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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?
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:
the gauge principle
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.
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?
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.
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.
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.
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 $U$?
Yes, sorry
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.
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