. Is it true that $\mbox{ker}(F)$ and $\mbox{coker}(F)$ are finitely ...
ernest's user avatar
  • 99
8 votes
1 answer
281 views

Failure of Fredholm property of elliptic PDE systems

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition. What can ...
Tom's user avatar
  • 101
8 votes
0 answers
400 views

Atiyah-Singer theorem in heat kernels and Dirac operators

I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne. I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the ...
user267839's user avatar
  • 5,948
8 votes
0 answers
111 views

Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action. I am trying to understand the Hopf bundle ($G=...
yths's user avatar
  • 181
7 votes
1 answer
516 views

preliminary reading recommendation before embarking on Connes non commutative geometry book?

I want to try to understand non commutative geometry by reading Connes's book ..and I am discovering it is a hard book to read :-) as I miss a lot of background specially in operator algebra and ...
JeffInVermont's user avatar
7 votes
1 answer
690 views

Parametrized Atiyah-Singer index theorem

Let $M$ be any smooth manifold (could be unorientable - I think). Let $E,F \to M$ be two complex vector bundles. Let $S$ be any compact space, and let $D_s:E\to F,s\in S$ be a continuous family ...
Thomas Kragh's user avatar
  • 2,560
7 votes
1 answer
2k views

Normal coordinates near the boundary

Let $M$ be an Riemannian manifold with boundary $\partial M$ and $e_n$ be a unit normal vector on $\partial M$. With respect to $e_n$, around a point $p$ on boundary, we have the usual normal ...
xiangsheng's user avatar
7 votes
1 answer
1k views

What's the relation between the heat kernel proof of the index theorem and deformation quantization?

In the book "Heat kernels and Dirac operators", I found the slogan by Quillen in the Introduction: "Dirac operators are a quantization of the theory of connections, and the supertrace of the heat ...
Zhaoting Wei's user avatar
  • 8,657
7 votes
0 answers
248 views

Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$

The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by $$ \eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
FractalScout's user avatar