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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2010
   • PermaLink
   Author: Urs
   Format: Markdownstarted [[formal dg-algebra]]

   started formal dg-algebra

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2010
   • PermaLink
   Author: Urs
   Format: Markdownexpanded a bit more In the old Sullivan reference the condition for A to be formal is that there is a quasi-iso <latex> A \to H^\bullet(A) </latex>. In newer references, such as the survey by Kathry Hess, it says instead that there is a span of quasi-isos between <latex>A</latex> and <latex> H^\bullet(A) </latex>. Since in the model structure on dg-algebras all objects are fibrant, I take it that this is equivalent to saying that <latex> A </latex> and <latex> H^\bullet(A)</latex> are isomorphic in the homotopy category. This is the way I state the definition now. But check if I am mixed up.

   expanded a bit more

   In the old Sullivan reference the condition for A to be formal is that there is a quasi-iso .

   In newer references, such as the survey by Kathry Hess, it says instead that there is a span of quasi-isos between and . Since in the model structure on dg-algebras all objects are fibrant, I take it that this is equivalent to saying that and are isomorphic in the homotopy category. This is the way I state the definition now. But check if I am mixed up.

3. • CommentRowNumber3.
   • CommentAuthorperezl.alonso
   • CommentTimeJun 19th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexGiven the use of Sullivan models for encoding quantization/eom's, what is the physical meaning of quantization described by nonformal spaces?

   Given the use of Sullivan models for encoding quantization/eom’s, what is the physical meaning of quantization described by nonformal spaces?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 19th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexLet's see. Essentially by definition it means that all Bianchi identities are of the standard untwisted form $\mathrm{d} F = 0$. So the system looks EoM-wise like a bunch of (possibly higher and/or self-dual but otherwise) ordinary free abelian gauge fields. Their flux quantization by a corresponding formal classifying space means to nevertheless enforce torsion-relations that may differ from those of standard free abelian gauge fields. Is that an answer to what you are asking?

   Let’s see. Essentially by definition it means that all Bianchi identities are of the standard untwisted form . So the system looks EoM-wise like a bunch of (possibly higher and/or self-dual but otherwise) ordinary free abelian gauge fields. Their flux quantization by a corresponding formal classifying space means to nevertheless enforce torsion-relations that may differ from those of standard free abelian gauge fields.

   Is that an answer to what you are asking?

5. • CommentRowNumber5.
   • CommentAuthorperezl.alonso
   • CommentTimeJun 19th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexSort of. In some (different) physics contexts, quasi-isomorphisms are sometimes interpreted as RG flow statements, what I was wondering is what such quasi-isomorphisms mean here.

   Sort of. In some (different) physics contexts, quasi-isomorphisms are sometimes interpreted as RG flow statements, what I was wondering is what such quasi-isomorphisms mean here.

6. • CommentRowNumber6.
   • CommentAuthorperezl.alonso
   • CommentTimeJun 19th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexpointer * [[Manuel Amann]]. *Non-formal Homogeneous Spaces* (2012). ([arXiv:1206.0786](https://arxiv.org/abs/1206.0786)). which I also added to [[Manuel Amann]] <a href="https://ncatlab.org/nlab/revision/diff/formal+dg-algebra/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/formal+dg-algebra/13">v13</a>, <a href="https://ncatlab.org/nlab/show/formal+dg-algebra">current</a>

   pointer

   • Manuel Amann. Non-formal Homogeneous Spaces (2012). (arXiv:1206.0786).

   which I also added to Manuel Amann

   diff, v13, current