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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 17th 2013
   • (edited Oct 17th 2013)
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   Author: Urs
   Format: MarkdownItexInspired by Matthew Ando's talk at the [Conference on twisted cohomology](http://wwwmath.uni-muenster.de/sfb/sfb878/2013/twists.html) that I am currently attending, I finally typed up a note on * _[[parameterized WZW models]]_

   Inspired by Matthew Ando’s talk at the Conference on twisted cohomology that I am currently attending, I finally typed up a note on

   • parameterized WZW models
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2013
   • (edited Oct 18th 2013)
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   Author: Urs
   Format: MarkdownItexI have now added what I think is a (simple) formal proof that a $Heis(\mathcal{L}_{WZW})$-structure on some $B$ induces a $G$-fiber $\infty$-bundle on $B$ equipped with a map to $\mathbf{B}^n U(1)_{conn}$ which restricts on each fiver to $\mathcal{L}_{WZW}$. (I still need an argument that this G-bundle is indeed G-principal). This is at * _[parameterized WZW model -- Globalization from a Heisenberg structure](http://ncatlab.org/nlab/show/parameterized+WZW+model#GlobalizationFromAHeisenbergStructure)_ . Leaving the WZW-terminology aside, this is completely general and formal and has nothing specifically to do with the special situation. I am using the following general abstract statements, which I hope I am not mixed up about: 1. dependent sum preserves fiber products (because limits over cospan diagram in the slice are computed as limits over the corresponding co-cone diagrams down in the base and the inclusion of a cospan diagram into its co-cone diagram is final) 1. by the same argument dependent sum preserves effective epimorphisms . Maybe some abstractly-minded reader can help me see why the argument at the above link also shows $G$-principality (if it does).

   I have now added what I think is a (simple) formal proof that a $Heis(\mathcal{L}_{WZW})$-structure on some $B$ induces a $G$-fiber $\infty$-bundle on $B$ equipped with a map to $\mathbf{B}^n U(1)_{conn}$ which restricts on each fiver to $\mathcal{L}_{WZW}$. (I still need an argument that this G-bundle is indeed G-principal).

   This is at

   • parameterized WZW model – Globalization from a Heisenberg structure .

   Leaving the WZW-terminology aside, this is completely general and formal and has nothing specifically to do with the special situation.

   I am using the following general abstract statements, which I hope I am not mixed up about:

   1. dependent sum preserves fiber products (because limits over cospan diagram in the slice are computed as limits over the corresponding co-cone diagrams down in the base and the inclusion of a cospan diagram into its co-cone diagram is final)

   2. by the same argument dependent sum preserves effective epimorphisms .

   Maybe some abstractly-minded reader can help me see why the argument at the above link also shows $G$-principality (if it does).

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 18th 2013
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   Author: DavidRoberts
   Format: MarkdownItexIsn't there a universal version of this over BG?

   Isn’t there a universal version of this over BG?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 19th 2013
   • (edited Oct 19th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOver $\mathbf{B} Heis(\mathcal{L}_{WZW})$. That's what I am trying to construct. So far I have explicit constructions in local models and I have a general formal argument that lifts from $G$-cocycles to $Heis(\mathcal{L}_{WZW})$ cocycles induced $G$-fiber $\infty$-bundles with an $n$-connection on the total space that restricts on each fiber to $\mathcal{L}_{WZW}$. What I am still lacking is a general formal argument that this $G$-fiber bundle is the $G$-principal bundle classified by the underlying $G$-cocycle.

   Over $\mathbf{B} Heis(\mathcal{L}_{WZW})$. That’s what I am trying to construct.

   So far I have explicit constructions in local models and I have a general formal argument that lifts from $G$-cocycles to $Heis(\mathcal{L}_{WZW})$ cocycles induced $G$-fiber $\infty$-bundles with an $n$-connection on the total space that restricts on each fiber to $\mathcal{L}_{WZW}$. What I am still lacking is a general formal argument that this $G$-fiber bundle is the $G$-principal bundle classified by the underlying $G$-cocycle.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 3rd 2015
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   Author: Urs
   Format: MarkdownItexProdded by discussion with Eric Sharpe, I have added to the entry pointers to the original articles by Jim Gates, such as [Gates-Siegel 88](http://ncatlab.org/nlab/show/parameterized+WZW+model#GatesSiegel88). Then I have replaced some of the discussion in the entry by pointers to the meanwhile more comprehensive note [[schreiber:Obstruction theory for parameterized higher WZW terms|cwzw]]. Eventually I'll go back and turn this into $n$Lab material. For the moment I have just expanded the [Idea](http://ncatlab.org/nlab/show/parameterized+WZW+model#Idea)-section a little bit, mentioning more of the story for the heterotic string. Much more needs to be done here.

   Prodded by discussion with Eric Sharpe, I have added to the entry pointers to the original articles by Jim Gates, such as Gates-Siegel 88.

   Then I have replaced some of the discussion in the entry by pointers to the meanwhile more comprehensive note cwzw. Eventually I’ll go back and turn this into $n$Lab material. For the moment I have just expanded the Idea-section a little bit, mentioning more of the story for the heterotic string.

   Much more needs to be done here.