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Added some formulas and a manifestly relativistic version to action functional.
I have also been reverting JA's changes to variant conventions of spelling and grammar.
edited the Idea-section of action functional and Euler-Lagrange equations
at action functional I tried to spell out in more detail (jet space) the definition of local action functionals.
Zoran, there is a bit by you with examples of local action functionals that maybe needs a new introductory sentence. I am not sure if I killed that or what happened. Sorry. Please have a look, I guess you can just add one sentence before the numbered item list.
Urs, thank you for being careful, but I really do not recall what I was saying there. Probably nothing important. If it is it will reemerge somewhere :)
Just for the record, these are very famous related references on geometry of action functionals and related BV geometry
Claude Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 4, 685–710, MR93b:58058, doi
Albert Schwarz, Geometry of Batalin-Vilkovisky quantization,, Commun. Math. Phys. 155, 249 (1993), euclid
Albert Schwarz, Semiclassical approximation in Batalin-Vilkovisky formalism, Comm. Math. Phys. 158 (1993), no. 2, 373–396, euclid
The $\infty$-categorical stuff seems a bit.. out of place, to say the least.
Also, shouldn’t a line object be the same thing as a ring, or local ring object?
I presume that the line object belongs to an abelian Lawvere theory, so that it has an underlying abelian group, and hence an action of $\mathbb{Z}$. And once your Lawvere theory contains monoids (i.e. contains an monoid operation), then the line object is, up to assuming the distributive law, a unital (non-commutative) ring. But I am curious as to where one would need to use the local ring part. Perhaps when dealing with homotopy kernels?
Also, shouldn’t a line object be the same thing as a ring, or local ring object?
Yes, line object in the sense described at that link. I made it “additive line object” in order to clarify.
(The whole entry is a stub. I am hoping to eventually bring it into shape.)
(The whole entry is a stub. I am hoping to eventually bring it into shape.)
I am a bit stuck right this moment, because I am not sure yet how to formulate the notion “critical locus” fully abstractly in a suitable oo-topos.
@Urs: In general, critical loci should be formulated in the same way as closed subschemes (this is how it’s done on algebraic geometry at least).
In general, critical loci should be formulated in the same way as closed subschemes (this is how it’s done on algebraic geometry at least).
I guess you are thinking something like the following:
for $X$ a scheme or something and $f : X \to \mathbb{A}^1$ a function, pass to the 1-form $d f$, regard it as a section $d f : X \to T^* X$ and then take the fiber of that morphism, i.e. the pullback along the 0-section $0 : X \to T^* X$.
This is clear enough. What I am thinking about is a good general abstract way to say this in an $\infty$-topos with a minimum of structural assumptions.
In a cohesive $\infty$-topos there is a general abstract way to take the differential of a function. But my first guess as to how produce the critical locus of the function by a homotopy fiber of that morphism was not quite right, though, and I haven’t made up my mind yet as to what the right way to go about it is.
Hi Tobias,
I think what the entry has is the standard definition of what standard literature means by “local action functional” (unless I am overlooking something), but it’s true that it doesn’t capture topological aspects of gauge theories, not to speak of higher gauge theories. Much of standard physics literature ignores all this, by default.
I have a few ideas of how to formulate things more intrinsically in a (higher) gauge theoretic context, some of which is in published articles. But this is not yet indicated in this entry here. If you are interested, I’d be happy to discuss this further.
am I supposed to announce all changes like this here on the nForum?
That’s the optimal way to go at least, as it gives everyone potentially interested here a chance to become aware of the changes and to react to them. That’s how a community process can start that might eventually lead to something good.
We also stopped, by and large, dropping query boxes in entries, for that kind of reason: discussion in query boxes is hard to follow by bystanders and even by those involved. Usually this forum here is the place where to discuss, and once stable (or semi-stable ) insights have been obtained that deserve to be archived in the $n$Lab one can move them there.
I have to run now, but one remark maybe to be discussed later: also local connections on a fixed bundle can be written as sections of a bundle. But maybe that’s not what you mean.
Yes, in a higher gauge theoretic context DW theory is actually even a sigma-model, meaning that this bundle is trivial. But its fibers are higher homotopy types:
a field configuration in G gauge theory on $\Sigma$ is a map $\Sigma \to \mathbf{B}G_{conn}$.
Sorry, my bus arrived this very second. Here $G$ is the gauge group and $\mathbf{B}G_{conn}$ is the moduli stack of $G$-principal connections. This is explained for instance at connection on a bundle a bit. For more see at geometry of physics.
For general $G$ this is configurations of Chern-Simons theory, Yang-Mills theory, etc. If $G$ is discrete as for DW theory, then $\mathbf{B}G_{conn} \simeq \mathbf{B}G$ collapses to the moduli stack for $G$-principal bundles. A field configuration of DW theory is a map $\Sigma \to \mathbf{B}G$, which is aequivalently a $G$-principal bundle over $\Sigma$.
The extended action functional of CS-theory/DW theory then is, by the way, just the refinement to to differential cohomology of the Pontrygin class / second Chern class
$\mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$(or mutliples of that, that’s the “level” of the theory).
This is the local Lagrangian of CS/DW theory in full gauge theoretic beauty.
The action evaluated on a field configuration is then the transgression
$\exp(i \int_\Sigma [\Sigma, \mathbf{L}] ) \;\;\colon\;\; [\Sigma , \mathbf{B}G_{conn}] \stackrel{[\Sigma, \mathbf{L}]}{\to} [\Sigma, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(i \int_{\Sigma} (-))}{\to} U(1) \,.$We disucss this a bit in our last article Extended higher cup-product Chern-Simons theories (schreiber).
Cool! I think I get the rough idea: a principal bundle (with connection) is a morphism $\Sigma\to \mathbf{B}G_{\mathrm{conn}}$, so that the collection of all these is the space of field configurations. But such a morphism can also just be regarded as a section of the trivial bundle with fiber $\mathbf{B}G_{\mathrm{conn}}$.
I was about to edit the action functional page, but now I see you’re working on it ;) By the way, the nLab is an incredible valuable resource to study higher category theory!
Yes, exactly!
(By the way, to turn on math here choose below the edit box the radio button labeled “markdown+itex”).
Yes, I am editing right now, putting in some of the remarks that I just made. I need just five more minutes or something. Eventually all this will be exposed in some detail at geometry of physics.
Okay, I have now expanded a bit at action functional in a new section Extended local action functionals in (higher) gauge theory.
I’d think this goes in the direction of replying to your query box, so I have taken the liberty and removed it. But let’s further discuss matters here. I can expand on various aspects, if you ask me to.
Thanks! I’m happy with this for the moment and will go back to studying DW-theory.
I guess I’ll see you and/or some other knowledgeable people in Nijmegen next week. Looking forward!
Ah, you’ll be here next week. Great, let’s talk then.
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