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1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeSep 13th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexI've been watching the discussion going on here for a while. It is WAAAAAAY over my head, but I occasionally see things that look neat, clever, etc. It is like watching a nail biting drama unfold. However, I have absolutely zero sense for what this could mean for physics. By physics, I mean something that models something that is observed in nature. NOT some hypothetical physical model. Does this stuff relate to anything observed in nature? Or is it more about developing tools that are hoped to one day apply to models of physics? Or is it pure mathematics (which would be fine of course)? Just trying to get a sense for the story I'm watching. PS: Even the 'Ideas" sections on the relevant pages on the nLab are over my head.

   I’ve been watching the discussion going on here for a while. It is WAAAAAAY over my head, but I occasionally see things that look neat, clever, etc. It is like watching a nail biting drama unfold.

   However, I have absolutely zero sense for what this could mean for physics. By physics, I mean something that models something that is observed in nature. NOT some hypothetical physical model.

   Does this stuff relate to anything observed in nature?

   Or is it more about developing tools that are hoped to one day apply to models of physics? Or is it pure mathematics (which would be fine of course)?

   Just trying to get a sense for the story I’m watching.

   PS: Even the ’Ideas” sections on the relevant pages on the nLab are over my head.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 13th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexOne thing it applies to is the [[D'Auria-Fre formulation of supergravity]] - which of course is not a model of observations to date. Indeed, beyond thinking of curvature of an ordinary connection as part of a 2-connection, I don't know (in my admittedly limited knowledge) what out there in the physical world even looks like a higher form. The candidates you are looking most likely come from solid state physics or something similarly 'down to earth' (i.e. not particle physics), but then I never got into anything more concrete than fundamental theoretical physics anyway, so I can't help much!

   One thing it applies to is the D’Auria-Fre formulation of supergravity - which of course is not a model of observations to date. Indeed, beyond thinking of curvature of an ordinary connection as part of a 2-connection, I don’t know (in my admittedly limited knowledge) what out there in the physical world even looks like a higher form. The candidates you are looking most likely come from solid state physics or something similarly ’down to earth’ (i.e. not particle physics), but then I never got into anything more concrete than fundamental theoretical physics anyway, so I can’t help much!

3. • CommentRowNumber3.
   • CommentAuthorEric
   • CommentTimeSep 13th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexThanks David. I guess the punch line is this: >The [[∞-Lie theory]]-reformulation exhibits the D'Auria-Fré-formalism as being secretly the realization of [[supergravity]] as a higher [[gauge theory]]. So although the jury is still out (as far as I know) whether higher gauge theory is relevant to models of nature, it is apparently relevant (at a minimum) to hypothetical(?) models of supergravity. Is it reasonable to think of this drama unfolding here on $\infty$-Chern Weil theory as being about tidying up the mathematics of $\infty$-Lie theory? That alone would be more than enough to keep me interested.

   Thanks David. I guess the punch line is this:

   The ∞-Lie theory-reformulation exhibits the D’Auria-Fré-formalism as being secretly the realization of supergravity as a higher gauge theory.

   So although the jury is still out (as far as I know) whether higher gauge theory is relevant to models of nature, it is apparently relevant (at a minimum) to hypothetical(?) models of supergravity.

   Is it reasonable to think of this drama unfolding here on $\infty$-Chern Weil theory as being about tidying up the mathematics of $\infty$-Lie theory? That alone would be more than enough to keep me interested.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 13th 2010
   • (edited Sep 13th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi Eric, > By physics, I mean something that models something that is observed in nature. NOT some hypothetical physical model. This is an important point that is not emphasized enough, generally: indeed, this huge undertaking of present-day fundamental theoretical physics, with all its topological field theories, supersymmetric Yang-Mills theories, 2-dimensional CFTs on higher genera, and all the other physical models that are well known _not_ to describe any present experimental data is a huge _exercise in holding your breath_ . I can understand if one loses patience with this. But I think there is no way aroung this for the purpose of actually understanding real-world fundamental physics. The problem is this: After, for instance, Newton figured out that differential equations are the right math that describes classical physics, people pretty soon (historically speaking) had a fairly solid understanding of the mathematical theory of differential equations and could apply it confidently all over the place. But after it was figured out later that quantum field theory is the right thing to describe quantum phyics, Almost a century later we are still pretty much in the dark about basic questions. It's getting much better recently, but really one is still in the process of just getting one's foot on the ground. And this is where this plethora of non-phenomenological models comes from. It's the analog of all the examples of differential equations none of which describe any observed physical process, but that you have to play with in order to be able to figure out a general theory of differential equations. So the community is on a vast full-circle tour aimed to come back to the real world, but it will still take some time. But I don't see any indication that there is a shortcut. And not for lack of trying. So this is where $\infty$-Chern-Weil theory comes in: it has become clear that gauge theory in its oberved form and in all its generalized forms is all about [[differential cohomology]]. So one needs to understand this. $\infty$-Chern-Weil theory is supposed to be one aspect of this (namely that which describes how nonabelian higher gauge theory ingteracts with the well-understood abelian case). > So although the jury is still out (as far as I know) whether higher gauge theory is relevant to models of nature Not quite. Remember Dirac's old argument about magnetic charge quantization? That was incomplete. The complete version is that magnetic charge is a differential 3-cocycle -- a connection on a $\mathbf{B}U(1)$-2-bundle ( a bundle gerbe). See the discussion at [[magnetic charge]]. And this is the template, it turns out, for all the higher anomaly cancellation phenomena, such as the Green-Schwarz mechanism with its [[Chern-Simons circle 3-bundle]] and the dual GS mechanism with its Chern-Simons circle 7-bundle. As you observe, neither of these have been shown to have direct relevance for observable physical data, but the general impression is that without understanding these there is no chance to understand fundamental physics.

   Hi Eric,

   By physics, I mean something that models something that is observed in nature. NOT some hypothetical physical model.

   This is an important point that is not emphasized enough, generally: indeed, this huge undertaking of present-day fundamental theoretical physics, with all its topological field theories, supersymmetric Yang-Mills theories, 2-dimensional CFTs on higher genera, and all the other physical models that are well known not to describe any present experimental data is a huge exercise in holding your breath .

   I can understand if one loses patience with this. But I think there is no way aroung this for the purpose of actually understanding real-world fundamental physics. The problem is this:

   After, for instance, Newton figured out that differential equations are the right math that describes classical physics, people pretty soon (historically speaking) had a fairly solid understanding of the mathematical theory of differential equations and could apply it confidently all over the place.

   But after it was figured out later that quantum field theory is the right thing to describe quantum phyics, Almost a century later we are still pretty much in the dark about basic questions. It’s getting much better recently, but really one is still in the process of just getting one’s foot on the ground. And this is where this plethora of non-phenomenological models comes from. It’s the analog of all the examples of differential equations none of which describe any observed physical process, but that you have to play with in order to be able to figure out a general theory of differential equations.

   So the community is on a vast full-circle tour aimed to come back to the real world, but it will still take some time. But I don’t see any indication that there is a shortcut. And not for lack of trying.

   So this is where $\infty$-Chern-Weil theory comes in: it has become clear that gauge theory in its oberved form and in all its generalized forms is all about differential cohomology. So one needs to understand this. $\infty$-Chern-Weil theory is supposed to be one aspect of this (namely that which describes how nonabelian higher gauge theory ingteracts with the well-understood abelian case).

   So although the jury is still out (as far as I know) whether higher gauge theory is relevant to models of nature

   Not quite. Remember Dirac’s old argument about magnetic charge quantization? That was incomplete. The complete version is that magnetic charge is a differential 3-cocycle – a connection on a $\mathbf{B}U(1)$-2-bundle ( a bundle gerbe). See the discussion at magnetic charge.

   And this is the template, it turns out, for all the higher anomaly cancellation phenomena, such as the Green-Schwarz mechanism with its Chern-Simons circle 3-bundle and the dual GS mechanism with its Chern-Simons circle 7-bundle.

   As you observe, neither of these have been shown to have direct relevance for observable physical data, but the general impression is that without understanding these there is no chance to understand fundamental physics.

5. • CommentRowNumber5.
   • CommentAuthorEric
   • CommentTimeSep 13th 2010
   • (edited Sep 13th 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexThanks Urs! That is pretty much the impression I got. >Not quite. Remember Dirac's old argument about magnetic charge quantization? That was incomplete. The complete version is that magnetic charge is a differential 3-cocycle -- a connection on a $\mathbf{B}U(1)$-2-bundle ( a bundle gerbe). See the discussion at [[magnetic charge]]. I remember discussing this description of magnetic charge at some point (maybe on the nCafe or even the String Coffee Table). I left the discussion not convinced that your conclusion was "right", i.e. I was not convinced that nature is like that. There seemed to be other possible explanations as well, e.g. perhaps spacetime is not a continuum. In fact, I suspect a lot of this $\infty$-stuff might turn out to be related to trying to keep track of "error terms" stemming from nice finite models posing as scary continuum models. When you develop a "toy" model that is finitary in nature and proceed to compare it to the "exact" continuum model, there will be differences of course. If you collect error terms to a certain "order", i.e. first order, second order, third order, these error terms form nice patterns and have nice properties that are probably described nicely by $\infty$-stuff. However, at the end of the day, these "error terms" could turn out to be errors in the continuum model whereas the finitary "toy" model is nice and clean and is truer to nature anyway. I wish you, of all people, would take this possibility more seriously, but I am patient :) Edit: I found the old discussion (it actually wasn't that old). [Re: Charges and Twisted Bundles, III: Anomalies](http://golem.ph.utexas.edu/category/2008/04/magnetic_charge_anomalies.html#c020957) My reservation didn't have anything to do with finitary vs continuum, but rather about assumptions about the topology related to geometrodynamics. I'd like to think about this some more. Time permitting... Edit^2: I must have been drunk on eggnog (as warned on the comment). Identifying points would not relate to magnetic charge in any way at all. The other identified point would merely be an electric charge of opposite sign. The thought was totally bogus. Still. My gut tells me something does not ring true even now. Need to think about it...

   Thanks Urs! That is pretty much the impression I got.

   Not quite. Remember Dirac’s old argument about magnetic charge quantization? That was incomplete. The complete version is that magnetic charge is a differential 3-cocycle – a connection on a $\mathbf{B}U(1)$-2-bundle ( a bundle gerbe). See the discussion at magnetic charge.

   I remember discussing this description of magnetic charge at some point (maybe on the nCafe or even the String Coffee Table). I left the discussion not convinced that your conclusion was “right”, i.e. I was not convinced that nature is like that. There seemed to be other possible explanations as well, e.g. perhaps spacetime is not a continuum.

   In fact, I suspect a lot of this $\infty$-stuff might turn out to be related to trying to keep track of “error terms” stemming from nice finite models posing as scary continuum models. When you develop a “toy” model that is finitary in nature and proceed to compare it to the “exact” continuum model, there will be differences of course. If you collect error terms to a certain “order”, i.e. first order, second order, third order, these error terms form nice patterns and have nice properties that are probably described nicely by $\infty$-stuff.

   However, at the end of the day, these “error terms” could turn out to be errors in the continuum model whereas the finitary “toy” model is nice and clean and is truer to nature anyway. I wish you, of all people, would take this possibility more seriously, but I am patient :)

   Edit: I found the old discussion (it actually wasn’t that old).

   Re: Charges and Twisted Bundles, III: Anomalies

   My reservation didn’t have anything to do with finitary vs continuum, but rather about assumptions about the topology related to geometrodynamics. I’d like to think about this some more. Time permitting…

   Edit^2: I must have been drunk on eggnog (as warned on the comment). Identifying points would not relate to magnetic charge in any way at all. The other identified point would merely be an electric charge of opposite sign. The thought was totally bogus. Still. My gut tells me something does not ring true even now. Need to think about it…

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeSep 13th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> I wish you, of all people, would take this possibility more seriously The reason we don't is that it's too vague! You are saying something like "But what if a basic assumption were dropped, wouldn't that mean that _everything_ came out different?" And, yeah, supposedly it would come out different. Which also means that probably it would not connect to what we already know it needs to connect to. Unless of course there is some good reason that it does. That said, I tink it is well possible that lrge parts of physics can be modeld by combinatorial structures. Effectively that's already happening! An $\infty$-groupoid is a combinatorial model for a continuous space, and we know well how lots of topological aspects of physics are modeled by these. It is also well possible that your dream is in fact true and that fundamentally spacetime is not modeled on $\mathbb{R}^n$ but on a lattice. All of what we are talking about in $\infty$-Chern-Weil theory would still go through, because it is formulated in arbitrary $\infty$-toposes, that do include geometry modeled on, say, cellular complexes. That's supposed to be part of the power of the general abstract theory: to make it independent of basic assumptions about the models. But given that, I don't expect that passing to geometry modeled on cellular spaces makes the subtle topological effects of higher gauge theory go awa by magic. > scary continuum models I don't find them scary. You should beware of thinking a model is wrong just because it is not as intuitively accessible as first.

   I wish you, of all people, would take this possibility more seriously

   The reason we don’t is that it’s too vague! You are saying something like “But what if a basic assumption were dropped, wouldn’t that mean that everything came out different?” And, yeah, supposedly it would come out different. Which also means that probably it would not connect to what we already know it needs to connect to. Unless of course there is some good reason that it does.

   That said, I tink it is well possible that lrge parts of physics can be modeld by combinatorial structures. Effectively that’s already happening! An $\infty$-groupoid is a combinatorial model for a continuous space, and we know well how lots of topological aspects of physics are modeled by these.

   It is also well possible that your dream is in fact true and that fundamentally spacetime is not modeled on $\mathbb{R}^n$ but on a lattice. All of what we are talking about in $\infty$-Chern-Weil theory would still go through, because it is formulated in arbitrary $\infty$-toposes, that do include geometry modeled on, say, cellular complexes. That’s supposed to be part of the power of the general abstract theory: to make it independent of basic assumptions about the models.

   But given that, I don’t expect that passing to geometry modeled on cellular spaces makes the subtle topological effects of higher gauge theory go awa by magic.

   scary continuum models

   I don’t find them scary. You should beware of thinking a model is wrong just because it is not as intuitively accessible as first.

7. • CommentRowNumber7.
   • CommentAuthorEric
   • CommentTimeSep 13th 2010
   • (edited Sep 13th 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexThanks again Urs. Sorry for thinking/dreaming out loud. It is a habit that is hard to break. >But given that, I don't expect that passing to geometry modeled on cellular spaces makes the subtle topological effects of higher gauge theory go away by magic. Thanks for your explanations. It does make me feel better knowing that the business with $\infty$-toposes will allow us (?) to pass to cellular complexes at the end of the day without much ado. If that is the case, might it not be easier to work out these topological subtleties in the combinatorial/cellular case first and then see what they imply for the continuum cases (if one cared to)? My head just begins to swirl when I see "Lie this" and "Lie that". My aversion to the continuum kicks in and poses a barrier for me to push forward. That is admittedly a serious weakness of mine.

   Thanks again Urs. Sorry for thinking/dreaming out loud. It is a habit that is hard to break.

   But given that, I don’t expect that passing to geometry modeled on cellular spaces makes the subtle topological effects of higher gauge theory go away by magic.

   Thanks for your explanations. It does make me feel better knowing that the business with $\infty$-toposes will allow us (?) to pass to cellular complexes at the end of the day without much ado. If that is the case, might it not be easier to work out these topological subtleties in the combinatorial/cellular case first and then see what they imply for the continuum cases (if one cared to)?

   My head just begins to swirl when I see “Lie this” and “Lie that”. My aversion to the continuum kicks in and poses a barrier for me to push forward. That is admittedly a serious weakness of mine.