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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 19th 2011
   • (edited Dec 19th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIf you think about it, there is a curious analogy between the notion of [[covariant phase space]] and the notion of [[Bohr-Sommerfeld leaf]] in geometric quantization. For an $n$-dimensional field theory, we have a 0-bundle in codimension 0, namely the action functional $$ \exp(i S(-)) : Configurations(\Sigma_n) \to U(1) \,. $$ Its curvature 1-form is $\exp(i S(-)) d S(-)$. The covariant phase space is the locus where this 0-bundle becomes flat -- which is the locus where the Euler-Lagramge equations are satisfied. Now in codimension 1 we have the [[prequantum line bundle]] on phase space $$ E : Config(\Sigma_{n-1}) \to \mathbf{B} U(1)_{conn} \,. $$ A [[Bohr-Sommerfeld leaf]] is a maximal-dimension locus where this line bundle with connection becomes trivial. Let's make this more systematic for Chern-Simons theory: let $$ \hat {\mathbf{c}} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} $$ be the [[Chern-Simons circle 3-bundle]] on the moduli stack of $G$-principal bundles with connection. 1) We transgress this to codimension 0 by forming the internal hom $$ [\Sigma_3 , \hat {\mathbf{c}}] : [\Sigma_3, \mathbf{B}G_{conn}] \to [\Sigma_3, \mathbf{B}^3 U(1)_{conn}] \stackrel{conc \circ \int_\Sigma}{\to} U(1) \,. $$ Taking now the critical locus of this 0-bundle with connection produces the covariant phase space. 2) We transgress this to codimension 1 by forming the internal hom $$ [\Sigma_2 , \hat {\mathbf{c}}] : [\Sigma_2, \mathbf{B}G_{conn}] \to [\Sigma_2, \mathbf{B}^3 U(1)_{conn}] \stackrel{conc \circ \int_\Sigma}{\to} \mathbf{B}U(1)_{conn} $$ and get the prequantum circle n-bundle. Its "critical loci" are Bohr-Sommerfeld leaves. It seems clear that this wants to be the beginning of a pattern of "extended geometric quantization". But I am not fully sure yet how to solidify the details of the story.

   If you think about it, there is a curious analogy between the notion of covariant phase space and the notion of Bohr-Sommerfeld leaf in geometric quantization.

   For an $n$-dimensional field theory, we have a 0-bundle in codimension 0, namely the action functional

   $$\exp(i S(-)) : Configurations(\Sigma_n) \to U(1) \,.$$

   Its curvature 1-form is $\exp(i S(-)) d S(-)$. The covariant phase space is the locus where this 0-bundle becomes flat – which is the locus where the Euler-Lagramge equations are satisfied.

   Now in codimension 1 we have the prequantum line bundle on phase space

   $$E : Config(\Sigma_{n-1}) \to \mathbf{B} U(1)_{conn} \,.$$

   A Bohr-Sommerfeld leaf is a maximal-dimension locus where this line bundle with connection becomes trivial.

   Let’s make this more systematic for Chern-Simons theory:

   let

   $$\hat {\mathbf{c}} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$$

   be the Chern-Simons circle 3-bundle on the moduli stack of $G$-principal bundles with connection.

   1) We transgress this to codimension 0 by forming the internal hom

   $$[\Sigma_3 , \hat {\mathbf{c}}] : [\Sigma_3, \mathbf{B}G_{conn}] \to [\Sigma_3, \mathbf{B}^3 U(1)_{conn}] \stackrel{conc \circ \int_\Sigma}{\to} U(1) \,.$$

   Taking now the critical locus of this 0-bundle with connection produces the covariant phase space.

   2) We transgress this to codimension 1 by forming the internal hom

   $$[\Sigma_2 , \hat {\mathbf{c}}] : [\Sigma_2, \mathbf{B}G_{conn}] \to [\Sigma_2, \mathbf{B}^3 U(1)_{conn}] \stackrel{conc \circ \int_\Sigma}{\to} \mathbf{B}U(1)_{conn}$$

   and get the prequantum circle n-bundle. Its “critical loci” are Bohr-Sommerfeld leaves.

   It seems clear that this wants to be the beginning of a pattern of “extended geometric quantization”. But I am not fully sure yet how to solidify the details of the story.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeDec 19th 2011
   • (edited Dec 19th 2011)
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   Author: zskoda
   Format: MarkdownItexI do not quite understand the framework but it reminds me strongly of the observation by Gerasimov (unpublished) that the Chern-Simons could be used to attempt the problem of finding a formula for the homotopy connection aka twisted cochain for homology of a principal bundle using the Feynman integral. Roughly speaking the homology class lifts by connection where one uses the Euler-Lagrange equations for the geodesics and for the [[twisting cochain]] one uses the effect on the amplitude which gives the monodromy. I think that story never fully materalized.

   I do not quite understand the framework but it reminds me strongly of the observation by Gerasimov (unpublished) that the Chern-Simons could be used to attempt the problem of finding a formula for the homotopy connection aka twisted cochain for homology of a principal bundle using the Feynman integral. Roughly speaking the homology class lifts by connection where one uses the Euler-Lagrange equations for the geodesics and for the twisting cochain one uses the effect on the amplitude which gives the monodromy. I think that story never fully materalized.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeDec 19th 2011
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   Author: zskoda
   Format: MarkdownItexExtended is in the sense to mimic the "extended TQFT" ?

   Extended is in the sense to mimic the “extended TQFT” ?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 19th 2011
   • (edited Dec 19th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes. What I am really after is this: I think the "differential characteristic maps" that we construct in "$\infty$-Chern-Simons theory" $$ \hat {\mathbf{c}} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} $$ are to be regarded as "extended" action functionals. In the sense that when we transgress them to mapping spaces out of an $n$-dimensional $\Sigma$, they become the actual action functionals, when we transgress them to mapping spaces out of an $(n-1)$-dimensional $\Sigma$ they become the prequantum line bundle, etc. So the classification of extended TQFTs suggests that to obtain such by quantization, we should take an "extended" action functional as above and apply some higher geometric quantization to it directly, without transgressing to anywhere. This should produce an $n$-vector space of states, and that $n$-space should already be the fully dualizable object that defines the whole TQFT. So I am trying to understand: what is the analog of a Bohr-Sommerfeld leaf for the case that a symplectic manifold is generalized to a moduli $\infty$-stack $\mathbf{B}G_{conn}$ equipped with the "prequantum circle $n$-bundle" $\hat {\mathbf{c}}$. As Chris Rogers noticed convincingly in the $n$-plectic case over a manifold: it should be true that the $n$-space of states is that of twisted $(n-1)$-vector bundles on these higher Bohr-Sommerfeld leaves (the higher analog of sections) of the restricted prequantum circle $n$-bundle. So, the prize question is: what is a _Bohr-Sommerfeld sub-$\infty$-stack_ in an $n$-plectic smooth $\infty$-groupoid of the form $$ \hat {\mathbf{c}} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} $$ ? Finding the right answer to this seemingly simple question will be a major thing.

   Yes.

   What I am really after is this:

   I think the “differential characteristic maps” that we construct in “$\infty$-Chern-Simons theory”

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

   are to be regarded as “extended” action functionals. In the sense that when we transgress them to mapping spaces out of an $n$-dimensional $\Sigma$, they become the actual action functionals, when we transgress them to mapping spaces out of an $(n-1)$-dimensional $\Sigma$ they become the prequantum line bundle, etc.

   So the classification of extended TQFTs suggests that to obtain such by quantization, we should take an “extended” action functional as above and apply some higher geometric quantization to it directly, without transgressing to anywhere. This should produce an $n$-vector space of states, and that $n$-space should already be the fully dualizable object that defines the whole TQFT.

   So I am trying to understand: what is the analog of a Bohr-Sommerfeld leaf for the case that a symplectic manifold is generalized to a moduli $\infty$-stack $\mathbf{B}G_{conn}$ equipped with the “prequantum circle $n$-bundle” $\hat {\mathbf{c}}$.

   As Chris Rogers noticed convincingly in the $n$-plectic case over a manifold: it should be true that the $n$-space of states is that of twisted $(n-1)$-vector bundles on these higher Bohr-Sommerfeld leaves (the higher analog of sections) of the restricted prequantum circle $n$-bundle.

   So, the prize question is: what is a Bohr-Sommerfeld sub-$\infty$-stack in an $n$-plectic smooth $\infty$-groupoid of the form

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

   ?

   Finding the right answer to this seemingly simple question will be a major thing.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 19th 2011
   • (edited Dec 19th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOne natural guess would be that it's as easy as this: **guess:** a Bohr-Sommerfeld sub-$\infty$-stack of $\hat {\mathbf{c}}$ is a [compact](http://ncatlab.org/nlab/show/compact+object+in+an+%28infinity%2C1%29-category) subobject $Q \hookrightarrow \mathbf{B}G_{conn}$ fitting into a homotopy diagram $$ \array{ Q &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G_{conn} &\stackrel{\hat {\mathbf{c}}}{\to}& \mathbf{B}^n U(1)_{conn} } \,. $$ This has a nice sound to it. Trouble is that I don't seem to know how to go about figuring out what the compact subobjects of $\mathbf{B}G_{conn}$ would be...

   One natural guess would be that it’s as easy as this:

   guess: a Bohr-Sommerfeld sub-$\infty$-stack of $\hat {\mathbf{c}}$ is a compact subobject $Q \hookrightarrow \mathbf{B}G_{conn}$ fitting into a homotopy diagram

   $$\array{ Q &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G_{conn} &\stackrel{\hat {\mathbf{c}}}{\to}& \mathbf{B}^n U(1)_{conn} } \,.$$

   This has a nice sound to it. Trouble is that I don’t seem to know how to go about figuring out what the compact subobjects of $\mathbf{B}G_{conn}$ would be…

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 19th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, and the above "guess" misses the maximality condtion that makes the isotropic $Q$ be Lagrangian. This, too, I don't know yet how to handle.

   Oh, and the above “guess” misses the maximality condtion that makes the isotropic $Q$ be Lagrangian. This, too, I don’t know yet how to handle.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeDec 19th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNice. The existence of Bohr-Sommerfeld leaves is a very strong condition on a system, sort of integrability. You must be then right it should work for extended Chern-Simons in extended sense. To make sure the question is understood, is there a definition of $n$-plectic infinity-groupoid in the $n$Lab ?

   Nice.

   The existence of Bohr-Sommerfeld leaves is a very strong condition on a system, sort of integrability. You must be then right it should work for extended Chern-Simons in extended sense. To make sure the question is understood, is there a definition of $n$-plectic infinity-groupoid in the $n$Lab ?

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeDec 19th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> the maximality condition that makes the isotropic Q be Lagrangian Do you have the correct categorification of the polarization for this case ? If so, this should give the "half dimension" like for the Heisenberg group.

   the maximality condition that makes the isotropic Q be Lagrangian

   Do you have the correct categorification of the polarization for this case ? If so, this should give the “half dimension” like for the Heisenberg group.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeDec 19th 2011
   • (edited Dec 19th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere is _[[symplectic infinity-groupoid]]_. A _pre-$n$-plectic_ structrure on a smooth $\infty$-groupoid $X$ would simply be a map $X \to \flat_{dR} \mathbf{B}^{n+1} \mathbb{R}$. Prequantization of this is a lift to a map $X \to \mathbf{B}^n U(1)_{conn}$. Now comes the question of 1. how to say that this is non-degenerate (pre-$n$-plectic refining to $n$-plectic); 1. what it means for a subject $Q \hookrightarrow X$ to be Lagrangian.

   There is symplectic infinity-groupoid.

   A pre-$n$-plectic structrure on a smooth $\infty$-groupoid $X$ would simply be a map $X \to \flat_{dR} \mathbf{B}^{n+1} \mathbb{R}$. Prequantization of this is a lift to a map $X \to \mathbf{B}^n U(1)_{conn}$.

   Now comes the question of

   1. how to say that this is non-degenerate (pre-$n$-plectic refining to $n$-plectic);

   2. what it means for a subject $Q \hookrightarrow X$ to be Lagrangian.

10. • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeDec 19th 2011
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    Author: zskoda
    Format: MarkdownItexBy the way, I do not see yet why the "compact subobject" condition.

    By the way, I do not see yet why the “compact subobject” condition.

11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeDec 19th 2011
    • (edited Dec 19th 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItex> what it means for a subject $Q \hookrightarrow X$ to be Lagrangian This is now interesting. Could one see it at the level of tangent algebroid ? It should be simpler there. Edit: some people study quasi-Hamiltonian systems which are related to Courant case. Maybe the hint to Lagrangean condition may be helped from those works.

    what it means for a subject $Q \hookrightarrow X$ to be Lagrangian

    This is now interesting. Could one see it at the level of tangent algebroid ? It should be simpler there.

    Edit: some people study quasi-Hamiltonian systems which are related to Courant case. Maybe the hint to Lagrangean condition may be helped from those works.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexThe "compact" is just a tribute to the definition of ordinary Bohr-Sommerfeld leaves, which are requireed to be compact submanifolds. As for "Lagrangian": maybe we have to go via $\infty$-Lie algebroids, yes. I was hoping that there is a more intrinsic condition. In particular since it would be nice to have a canonical good notion of "Lagrangian subspace" also for the $n$-plectic case on an ordinary manifold.

    The “compact” is just a tribute to the definition of ordinary Bohr-Sommerfeld leaves, which are requireed to be compact submanifolds.

    As for “Lagrangian”: maybe we have to go via $\infty$-Lie algebroids, yes. I was hoping that there is a more intrinsic condition. In particular since it would be nice to have a canonical good notion of “Lagrangian subspace” also for the $n$-plectic case on an ordinary manifold.