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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 11th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAt what point in physics does it matter that I consider the groupoid $$ \mathbf{Riem}(\Sigma)//\mathbf{Diff}(\Sigma) $$ rather than the set $$ \mathbf{Riem}(\Sigma)/\mathbf{Diff}(\Sigma)? $$ At [[general covariance]] it says the latter is sufficient >to perform variational calculus and hence derive the equations of motion of the theory. So what does the groupoid add? Is there an issue of being able to reconstruct $\Sigma$ in each case?

   At what point in physics does it matter that I consider the groupoid

   $$\mathbf{Riem}(\Sigma)//\mathbf{Diff}(\Sigma)$$

   rather than the set

   $$\mathbf{Riem}(\Sigma)/\mathbf{Diff}(\Sigma)?$$

   At general covariance it says the latter is sufficient

   to perform variational calculus and hence derive the equations of motion of the theory.

   So what does the groupoid add?

   Is there an issue of being able to reconstruct $\Sigma$ in each case?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2013
   • (edited Nov 11th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexGood question. One main answer is: _passing from homotopy quotients to naive quotients destroys locality in field theory, as in "[[local field theory]]"_ This is just the main property of stacks translated to moduli stacks of physical fields: if one forgets the (auto-)gauge transformations, then it is impossible, in general, to reconstruct global field configurations from local ones. This is a point that needs to be emphasized more. These days in the blogosphere one sees that the meme is spreading among supposedly big-shot theoretical physicists that "gauge invariance is just a redundancy" and that one can happily quotient out gauge equivalence. Notably the work by Arkani-Hamed et al on scattering amplitudes has often been accompanied by such statements. This is true only if one sacrifices locality. But these are the two fundamental principles of modern physics: 1. the gauge principle 1. the principle of locality. The first implies that the world is described by homotopy theory ($\infty$-groupoids). The second that it is described in fact by geometric homotopy theory ($\infty$-stacks). Neither should be thrown out of the window if one is after the full picture.

   Good question. One main answer is:

   passing from homotopy quotients to naive quotients destroys locality in field theory, as in “local field theory”

   This is just the main property of stacks translated to moduli stacks of physical fields: if one forgets the (auto-)gauge transformations, then it is impossible, in general, to reconstruct global field configurations from local ones.

   This is a point that needs to be emphasized more. These days in the blogosphere one sees that the meme is spreading among supposedly big-shot theoretical physicists that “gauge invariance is just a redundancy” and that one can happily quotient out gauge equivalence. Notably the work by Arkani-Hamed et al on scattering amplitudes has often been accompanied by such statements.

   This is true only if one sacrifices locality. But these are the two fundamental principles of modern physics:

   1. the gauge principle

   2. the principle of locality.

   The first implies that the world is described by homotopy theory ($\infty$-groupoids). The second that it is described in fact by geometric homotopy theory ($\infty$-stacks). Neither should be thrown out of the window if one is after the full picture.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 11th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThanks! So from dcct this is here: >Another cause is that often the nature of the gauge principle is actively misunderstood: often one sees texts claiming that gauge invariance is just a "redundancy" in the description of a physics, insinuating that one might just as well pass to the set of gauge equivalence classes. And this is not true: passing to gauge equivalence classes leads to violation of the other principle of modern physics, the principle of locality. For reconstructing non-trivial global gauge field configurations (often known as "instantons" in the physics literature) from local data, it is crucial to retain all the information about the gauge equivalences, for it is the way in which these serve to glue local gauge field data to global data that determines the global field content. And there's relevant material in 3.6.10.1 and 3.9.14.1.2. Can we see anything of this reconstructing of non-trivial global gauge field configurations in the baby example I'm working out [here](http://golem.ph.utexas.edu/category/2013/10/the_hott_approach_to_physics.html#c044962), or do we need a more geometric example?

   Thanks! So from dcct this is here:

   Another cause is that often the nature of the gauge principle is actively misunderstood: often one sees texts claiming that gauge invariance is just a “redundancy” in the description of a physics, insinuating that one might just as well pass to the set of gauge equivalence classes. And this is not true: passing to gauge equivalence classes leads to violation of the other principle of modern physics, the principle of locality. For reconstructing non-trivial global gauge field configurations (often known as “instantons” in the physics literature) from local data, it is crucial to retain all the information about the gauge equivalences, for it is the way in which these serve to glue local gauge field data to global data that determines the global field content.

   And there’s relevant material in 3.6.10.1 and 3.9.14.1.2.

   Can we see anything of this reconstructing of non-trivial global gauge field configurations in the baby example I’m working out here, or do we need a more geometric example?

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 11th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexDiscussion continues of all these things. It occurred to me to raise the issue of active and passive transformations. Is there a reason why we don't have anything on this? The Wikipedia [entry](http://en.wikipedia.org/wiki/Active_and_passive_transformation) gives an example where we can see a particular transformation either way.

   Discussion continues of all these things. It occurred to me to raise the issue of active and passive transformations. Is there a reason why we don’t have anything on this? The Wikipedia entry gives an example where we can see a particular transformation either way.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 15th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex>if one forgets the (auto-)gauge transformations, then it is impossible, in general, to reconstruct global field configurations from local ones. Does this mean that if a spacetime $\Sigma$ is the union of two others $\Sigma_i, i = 1,2$, then from the $$ [\Sigma_i, \mathbf{Fields}]//\mathbf{B} Aut(\Sigma_i) $$ it is possible to construct $$ [\Sigma, \mathbf{Fields}]//\mathbf{B} Aut(\Sigma)? $$ Perhaps some gluing data needs to be added.

   if one forgets the (auto-)gauge transformations, then it is impossible, in general, to reconstruct global field configurations from local ones.

   Does this mean that if a spacetime $\Sigma$ is the union of two others $\Sigma_i, i = 1,2$, then from the

   $$[\Sigma_i, \mathbf{Fields}]//\mathbf{B} Aut(\Sigma_i)$$

   it is possible to construct

   $$[\Sigma, \mathbf{Fields}]//\mathbf{B} Aut(\Sigma)?$$

   Perhaps some gluing data needs to be added.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 15th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe spacetime example is a subtle one to discuss this point. Let's first look at an easier one of fields _on_ spacetime. A gauge field theory assigns to subsets $U$ of $\Sigma$ the groupoid of $G$-gauge fields on $\Sigma$. This assignment is a stack, and the descent property of this stack says precisely that and how the local field assignsments glue together to give the global field assignsments. If we 0-truncate this stack by sending each $U$ to the set of gauge equivalence classes of fields this breaks down.

   The spacetime example is a subtle one to discuss this point. Let’s first look at an easier one of fields on spacetime.

   A gauge field theory assigns to subsets $U$ of $\Sigma$ the groupoid of $G$-gauge fields on $\Sigma$. This assignment is a stack, and the descent property of this stack says precisely that and how the local field assignsments glue together to give the global field assignsments.

   If we 0-truncate this stack by sending each $U$ to the set of gauge equivalence classes of fields this breaks down.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 17th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexBack online. >A gauge field theory assigns to subsets $U$ of $\Sigma$ the groupoid of $G$-gauge fields on $\Sigma$. You mean >A gauge field theory assigns to subsets $U$ of $\Sigma$ the groupoid of $G$-gauge fields on $U$?

   Back online.

   A gauge field theory assigns to subsets $U$ of $\Sigma$ the groupoid of $G$-gauge fields on $\Sigma$.

   You mean

   A gauge field theory assigns to subsets $U$ of $\Sigma$ the groupoid of $G$-gauge fields on $U$?

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeNov 17th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, sorry

   Yes, sorry

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 18th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOk good, but still it's going to be difficult in the general covariant case of #5, isn't it? I could perhaps see that one could stitch together patches to give diffeomorphisms on the whole of spacetime which are 'close to the identity', in the sense of leaving the submanifolds invariant.

   Ok good, but still it’s going to be difficult in the general covariant case of #5, isn’t it? I could perhaps see that one could stitch together patches to give diffeomorphisms on the whole of spacetime which are ’close to the identity’, in the sense of leaving the submanifolds invariant.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 5th 2017
    • (edited Nov 5th 2017)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexGiven what you said about redundancy of gauge symmetry in #2 and the quotation in #3, perhaps there's another case by Witten in [Symmetry and Emergence](https://arxiv.org/abs/1710.01791): >The meaning of global symmetries is clear: they act on physical observables. Gauge symmetries are more elusive as they typically do not act on physical observables. Gauge symmetries are redundancies in the mathematical description of a physical system rather than properties of the system itself. Maybe, though, you could still maintain that they are necessary parts of the description, and not to be factored out. In view of the extraordinary dualities in string theory, isn't it rather suspect to distinguish what is part of the mathematical description and what is part of the physical system itself?

    Given what you said about redundancy of gauge symmetry in #2 and the quotation in #3, perhaps there’s another case by Witten in Symmetry and Emergence:

    The meaning of global symmetries is clear: they act on physical observables. Gauge symmetries are more elusive as they typically do not act on physical observables. Gauge symmetries are redundancies in the mathematical description of a physical system rather than properties of the system itself.

    Maybe, though, you could still maintain that they are necessary parts of the description, and not to be factored out. In view of the extraordinary dualities in string theory, isn’t it rather suspect to distinguish what is part of the mathematical description and what is part of the physical system itself?

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2017
    • (edited Nov 5th 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexYeah, I have seen this article, and people have raised that question. What can I say. I suppose we are faced with an informal text that glosses over details. Gauge symmetries for $G$-CS theory on a manifold with boundary leave their imprint on the boundary, giving rise to $G$-WZW theory theory. For the WZW theory the $G$ is manifestly not a redundancy, but its very target space. Also, I'd need to look up the homotopy groups again, but I suppose for the example that is mentioned in the article, S-duality between $SO$- and $Sp$-gauge theory, the instanton sectors $H^1(\Sigma^\ast, B SO)$ and $H^1(\Sigma^\ast, B Sp)$ may indeed differ. So until shown a convincing counter-argument more mathematically detailed than an informal essay, I will maintain that it is wrong to say that "gauge symmetry is just a redundancy", unless a bunch of simplifying assumptions are made.

    Yeah, I have seen this article, and people have raised that question. What can I say. I suppose we are faced with an informal text that glosses over details.

    Gauge symmetries for $G$-CS theory on a manifold with boundary leave their imprint on the boundary, giving rise to $G$-WZW theory theory. For the WZW theory the $G$ is manifestly not a redundancy, but its very target space.

    Also, I’d need to look up the homotopy groups again, but I suppose for the example that is mentioned in the article, S-duality between $SO$- and $Sp$-gauge theory, the instanton sectors $H^1(\Sigma^\ast, B SO)$ and $H^1(\Sigma^\ast, B Sp)$ may indeed differ.

    So until shown a convincing counter-argument more mathematically detailed than an informal essay, I will maintain that it is wrong to say that “gauge symmetry is just a redundancy”, unless a bunch of simplifying assumptions are made.

12. • CommentRowNumber12.
    • CommentAuthorJohn Dougherty
    • CommentTimeNov 5th 2017
    • PermaLink
    Author: John Dougherty
    Format: MarkdownItexFor what it's worth, I think that you can make sense of some of the Witten quote in #10 in a way that's consistent with the nPOV. Suppose that we read "redundancy" as "identity" (in the HoTT sense) or "equivalence" (in the 1-category sense) and "symmetry" as "observable-preserving bijection". Then Witten's claim is that there are sometimes observable-preserving bijections between non-isomorphic configurations in gauge theories. And this is true: for example, in the asymptotically trivial sector of QCD a gauge transformation is only an isomorphism if it can be smoothly extended to the boundary, but it will preserve the classical physical observables regardless of what's going on at the boundary. It won't preserve the instanton number, but it's still a symmetry transformation with respect to the all the observables that were around before moving to the asymptotically trivial sector. It seems to me that this is a case where the nPOV can be helpful in clearing up a conceptual confusion. (In fact, I have a paper under review at the moment arguing this.) Naively, the configuration space of asymptotically trivial QCD is the subspace of configurations that are asymptotically trivial---the ordinary equalizer. But from the nPOV, we should instead use the homotopy equalizer, and this gives the correct instanton sectors. The isomorphisms in the asymptotically trivial sector are then the isomorphisms of the gauge transformations that smoothly extend to the boundary, not all the gauge transformations. I read claims like Witten's in #10 as an attempt to articulate the distinction between "isomorphism" in the sense of "observable-preserving bijection" and "isomorphism" in the sense of "equivalence in a 1-category". In the total configuration space of QCD these coincide, but the homotopy equalizer produces a configuration groupoid in which some observable-preserving bijections aren't equivalences, because they don't preserve the boundary. It's true that "redundancy" is probably the wrong word for this. In one sense it's appropriate because gauge-equivalent configurations aren't distinct, while symmetry-related configurations can be. If you count using ordinary equality, then there are distinct objects representing the same physical state of affairs. But it also suggests that it's purely a feature of the mathematical representation and that we wouldn't lose anything physical if we got rid of it, and that's wrong. So I think this attitude that "gauge = redundancy" isn't totally right, but it's a plausible compromise if you don't use the nPOV resources.

    For what it’s worth, I think that you can make sense of some of the Witten quote in #10 in a way that’s consistent with the nPOV. Suppose that we read “redundancy” as “identity” (in the HoTT sense) or “equivalence” (in the 1-category sense) and “symmetry” as “observable-preserving bijection”. Then Witten’s claim is that there are sometimes observable-preserving bijections between non-isomorphic configurations in gauge theories. And this is true: for example, in the asymptotically trivial sector of QCD a gauge transformation is only an isomorphism if it can be smoothly extended to the boundary, but it will preserve the classical physical observables regardless of what’s going on at the boundary. It won’t preserve the instanton number, but it’s still a symmetry transformation with respect to the all the observables that were around before moving to the asymptotically trivial sector.

    It seems to me that this is a case where the nPOV can be helpful in clearing up a conceptual confusion. (In fact, I have a paper under review at the moment arguing this.) Naively, the configuration space of asymptotically trivial QCD is the subspace of configurations that are asymptotically trivial—the ordinary equalizer. But from the nPOV, we should instead use the homotopy equalizer, and this gives the correct instanton sectors. The isomorphisms in the asymptotically trivial sector are then the isomorphisms of the gauge transformations that smoothly extend to the boundary, not all the gauge transformations. I read claims like Witten’s in #10 as an attempt to articulate the distinction between “isomorphism” in the sense of “observable-preserving bijection” and “isomorphism” in the sense of “equivalence in a 1-category”. In the total configuration space of QCD these coincide, but the homotopy equalizer produces a configuration groupoid in which some observable-preserving bijections aren’t equivalences, because they don’t preserve the boundary.

    It’s true that “redundancy” is probably the wrong word for this. In one sense it’s appropriate because gauge-equivalent configurations aren’t distinct, while symmetry-related configurations can be. If you count using ordinary equality, then there are distinct objects representing the same physical state of affairs. But it also suggests that it’s purely a feature of the mathematical representation and that we wouldn’t lose anything physical if we got rid of it, and that’s wrong. So I think this attitude that “gauge = redundancy” isn’t totally right, but it’s a plausible compromise if you don’t use the nPOV resources.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexThat sounds good! Might you have a preprint of your paper to share?

    That sounds good! Might you have a preprint of your paper to share?

14. • CommentRowNumber14.
    • CommentAuthorJohn Dougherty
    • CommentTimeNov 6th 2017
    • PermaLink
    Author: John Dougherty
    Format: MarkdownItexThe first draft of it is available [here](http://www.johnedougherty.com/papers/size.pdf), but I'm in the midst of heavy revisions in light of reviewer feedback at the moment, so there'll be a better version of the paper available at that link after the 19th.

    The first draft of it is available here, but I’m in the midst of heavy revisions in light of reviewer feedback at the moment, so there’ll be a better version of the paper available at that link after the 19th.