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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeApr 18th 2010
   • (edited Apr 18th 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownThere has been much attention in the nlab on groupoid cardinality of Leinster/Berger and Euler characteristics of a category of Leinster; and Baez's work with collaborators on groupoidification and his earlier talks and posts on cardinality. Urs noticed that it fits with Freed's ideas on Feynman path integrals and came up with Freed-Schreiber-Ulm "kantization" formulas. While this works well for some finite and TQFT situations, I would like to know what happens in general. Tom and Simon Willerton have been doing infinite extensions to metric spaces and heat-kernel like expressions were important there. This reminded me of the equivariant localization formulas of [[Atiyah-Bott fixed point formula|Atiyah-Bott]], Duisterman-Heckmann, Witten and others which are used in a number of situations but also computing the first term in heat-like expansions for Feyman integrals. In nice examples, like WZNW model, TQFTs, Chern-Simons, the semiclassical expressions give exact result. That is why the "kantization" in its present version gievs a good result. But we should go beyond. Thus we should understand similar expansions from nPOV. So I started creating some elementary background entries (for a while) like [[semiclassical approximation]] and now something closer to topologically oriented people on the blog: [[Lefschetz trace point formula]]. Soon there will appear various related index formulas and equivariant index formulas. I should tell in advance: the usual Lefschetz formulas are for the traces for one mapping; the equivariant ones are for family index by elements of a group. So it is not a number but a numbered valued function on the group. Thus we are arriving to a character. How now about the case when the group is categorified and we have categorified traces ? In that case we should formulate an appropriate ellipticity notion for a complex of operators on 2-bundles, and come after a categorified index formula. And then to get the G-equivariant version for G a 2-group. Some good kantization formulas should come from index formulas of that kind. By transgression, of course, it should be related to ideas like index formula on loop space, like Witten's index theorem; and eventually also to elliptic cohomology. Right ? Edit: yet another thing are anomalies. We took some formulation of anomaly cancellation directly from geometric condition on equipping the space with a particular structure, which then boils down to lift and voila some (nonabelian) obstruction. But originally one looks at amplitudes in QFT, does various standard things to them like zeta function regularization and finds obstruction from there. This took some development in works of Alvarez-Gaume, Jackiw, Stora, Witten and so on, with the role of the geometry of determinant line bundle emphasised by Quillen, Atiyah-Singer, Freed...We did not really go to these origins, and we should I think.

   There has been much attention in the nlab on groupoid cardinality of Leinster/Berger and Euler characteristics of a category of Leinster; and Baez's work with collaborators on groupoidification and his earlier talks and posts on cardinality. Urs noticed that it fits with Freed's ideas on Feynman path integrals and came up with Freed-Schreiber-Ulm "kantization" formulas. While this works well for some finite and TQFT situations, I would like to know what happens in general. Tom and Simon Willerton have been doing infinite extensions to metric spaces and heat-kernel like expressions were important there. This reminded me of the equivariant localization formulas of Atiyah-Bott, Duisterman-Heckmann, Witten and others which are used in a number of situations but also computing the first term in heat-like expansions for Feyman integrals. In nice examples, like WZNW model, TQFTs, Chern-Simons, the semiclassical expressions give exact result. That is why the "kantization" in its present version gievs a good result. But we should go beyond. Thus we should understand similar expansions from nPOV. So I started creating some elementary background entries (for a while) like semiclassical approximation and now something closer to topologically oriented people on the blog: Lefschetz trace point formula. Soon there will appear various related index formulas and equivariant index formulas.

   I should tell in advance: the usual Lefschetz formulas are for the traces for one mapping; the equivariant ones are for family index by elements of a group. So it is not a number but a numbered valued function on the group. Thus we are arriving to a character. How now about the case when the group is categorified and we have categorified traces ? In that case we should formulate an appropriate ellipticity notion for a complex of operators on 2-bundles, and come after a categorified index formula. And then to get the G-equivariant version for G a 2-group. Some good kantization formulas should come from index formulas of that kind. By transgression, of course, it should be related to ideas like index formula on loop space, like Witten's index theorem; and eventually also to elliptic cohomology. Right ?

   Edit: yet another thing are anomalies. We took some formulation of anomaly cancellation directly from geometric condition on equipping the space with a particular structure, which then boils down to lift and voila some (nonabelian) obstruction. But originally one looks at amplitudes in QFT, does various standard things to them like zeta function regularization and finds obstruction from there. This took some development in works of Alvarez-Gaume, Jackiw, Stora, Witten and so on, with the role of the geometry of determinant line bundle emphasised by Quillen, Atiyah-Singer, Freed...We did not really go to these origins, and we should I think.

2. • CommentRowNumber2.
   • CommentAuthordomenico_fiorenza
   • CommentTimeApr 18th 2010
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexThanks Zoran. I agree with all of the above. at present I'll still be playing for a while with toy models (I'm planning to write a few lines on the Yetter model from the topological point of view I've been writing about the DW model lately), but I'd be really happy if this could be just the first step towards an understanding the large picture you're sketching above. In particular anomalies and determinant bundles are something I've been hinting to in a few posts related to kantization here on the forum (there was nothing particularly interesting there, so it's not worth going and searching for them). the idea is that from the correct categorical point of view, anomaly is something we should expect, and not something that happens to be there only since we chose that particular way to compute the path integral (e.g., by zeta function regularization). in other words, when seen fom the correct perspective, we should a priory know that the result will be a section of a nontrivia bundle, and only when some miracle occurs, this bundle will turn out to be trivial. the Yetter model, now.. :)

   Thanks Zoran. I agree with all of the above. at present I’ll still be playing for a while with toy models (I’m planning to write a few lines on the Yetter model from the topological point of view I’ve been writing about the DW model lately), but I’d be really happy if this could be just the first step towards an understanding the large picture you’re sketching above.

   In particular anomalies and determinant bundles are something I’ve been hinting to in a few posts related to kantization here on the forum (there was nothing particularly interesting there, so it’s not worth going and searching for them). the idea is that from the correct categorical point of view, anomaly is something we should expect, and not something that happens to be there only since we chose that particular way to compute the path integral (e.g., by zeta function regularization). in other words, when seen fom the correct perspective, we should a priory know that the result will be a section of a nontrivia bundle, and only when some miracle occurs, this bundle will turn out to be trivial.

   the Yetter model, now.. :)

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeApr 18th 2010
   • PermaLink
   Author: zskoda
   Format: Markdown> particular way to compute the path integral (e.g., by zeta function regularization) to DEFINE the path integral; what generality will make the tests for good definitions ? so far from kantization recipes I see no handle of quantization conditions, even in 0-dimensional case of usual QM, when one should at least come to the Maslov class...

   particular way to compute the path integral (e.g., by zeta function regularization)

   to DEFINE the path integral; what generality will make the tests for good definitions ? so far from kantization recipes I see no handle of quantization conditions, even in 0-dimensional case of usual QM, when one should at least come to the Maslov class...

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a brief paragraph relating to the Weil conjectures to [[Lefschetz fixed point theorem]], and a reference

   added a brief paragraph relating to the Weil conjectures to Lefschetz fixed point theorem, and a reference

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 3rd 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThere's a g+ [discussion](https://plus.google.com/u/0/+CodyRoux/posts/L5dreTZLGUQ) happening about whether some form of Lefschetz fixed-point theorem can be established in Homotopy Type Theory.

   There’s a g+ discussion happening about whether some form of Lefschetz fixed-point theorem can be established in Homotopy Type Theory.