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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.
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  • $\begingroup$ Could you include a reference or give a little more context for this "regularized euler class" ? $\endgroup$
    – kiran
    Commented Dec 24, 2020 at 19:14
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    $\begingroup$ 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. $\endgroup$
    – David Ben-Zvi
    Commented Sep 10, 2021 at 2:06
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    $\begingroup$ 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 $\endgroup$
    – David Ben-Zvi
    Commented Sep 10, 2021 at 2:09
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    $\begingroup$ 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. $\endgroup$
    – domenico fiorenza
    Commented Feb 5, 2023 at 16:01
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    $\begingroup$ 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 $\endgroup$
    – domenico fiorenza
    Commented Feb 6, 2023 at 17:16

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