31
$\begingroup$

Let $X$ be a relatively nice scheme or topological space.

In various physics papers I've come accross, the Todd class $\text{Td}(T_X)$ is viewed as the Euler class of the normal bundle to $X\to LX$. Here $LX$ is the loop space of $X$. They usually call it $e_{reg}$, the regularised Euler class.

Question: Is there a way of making this notion of regularised Euler class rigorous?

Whatever the correct definition is, it should work for

  • finite dimension vector bundles, where it reduces to usual notion of Euler class,
  • loop spaces, where it reduces to the Todd class.
Improve this question
$\endgroup$
14
  • $\begingroup$ Could you include a reference or give a little more context for this "regularized euler class" ? $\endgroup$ Commented Dec 24, 2020 at 19:14
  • 7
    $\begingroup$ 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. $\endgroup$ Commented Sep 10, 2021 at 2:06
  • 2
    $\begingroup$ 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 $\endgroup$ Commented Sep 10, 2021 at 2:09
  • 2
    $\begingroup$ 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. $\endgroup$ Commented Feb 5, 2023 at 16:01
  • 1
    $\begingroup$ 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 $\endgroup$ Commented Feb 6, 2023 at 17:16

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.