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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 22nd 2010
   • (edited Sep 22nd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt is clear that [[infinity-Chern-Weil theory]] will induce lots of examples of _oo-Chern-Simons theory_ : for every [[Chern-Simons element]] on an $\infty$-Lie algebroid $\mathfrak{a}$, there is the corresponding generalized Chern-Simons action functional on the space of $\mathfrak{a}$-valued connections/forms. I have started now listing all the familiar QFTs that are obtained as special cases this way. This is a joint project I am doing with Chris Rogers. So I started that list with comments and proofs at [[Chern-Simons element]] and began creating auxiliary entries as the need was. So there are now some stubs on * [[topological Yang-Mills theory]], * [[BF-theory]] * [[cosmological constant]] (coupling these three yields the 2-Chern-Simons theory for the canonical invariant polynomial on a strict Lie 2-algebra !) also did * [[fivebrane Lie 6-algebra]] (that entry was due a long time ago)

   It is clear that infinity-Chern-Weil theory will induce lots of examples of oo-Chern-Simons theory : for every Chern-Simons element on an $\infty$-Lie algebroid $\mathfrak{a}$, there is the corresponding generalized Chern-Simons action functional on the space of $\mathfrak{a}$-valued connections/forms.

   I have started now listing all the familiar QFTs that are obtained as special cases this way. This is a joint project I am doing with Chris Rogers.

   So I started that list with comments and proofs at Chern-Simons element and began creating auxiliary entries as the need was. So there are now some stubs on

   • topological Yang-Mills theory,

   • BF-theory

   • cosmological constant

   (coupling these three yields the 2-Chern-Simons theory for the canonical invariant polynomial on a strict Lie 2-algebra !)

   also did

   • fivebrane Lie 6-algebra

   (that entry was due a long time ago)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 23rd 2010
   • (edited Sep 23rd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI think I have shown that the [[AKSZ theory]] Lagrangian for any symplectic target Lie n-algebroid $(\mathfrak{a}, \omega)$ is precisely the $\infty$-Chern-Simons theory action functional corresponding to the [[Chern-Simons element]] of $\omega$. The proof is <a href="http://ncatlab.org/nlab/show/Chern-Simons+element#Symplectic">here</a>. So if I didn't make a mistake, this means that the fom of the AKSZ-Lagrangian, that sum of two terms, need not be decreed by hand but follows form $\infty$-Chern-Simons theory.

   I think I have shown that the AKSZ theory Lagrangian for any symplectic target Lie n-algebroid $(\mathfrak{a}, \omega)$ is precisely the $\infty$-Chern-Simons theory action functional corresponding to the Chern-Simons element of $\omega$.

   The proof is here.

   So if I didn’t make a mistake, this means that the fom of the AKSZ-Lagrangian, that sum of two terms, need not be decreed by hand but follows form $\infty$-Chern-Simons theory.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 23rd 2010
   • (edited Sep 23rd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexA maybe interesting aspect of this is that this kind of diagram that encodes on the one hand the structure of $\infty$-connections \[ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& transition\;function \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &&& connection \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle -\rangle}{\leftarrow}& inv(\mathfrak{g}) &&& curvature\;characteristic\;form } \] and also on the other hand encodes transgression between oo-Lie algebra cocycles and invariant polynomials via Chern-Simons elements \[ \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1} \mathbb{R}) &&& cocycle \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1} \mathbb{R}) &&& Chern-Simons\;element \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& inv(b^{n-1} \mathbb{R}) &&& invariant polynomial } \,. \] thirdly, by the above, also enocodes, as a special case, the central ingredients of _mechanics_ \[ \array{ CE(\mathfrak{P}) &\stackrel{H}{\leftarrow}& CE(b^{n}\mathbb{R}) &&& Hamiltonian \\ \uparrow && \uparrow \\ W(\mathfrak{P}) &\stackrel{L}{\leftarrow}& W(b^n \mathbb{R}) &&& Lagrangian \\ \uparrow && \uparrow \\ inv(\mathfrak{P}) &\stackrel{\omega}{\leftarrow}& inv(b^n \mathbb{R}) &&& symplectic\;structure } \] (where $\mathfrak{P}$ here is an $\infty$-Lie algebroid that is an [[n-symplectic manifold]]).

   A maybe interesting aspect of this is that this kind of diagram that encodes on the one hand the structure of $\infty$-connections

   $$\array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& transition\;function \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &&& connection \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle -\rangle}{\leftarrow}& inv(\mathfrak{g}) &&& curvature\;characteristic\;form }$$

   and also on the other hand encodes transgression between oo-Lie algebra cocycles and invariant polynomials via Chern-Simons elements

   $$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1} \mathbb{R}) &&& cocycle \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1} \mathbb{R}) &&& Chern-Simons\;element \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}& inv(b^{n-1} \mathbb{R}) &&& invariant polynomial } \,.$$

   thirdly, by the above, also enocodes, as a special case, the central ingredients of mechanics

   $$\array{ CE(\mathfrak{P}) &\stackrel{H}{\leftarrow}& CE(b^{n}\mathbb{R}) &&& Hamiltonian \\ \uparrow && \uparrow \\ W(\mathfrak{P}) &\stackrel{L}{\leftarrow}& W(b^n \mathbb{R}) &&& Lagrangian \\ \uparrow && \uparrow \\ inv(\mathfrak{P}) &\stackrel{\omega}{\leftarrow}& inv(b^n \mathbb{R}) &&& symplectic\;structure }$$

   (where $\mathfrak{P}$ here is an $\infty$-Lie algebroid that is an n-symplectic manifold).

4. • CommentRowNumber4.
   • CommentAuthorEric
   • CommentTimeSep 25th 2010
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   Author: Eric
   Format: MarkdownItexIs this related to the conjecture in Witten's [A note on the antibracket formalism](http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?199004090)?

   Is this related to the conjecture in Witten’s A note on the antibracket formalism?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 27th 2010
   • (edited Sep 27th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat's different: that note on the anti-bracket formalism is about observing that the "master equation" $\Delta S = \{S , S\}$ in BV-formalism is nothing but the image of $d S = 0$ under the isomorphism between differential forms and multi-vectorfields that is induced by any choice of volume form. There are two aspects to [[AKSZ theory]]: one concerns the structure of the Lagrangian, the other the inclusion of ghosts and anti-fields. What I said above concerns the first aspect: I am claiming that there is an even more fundamental reason for the structure of these AKSZ-Lagrangians: they are just Chern-Simons Lagrangians for $\infty$-Lie algbroids. The second aspect will also have a nice description: the BV-BRST complex of fields on $\Sigma$ with values in the target $\infty$-Lie algebroid $\mathfrak{a}$ is just the internal hom $[T \Sigma, \mathfrak{a}]$ in the category of _derived_ $\infty$-stacks (compare <a href="http://www.ncatlab.org/nlab/show/derived+infinity-Lie+algebroid">derived oo-Lie algebroid</a>). We once had a long discussion about this on the nCaf&eacute;. There my puzzlement had been that the internal hom of cochain complexes takes us out of the non-negatively graded ones that come under dold-Kan from oo-groupoids. One suggestion back then had been to use "$\mathbb{Z}$-groupoids" instead, to get the unbounded grading. But that's not the answer. The answer is that one has to think of this in derived $\infty$-stacks: by the theory of [[function algebras on infinity-stacks]] a simplicial presheaf on simplicial algebras maps to a cosimplicial simplicial algbra, which then under Dold-Kan maps to an _unbounded_ dg-algebra.

   That’s different: that note on the anti-bracket formalism is about observing that the “master equation” $\Delta S = \{S , S\}$ in BV-formalism is nothing but the image of $d S = 0$ under the isomorphism between differential forms and multi-vectorfields that is induced by any choice of volume form.

   There are two aspects to AKSZ theory: one concerns the structure of the Lagrangian, the other the inclusion of ghosts and anti-fields. What I said above concerns the first aspect: I am claiming that there is an even more fundamental reason for the structure of these AKSZ-Lagrangians: they are just Chern-Simons Lagrangians for $\infty$-Lie algbroids.

   The second aspect will also have a nice description: the BV-BRST complex of fields on $\Sigma$ with values in the target $\infty$-Lie algebroid $\mathfrak{a}$ is just the internal hom $[T \Sigma, \mathfrak{a}]$ in the category of derived $\infty$-stacks (compare derived oo-Lie algebroid).

   We once had a long discussion about this on the nCafé. There my puzzlement had been that the internal hom of cochain complexes takes us out of the non-negatively graded ones that come under dold-Kan from oo-groupoids. One suggestion back then had been to use “$\mathbb{Z}$-groupoids” instead, to get the unbounded grading. But that’s not the answer. The answer is that one has to think of this in derived $\infty$-stacks:

   by the theory of function algebras on infinity-stacks a simplicial presheaf on simplicial algebras maps to a cosimplicial simplicial algbra, which then under Dold-Kan maps to an unbounded dg-algebra.

6. • CommentRowNumber6.
   • CommentAuthorEric
   • CommentTimeSep 27th 2010
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   Author: Eric
   Format: MarkdownItexSorry. I should have been more explicit. The main material about the master equation is not the conjecture I was referring to. From the bottom of page 6 he points out a formal similarity between the symmetry of the master equation and that from open string theory. In the third paragraph on page 7 he says: >It is tempting to believe that in a suitable context, one could find an integration law in the antibracket formalism, and find a Lagrangian $\mathcal{L}(S)$ whose variational equation would be the quantum master equation. This $\mathcal{L}$ would be some sort of abstract Chern-Simons Lagrangian, and would play the role of a string field theory action. Such a framework for string field theory, if it exists, would very likely be far more attractive than what we now know. Did you find that Lagrangian?

   Sorry. I should have been more explicit. The main material about the master equation is not the conjecture I was referring to. From the bottom of page 6 he points out a formal similarity between the symmetry of the master equation and that from open string theory.

   In the third paragraph on page 7 he says:

   It is tempting to believe that in a suitable context, one could find an integration law in the antibracket formalism, and find a Lagrangian $\mathcal{L}(S)$ whose variational equation would be the quantum master equation. This $\mathcal{L}$ would be some sort of abstract Chern-Simons Lagrangian, and would play the role of a string field theory action. Such a framework for string field theory, if it exists, would very likely be far more attractive than what we now know.

   Did you find that Lagrangian?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 27th 2010
   • (edited Sep 27th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOther people back then found that Lagrangian. I may be mixing up the dates, I thought that was essentially Witten himself, later this was developed intensively by Zwiebach. In the hands of Zwiebach the BV-BRST-formalism found one of its more striking applications in bosonic string field theory and this success considerably contributed to the interest in BV-theory applied to other gauge theories. No, what I am claiming to add here is a deeper reason for all the Chern-Simons-like Lagrangians around us. I am claiming that there is a notion of connection on $\infty$-bundles and a generalization of the refined Chern-Weil homomorphism to these such that all those CS-like action functionals in physics are the "secondary characteristic classes" of such, for various choices of target spaces. I expect this should also apply to (bosonic) string field theory, but that I haven't thought about enough.

   Other people back then found that Lagrangian. I may be mixing up the dates, I thought that was essentially Witten himself, later this was developed intensively by Zwiebach. In the hands of Zwiebach the BV-BRST-formalism found one of its more striking applications in bosonic string field theory and this success considerably contributed to the interest in BV-theory applied to other gauge theories.

   No, what I am claiming to add here is a deeper reason for all the Chern-Simons-like Lagrangians around us. I am claiming that there is a notion of connection on $\infty$-bundles and a generalization of the refined Chern-Weil homomorphism to these such that all those CS-like action functionals in physics are the “secondary characteristic classes” of such, for various choices of target spaces.

   I expect this should also apply to (bosonic) string field theory, but that I haven’t thought about enough.

8. • CommentRowNumber8.
   • CommentAuthorEric
   • CommentTimeSep 27th 2010
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   Author: Eric
   Format: MarkdownItexBut even if they found the Lagrangian back then, it seems it was somewhat mysterious and seems somewhat ad hoc in nature, e.g. adding two terms by hand. In a way, you found the "right" Lagrangian that reproduces what other have found before in various contexts. Is that fair to say?

   But even if they found the Lagrangian back then, it seems it was somewhat mysterious and seems somewhat ad hoc in nature, e.g. adding two terms by hand. In a way, you found the “right” Lagrangian that reproduces what other have found before in various contexts. Is that fair to say?

9. • CommentRowNumber9.
   • CommentAuthorEric
   • CommentTimeSep 27th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexBy the way, which target space would reproduce "physics", i.e. the kind that can be observed directly by current experimental methods? I'm getting the sense that you and others are developing very general mathematical weapons the extend known physics on certain simple target spaces to similar physics on more general spaces. One challenge then is to find the "right" target space.

   By the way, which target space would reproduce “physics”, i.e. the kind that can be observed directly by current experimental methods?

   I’m getting the sense that you and others are developing very general mathematical weapons the extend known physics on certain simple target spaces to similar physics on more general spaces. One challenge then is to find the “right” target space.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2010
    • (edited Sep 27th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> Is that fair to say? I'd say it like this: it wasn't noticed before that a large number of Lagrangians are all special cases of the same principle: they are all Chern-Simons Lagrangians, but for Chern-Simons forms on general $\infty$-Lie algebroids. In AKSZ theory this statement is almost there: they notice that the Lagrangian is built from two pieces that are both naturally obtained by transgression of the "symplectic form" on the target $\infty$-Lie algebroid. What I observe is: this "symplectic form" ought to be thought of as an invariant polynomial, and that these two pieces then constitute the corresponding Chern-Simons element, and that this is a construction that is induced from some very general abstract theory and involves "no human intervention". > By the way, which target space would reproduce "physics", i.e. the kind that can be observed directly by current experimental methods? That's a good questions. All these $\infty$-Chern-Simons theories have the flavor of topological fielld theories. But 1. we sort of know that various "physical" theories are the boundary theories of these topological CS-theories. Notably the boundary theory of ordinary CS theory is the physical WZW model describing the string on a group manifold. And maybe more importantly: Kontsevich/Cattaneo-Felder in effect showed that the boundary theory of the 2d Poisson $\sigma$-model encodes on its boundary the quantum theory of the ordinary particle whose phase space gives the given Poisson Lie algebroid. Witten has recently published more articles along these lines, getting physical 1d QM from topological strings. There should be a general such "holography" mechanism by which the physical theories sit on the boundary of topological ones. But I don't understand this well enough yet. 1. Another phenomenon is that physical theories arise as In&ouml;n&uuml;-Wigner contractions of Chern-Simons theories. This is the point that Zanelli explores <a href="http://ncatlab.org/nlab/show/Chern-Simons+element#Zanelli">here</a>: how theories of gravity arise as limits of higher Chern-Simons theories.

    Is that fair to say?

    I’d say it like this: it wasn’t noticed before that a large number of Lagrangians are all special cases of the same principle: they are all Chern-Simons Lagrangians, but for Chern-Simons forms on general $\infty$-Lie algebroids.

    In AKSZ theory this statement is almost there: they notice that the Lagrangian is built from two pieces that are both naturally obtained by transgression of the “symplectic form” on the target $\infty$-Lie algebroid.

    What I observe is: this “symplectic form” ought to be thought of as an invariant polynomial, and that these two pieces then constitute the corresponding Chern-Simons element, and that this is a construction that is induced from some very general abstract theory and involves “no human intervention”.

    By the way, which target space would reproduce “physics”, i.e. the kind that can be observed directly by current experimental methods?

    That’s a good questions. All these $\infty$-Chern-Simons theories have the flavor of topological fielld theories. But

    1. we sort of know that various “physical” theories are the boundary theories of these topological CS-theories. Notably the boundary theory of ordinary CS theory is the physical WZW model describing the string on a group manifold. And maybe more importantly: Kontsevich/Cattaneo-Felder in effect showed that the boundary theory of the 2d Poisson $\sigma$-model encodes on its boundary the quantum theory of the ordinary particle whose phase space gives the given Poisson Lie algebroid. Witten has recently published more articles along these lines, getting physical 1d QM from topological strings. There should be a general such “holography” mechanism by which the physical theories sit on the boundary of topological ones. But I don’t understand this well enough yet.

    2. Another phenomenon is that physical theories arise as Inönü-Wigner contractions of Chern-Simons theories. This is the point that Zanelli explores here: how theories of gravity arise as limits of higher Chern-Simons theories.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2010
    • (edited Sep 28th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItexcreated on my personal web <a href="http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory">oo-Chern-Simons theory</a> showing how the generalized oo-Chern-Simons action functionals (including DW-theory, ordinary CS theory, higher CS theory, all AKSZ theories, BF-theory, supergravity) arise from first principles. This builds on our old discussion at [[Dijkgraaf-Witten theory]], observing that this generalizes.

    created on my personal web oo-Chern-Simons theory showing how the generalized oo-Chern-Simons action functionals (including DW-theory, ordinary CS theory, higher CS theory, all AKSZ theories, BF-theory, supergravity) arise from first principles.

    This builds on our old discussion at Dijkgraaf-Witten theory, observing that this generalizes.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2010
    • (edited Sep 29th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI will now claim something: I claim that the Lagrangian of 11d supergravity is a degree 11 [[Chern-Simons element]] on the _supergravity Lie 6-algebra_ , hence that 11d sugra is an example of $\infty$-Chern-Simons theory. More generally, I claim that what D'Auria-Fre call -- curiously -- the _cosmo-cocycle condition_ (for instance the system of equations 4.2 <a href="http://ncatlab.org/nlab/files/GeometricSupergravity.pdf#page=20">here</a>) on a Lagrangian is precisely the statemement that this Lagrangian is a Chern-Simons element (under the condition that the pure curvature term is $d_W$-closed, which they don't seem to require explicitly, but which is true in their examples!) So that makes all other such examples of supergravity theories in their book be examples of oo-Chern-Simons action functionals. I'll start to spell that out now in the nLab entry. But it's late here in Vienna, and I should rather sleep. So maybe to be completed tmorrow.

    I will now claim something:

    I claim that the Lagrangian of 11d supergravity is a degree 11 Chern-Simons element on the supergravity Lie 6-algebra , hence that 11d sugra is an example of $\infty$-Chern-Simons theory.

    More generally, I claim that what D’Auria-Fre call – curiously – the cosmo-cocycle condition (for instance the system of equations 4.2 here) on a Lagrangian is precisely the statemement that this Lagrangian is a Chern-Simons element (under the condition that the pure curvature term is $d_W$-closed, which they don’t seem to require explicitly, but which is true in their examples!) So that makes all other such examples of supergravity theories in their book be examples of oo-Chern-Simons action functionals.

    I’ll start to spell that out now in the nLab entry. But it’s late here in Vienna, and I should rather sleep. So maybe to be completed tmorrow.

13. • CommentRowNumber13.
    • CommentAuthorEric
    • CommentTimeSep 29th 2010
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    Author: Eric
    Format: MarkdownItexThat (#12) sounds like a very interesting result. It is not clear to me from your comment #10, but is there some way to obtain the standard model from Chern-Simons theory somehow?

    That (#12) sounds like a very interesting result.

    It is not clear to me from your comment #10, but is there some way to obtain the standard model from Chern-Simons theory somehow?

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2010
    • (edited Sep 29th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> It is not clear to me from your comment #10, but is there some way to obtain the standard model from Chern-Simons theory somehow? Right, so the question is: to which extent is the action functional of standard Yang-Mills theory also a special case of a Chern-Simons action functional for a suitable Lie $n$-algebra? Notice that this will necessarily have an indirect answer, because everything coming out of $\infty$-CS theory will be in "first order formalism" (as in: [[first order formulation of gravity]]) and so in particular all terms in the action that would involve the Hodge star operator cannot appear directly, but their equations of motion can appear as the equations of motion once those of certain auxiliary fields are solved. As an example, in the case of 11d supergravity discussed so far, there is the supergravity $C$-field that is a 3-form whose equations o motion follow from the standard Yang-Mills type action functional $$ \mathcal{L}(C) = d C \wedge \star d C + \cdots \,. $$ But the same equations of motion also follow from the first order Lagrangian of 11d-supergravity. This contains more fields, but for some of them the equation of motion is just an algebraic constraint that identifies some of them with $\star d C$, even though the Hodge star does not appear explicitly in the action. It appears "dynamically". (In fact I need to add this discussion to the nLab entry. Because strictly speaking to get these terms for the $C$-field DAuria-Fre pass from using the [[supergravity Lie 6-algebra]] to a Lie-6-algebr_oid_ .) I'll have to check how this story goes in other dimensions. D'Auria-Fre advertize the article * P. Fré, Nucl. Phys. B186 (1981) 44 for a discussion of the abelian Yang-Mills term in $N=2$ and $N=3$ sugra along their lines. But I haven't checked that yet. But apart from these abelian Yang-Mills terms, where do the nonabelian ones come from? One possibility is: they are not fundamental, but appear by the Kaluza-Klein mechanism. Take a Lagrangian for purely gravitational fields on a $d+n$-dimensional manifold, and consider the case that this is of the form $X \times Y$ for $X$ being $d$-dimensional. One may think of the fields on $X \times Y$ with tensor components along $Y$ as more fields, just on $X$. If one then looks at solutuions of the gravity equations of motion thaat describe a product metric on $X \times Y$ such that there is a group $G$ of diffeomorphisms of $Y$, then in terms of fields on $X$ the resulting dynamics looks like gravity on $X$ coupled to Yang-Mills theory for gauge group $G$ and coupled to a bunch of scalar fields. This is one way to get YM-type dynamics. There are others, but this is maybe the "purest" in as far as ordinary field theory goes.

    It is not clear to me from your comment #10, but is there some way to obtain the standard model from Chern-Simons theory somehow?

    Right, so the question is: to which extent is the action functional of standard Yang-Mills theory also a special case of a Chern-Simons action functional for a suitable Lie $n$-algebra?

    Notice that this will necessarily have an indirect answer, because everything coming out of $\infty$-CS theory will be in "first order formalism" (as in: first order formulation of gravity) and so in particular all terms in the action that would involve the Hodge star operator cannot appear directly, but their equations of motion can appear as the equations of motion once those of certain auxiliary fields are solved.

    As an example, in the case of 11d supergravity discussed so far, there is the supergravity $C$-field that is a 3-form whose equations o motion follow from the standard Yang-Mills type action functional

    $$\mathcal{L}(C) = d C \wedge \star d C + \cdots \,.$$

    But the same equations of motion also follow from the first order Lagrangian of 11d-supergravity. This contains more fields, but for some of them the equation of motion is just an algebraic constraint that identifies some of them with $\star d C$, even though the Hodge star does not appear explicitly in the action. It appears "dynamically".

    (In fact I need to add this discussion to the nLab entry. Because strictly speaking to get these terms for the $C$-field DAuria-Fre pass from using the supergravity Lie 6-algebra to a Lie-6-algebr_oid_ .)

    I’ll have to check how this story goes in other dimensions. D’Auria-Fre advertize the article

    • P. Fré, Nucl. Phys. B186 (1981) 44

    for a discussion of the abelian Yang-Mills term in $N=2$ and $N=3$ sugra along their lines. But I haven’t checked that yet.

    But apart from these abelian Yang-Mills terms, where do the nonabelian ones come from? One possibility is: they are not fundamental, but appear by the Kaluza-Klein mechanism.

    Take a Lagrangian for purely gravitational fields on a $d+n$-dimensional manifold, and consider the case that this is of the form $X \times Y$ for $X$ being $d$-dimensional. One may think of the fields on $X \times Y$ with tensor components along $Y$ as more fields, just on $X$. If one then looks at solutuions of the gravity equations of motion thaat describe a product metric on $X \times Y$ such that there is a group $G$ of diffeomorphisms of $Y$, then in terms of fields on $X$ the resulting dynamics looks like gravity on $X$ coupled to Yang-Mills theory for gauge group $G$ and coupled to a bunch of scalar fields.

    This is one way to get YM-type dynamics. There are others, but this is maybe the "purest" in as far as ordinary field theory goes.

15. • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 29th 2010
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    Author: DavidRoberts
    Format: MarkdownItexThis is pretty awesome stuff. It is what I originally wanted to do my thesis on, but the state of the art was waiting for someone with smarts to come along.... ;-) If it's taken Urs (+others) 5-6 years to get to this point, I'm glad I didn't blindly stick to my guns. But more seriously, >This contains more fields, but for some of them the equation of motion is just an algebraic constraint that identifies some of them with $\star d C$, even though the Hodge star does not appear explicitly in the action. It appears "dynamically". what sort of thing would happen in the quantum theory? Will the formalism still force this identification? I'm being dense because my Lagrangian-foo is gone out the window.

    This is pretty awesome stuff. It is what I originally wanted to do my thesis on, but the state of the art was waiting for someone with smarts to come along…. ;-) If it’s taken Urs (+others) 5-6 years to get to this point, I’m glad I didn’t blindly stick to my guns.

    But more seriously,

    This contains more fields, but for some of them the equation of motion is just an algebraic constraint that identifies some of them with $\star d C$, even though the Hodge star does not appear explicitly in the action. It appears "dynamically".

    what sort of thing would happen in the quantum theory? Will the formalism still force this identification? I’m being dense because my Lagrangian-foo is gone out the window.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItex> This is pretty awesome stuff. Yeah, I am pretty fond of it. I am getting to the point that I will run around saying "_Everything_ is [[schreiber:infinity-Chern-Simons theory]]". > If it's taken Urs (+others) 5-6 years to get to this point, I'm glad I didn't blindly stick to my guns. It's a curious exercise: the operations involved are elementary and straightforward, but what takes so much time is to get used to the yoga of the operations with the symbols to the extent that you can see the matrix behind the strings of green symbols. You can tell that D'Auria and Fré themselves tried hard to understand what it is they are doing _conceptually_ : their attempt to clarify this is reflected in their invention of terms like "soft group manifold", "soft form" etc. But this is not the answer. Moreover, I think some of the symbols they write to paper do not actually literally parse as advertized. This stopped me from understanding "rheonomy" for a long time: it is derived using would-be operations on differential forms that do not strictly speaking make sense! But the thing is that the operations do make sense and all their math is all fine if we think of them as images of operations on the Weil algebra under a homomorphism to forms. Things like that. On the standard mechanism of first order formalism and auxiliary fields: >> This contains more fields, but for some of them the equation of motion is just an algebraic constraint that identifies some of them with $\star d C$, even though the Hodge star does not appear explicitly in the action. It appears "dynamically". > what sort of thing would happen in the quantum theory? Will the formalism still force this identification? I'm being dense because my Lagrangian-foo is gone out the window. The standard path-integral lore says that algebraic field equations remain true on the nose after quantization. You think of the algebraic constraint $C(\phi) = 0$ as being encoded in the action by a Lagrange multiplier $\lambda$ $$ \mathcal{L}(\phi,\lambda) = C(\phi) \lambda $$ and then argue that the path integral over $\lambda$ is the Fourier-transformation of the exponential $\int \exp(i C \lambda) d \lambda$ that produces the $\delta$-distribution supported at the constraint surface $C(\phi) = 0$. (There is also a more sophisticated way of saying the same using BV-BRST formalism, but this I can't do while at breakfast, as I am now.)

    This is pretty awesome stuff.

    Yeah, I am pretty fond of it. I am getting to the point that I will run around saying “Everything is infinity-Chern-Simons theory (schreiber)”.

    If it’s taken Urs (+others) 5-6 years to get to this point, I’m glad I didn’t blindly stick to my guns.

    It’s a curious exercise: the operations involved are elementary and straightforward, but what takes so much time is to get used to the yoga of the operations with the symbols to the extent that you can see the matrix behind the strings of green symbols.

    You can tell that D’Auria and Fré themselves tried hard to understand what it is they are doing conceptually : their attempt to clarify this is reflected in their invention of terms like “soft group manifold”, “soft form” etc. But this is not the answer. Moreover, I think some of the symbols they write to paper do not actually literally parse as advertized. This stopped me from understanding “rheonomy” for a long time: it is derived using would-be operations on differential forms that do not strictly speaking make sense! But the thing is that the operations do make sense and all their math is all fine if we think of them as images of operations on the Weil algebra under a homomorphism to forms. Things like that.

    On the standard mechanism of first order formalism and auxiliary fields:

    This contains more fields, but for some of them the equation of motion is just an algebraic constraint that identifies some of them with $\star d C$, even though the Hodge star does not appear explicitly in the action. It appears “dynamically”.

    what sort of thing would happen in the quantum theory? Will the formalism still force this identification? I’m being dense because my Lagrangian-foo is gone out the window.

    The standard path-integral lore says that algebraic field equations remain true on the nose after quantization. You think of the algebraic constraint $C(\phi) = 0$ as being encoded in the action by a Lagrange multiplier $\lambda$

    $$\mathcal{L}(\phi,\lambda) = C(\phi) \lambda$$

    and then argue that the path integral over $\lambda$ is the Fourier-transformation of the exponential $\int \exp(i C \lambda) d \lambda$ that produces the $\delta$-distribution supported at the constraint surface $C(\phi) = 0$.

    (There is also a more sophisticated way of saying the same using BV-BRST formalism, but this I can’t do while at breakfast, as I am now.)

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexWorked further on the <a href="http://ncatlab.org/nlab/show/D%27Auria-Fre+formulation+of+supergravity#ChernSimonsElements">section</a> with the claim about how the sugra action functional is an $\infty$-CS-term. More polishing is certainly possible, but I have to run now.

    Worked further on the section with the claim about how the sugra action functional is an $\infty$-CS-term. More polishing is certainly possible, but I have to run now.

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2010
    • (edited Sep 29th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI need to correct something: of course the D'Auria-Fre "cosmo-cocycle condition" only asserts precisely that in $d_W cs$ the terms _linear_ in the curvatures vanish. It does not make a statement about the terms of higher order in the curvature. So I still need to check if for their Lagrangians these actually do vanish (except possibly for the pure curvature term).

    I need to correct something:

    of course the D’Auria-Fre “cosmo-cocycle condition” only asserts precisely that in $d_W cs$ the terms linear in the curvatures vanish. It does not make a statement about the terms of higher order in the curvature.

    So I still need to check if for their Lagrangians these actually do vanish (except possibly for the pure curvature term).

19. • CommentRowNumber19.
    • CommentAuthorEric
    • CommentTimeSep 29th 2010
    • PermaLink
    Author: Eric
    Format: MarkdownItex>I am getting to the point that I will run around saying "_Everything_ is [[schreiber:infinity-Chern-Simons theory]]". Yep yep. That is what I see happening as well. Getting geometry from topology, i.e. YM from CS, seems tricky, but not impossible I suppose. I wish I understood how it works. You suggested a few avenues that sound plausible, but nothing yet that makes me think, "Ah yes. Nature must be like that." (As if I knew what nature should be like!) When these kinds of unifications occur, it definitely means you're on the right track. Very cool.

    I am getting to the point that I will run around saying “Everything is infinity-Chern-Simons theory (schreiber)”.

    Yep yep. That is what I see happening as well.

    Getting geometry from topology, i.e. YM from CS, seems tricky, but not impossible I suppose. I wish I understood how it works. You suggested a few avenues that sound plausible, but nothing yet that makes me think, “Ah yes. Nature must be like that.” (As if I knew what nature should be like!)

    When these kinds of unifications occur, it definitely means you’re on the right track. Very cool.

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexDomenico, are you still reading here? Here is a some observation: As we have been discussing, we get an $\infty$-Chern-Simons Lagrangean from every characteristic class $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ simply by postcomposing with the universal curvature characteristic form $$ \mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) $$ and then lifting to differential cohomology $\hat \mathbf{c}$. Note in passing that of interest are the homotopy fibers of $\hat \mathbf{c}$, encoding differential obstruction classes like differential string structures, etc. Now, as we have discussed, by a general abstract integration-without-integration proces we should get the corresponding action functional $$ \exp(i S(-))_{\mathbf{c}} : [\Sigma,\mathbf{B}G_{conn}] \to \mathbf{B}^{n-dim \Sigma}U(1) \,. $$ Observe that in BRST-BV quantization we want to pass to the derived critical locus of this, the derived 0-locus of $d \exp(i S)$. What does this mean in the above language? Curiously, it means repeating the previous step: we form $d \exp(i S)$ in codimension 0 by postcomposition with the universal 1-curvature characteristic form of U(1), also known as the Maurer-Cartan form $\theta : U(1) \to \mathbf{\flat}_{dR} \mathbf{B}U(1)$ to get $$ d \exp(i S)_{\mathbf{c}} : [\Sigma,\mathbf{B}G_{conn}] \stackrel{\exp(i S)}{\to} \mathbf{B}^{n-dim\Sigma} U(1) \stackrel{d}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1 - dim \Sigma} U(1) \,. $$ Taking the derived 0-locus of this just means again to form its homotopy fiber --- albeit while regarding it as a morphism in the ambient derived $\infty$-topos. I am not sure yet what it is telling me, but it seems like something to take notice of: the derived phase space of a field theory action functional is precisely analogous to a differential twisted cohomology (such a differential string structures) for a classical Lagrangean. Do you see what I mean? Have to run now to catch a bus.

    Domenico, are you still reading here?

    Here is a some observation:

    As we have been discussing, we get an $\infty$-Chern-Simons Lagrangean from every characteristic class $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ simply by postcomposing with the universal curvature characteristic form

    $$\mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)$$

    and then lifting to differential cohomology $\hat \mathbf{c}$. Note in passing that of interest are the homotopy fibers of $\hat \mathbf{c}$, encoding differential obstruction classes like differential string structures, etc.

    Now, as we have discussed, by a general abstract integration-without-integration proces we should get the corresponding action functional

    $$\exp(i S(-))_{\mathbf{c}} : [\Sigma,\mathbf{B}G_{conn}] \to \mathbf{B}^{n-dim \Sigma}U(1) \,.$$

    Observe that in BRST-BV quantization we want to pass to the derived critical locus of this, the derived 0-locus of $d \exp(i S)$. What does this mean in the above language? Curiously, it means repeating the previous step: we form $d \exp(i S)$ in codimension 0 by postcomposition with the universal 1-curvature characteristic form of U(1), also known as the Maurer-Cartan form $\theta : U(1) \to \mathbf{\flat}_{dR} \mathbf{B}U(1)$ to get

    $$d \exp(i S)_{\mathbf{c}} : [\Sigma,\mathbf{B}G_{conn}] \stackrel{\exp(i S)}{\to} \mathbf{B}^{n-dim\Sigma} U(1) \stackrel{d}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1 - dim \Sigma} U(1) \,.$$

    Taking the derived 0-locus of this just means again to form its homotopy fiber — albeit while regarding it as a morphism in the ambient derived $\infty$-topos.

    I am not sure yet what it is telling me, but it seems like something to take notice of: the derived phase space of a field theory action functional is precisely analogous to a differential twisted cohomology (such a differential string structures) for a classical Lagrangean.

    Do you see what I mean? Have to run now to catch a bus.

21. • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexTo amplify one technical point that I used in the above argument: I think <a href="http://www.ncatlab.org/schreiber/show/infinity-Chern-Simons+theory#TheActionFunctional">our theorem</a> on the intrinsic exponentiated integraton $exp(i \int_\Sigma L)$ of the $\infty$-Chern-Simons Lagrangian $L$ refines from a function on the discrete $\infty$-groupoid of field configurations to a smooth function on the smooth $\infty$-groupoid of field configurations. more precisely, hitting $$ L : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} $$ with the internal hom out of a manifold $\Sigma$ $$ [\Sigma, \mathbf{B}G_{conn}] \to [\Sigma,\mathbf{B}^n U(1)_{conn}] $$ and then postcomposing with 0-truncation _and then_ with concretization $$ \cdots \to [\Sigma,\mathbf{B}^n U(1)_{conn}] \to \tau_{\leq 0 } [\Sigma,\mathbf{B}^n U(1)_{conn}] \to Conc \tau_{\leq 0 } [\Sigma,\mathbf{B}^n U(1)_{conn}] $$ is equivalently a morphism $$ \exp(i \int_\Sigma L) : [\Sigma, \mathbf{B}G_{conn}] \to U(1) $$ form the _smooth_ $\infty$-groupoid of field configurations to the smooth manifold $U(1)$. I think the argument for that is a pretty straightforwatd generalization of the previous argument for the external version: The internal hom assigns $$ [\Sigma, \mathbf{B}^n U(1)_{conn}] : V \mapsto \mathbf{H}(V \times \Sigma, \mathbf{B}^n U(1)_{conn}) $$ but concretization sends this to its set of point evaluations along points $\{v\} \to V$ $$ \{v\} \mapsto \mathbf{H}(\{v\} \times \Sigma, \mathbf{B}^n U(1)_{conn}) \,. $$ For these the previous external argument applies, to give $$ \cdots \simeq \mathbf{H}(\{v\} \times \Sigma, \mathbf{\flat} \mathbf{B}^n U(1)) $$ (for dimensional reasons due to $dim \Sigma \leq n$) and then $$ \cdots \simeq \infty Grpd(\Pi(\Sigma), B^n U(1)) $$ and then with the universal coefficient theorem $$ \stackrel{\tau_{\leq 0}}{\mapsto} U(1) \,. $$ In fact, looking a little closer at the presentation of $\mathbf{B}^n U(1)_{conn}$ one can show (and this is a necessary statement for the whole story anyway) that under this identification indeed a closed degree-$dim \Sigma$ differential form $\omega$ on $\Sigma$ maps to $\exp(i \int_\Sigma \omega)$. Using this, one sees then that in the concretization $V$ maps to the set of smooth $U(1)$-valued functions on $V$. I am not saying this well, will try to write out a better formulation in the entry on $\infty$-Chern-Simons theory, but I think the upshot is that at least for $dim \Sigma = n$ we have $$ Conc \tau_{\leq 0}[\Sigma, \mathbf{B}^n U(1)] \simeq U(1) $$ as _smooth_ spaces, and hence the intrinsic action functional $$ \exp(i S) : [\Sigma, \mathbf{B}G_{conn}] \to U(1) $$ indeed as a morphism in $Smooth \infty Grpd$.

    To amplify one technical point that I used in the above argument:

    I think our theorem on the intrinsic exponentiated integraton $exp(i \int_\Sigma L)$ of the $\infty$-Chern-Simons Lagrangian $L$ refines from a function on the discrete $\infty$-groupoid of field configurations to a smooth function on the smooth $\infty$-groupoid of field configurations.

    more precisely, hitting

    $$L : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}$$

    with the internal hom out of a manifold $\Sigma$

    $$[\Sigma, \mathbf{B}G_{conn}] \to [\Sigma,\mathbf{B}^n U(1)_{conn}]$$

    and then postcomposing with 0-truncation and then with concretization

    $$\cdots \to [\Sigma,\mathbf{B}^n U(1)_{conn}] \to \tau_{\leq 0 } [\Sigma,\mathbf{B}^n U(1)_{conn}] \to Conc \tau_{\leq 0 } [\Sigma,\mathbf{B}^n U(1)_{conn}]$$

    is equivalently a morphism

    $$\exp(i \int_\Sigma L) : [\Sigma, \mathbf{B}G_{conn}] \to U(1)$$

    form the smooth $\infty$-groupoid of field configurations to the smooth manifold $U(1)$.

    I think the argument for that is a pretty straightforwatd generalization of the previous argument for the external version:

    The internal hom assigns

    $$[\Sigma, \mathbf{B}^n U(1)_{conn}] : V \mapsto \mathbf{H}(V \times \Sigma, \mathbf{B}^n U(1)_{conn})$$

    but concretization sends this to its set of point evaluations along points $\{v\} \to V$

    $$\{v\} \mapsto \mathbf{H}(\{v\} \times \Sigma, \mathbf{B}^n U(1)_{conn}) \,.$$

    For these the previous external argument applies, to give

    $$\cdots \simeq \mathbf{H}(\{v\} \times \Sigma, \mathbf{\flat} \mathbf{B}^n U(1))$$

    (for dimensional reasons due to $dim \Sigma \leq n$) and then

    $$\cdots \simeq \infty Grpd(\Pi(\Sigma), B^n U(1))$$

    and then with the universal coefficient theorem

    $$\stackrel{\tau_{\leq 0}}{\mapsto} U(1) \,.$$

    In fact, looking a little closer at the presentation of $\mathbf{B}^n U(1)_{conn}$ one can show (and this is a necessary statement for the whole story anyway) that under this identification indeed a closed degree-$dim \Sigma$ differential form $\omega$ on $\Sigma$ maps to $\exp(i \int_\Sigma \omega)$.

    Using this, one sees then that in the concretization $V$ maps to the set of smooth $U(1)$-valued functions on $V$.

    I am not saying this well, will try to write out a better formulation in the entry on $\infty$-Chern-Simons theory, but I think the upshot is that at least for $dim \Sigma = n$ we have

    $$Conc \tau_{\leq 0}[\Sigma, \mathbf{B}^n U(1)] \simeq U(1)$$

    as smooth spaces, and hence the intrinsic action functional

    $$\exp(i S) : [\Sigma, \mathbf{B}G_{conn}] \to U(1)$$

    indeed as a morphism in $Smooth \infty Grpd$.

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI've been editing [[schreiber:infinity-Chern-Simons theory]], expanding the existing discussion here and there and then in particular adding something along the lines of the above two comments.

    I’ve been editing infinity-Chern-Simons theory (schreiber), expanding the existing discussion here and there and then in particular adding something along the lines of the above two comments.

23. • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have now begun working on the section <a href="http://www.ncatlab.org/schreiber/show/infinity-Chern-Simons+theory#ASKZTheory">AKSZ theory</a>. Mainly I copied over from [[Chern-Simons element]] and then expanded the proof that the AKSZ action functional is the $\infty$-Chern-Simons action functional that corresponds to the canonical invariant polynomial on a symplectic Lie $n$-algebroid. I think back when I wrote out the proof first I had a wrong remnant relative prefactor of $1/(n+2)$ in the two summands of the Chern-Simons element. I believe I have corrected this now, but as usual with the dg-yoga, it is easy to get combinatorial prefactors and signs mixed up. Luckily it does not matter for the statement either way, since it only affects the normalization of the invariant polynomial that the Poisson-tensor cocycle transgresses to. But if somebody (Domenico, Chris?) could check my computation, i'd appreciate it. I've written out many more intermediate steps now.

    I have now begun working on the section AKSZ theory.

    Mainly I copied over from Chern-Simons element and then expanded the proof that the AKSZ action functional is the $\infty$-Chern-Simons action functional that corresponds to the canonical invariant polynomial on a symplectic Lie $n$-algebroid.

    I think back when I wrote out the proof first I had a wrong remnant relative prefactor of $1/(n+2)$ in the two summands of the Chern-Simons element. I believe I have corrected this now, but as usual with the dg-yoga, it is easy to get combinatorial prefactors and signs mixed up. Luckily it does not matter for the statement either way, since it only affects the normalization of the invariant polynomial that the Poisson-tensor cocycle transgresses to. But if somebody (Domenico, Chris?) could check my computation, i’d appreciate it. I’ve written out many more intermediate steps now.

24. • CommentRowNumber24.
    • CommentAuthordomenico_fiorenza
    • CommentTimeFeb 27th 2011
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItex> Domenico, are you still reading here? Here I am :) starting to read now yesterday posts

    Domenico, are you still reading here?

    Here I am :) starting to read now yesterday posts

25. • CommentRowNumber25.
    • CommentAuthordomenico_fiorenza
    • CommentTimeFeb 27th 2011
    • (edited Feb 27th 2011)
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItex>What does this mean in the above language? Curiously, it means repeating the previous step: we form $d \exp(i S)$ in codimension 0 by postcomposition with the universal 1-curvature characteristic form of U(1), also known as the Maurer-Cartan form $\theta : U(1) \to \mathbf{\flat}_{dR} \mathbf{B}U(1)$ This is brilliant! to me, reading that has been one of the most totally-trivial-and-totally-mind-changing remarks I've ever come onto! let me expand that in a totally classical and down to earth version for the sake of eventual interested readers. Let us begin with a smooth function real valued function $f:M\to \mathbb{R}$ on a smooth manifold $M$. What is the 1-form $d f$, then? we can consider $\mathbb{R}$ as a smmoth manifold itself, and consider the identity $id_\mathbb{R}:\mathbb{R}\to \mathbb{R}$ as a smooth real valued function on $\mathbb{R}$. Let us consider the differential of this very special function, and call it $d x$. It can be given a completely intrinsic Lie theoretic characterization by noticing that under the canonical identification of the tangent space $T_0\mathbb{R}$ with $\mathbb{R}$, $d x$ is the unique translation invariant $T_0\mathbb{R}$-valued $1-form$ on $\mathbb{R}$ inducing the identity on $T_0\mathbb{R}$. In short, $d x$ is the Maurer-Cartan form of the Lie group $\mathbb{R}$. Now, let us come back to the differential of $f: M\to \mathbb{R}$. We clearly have $f=f^*(id_\mathbb{R})$ so that $d f=d f^*(id_\mathbb{R})= f^*(dx)$, i.e., $d f$ is the pull-back of the Maurer-Cartan form of $\mathbb{R}$! and if we look at $d x$ as a morphism $d x: \mathbb{R}\to \flat_{dR}\mathbb{B}^1\mathbb{R}$, then what we are doing is nothing but a composition: $$ M\stackrel{f}{\to}\mathbb{R}\stackrel{d x}{\to} \flat_{dR}\mathbf{B}^1\mathbb{R} $$ One immediately sees how this adapts to the problem of describing the critical locus of a smooth function $f: M\to U(1)$. Here, classically one takes a lift $S: M\to \mathbb{R}$, i.e., writes $f=e^{i S}$ and then looks at the zero locus of $d S$. but by what we have just said $d S$ is nothing but the pull-back $S^*(d x)$ which in turn is $f^*(d \theta)$, where $d \theta$ is the Maurer-Cartan form on $U(1)$. So, in the end we are interested in the fiber (homotopy fiber, which else?) of $$ M\stackrel{f}{\to}U(1)\stackrel{d \theta}{\to} \flat_{dR}\mathbf{B}^1\mathbb{R} $$ over $0$.

    What does this mean in the above language? Curiously, it means repeating the previous step: we form $d \exp(i S)$ in codimension 0 by postcomposition with the universal 1-curvature characteristic form of U(1), also known as the Maurer-Cartan form $\theta : U(1) \to \mathbf{\flat}_{dR} \mathbf{B}U(1)$

    This is brilliant! to me, reading that has been one of the most totally-trivial-and-totally-mind-changing remarks I’ve ever come onto! let me expand that in a totally classical and down to earth version for the sake of eventual interested readers. Let us begin with a smooth function real valued function $f:M\to \mathbb{R}$ on a smooth manifold $M$. What is the 1-form $d f$, then? we can consider $\mathbb{R}$ as a smmoth manifold itself, and consider the identity $id_\mathbb{R}:\mathbb{R}\to \mathbb{R}$ as a smooth real valued function on $\mathbb{R}$. Let us consider the differential of this very special function, and call it $d x$. It can be given a completely intrinsic Lie theoretic characterization by noticing that under the canonical identification of the tangent space $T_0\mathbb{R}$ with $\mathbb{R}$, $d x$ is the unique translation invariant $T_0\mathbb{R}$-valued $1-form$ on $\mathbb{R}$ inducing the identity on $T_0\mathbb{R}$. In short, $d x$ is the Maurer-Cartan form of the Lie group $\mathbb{R}$. Now, let us come back to the differential of $f: M\to \mathbb{R}$. We clearly have $f=f^*(id_\mathbb{R})$ so that $d f=d f^*(id_\mathbb{R})= f^*(dx)$, i.e., $d f$ is the pull-back of the Maurer-Cartan form of $\mathbb{R}$! and if we look at $d x$ as a morphism $d x: \mathbb{R}\to \flat_{dR}\mathbb{B}^1\mathbb{R}$, then what we are doing is nothing but a composition:

    $$M\stackrel{f}{\to}\mathbb{R}\stackrel{d x}{\to} \flat_{dR}\mathbf{B}^1\mathbb{R}$$

    One immediately sees how this adapts to the problem of describing the critical locus of a smooth function $f: M\to U(1)$. Here, classically one takes a lift $S: M\to \mathbb{R}$, i.e., writes $f=e^{i S}$ and then looks at the zero locus of $d S$. but by what we have just said $d S$ is nothing but the pull-back $S^*(d x)$ which in turn is $f^*(d \theta)$, where $d \theta$ is the Maurer-Cartan form on $U(1)$. So, in the end we are interested in the fiber (homotopy fiber, which else?) of

    $$M\stackrel{f}{\to}U(1)\stackrel{d \theta}{\to} \flat_{dR}\mathbf{B}^1\mathbb{R}$$

    over $0$.

26. • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2011
    • (edited Feb 27th 2011)
    • PermaLink
    Author: Urs
    Format: MarkdownItexHi Domenico, thanks for the reaction and feedback! There is now an obvious generalization and then an obvious question: I think one can show, using a semi-evident notion of "n-concreteness", that in general codimension $n - dim \Sigma$ we have the extended action functional $$ \exp(i S(-)) : [\Sigma, A_{conn}] \to \mathbf{B}^{n-dim \Sigma} U(1) \,. $$ For each $\Sigma$ this has a differential refinement to $$ d \exp(i S(-)) : [\Sigma, A_{conn}] \stackrel{\exp(i S(-))}{\to} \mathbf{B}^{n - \dim \Sigma} U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1-dim \Sigma} U(1) $$ And all these have homotopy fibers. As we have seen, they deserve to tbe thought of as the "extended covariant phase spaces" of the theory. The question is: 1. is such an "extended covariant phase space" (in the sense of "extended QFT") something that has secretly been discussed in other terms in the literature before? 1. How does the extended covariant phase space relate to the extended quantization of the theory?

    Hi Domenico,

    thanks for the reaction and feedback!

    There is now an obvious generalization and then an obvious question:

    I think one can show, using a semi-evident notion of “n-concreteness”, that in general codimension $n - dim \Sigma$ we have the extended action functional

    $$\exp(i S(-)) : [\Sigma, A_{conn}] \to \mathbf{B}^{n-dim \Sigma} U(1) \,.$$

    For each $\Sigma$ this has a differential refinement to

    $$d \exp(i S(-)) : [\Sigma, A_{conn}] \stackrel{\exp(i S(-))}{\to} \mathbf{B}^{n - \dim \Sigma} U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1-dim \Sigma} U(1)$$

    And all these have homotopy fibers. As we have seen, they deserve to tbe thought of as the “extended covariant phase spaces” of the theory.

    The question is:

    1. is such an “extended covariant phase space” (in the sense of “extended QFT”) something that has secretly been discussed in other terms in the literature before?

    2. How does the extended covariant phase space relate to the extended quantization of the theory?

27. • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexMaybe more urgently and as a first step: we should write out something about variational calculus in the sheaf-theoretic picture. Usually it is discussed in terms of infinite-dimensional manifold theory, but I guess that's not the natural formulation. We can do this for very simple examples: $X$ a smooth Riemannian manifold, $[\Sigma,X]$ the diffeological space of smooth curves, for $\Sigma = [0,1]$ the standard interval, $S : [\Sigma,X] \to \mathbb{R}$ the standard kinetic action functional/energy functional regarded as a morphism of sheaves and then $d S : [\Sigma,X] \to \mathbb{R} \stackrel{\theta}{\to} \mathbf{\flat}_{dR} \mathbf{B} \mathbb{R}$ its differential refinement: compute the sheaf-theoretic fiber and show that it is the sheaf of solutions to the Euler-Lagrange equations. Unwinding this a bit, we have that $\mathbf{\flat}_{dR} \mathbf{B} \mathbb{R} \simeq \Omega_{cl}^1(-)$ is just the sheaf of closed 1-forms (on $CartSp$). So on plots the sheaf morphism $d S$ reads in a smooth funcion $\gamma : \Sigma \times U \to X$ (a smooth $U$-parameterized family of curves in $X$) and then first sends it to the smooth familiy $$ S(\gamma) : U \to \mathbb{R} $$ of real numbers, at each point of $U$ being the value of the action $S$ on a path $\gamma_u$, and then this function is sent to its differential 1-form $$ d S(\gamma) \in \Omega^1_{cl}(U) \,. $$ For variational calculus we will want to restrict this along an inclusion $$ [\Sigma,X]_{x,y} \hookrightarrow [\Sigma,X] $$ of the space of those paths with boundary on $\{x\} \cup \{y\}$. So what's the fiber over 0? It's the sheafificationn of the presheaf that sends $U$ to the set of $U$-parameterized families of paths whose action is extremal _among all $U$-parameterized paths_ . The computation is the standard one for Euler-Lagrange theory, only that it is actually finite dimensional over $U$, $$ \begin{aligned} d S_u & = d u^i \frac{\partial}{\partial u^i} \int_{[0,1]} L(\gamma_u(t), \gamma'_u(t)) d t \\ & = d u^i \int_{[0,1]} \left( \frac{\partial}{\partial \gamma_u(t)} L(\gamma_u(t), \gamma'_u(t)) - \frac{d}{d t} \frac{\partial}{\partial \gamma'_u(t)} L(\gamma_u(t), \gamma'_u(t)) \right) \frac{\partial \gamma(t)_u}{\partial u^i} d t \end{aligned} $$ So it is clear tha the fiber presheaf of $d S$ contains all the standard solutions to the EL-equations. It requires maybe a bit more discussion that it does not contain anything else. Heustistically this should be clear, using density of smooth functions, but I'd have to think. Possibly Andrew has thought about this: variational calculus in the category of diffeological and/or Frölicher spaces?

    Maybe more urgently and as a first step: we should write out something about variational calculus in the sheaf-theoretic picture. Usually it is discussed in terms of infinite-dimensional manifold theory, but I guess that’s not the natural formulation.

    We can do this for very simple examples: $X$ a smooth Riemannian manifold, $[\Sigma,X]$ the diffeological space of smooth curves, for $\Sigma = [0,1]$ the standard interval, $S : [\Sigma,X] \to \mathbb{R}$ the standard kinetic action functional/energy functional regarded as a morphism of sheaves and then $d S : [\Sigma,X] \to \mathbb{R} \stackrel{\theta}{\to} \mathbf{\flat}_{dR} \mathbf{B} \mathbb{R}$ its differential refinement: compute the sheaf-theoretic fiber and show that it is the sheaf of solutions to the Euler-Lagrange equations.

    Unwinding this a bit, we have that $\mathbf{\flat}_{dR} \mathbf{B} \mathbb{R} \simeq \Omega_{cl}^1(-)$ is just the sheaf of closed 1-forms (on $CartSp$).

    So on plots the sheaf morphism $d S$ reads in a smooth funcion $\gamma : \Sigma \times U \to X$ (a smooth $U$-parameterized family of curves in $X$) and then first sends it to the smooth familiy

    $$S(\gamma) : U \to \mathbb{R}$$

    of real numbers, at each point of $U$ being the value of the action $S$ on a path $\gamma_u$, and then this function is sent to its differential 1-form

    $$d S(\gamma) \in \Omega^1_{cl}(U) \,.$$

    For variational calculus we will want to restrict this along an inclusion

    $$[\Sigma,X]_{x,y} \hookrightarrow [\Sigma,X]$$

    of the space of those paths with boundary on $\{x\} \cup \{y\}$.

    So what’s the fiber over 0? It’s the sheafificationn of the presheaf that sends $U$ to the set of $U$-parameterized families of paths whose action is extremal among all $U$-parameterized paths .

    The computation is the standard one for Euler-Lagrange theory, only that it is actually finite dimensional over $U$,

    $$\begin{aligned} d S_u & = d u^i \frac{\partial}{\partial u^i} \int_{[0,1]} L(\gamma_u(t), \gamma'_u(t)) d t \\ & = d u^i \int_{[0,1]} \left( \frac{\partial}{\partial \gamma_u(t)} L(\gamma_u(t), \gamma'_u(t)) - \frac{d}{d t} \frac{\partial}{\partial \gamma'_u(t)} L(\gamma_u(t), \gamma'_u(t)) \right) \frac{\partial \gamma(t)_u}{\partial u^i} d t \end{aligned}$$

    So it is clear tha the fiber presheaf of $d S$ contains all the standard solutions to the EL-equations. It requires maybe a bit more discussion that it does not contain anything else. Heustistically this should be clear, using density of smooth functions, but I’d have to think.

    Possibly Andrew has thought about this: variational calculus in the category of diffeological and/or Frölicher spaces?

28. • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexPatrick Iglesias-Zemmour discusses this on <a href="http://math.huji.ac.il/~piz/documents/Diffeology.pdf#page=64">pages 64-65</a> of his book

    Patrick Iglesias-Zemmour discusses this on pages 64-65 of his book

29. • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have further expanded the proof and computation in the <a href="http://www.ncatlab.org/schreiber/show/infinity-Chern-Simons+theory#ASKZTheory">section</a> on AKSZ theory. There are still signs to be taken care of, but I have to interrupt now.

    I have further expanded the proof and computation in the section on AKSZ theory. There are still signs to be taken care of, but I have to interrupt now.

30. • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2011
    • (edited Mar 1st 2011)
    • PermaLink
    Author: Urs
    Format: MarkdownItexbased on behind-the-scenes discussion with Domenico, I have reworked the exposition of the proof of the intrinsic-action-functional-by-integration-without-integrationat <a href="http://www.ncatlab.org/schreiber/show/infinity-Chern-Simons+theory#TheActionFunctional">oo-Chern-Simons theory -- action functionals</a> in an attempt to get across more clearly what's going on and how manifest integration appears, via the universal coefficient theorem

    based on behind-the-scenes discussion with Domenico, I have reworked the exposition of the proof of the intrinsic-action-functional-by-integration-without-integrationat oo-Chern-Simons theory – action functionals in an attempt to get across more clearly what’s going on and how manifest integration appears, via the universal coefficient theorem

31. • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeApr 22nd 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexWrote out statement and proof of the equations of motion of $\infty$-Chern-Simons theory: <a href="http://nlab.mathforge.org/schreiber/show/infinity-Chern-Simons+theory#EquationsOfMotion">oo-Chern-Simons theory -- Equations of motion</a> (on my personal web).

    Wrote out statement and proof of the equations of motion of $\infty$-Chern-Simons theory: oo-Chern-Simons theory – Equations of motion (on my personal web).

32. • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded statement and proof of the presymplectic structure on the covariant phase space of $\infty$-Chern-Simons theory: in [Equations of motion and presymplectic covarian phase space](http://nlab.mathforge.org/schreiber/show/infinity-Chern-Simons+theory#EquationsOfMotion). Also added to the section [Supergravity](http://nlab.mathforge.org/schreiber/show/infinity-Chern-Simons+theory#Supergravity) the statement that the weaker condition of not having a [[Chern-Simons element]] but still a "cosmo cocyle" ensures that vanishing $\infty$-curvature $$ F_A = 0 $$ is still a _sufficient_ condition for the field configuration $A$ to solve the Euler-Lagrange equations (though in general not necessary. This is not even in general necessary anymore for higher Chern-Simons theory. )

    added statement and proof of the presymplectic structure on the covariant phase space of $\infty$-Chern-Simons theory: in Equations of motion and presymplectic covarian phase space.

    Also added to the section Supergravity the statement that the weaker condition of not having a Chern-Simons element but still a “cosmo cocyle” ensures that vanishing $\infty$-curvature

    $$F_A = 0$$

    is still a sufficient condition for the field configuration $A$ to solve the Euler-Lagrange equations (though in general not necessary. This is not even in general necessary anymore for higher Chern-Simons theory. )

33. • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexHi Domenico, in case you are reading this thread here: I was thinking a bit about the canonical [[presymplectic structure]] on [[covariant phase space]] of $\infty$-Chern-Simons theory. (A really nice reference is the old article by Zuckerman [here](http://nlab.mathforge.org/nlab/files/ZuckermanVariation.pdf)). There one reads off a 1-form on the space of field configurations from the boundary term in the variation of the action functional. But I am thinking: there is a much more direct and more general abstract reason why this 1-form on field space exists: it is the transgression of the _curvature_ of the Lagrangian to field space. In the general abstract language: if $G$ is a smooth $\infty$-group and $$ \mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R} $$ is an (unrefined) $\infty$-Chern-Weil homomorphism, hence the curvature of a Lagrangian of the corresponding $\infty$-Chern-Simons theory, then for $n$-dimensional $\Sigma$ form the internal hom $$ [\Sigma, \mathbf{B}G_{conn}] \to [\Sigma, \mathbf{\flat}_{dR} \mathbf{B}^{n+1}] $$ We can evaluate this morphism of $\infty$-sheaves on the interval and then apply our transgression theorem to get the function. $$ \mathbf{H}(\Sigma \times I, \mathbf{B}G_{conn}) \to \mathbb{R} \,. $$ This is the parallel transport of the transgressed 1-form: on the left we have paths of $G$-connections on $\Sigma$. So the derivative of this function at the origin of the interval is the 1-form in question. I'll unwind this: for $A$ a form on $\Sigma$ with values in the $L_\infty$-algebra $\mathfrak{g}$ and $\langle -\rangle$ the invariant polynomial that controls the above $\infty$-Chern-Weil homomorphism, with Chern-Simons element $cs$, let $\hat A$ be a $t \in [0,1]$-dependent path of such forms. Then $$ \begin{aligned} \int_\Sigma \langle F_{\hat A}\rangle & = \int d CS(\hat A) \\ & = \frac{d}{d t} \int_\Sigma CS(\hat A) d t + \int_\Sigma d_\Sigma CS(\hat A) \end{aligned} \,. $$ The first term vanishes on shell. Using that $F_{\hat A}|_t = F_{A(t)} + \frac{d}{d t}\hat A \wedge d t$ the second term is $$ \cdots = \int_{\partial \Sigma}CS_{lincurv}(A\wedge \cdots \wedge A \wedge \frac{d}{d t}A) d t \,, $$ where $CS_{lincurv}$ is the component of the Chern-Simons element that is linear in the curvature (in the shifted component of the Weil algebra). So the transgressed 1-form in question is $$ \Theta : \delta A \mapsto \int_{\partial \Sigma} CS_{lincurv}(A\wedge \cdots \wedge A \wedge \delta A) \,. $$ That's indeed the Zuckerman-Witten presymplectic form for the given theory, as I have spelled out at [[schreiber:infinity-Chern-Simons theory]]. For ordinary CS theory this is $$ \Theta(\delta A) = \int_{\partial \Sigma} \langle A \wedge \delta A \rangle $$ and hence the presymplectic form is $$ \omega = d \theta : (\delta A_1, \delta A_2) \mapsto \int_{\partial \Sigma} \langle \delta A_1 \wedge \delta A_2\rangle \,. $$ In this form one can find this in the literature, relevant references are now at [[Chern-Simons theory]].

    Hi Domenico,

    in case you are reading this thread here:

    I was thinking a bit about the canonical presymplectic structure on covariant phase space of $\infty$-Chern-Simons theory. (A really nice reference is the old article by Zuckerman here). There one reads off a 1-form on the space of field configurations from the boundary term in the variation of the action functional.

    But I am thinking: there is a much more direct and more general abstract reason why this 1-form on field space exists: it is the transgression of the curvature of the Lagrangian to field space.

    In the general abstract language:

    if $G$ is a smooth $\infty$-group and

    $$\mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}$$

    is an (unrefined) $\infty$-Chern-Weil homomorphism, hence the curvature of a Lagrangian of the corresponding $\infty$-Chern-Simons theory, then for $n$-dimensional $\Sigma$ form the internal hom

    $$[\Sigma, \mathbf{B}G_{conn}] \to [\Sigma, \mathbf{\flat}_{dR} \mathbf{B}^{n+1}]$$

    We can evaluate this morphism of $\infty$-sheaves on the interval and then apply our transgression theorem to get the function.

    $$\mathbf{H}(\Sigma \times I, \mathbf{B}G_{conn}) \to \mathbb{R} \,.$$

    This is the parallel transport of the transgressed 1-form: on the left we have paths of $G$-connections on $\Sigma$.

    So the derivative of this function at the origin of the interval is the 1-form in question.

    I’ll unwind this: for $A$ a form on $\Sigma$ with values in the $L_\infty$-algebra $\mathfrak{g}$ and $\langle -\rangle$ the invariant polynomial that controls the above $\infty$-Chern-Weil homomorphism, with Chern-Simons element $cs$, let $\hat A$ be a $t \in [0,1]$-dependent path of such forms. Then

    $$\begin{aligned} \int_\Sigma \langle F_{\hat A}\rangle & = \int d CS(\hat A) \\ & = \frac{d}{d t} \int_\Sigma CS(\hat A) d t + \int_\Sigma d_\Sigma CS(\hat A) \end{aligned} \,.$$

    The first term vanishes on shell. Using that $F_{\hat A}|_t = F_{A(t)} + \frac{d}{d t}\hat A \wedge d t$ the second term is

    $$\cdots = \int_{\partial \Sigma}CS_{lincurv}(A\wedge \cdots \wedge A \wedge \frac{d}{d t}A) d t \,,$$

    where $CS_{lincurv}$ is the component of the Chern-Simons element that is linear in the curvature (in the shifted component of the Weil algebra).

    So the transgressed 1-form in question is

    $$\Theta : \delta A \mapsto \int_{\partial \Sigma} CS_{lincurv}(A\wedge \cdots \wedge A \wedge \delta A) \,.$$

    That’s indeed the Zuckerman-Witten presymplectic form for the given theory, as I have spelled out at infinity-Chern-Simons theory (schreiber).

    For ordinary CS theory this is

    $$\Theta(\delta A) = \int_{\partial \Sigma} \langle A \wedge \delta A \rangle$$

    and hence the presymplectic form is

    $$\omega = d \theta : (\delta A_1, \delta A_2) \mapsto \int_{\partial \Sigma} \langle \delta A_1 \wedge \delta A_2\rangle \,.$$

    In this form one can find this in the literature, relevant references are now at Chern-Simons theory.

34. • CommentRowNumber34.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 4th 2011
    • PermaLink
    Author: domenico_fiorenza
    Format: MarkdownItexHi Urs, thanks for pointing my attention to this, too!

    Hi Urs,

    thanks for pointing my attention to this, too!

35. • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added to [[schreiber:infinity-Chern-Simons theory]] a section "Introduction and overview". I have also added some subsections to the various Examples-sections and added some statements here and there. Also created a section for "magnetic dual ordinary Chern-Simons theory". More to be done, but I have to rush now to get some breakfast.

    I have added to infinity-Chern-Simons theory (schreiber) a section “Introduction and overview”.

    I have also added some subsections to the various Examples-sections and added some statements here and there. Also created a section for “magnetic dual ordinary Chern-Simons theory”.

    More to be done, but I have to rush now to get some breakfast.

36. • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have split off the section with examples to a new entry [[schreiber:infinity-Chern-Simons theory -- examples]] because the entry was getting too long (probably for the reader, but also for the software: it woundn't save anymore)

    I have split off the section with examples to a new entry infinity-Chern-Simons theory – examples (schreiber) because the entry was getting too long (probably for the reader, but also for the software: it woundn’t save anymore)

37. • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexIn the [Introduction and overview](http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory#introduction_and_overview_3) at [[schreiber:infinity-Chern-Simons theory]] I have added one subsection [Deficiencies of classical Chern-Weil theory](http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory#ChernWeilDeficiencies) in order to lead over from the review of classical CW-theory in the subsection above to the necessity for considering the oo-version in the subsection below.

    In the Introduction and overview at infinity-Chern-Simons theory (schreiber) I have added one subsection

    Deficiencies of classical Chern-Weil theory

    in order to lead over from the review of classical CW-theory in the subsection above to the necessity for considering the oo-version in the subsection below.

38. • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added two aspects: * at [[schreiber:infinity-Chern-Simons theory -- covariant phase space]] I discuss that for $\infty$-Chern-Simons theory coming from binary and non-degenerate invariant polynomials, all diffeomorphisms connected to the identity are connected by a gauge transformation to the identity (this means among other things that in these cases the $\infty$-Lie algebroid of the configuration $\infty$-groupoid $[\Sigma, A_{comm}]$ is already the correct BRST complex, whereas otherwise there may be diffeomorphism-ghosts to be added). * at [[schreiber:infinity-Chern-Simons theory -- action functionals]] I have amplified the existence of the smooth refinement if the action functional more, using the new discussion at [[concrete smooth infinity-groupoid]]

    I have added two aspects:

    • at infinity-Chern-Simons theory – covariant phase space (schreiber) I discuss that for $\infty$-Chern-Simons theory coming from binary and non-degenerate invariant polynomials, all diffeomorphisms connected to the identity are connected by a gauge transformation to the identity

      (this means among other things that in these cases the $\infty$-Lie algebroid of the configuration $\infty$-groupoid $[\Sigma, A_{comm}]$ is already the correct BRST complex, whereas otherwise there may be diffeomorphism-ghosts to be added).

    • at infinity-Chern-Simons theory – action functionals (schreiber) I have amplified the existence of the smooth refinement if the action functional more, using the new discussion at concrete smooth infinity-groupoid

39. • CommentRowNumber39.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 12th 2013
    • PermaLink
    Author: jim_stasheff
    Format: TextHas there been any discussion of The calssical master equation of Felder-Kazhdan? Searching for `master equation' under topics produced no results???

    Has there been any discussion of The calssical master equation of Felder-Kazhdan?
    Searching for `master equation' under topics produced no results???
40. • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2013
    • (edited Jan 12th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexWe haven't discussed it here on the $n$Forum, if that's what you mean. The article * Giovanni Felder, David Kazhdan, _The classical master equation_ ([arXiv:1212.1631](http://arxiv.org/abs/1212.1631)) is now referenced at _[BV-BRST formalism -- References -- For Lagranian BV](http://ncatlab.org/nlab/show/BV-BRST+formalism#ReferencesForLagrangianBVForLagrangianTheories)_ I am not sure if I would follow you in speaking of "The calssical master equation of Felder-Kazhdan" because what they discuss is the standard classical master equation, but they aim to provide a more systematic statement about its existence and uniqueness than has maybe been done before. If you'd aske me, I'd say that one still needs to handle the higher geometry involved systematically. Specifically in "derived algebraic geometry" this is being investigated in the ongoing program * Gabriele Vezzosi, _Derived critical loci I - Basics_, ([arxiv/1109.5213](http://arxiv.org/abs/1109.5213)) * Tony Pantev, Bertrand Toen, M. Vaquie, G. Vezzosi, _Quantization and derived moduli spaces I: shifted symplectic structures_, ([arxiv/1111.3209](http://arxiv.org/abs/1111.3209)) Earlier I had made some comments along these lines at * [[Urs Schreiber]], _[[schreiber:derived critical locus]]_ . At that entry there are now some pointers to a fully general formalization.

    We haven’t discussed it here on the $n$Forum, if that’s what you mean. The article

    • Giovanni Felder, David Kazhdan, The classical master equation (arXiv:1212.1631)

    is now referenced at BV-BRST formalism – References – For Lagranian BV

    I am not sure if I would follow you in speaking of “The calssical master equation of Felder-Kazhdan” because what they discuss is the standard classical master equation, but they aim to provide a more systematic statement about its existence and uniqueness than has maybe been done before.

    If you’d aske me, I’d say that one still needs to handle the higher geometry involved systematically. Specifically in “derived algebraic geometry” this is being investigated in the ongoing program

    • Gabriele Vezzosi, Derived critical loci I - Basics, (arxiv/1109.5213)

    • Tony Pantev, Bertrand Toen, M. Vaquie, G. Vezzosi, Quantization and derived moduli spaces I: shifted symplectic structures, (arxiv/1111.3209)

    Earlier I had made some comments along these lines at

    • Urs Schreiber, derived critical locus (schreiber) .

    At that entry there are now some pointers to a fully general formalization.

41. • CommentRowNumber41.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 13th 2013
    • PermaLink
    Author: jim_stasheff
    Format: TextThanks for that link - but why: the subspace on which the ??de Rham?? differential of S vanishes as for: “The calssical master equation of Felder-Kazhdan” in addition to the obvious typo, the " is misplaced - I meant “The classical master equation" of Felder-Kazhdan meaning the paper It's done in an algebraic-geometric setting though it seems to me smooth is enough and the methods should also handle some singularities - anyone care to comment/discuss?

    Thanks for that link - but why:
    the subspace on which the ??de Rham?? differential of S vanishes
    
    as for:
    “The calssical master equation of Felder-Kazhdan” in addition to the obvious typo, the " is misplaced - I meant
    “The classical master equation" of Felder-Kazhdan meaning the paper
    
    It's done in an algebraic-geometric setting though it seems to me smooth is enough and the methods should also handle some singularities - anyone care to comment/discuss?
42. • CommentRowNumber42.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 13th 2013
    • PermaLink
    Author: jim_stasheff
    Format: TextFelder confirms: Yes everything works in the smooth, in the complex geometric and in the algebraic setting. We wrote the proofs in the algebraic case; in the other cases the proofs are the same (or simpler since you have local coordinates).

    Felder confirms:
    Yes everything works in the smooth, in the complex geometric and
    in the algebraic setting. We wrote the proofs in the algebraic case;
    in the other cases the proofs are the same (or simpler since you
    have local coordinates).
43. • CommentRowNumber43.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 13th 2013
    • PermaLink
    Author: jim_stasheff
    Format: TextIn re: Urs Schreiber, derived critical locus (schreiber) . Sorry to have overlooked this earlier - some nice insights BUT free O(C)-algebra - graded commutative understood? `the differential of th BRST-complex of $mathfrak c$, the latter being? gauge symms close off shell - how can they fail? the bracket of symms is a symm at the end, no further comment on higher order stuff? realtions, homtopy Jacobi, etc.? the infty-Lie algebrodi IS the BRST complex - they are in the same cat?

    In re:
    Urs Schreiber, derived critical locus (schreiber) .
    
    Sorry to have overlooked this earlier - some nice insights
    BUT
    
    free O(C)-algebra - graded commutative understood?
    
    `the differential of th BRST-complex of $mathfrak c$, the latter being?
    
    gauge symms close off shell - how can they fail? the bracket of symms is a symm
    
    at the end, no further comment on higher order stuff? realtions, homtopy Jacobi, etc.?
    
    the infty-Lie algebrodi IS the BRST complex - they are in the same cat?
44. • CommentRowNumber44.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2013
    • (edited Jan 13th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> free O(C)-algebra - graded commutative understood? Yes. > gauge symms close off shell - how can they fail? the bracket of symms is a symm Yes, that's an important subtopic in BRST where people don't use the full algebra of symmetries (which of course closes) but only a subalgebra, which still closes on shell. It's discussed in generality in the standard texts. But the main application is, as far as I can see, various setups in supergravity, where the supersymmetry transformations are often implemented only on-shell. > the infty-Lie algebrodi IS the BRST complex - they are in the same cat? The BRST complex is the "formal dual" of the $\infty$-Lie algebroid, equivalently the algebra of functions on it, or its Chevalley-Eilenberg algebra, yes.

    free O(C)-algebra - graded commutative understood?

    Yes.

    gauge symms close off shell - how can they fail? the bracket of symms is a symm

    Yes, that’s an important subtopic in BRST where people don’t use the full algebra of symmetries (which of course closes) but only a subalgebra, which still closes on shell. It’s discussed in generality in the standard texts. But the main application is, as far as I can see, various setups in supergravity, where the supersymmetry transformations are often implemented only on-shell.

    the infty-Lie algebrodi IS the BRST complex - they are in the same cat?

    The BRST complex is the “formal dual” of the $\infty$-Lie algebroid, equivalently the algebra of functions on it, or its Chevalley-Eilenberg algebra, yes.

45. • CommentRowNumber45.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 14th 2013
    • PermaLink
    Author: jim_stasheff
    Format: Text@But the main application is, as far as I can see, various setups in supergravity, of course, as i know you know, the entire BV formalism is driven by the `open' algebra distinction

    @But the main application is, as far as I can see, various setups in supergravity,
    
    of course, as i know you know, the entire BV formalism is driven by the `open' algebra distinction
46. • CommentRowNumber46.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2013
    • (edited Jan 14th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> the entire BV formalism is driven by the `open' algebra distinction I find this a secondary aspect. To me the central point is that it models, under some conditions, the derived critical locus of an action functional defined on the off-shell moduli stack of fields. For this perspective the algebra always closes.

    the entire BV formalism is driven by the ‘open’ algebra distinction

    I find this a secondary aspect. To me the central point is that it models, under some conditions, the derived critical locus of an action functional defined on the off-shell moduli stack of fields. For this perspective the algebra always closes.

47. • CommentRowNumber47.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 15th 2013
    • PermaLink
    Author: jim_stasheff
    Format: Textagreed, especially `that it models' but the BV formalism is a specific model, which is not needed if there is closure on shell

    agreed, especially `that it models' but the BV formalism is a specific model, which is not needed if there is closure on shell
48. • CommentRowNumber48.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2013
    • PermaLink
    Author: Urs
    Format: Markdown> not needed if there is closure on shell Not sure what you mean here. In the quantization of most systems the gauge algebra closes off- shell, notably in Yang-Mills theory. But certainly one needs [[BV-BRST formalism]] to discuss the (perturbative) quantization of these systems.

    not needed if there is closure on shell

    Not sure what you mean here. In the quantization of most systems the gauge algebra closes off- shell, notably in Yang-Mills theory. But certainly one needs BV-BRST formalism to discuss the (perturbative) quantization of these systems.

49. • CommentRowNumber49.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 16th 2013
    • PermaLink
    Author: jim_stasheff
    Format: TextI meant the BV-BRST formalism is not needed if there is closure, i.e. structure constants, of the gauge algebra on shell

    I meant the BV-BRST formalism is not needed if there is closure, i.e. structure constants, of the gauge algebra on shell
50. • CommentRowNumber50.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownYes, but I am still not sure why you say this. The BV-BRST formalism is needed notably in the perturbative quantization of Yang-Mills theory with off-shell closing gauge algebra.

    Yes, but I am still not sure why you say this. The BV-BRST formalism is needed notably in the perturbative quantization of Yang-Mills theory with off-shell closing gauge algebra.

51. • CommentRowNumber51.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 17th 2013
    • PermaLink
    Author: jim_stasheff
    Format: Textor any other with off-shell closing without on-shell closing

    or any other with off-shell closing without on-shell closing
52. • CommentRowNumber52.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2013
    • (edited Jan 17th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexIf the bracket closes off-shell then it also closes on-shell. There is a space of fields $Fields(X)$ and there is the subspace $PhaseSpace(X) \hookrightarrow Fields(X)$ of those fields that satisfy the equations of motion. One says that $PhaseSpace(X)$ is "the shell". (This comes from the example where $X = \mathbb{R}$ and $Fields(X) = Maps(\mathbb{R}^1, \mathbb{R}^{4})$ and where the action functional is that of the free relativistic particle of some mass. In this case the equations of motion assert that $p^2 = m^2$, hence that the eigenmomentum "sits on the mass shell".) So if some thing exists on $Fields(X)$ then it has a restriction to $PhaseSpace(X)$. Hence off-shell anything implies on-shell anything. But the thing is that some structures may not exist on all of $FIelds(X)$ but only on $PhaseSpace(X)$. Then one says that they exist "only on-shell".

    If the bracket closes off-shell then it also closes on-shell.

    There is a space of fields $Fields(X)$ and there is the subspace $PhaseSpace(X) \hookrightarrow Fields(X)$ of those fields that satisfy the equations of motion. One says that $PhaseSpace(X)$ is “the shell”.

    (This comes from the example where $X = \mathbb{R}$ and $Fields(X) = Maps(\mathbb{R}^1, \mathbb{R}^{4})$ and where the action functional is that of the free relativistic particle of some mass. In this case the equations of motion assert that $p^2 = m^2$, hence that the eigenmomentum “sits on the mass shell”.)

    So if some thing exists on $Fields(X)$ then it has a restriction to $PhaseSpace(X)$. Hence off-shell anything implies on-shell anything.

    But the thing is that some structures may not exist on all of $FIelds(X)$ but only on $PhaseSpace(X)$. Then one says that they exist “only on-shell”.

53. • CommentRowNumber53.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 18th 2013
    • PermaLink
    Author: jim_stasheff
    Format: TextI had in mind e.g. that constraints close on a Lie algebra in the sense of structure functions but the real vector spanned by the constraints may not may not `close on a Lie algebra'

    I had in mind e.g. that constraints close on a Lie algebra in the sense of structure functions but the real vector spanned by the constraints may not may not `close on a Lie algebra'