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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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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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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
15 votes
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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
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13 votes
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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
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12 votes
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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
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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
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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
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10 votes
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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
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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
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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
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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
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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
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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
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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
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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
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162 views

Vanishing of K-theoretic index and positive scalar curvature

I'm confused about a seemingly basic point about a classical result on positive scalar curvature and would appreciate it if an expert could help me out. Let $M^n$ be a closed spin manifold with ...
geometricK's user avatar
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7 votes
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344 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 $...
Magnus Goffeng's user avatar
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K-homology fundamental class for singular varieties?

Given a smooth $\text{Spin}^c$ compact manifold without boundary $M$, a suitable normalization of the Dirac operator defines the fundamental class of $M$ in Kasparov's $KK(\mathbb{C}, C^0(M))$. This ...
Shaoyun Bai's user avatar
6 votes
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337 views

Atiyah–Singer Index theorem for the pedestrian / layperson

So I came across the so-called Atiyah–Singer Index Theorem (ASIT) and claims of it being an extremely powerful and versatile tool. Question. What is a truly simple application of the ASIT to obtain a ...
dohmatob's user avatar
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6 votes
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160 views

Elliptic operators with with same index but non homotopic symbols

Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$. Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold. In Atiyah-Singer "the index of ...
Overflowian's user avatar
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Spectral flow of Dirac operator twisted by instanton

Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...
Gorapada Bera's user avatar
6 votes
0 answers
325 views

Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?

I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields. For example, the first Chern class of a complex line ...
Joe's user avatar
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209 views

Variation Formula of APS $\eta$-Invariant and Chern-Simons Theory

In Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, the author proved the variation formula of the APS $\eta$-invariant. I have a few questions about their proof. I will ...
Valac's user avatar
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0 answers
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The space of Fredholm operators as a classifying space

Is it true that the space of Fredholm operators on a separable Hilbert space is the classifying space for K-theory in the category of paracompact spaces? Everyone quotes the theorem of Atiyah-Janich ...
user58951's user avatar
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0 answers
284 views

Relative index theorem for Clifford linear Dirac operators

Dear community, there is relative index theorem due to Gromov and Lawson (Thm. 4.18 in POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS) which states that \begin{...
Paul Meier's user avatar
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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,...
Isacu's user avatar
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Integrand in equivariant Atiyah-Patodi-Singer index theorem

I am trying to determine the integrand in the equivariant Atiyah-Patodi-Singer index theorem by Donnelli ("Eta invariants for G-spaces") for spin manifolds if one only has isolated fixed points. In "...
PR_'s user avatar
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0 answers
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About mod 2 Index of Dirac Operators in 3D on Non-Orientable Manifold

I was reading Witten's paper "Fermion Path Integrals and Topological Phases" (https://arxiv.org/abs/1508.04715). He claimed that Indeed, on an orientable 3-manifold, the eigenvalues of the Dirac ...
Weicheng Ye's user avatar
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0 answers
138 views

Local family index theorem, but with Chern class?

Let $\pi:X\to B$ be a proper submersion with spin fibers, and $E\to X$ a Hermitian vector bundle with a unitary connection $\nabla$. Then the local family index theorem for spin Dirac operator twisted ...
Ho Man-Ho's user avatar
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232 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 ...
Brennan's user avatar
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0 answers
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Submodules of a Hilbert space with finite Jones index with respect to a von Neumann algebra

While studying some basic theory of Cartan subalgebras of von Neumann algebras I found the following fact that I couldn't prove: Let $H$ be a Hilbert space, $A$ and $B$ trace von Neumann subalgebras ...
John N.'s user avatar
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How does the Atiyah-Singer index theorem in a relative setting related to "ringed spaces and pseudocoherent complexes of finite tor-dimension"?

I come across the following paragraph from the article Reminiscences of Grothendieck and His School, here is from the part of the interview by Luc Illusie,: " I was indeed looking for an Atiyah-...
Bombyx mori's user avatar
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5 votes
0 answers
710 views

Local Index Formula vs Atiyah Singer Index Theorem

I have a question concerning the so called Local Index Formula by Connes in noncommutative geometry. First issue: why it is called Index Formula? I spoke to one person about this and he gave me the ...
truebaran's user avatar
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4 votes
0 answers
63 views

Pfaffian elements and anomalies

If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...
domenico fiorenza's user avatar
4 votes
1 answer
143 views

Size of Hilbert space in geometric quantization from index theorem

In these notes on geometric quantization by Nair, on page 24, the Bohr-Sommerfeld rule in quantum mechanics is interpreted in terms of the Atiyah-Singer index theorem. To be precise, the polarization ...
Mtheorist's user avatar
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4 votes
1 answer
229 views

Comments and reference-request on books for KK-theory

I am looking for a good reference to learn Kasparov's KK-theory, where my motivation is to understand (and hopefully can do something about) the Atiyah-Singer index theorem in terms of KK-theory. I ...
Ho Man-Ho's user avatar
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4 votes
0 answers
140 views

Push forward of Chern character and index theorem

I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998). I expose here the setup for my ...
BinAcker's user avatar
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4 votes
0 answers
107 views

Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...
BinAcker's user avatar
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4 votes
0 answers
111 views

Index of glued operator

Suppose $X_1$ is a manifold which has a tubular end $\mathbb R^+\times Y$, and $X_2$ is a manifold which has a tubular end $\mathbb R^-\times Y$. Here, $X_1,X_2$ are orientable manifolds and $Y$ is a ...
Mohan Swaminathan's user avatar
4 votes
0 answers
142 views

Definition of the $G$-equivariant index map

My question concerns a statement on page 12 of the following paper of Baum, Connes, and Higson: http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf about the definition ...
geometricK's user avatar
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4 votes
0 answers
243 views

Does a "symbolically elliptic" sequence of operators have an analytic index?

Does a symbolically elliptic sequence of differential operators have an analytic index? cohomology? For example, is there any concrete meaning of the Todd genus of an almost complex manifold in terms ...
Ben Wieland's user avatar
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4 votes
0 answers
310 views

The proof of the splitting principle in equivariant K-theory via flag manifolds

In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely: Let $j: T\...
Zhaoting Wei's user avatar
  • 8,657
3 votes
0 answers
141 views

A question about index of Dirac operator

Let $\Phi: M\to S^n$ be a map from an even-dimensional, $\dim M=n$, spin manifold $M$ with the boundary $\partial M$ to a unit sphere. And $\Phi$ is locally constant near $\partial M$. If we take a ...
Radeha Longa's user avatar
3 votes
0 answers
70 views

A confusion about an assumption in the setting of the local family index theorem

Let $\pi:M\to B$ be a proper submersion with closed, oriented and spin fibers. Then one can state the local family index theorem as an equality of differential forms (ignoring the details here). I am ...
Ho Man-Ho's user avatar
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3 votes
0 answers
50 views

Dependence of Roe algebra and coarse index on the Riemannian metric

Let $(M,g)$ be a spin Riemannian manifold. The coarse index of the Dirac operator $D$ lies in the $K$-theory of the Roe algebra, which I will denote by $C^*(M,g)$ since its construction uses $g$. I ...
geometricK's user avatar
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3 votes
0 answers
311 views

Discrete spectrum of Dirac operator

It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete. For example at least for $d=4$, this ...
annie marie cœur's user avatar
3 votes
0 answers
72 views

Analyticity of the regularized $\eta$-invariant

The APS $\eta$- invariant of an operator $B$ with eigenvalues $\lambda$ is defined as $$\eta = \sum_\lambda sgn (\lambda)$$ which is a divergent sum and it can be regularized as follows: $$\eta(s) = \...
Tom's user avatar
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3 votes
0 answers
83 views

Dixmier trace, Wodzicki residue and topological index

There are a well-known facts about Dixmier trace and Wodzicki residue. Let $P$ be an elliptic pseudodifferential operator of degree $−n$ on a compact Riemannian manifold $(M,g)$, than its Dixmier ...
Aleksandr Alekseev's user avatar
3 votes
0 answers
139 views

$\mathbb{Z}_2$-grading by Hodge star operator (for signature theorem)

This question may be a bit low level for MO but I have not received any attention from the SE post. Consider the algebra of exterior forms $\bigwedge T^*M$ on an even dimensional $n$-manifold $M$. We ...
Guest123412341234's user avatar