Timeline for Todd class as an Euler class
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Feb 11, 2023 at 19:01 | comment | added | Pulcinella | You can define $i_*$ for any closed embedding between smooths actually, and $i^*i_*=e$ is still true. | |
| Feb 11, 2023 at 12:46 | comment | added | domenico fiorenza | @Pulcinella The two questions are indeed related: in finite dimensions in the equivariant setting, after localization both $e(N_i)$ and $i^*$ become invertible, so that one can define $i_*$ from the formula $i^*i_*=e(N_i)$. This is another of the reasons inducing me to think that the equivariant setting is the correct one to handle the Euler class of the normal bundle to $i\colon X\to LX$. For Gysin I admit I have never thought to that: it is a very good question! | |
| Feb 9, 2023 at 10:21 | comment | added | Pulcinella | Also I'd be curious whether you think it's necessary to work equivariantly to get an answer. The Atiyah paper worked non equivariantly, but a lot of their intermediate algebra makes it look like equivariant setup is the one to use. | |
| Feb 9, 2023 at 8:26 | comment | added | Pulcinella | @domenicofiorenza That's a good question! It was part of what I was getting at originally. A question I might have for you is if you know whether 1. this regularised Euler class might fit into a Gysin sequence, or 2. if it satisfies a formula like $i^*i_*=e(N_i)$ for $i:X\to LX$. Obviously $i_*$ is undefined at the moment, but ignore that for now... | |
| Feb 9, 2023 at 7:14 | comment | added | domenico fiorenza | @Pulcinella Well, in the finite dimensional case that is the definition of equivariant Chern classes. In the infinite dimensional case the finite dimensional formulas become infinite products and one has to make sense of them. So the real question, I guess, is what singles out the zeta-regularization among the possible ways of giving a meaning to those infinite products? Surely zeta-regularization has good properties (see, e.g., the paper by Quine,Heydari and Song), but I do not know how to completely avoid an element of choice making it absolutely canonical. | |
| Feb 8, 2023 at 5:57 | comment | added | Pulcinella | @domenicofiorenza I've had a look and I'm a bit worried by the fact that your top chern classes (of the normal bundle of $X$ inside $LX$) are defined by hand like in Atiyah's original paper, rather than follow from some more general notion of euler class, from which the formal Weierstrass-regularisation type formulas also fall out. | |
| Feb 6, 2023 at 17:16 | comment | added | domenico fiorenza | I'd say Sections 2 and 6. In Section 6 you'll have to replace every occurrence of $\mathbb{C}/\Lambda$ with $\mathbb{R}/\Lambda$ or ask Mattia for a copy of his thesis, where you can find the computation spelled out in detail | |
| Feb 5, 2023 at 22:47 | comment | added | Pulcinella | @domenicofiorenza Thanks! Could you point to the most relevant section(s)? | |
| Feb 5, 2023 at 16:01 | comment | added | domenico fiorenza | Always unpleasant to mention a paper one has contributed to, but something very close to an answer to the original question can be found in arxiv.org/pdf/2106.14945.pdf . There, only the Witten class case is treated in detail as we considered Atiyah's treatment in "Circular symmetry and stationary-phase approximation" to be complete for the Todd class. Yet, the details on the Todd class case in the spirit of our paper can be found in Mattia Coloma's thesis. | |
| S Sep 17, 2021 at 20:06 | history | bounty ended | CommunityBot | ||
| S Sep 17, 2021 at 20:06 | history | notice removed | CommunityBot | ||
| Sep 10, 2021 at 2:09 | comment | added | David Ben-Zvi | You might also want to look at the closely related literature deriving the Witten and A^ genus from [rigorous mathematical] quantum field theory - see Costello on the Witten genus arxiv.org/abs/1006.5422, and Grady-Gwilliam and Grady on the A^ genus arxiv.org/abs/1110.3533 and arxiv.org/abs/1211.6816 | |
| Sep 10, 2021 at 2:06 | comment | added | David Ben-Zvi | There's a version of this question in derived algebraic geometry (explaining the Todd genus and Grothendieck-Riemann-Roch via derived versions of loop spaces) that's well understood - the idea is due to a visionary but hard to understand paper of Markarian arxiv.org/abs/math/0610553, a related story is explained in arxiv.org/abs/1305.7175 and the final derivation of the Todd genus is due to Kondyrev and Prikhodko arxiv.org/abs/1906.00172 --- see also closely related arxiv.org/abs/1511.03589 and arxiv.org/abs/1804.00879. | |
| S Sep 9, 2021 at 18:40 | history | bounty started | Pulcinella | ||
| S Sep 9, 2021 at 18:40 | history | notice added | Pulcinella | Authoritative reference needed | |
| Feb 7, 2021 at 22:39 | comment | added | Pulcinella | I suspect this might still be unknown: see Atiyah's paper "Circular symmetry and stationary-phase approximation", where it's apparent that these sorts of questions are still not rigorously solved (and Brylinski's more recent book on loop spaces where he repeats this). | |
| Dec 24, 2020 at 21:57 | comment | added | Pulcinella | @kiran Of course. This comes up a lot in papers of Rongmin Lu, e.g. "Regularized Equivariant Euler Classesand Gamma Functions", section 6 digital.library.adelaide.edu.au/dspace/bitstream/2440/50479/8/… | |
| Dec 24, 2020 at 19:14 | comment | added | kiran | Could you include a reference or give a little more context for this "regularized euler class" ? | |
| Dec 23, 2020 at 11:29 | history | asked | Pulcinella | CC BY-SA 4.0 |