Timeline for Calculation of the top Chern class of spinor bundle over $S^{2n}$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 10, 2021 at 5:37 | comment | added | Radeha Longa | Excuse me, how about my calculation in the link? Thanks in advance @Jonas mathoverflow.net/questions/410288/… | |
| Nov 18, 2021 at 9:56 | comment | added | Jonas | I added a reference and an alternative calculation to my answer. I also noticed some sign trouble if $2n \neq 0$ mod $4$. I am thankful for any proposed corrections. For "the spin operator twisted by..." checkout Chapter II, § 5,6,7 of the reference. | |
| Nov 18, 2021 at 9:52 | history | edited | Jonas | CC BY-SA 4.0 |
Added alternative, corrected sign mistake
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| Nov 17, 2021 at 16:42 | comment | added | Radeha Longa | The top Chern class of any complex spinor bundle over the sphere is zero? How to see it in your above proof?@Jonas | |
| Nov 17, 2021 at 15:36 | comment | added | Radeha Longa | I'm a little confused. Could you please explain the nontrivial intermediate step more explicitly? What's the meaning that the spin operator is twisted by $\Sigma^+-\Sigma^-$? Is there some reference to this? Thanks | |
| Nov 17, 2021 at 15:28 | comment | added | Jonas | The Chern-Gauss-Bonnet theorem can be derived by plugging in the Hodge-de Rham operator $d + d^*$ (mapping from even to odd forms). The (maybe nontrivial) intermediate step is, that for spin manifolds the Hodge-de Rham operator is the spin dirac operator twisted by $\Sigma^+ - \Sigma^-$. I will add a nice reference for this to my answer. I believe, the equation $\hat{A}(TM) ^{-1} e(TM) = ch(\Sigma^+ - \Sigma^-)$ is 'pre-Index Theorem' and probably due to Hirzebruch (maybe in the differentiable Riemann-Roch theorems). | |
| Nov 17, 2021 at 13:56 | comment | added | Radeha Longa | We all know by Atiyah-Singer Index that $\chi(M)=\int_M e(TM)$, and it actually the Chern -Gauss-Bonnet formula, but I'm a little confused with $\chi(M)=\int_M\widehat{A}(TM)ch(\Sigma^+-\Sigma^-)$, why does it hold? Is it also from the Atiyah-Singer index theorem? If so, is here $\chi(M)$ the analytical index of which operator? | |
| S Nov 16, 2021 at 18:11 | review | First answers | |||
| Nov 16, 2021 at 18:22 | |||||
| S Nov 16, 2021 at 18:11 | history | answered | Jonas | CC BY-SA 4.0 |