All Questions
19 questions
14
votes
1
answer
573
views
Different proof techniques of the Atiyah-Singer index theorem
I am aware of the usual K-theoretical (cobordism, operator algebras) and heat kernel proofs of the index theorem, as answered in other questions in this site, e.g. here.
However, I recently read this ...
1
vote
0
answers
151
views
Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character
In Witten's 1989 QFT and Jones polynomial paper,
he wrote in eq.2.22 that
Atiyah Patodi Singer theorem says that the combination:
$$
\frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi}
$$
is a ...
- 447
5
votes
0
answers
280
views
Was an index theorem for manifold with local boundary condition proven?
I would like to ask a question on the bibliography of the index theorems on manifold with boundary.
Before my bibliographical research my understanding of the field was that for manifold with boundary,...
- 51
15
votes
1
answer
1k
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 ...
- 355
7
votes
0
answers
253
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|^...
- 91
2
votes
1
answer
140
views
Could an inverse of (weak) Morse inequality exists in some special case?
Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:
Problem
$M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse ...
- 697
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 ...
- 2,995
5
votes
0
answers
238
views
Tensor product of "difference bundles" ( Atiyah construction)
There is a well-known in index theory "difference bundle" construction of Atiyah( see for example the original paper "Clifford modules"). And also there is a corresponding formula for the tensor ...
- 51
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 ...
- 9,870
2
votes
0
answers
105
views
Multiplicativity of the analytic index (or of kernel bundle)
What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators.
In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
- 1,173
7
votes
0
answers
359
views
Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices
Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $...
- 186
9
votes
0
answers
268
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 ...
- 405
62
votes
3
answers
6k
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 ...
- 9,330
4
votes
1
answer
261
views
Lifting a differential operator
Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over compact manifold $M$. If $M$ is not simply connected one can construct the universal ...
- 9,330
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 $d+d^*$ (...
- 9,330
10
votes
0
answers
739
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 ...
- 9,019
9
votes
0
answers
408
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 ...
- 6,829
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 ...
- 1,415
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 (...
- 405