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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeFeb 3rd 2010
   • (edited Feb 3rd 2010)
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   Author: domenico_fiorenza
   Format: HtmlI'd like to create an entry [[gauge fixing]]. the usual method I use to create new entries (typing their name in the search form) does not work, since search redirects me to [[examples for Lagrangian BV]]. now I'm trying to put a link here to see if this works (but I'm skeptic..). should it not work, how do I create the new entry? thanks edit: it works!!!! :-) I'll now start writing the entry

   I'd like to create an entry gauge fixing. the usual method I use to create new entries (typing their name in the search form) does not work, since search redirects me to examples for Lagrangian BV. now I'm trying to put a link here to see if this works (but I'm skeptic..). should it not work, how do I create the new entry?
   
   thanks
   
   edit: it works!!!! :-)
   
   I'll now start writing the entry
2. • CommentRowNumber2.
   • CommentAuthordomenico_fiorenza
   • CommentTimeFeb 3rd 2010
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   Author: domenico_fiorenza
   Format: Htmlcreated a stubby entry [[gauge fixing]] (nothing in mathematics makes sense except in the light of higher category theory).

   created a stubby entry gauge fixing (nothing in mathematics makes sense except in the light of higher category theory).
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 3rd 2010
   • (edited Feb 3rd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownThanks. I think it is really good to give this category-theoretic perspective on what gauge fixing really is, yes. Thanks for starting this entry. I added an "Idea"-section with some introductory words and also started an "Examples"-section.

   Thanks.

   I think it is really good to give this category-theoretic perspective on what gauge fixing really is, yes. Thanks for starting this entry.

   I added an "Idea"-section with some introductory words and also started an "Examples"-section.

4. • CommentRowNumber4.
   • CommentAuthordomenico_fiorenza
   • CommentTimeFeb 3rd 2010
   • PermaLink
   Author: domenico_fiorenza
   Format: HtmlI'm glad you liked the entry. now I'm trying to think to what is anomaly from this perspective. it is a classical fact that by hand computations of path integrals produce a section of a line bundle on the moduli space of the theory rather than a function. it would be nice to present things in such a way that this is not a surprise but exactly what one should expect, i.e. that it does not depend on the trick one used to give a meaning to the path integral, but it is the "true nature" of Z (so the surprise is when the theory is non-anomalous..)

   I'm glad you liked the entry. now I'm trying to think to what is anomaly from this perspective. it is a classical fact that by hand computations of path integrals produce a section of a line bundle on the moduli space of the theory rather than a function. it would be nice to present things in such a way that this is not a surprise but exactly what one should expect, i.e. that it does not depend on the trick one used to give a meaning to the path integral, but it is the "true nature" of Z (so the surprise is when the theory is non-anomalous..)
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 3rd 2010
   • PermaLink
   Author: Urs
   Format: MarkdownYes, so the anomaly business comes in when we do not start with having an action-function/functor on configuration space, but instead have just a line bundle (with connection) and only know that the action is some flat non-trivial section of it. By the way, we have stub entries like [[quantum anomaly]] and [[Green-Schwarz mechanism]] that talk a little about this. I have meant to give a fully formal nPOV description of this, with the relevant differential cocycles accordingly modeled as functors etc, but haven't really gotten around to doing that. I did talk about it a bit on the blog, once, though, for instance in <a href="http://golem.ph.utexas.edu/category/2008/04/magnetic_charge_anomalies.html">Charges and Twisted Bundles, III: Anomalies</a> There is a very nice general abstract story to be told here. It's good that you are pushing me. Maybe together we can write some decent stuff in some nice entries.

   Yes, so the anomaly business comes in when we do not start with having an action-function/functor on configuration space, but instead have just a line bundle (with connection) and only know that the action is some flat non-trivial section of it.

   By the way, we have stub entries like quantum anomaly and Green-Schwarz mechanism that talk a little about this.

   I have meant to give a fully formal nPOV description of this, with the relevant differential cocycles accordingly modeled as functors etc, but haven't really gotten around to doing that.

   I did talk about it a bit on the blog, once, though, for instance in Charges and Twisted Bundles, III: Anomalies

   There is a very nice general abstract story to be told here. It's good that you are pushing me. Maybe together we can write some decent stuff in some nice entries.

6. • CommentRowNumber6.
   • CommentAuthordomenico_fiorenza
   • CommentTimeFeb 3rd 2010
   • (edited Feb 3rd 2010)
   • PermaLink
   Author: domenico_fiorenza
   Format: Htmlyes, and anomaly can come in even when one started with what looked like an honest gauge-invariant action-function on the space of fields: think of conformal anomaly in the path-integral quantization of the bosonic string. there there is this very nice phenomenon: I would like to integrate over the space of all metrics and maps to <latex>\mathbb{R}^N</latex>, i.e., I want to push forward <latex>e^{iS}</latex> along <latex>metrics\times maps\to \{point\}</latex>. this factors as <latex>metrics\times maps\to metrics \to moduli \to \{point\}</latex> and anomaly happens in the first push-forward, since the integral over all maps to <latex>\mathbb{R}^N</latex> depends on the metric, and not only on the isomorphism class of the metric (under the <latex>Diff\times Weyl</latex> action). but what is really remarkable is that though we do not have <latex>Diff\times Weyl</latex>-invariance anymore, the <latex>Diff\times Weyl</latex>-dependence is still functorial! and it is precisely this functoriality (in a way depending on <latex>N</latex>) to say that we can interpret the result of integration on the metrics as the section of a line bundle on moduli. and when this line bundle is trivial we can further integrate on the moduli to get a number. so the magic here appears to be the fact that after push-forwarding we still have a functor, and this is automatically ensured (by definition) by a Kan extension-type path itegration. still very confused.. I'll try to think and fix this

   yes, and anomaly can come in even when one started with what looked like an honest gauge-invariant action-function on the space of fields: think of conformal anomaly in the path-integral quantization of the bosonic string. there there is this very nice phenomenon: I would like to integrate over the space of all metrics and maps to , i.e., I want to push forward along . this factors as
   
   
   
   and anomaly happens in the first push-forward, since the integral over all maps to depends on the metric, and not only on the isomorphism class of the metric (under the action). but what is really remarkable is that though we do not have -invariance anymore, the -dependence is still functorial! and it is precisely this functoriality (in a way depending on ) to say that we can interpret the result of integration on the metrics as the section of a line bundle on moduli. and when this line bundle is trivial we can further integrate on the moduli to get a number. so the magic here appears to be the fact that after push-forwarding we still have a functor, and this is automatically ensured (by definition) by a Kan extension-type path itegration.
   
   still very confused.. I'll try to think and fix this
7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 3rd 2010
   • (edited Feb 3rd 2010)
   • PermaLink
   Author: Urs
   Format: Markdownyou know, I had been thinking about similar lines but never quite felt that I understood this to the point that it became a general abstract tautology, if you see what I mean. I think for gauge theories, of the kind described in Freed's article, I'd know how to take what Freed writes and reformulate it in a completely abstract way, such that it boils down to turning a crank on an abstract machinery. But I am nott sure how to do this with things lilke the Polyakov action. At some point this made me think the following: I was thinking that maybe we ought to try to think of _every_ physics theory as a gauge theory, thereby realizing it in the realm where we have good control of the abstract whereabouts. Of course this is not a new idea, but let me make it explicit here: we really want to be talking about the superstring. It's action is, as you know, really a worldsheet supergravity coupled to a bunch of worldsheet "matter" fields (which are of course the sigma-model embedding fields from the target space perspective). Now there are various attempts to formulate plain gravity as a gauge theory with constraints. They all don't seem entirely satisfactory. But here we are talking supergravity. And I find it striking that for supergravity the situation is the reverse: the formulationns of supergravity that do _not_ formulate it as a gauge theory seem awkward. I don't know, did you ever have had a look at the original articlle Cremmer-Julia-Scherk on D=11 SUGRA? That whole article, impressive as it is, reads like one big lesson in awkwardness. All of standard supergravity does. But then, there is the [[D'Auria-Fre formulation of supergravity]] and, lo and behold, it is very elegant and manages to show how all the awkward details drop out from turning a nice crank. And: it is crucualliay formuated as a gauge theory -- a higher gauge theory even. (I mean, the original authors did not exactly realize this, but it is evident once one knows what a higher gauge theory looks like). This is not a proof of anything, of course. But it made me wonder. Maybe we shouldn't be looking too much at the bosonic Polyakov action if we want to come from an abstract point of view. Maybe we should take the RNS worldsheet superstring action before the diffomorphism gauge has been fixed, which is worldsheet supergravity, and then think of that as a gauge theory using standard first order formalism and guidance from D'Auria-Fre. Maybe then we will see how the conformal Weyl-anomaly fits in exactly into the general absstract picture we had before. I don't know, but this is something I would like to find time to look into.

   you know, I had been thinking about similar lines but never quite felt that I understood this to the point that it became a general abstract tautology, if you see what I mean.

   I think for gauge theories, of the kind described in Freed's article, I'd know how to take what Freed writes and reformulate it in a completely abstract way, such that it boils down to turning a crank on an abstract machinery.

   But I am nott sure how to do this with things lilke the Polyakov action.

   At some point this made me think the following: I was thinking that maybe we ought to try to think of every physics theory as a gauge theory, thereby realizing it in the realm where we have good control of the abstract whereabouts.

   Of course this is not a new idea, but let me make it explicit here: we really want to be talking about the superstring. It's action is, as you know, really a worldsheet supergravity coupled to a bunch of worldsheet "matter" fields (which are of course the sigma-model embedding fields from the target space perspective).

   Now there are various attempts to formulate plain gravity as a gauge theory with constraints. They all don't seem entirely satisfactory. But here we are talking supergravity. And I find it striking that for supergravity the situation is the reverse: the formulationns of supergravity that do not formulate it as a gauge theory seem awkward.

   I don't know, did you ever have had a look at the original articlle Cremmer-Julia-Scherk on D=11 SUGRA? That whole article, impressive as it is, reads like one big lesson in awkwardness. All of standard supergravity does.

   But then, there is the D'Auria-Fre formulation of supergravity and, lo and behold, it is very elegant and manages to show how all the awkward details drop out from turning a nice crank. And: it is crucualliay formuated as a gauge theory -- a higher gauge theory even. (I mean, the original authors did not exactly realize this, but it is evident once one knows what a higher gauge theory looks like).

   This is not a proof of anything, of course. But it made me wonder. Maybe we shouldn't be looking too much at the bosonic Polyakov action if we want to come from an abstract point of view. Maybe we should take the RNS worldsheet superstring action before the diffomorphism gauge has been fixed, which is worldsheet supergravity, and then think of that as a gauge theory using standard first order formalism and guidance from D'Auria-Fre.

   Maybe then we will see how the conformal Weyl-anomaly fits in exactly into the general absstract picture we had before.

   I don't know, but this is something I would like to find time to look into.

8. • CommentRowNumber8.
   • CommentAuthordomenico_fiorenza
   • CommentTimeFeb 4th 2010
   • PermaLink
   Author: domenico_fiorenza
   Format: HtmlI see your point (so I'll now be looking at D'Auria-Fre formulation of sugra, and this will keep me busy for a while..). but let me say that we could learn a lot already from Polyakov action, too. in the rigorous mathemathical descriptions of path-integral quantization of Polyakov action I'm aware of (Bost's Seminaire Bourbaki is my favourite reference), the path integral is computed for closed surfaces. this misses an important point: that surfaces should be swept between strings, so that they should be thought of not as objects but as morphisms. and even strings, maybe one should always think of closed strings as a particular instance of open strings. you already see where I'm pointing to: let us assume there is an ideal extended string theory, with 0-,1- and 2-morphisms given by points, strings and surfaces. this theory would not be purely toplogical, since we would have the datum of a metric on the surfaces (and maybe a G-bundle attached to strings, what I guess string theorists call the Chan-Paton factors, but let us not mess up things at this level and forget about this for the moment). 3-morphisms would then consist not only of diffeomorphisms between surfaces (as in an extended TQFT) but also of Weyl rescalings of the metric. ideally, this continues to give a <latex>(\infty,2)</latex>-category of strings (a close relative of 2Bord). let us calla this <latex>2Bord_{Str}</latex> (waiting for a better name) adding a target space X produces the relative version <latex>2Bord_{Str}(X)</latex>. for instance in the basic Polyakov action one deals with <latex>2Bord_{Str}(\mathbb{R}^N)</latex>. now, integration on maps is push-forward along <latex>2Bord_{Str}(X)\to 2Bord_{Str}</latex>. so I can think of path-integral quantization of Polyakov action for closed surfaces in these terms: I want to compute this path-integral. there are presumibly many ways of achieving this. but if I think closed surfaces as a part of a bigger picture involving 2Vect, then there would be a canonical path integration of the big picture, and the restriction of this to closed surfaces would be a "canonical" path integration of the small picture. this should tell us which kind of "thing" should be the partition function at the level of moduli spaces (I'm quite optimistic: I'm confident the abstract nonsense, one worked out, will tell us we should expect a section of a line bundle). if we are lucky we could even arrive to a receipt: "if the bigger picture exists, then the little path-integral is the following (independent of the details of the bigger picture)". so even in absence of the bigger picture it would be extremely reasonable to take this as a definition of the path integral over closed surfaces. maybe <latex>2Bord_{Str}</latex> could have some interest on its own (what are 4-morphisms?). I'll try to think to this.

   I see your point (so I'll now be looking at D'Auria-Fre formulation of sugra, and this will keep me busy for a while..). but let me say that we could learn a lot already from Polyakov action, too. in the rigorous mathemathical descriptions of path-integral quantization of Polyakov action I'm aware of (Bost's Seminaire Bourbaki is my favourite reference), the path integral is computed for closed surfaces. this misses an important point: that surfaces should be swept between strings, so that they should be thought of not as objects but as morphisms. and even strings, maybe one should always think of closed strings as a particular instance of open strings. you already see where I'm pointing to: let us assume there is an ideal extended string theory, with 0-,1- and 2-morphisms given by points, strings and surfaces. this theory would not be purely toplogical, since we would have the datum of a metric on the surfaces (and maybe a G-bundle attached to strings, what I guess string theorists call the Chan-Paton factors, but let us not mess up things at this level and forget about this for the moment). 3-morphisms would then consist not only of diffeomorphisms between surfaces (as in an extended TQFT) but also of Weyl rescalings of the metric. ideally, this continues to give a -category of strings (a close relative of 2Bord). let us calla this (waiting for a better name) adding a target space X produces the relative version . for instance in the basic Polyakov action one deals with .
   
   now, integration on maps is push-forward along . so I can think of path-integral quantization of Polyakov action for closed surfaces in these terms: I want to compute this path-integral. there are presumibly many ways of achieving this. but if I think closed surfaces as a part of a bigger picture involving 2Vect, then there would be a canonical path integration of the big picture, and the restriction of this to closed surfaces would be a "canonical" path integration of the small picture. this should tell us which kind of "thing" should be the partition function at the level of moduli spaces (I'm quite optimistic: I'm confident the abstract nonsense, one worked out, will tell us we should expect a section of a line bundle). if we are lucky we could even arrive to a receipt: "if the bigger picture exists, then the little path-integral is the following (independent of the details of the bigger picture)". so even in absence of the bigger picture it would be extremely reasonable to take this as a definition of the path integral over closed surfaces.
   
   maybe could have some interest on its own (what are 4-morphisms?). I'll try to think to this.
9. • CommentRowNumber9.
   • CommentAuthordomenico_fiorenza
   • CommentTimeFeb 4th 2010
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   Author: domenico_fiorenza
   Format: HtmlI've now gone through D'Auria-Fre formulation of sugra as it is presented in the nLab. So let me see if I got the general sense of it: i) gauge theory is the theory of connections on principal bundles ii) higher gauge theory is the theory of <latex>\infty</latex>-connections on <latex>\infty</latex>-principal bundles (as special cases we may have <latex>n</latex>-connections on <latex>n</latex>-principal bundles) iii) example: sugra is an higher gauge theory (idea: any reasonable field theory should be an higher gauge theory) iv) a connection is a representation of path groupoid into Vect v) an <latex>\infty</latex>-connection is a representation of <latex>\infty</latex>-path groupoid into <latex>\infty</latex>Vect vi) a path-integral is a push-forward (a Kan extension?) of this representation

   I've now gone through D'Auria-Fre formulation of sugra as it is presented in the nLab. So let me see if I got the general sense of it:
   
   i) gauge theory is the theory of connections on principal bundles
   ii) higher gauge theory is the theory of -connections on -principal bundles (as special cases we may have -connections on -principal bundles)
   iii) example: sugra is an higher gauge theory (idea: any reasonable field theory should be an higher gauge theory)
   iv) a connection is a representation of path groupoid into Vect
   v) an -connection is a representation of -path groupoid into Vect
   vi) a path-integral is a push-forward (a Kan extension?) of this representation
10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownYes. Well, there are some technical subtleties here and there, but this is certainly the picture, yes.

    Yes.

    Well, there are some technical subtleties here and there, but this is certainly the picture, yes.

11. • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 4th 2010
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    Author: TobyBartels
    Format: Markdown@ domenico #1 Try creating a link on a relevant page (or the [[Sandbox]]). See also [HowTo#newpage](http://ncatlab.org/nlab/show/HowTo#newpage).

    @ domenico #1

    Try creating a link on a relevant page (or the Sandbox).

    See also HowTo#newpage.