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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeOct 9th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea A notion of connection for [[principal 2-bundles]] that does not impose the fake flatness condition. ## Related concepts * [[adjusted Weil algebra]] ## References See also the references at [[adjusted Weil algebra]]. Definition of adjusted connections in terms of [[cocycle data]]: * [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], _Explicit Non-Abelian Gerbes with Connections_, [arXiv:2203.00092](https://arxiv.org/abs/2203.00092) Examples for T-duality: * [[Hyungrok Kim]], [[Christian Saemann]], _T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections_, [arXiv:2303.16162](https://arxiv.org/abs/2303.16162) Further references for T-duality: * [[Konrad Waldorf]], _Geometric T-duality: Buscher rules in general topology_, [arXiv:2207.11799](https://arxiv.org/abs/2207.11799). * [[Thomas Nikolaus]], [[Konrad Waldorf]], _Higher geometry for non-geometric T-duals_, [arXiv:1804.00677](https://arxiv.org/abs/1804.00677). <a href="https://ncatlab.org/nlab/revision/adjusted+connection/1">v1</a>, <a href="https://ncatlab.org/nlab/show/adjusted+connection">current</a>

   Created:

   Idea

   A notion of connection for principal 2-bundles that does not impose the fake flatness condition.

   Related concepts

   • adjusted Weil algebra

   References

   See also the references at adjusted Weil algebra.

   Definition of adjusted connections in terms of cocycle data:

   • Dominik Rist, Christian Saemann, Martin Wolf, Explicit Non-Abelian Gerbes with Connections, arXiv:2203.00092

   Examples for T-duality:

   • Hyungrok Kim, Christian Saemann, T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections, arXiv:2303.16162

   Further references for T-duality:

   • Konrad Waldorf, Geometric T-duality: Buscher rules in general topology, arXiv:2207.11799.

   • Thomas Nikolaus, Konrad Waldorf, Higher geometry for non-geometric T-duals, arXiv:1804.00677.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 10th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the original references, with brief commentary <a href="https://ncatlab.org/nlab/revision/diff/adjusted+connection/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/adjusted+connection/2">v2</a>, <a href="https://ncatlab.org/nlab/show/adjusted+connection">current</a>

   added the original references, with brief commentary

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeOct 10th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexThanks a lot for providing the original sources. Is the “modified Weil algebra” the same as “adjusted Weil algebra”? Is there a source that discusses adjusted connections in full generality of Lie ∞-groups? Rist–Saemann–Wolf only discuss Lie 2-algebras, whereas your papers seem to talk only about String-like extensions, if I understood them correctly.

   Thanks a lot for providing the original sources. Is the “modified Weil algebra” the same as “adjusted Weil algebra”?

   Is there a source that discusses adjusted connections in full generality of Lie ∞-groups?

   Rist–Saemann–Wolf only discuss Lie 2-algebras, whereas your papers seem to talk only about String-like extensions, if I understood them correctly.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 11th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexWe actually called it the *Chern-Simons $L_\infty$-algebra* ([p. 48](https://arxiv.org/pdf/0801.3480.pdf#page=48)), because it brings out the expected Chern-Simons-terms in the $L_\infty$ -- which is what the whole "adjustment"-process is about. There is no sense of doing this for non-string-like extensions: These higher central extensions are what is classified by invariant polynomials transgressing to Chern-SImons elements. In accounts you may be looking at this may be hidden in the assumption of a bilinear invariant pairing: This is the binary invariant polynomial which classifies the String-like extension. Without such, there is no sense to adjust to any Chern-Simons terms. The delooping $\mathbf{B}G$ of the smooth $\infty$-group Lie integrating an $L_\infty$-algebra $\mathfrak{g}$ is a truncation of the stackification of $$ exp(\mathfrak{g}) \,\colon\, (U, [k]) \,\mapsto\, Hom_{dgAlg}\big( CE(\mathfrak{g}),\, \Omega^\bullet_{dR, vert}( U \times \Delta^k_{smth} ) \big) \,. $$ This is the stacky formulation of Henriques's notion of integration of $L_\infty$-algebras. For Lie 1-algebras and using 1-truncation this reduces to the "path method" for Lie integration, which is for instance implicit in [Brylinski & McLaughlin 1996](https://ncatlab.org/nlab/show/adjusted+connection#BrylinskiMcLaughlin96). The process we described enhances this Lie integration (in a way I have recalled when we discussed this last time) to a stack $\mathbf{B}G_{conn}$ such that maps $X \to \mathbf{B}G_{conn}$ are (cocycles for) $G$-principal $\infty$-bundles on $X$ with non-fake flat $\infty$-connection given by the expected Chern-Simons terms.

   We actually called it the Chern-Simons $L_\infty$-algebra (p. 48), because it brings out the expected Chern-Simons-terms in the $L_\infty$ – which is what the whole “adjustment”-process is about.

   There is no sense of doing this for non-string-like extensions: These higher central extensions are what is classified by invariant polynomials transgressing to Chern-SImons elements. In accounts you may be looking at this may be hidden in the assumption of a bilinear invariant pairing: This is the binary invariant polynomial which classifies the String-like extension. Without such, there is no sense to adjust to any Chern-Simons terms.

   The delooping $\mathbf{B}G$ of the smooth $\infty$-group Lie integrating an $L_\infty$-algebra $\mathfrak{g}$ is a truncation of the stackification of

   $$exp(\mathfrak{g}) \,\colon\, (U, [k]) \,\mapsto\, Hom_{dgAlg}\big( CE(\mathfrak{g}),\, \Omega^\bullet_{dR, vert}( U \times \Delta^k_{smth} ) \big) \,.$$

   This is the stacky formulation of Henriques’s notion of integration of $L_\infty$-algebras. For Lie 1-algebras and using 1-truncation this reduces to the “path method” for Lie integration, which is for instance implicit in Brylinski & McLaughlin 1996.

   The process we described enhances this Lie integration (in a way I have recalled when we discussed this last time) to a stack $\mathbf{B}G_{conn}$ such that maps $X \to \mathbf{B}G_{conn}$ are (cocycles for) $G$-principal $\infty$-bundles on $X$ with non-fake flat $\infty$-connection given by the expected Chern-Simons terms.