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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 15th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor completeness, to go with *[[U(ℋ)]]*, for the moment mainly in order to record references, such as: * [[David John Simms]], *Topological aspects of the projective unitary group*, Math. Proc. Camb. Phil. Soc. **68** 1 (1970) 57-60 ([doi:10.1017/S0305004100001043](https://doi.org/10.1017/S0305004100001043)) <a href="https://ncatlab.org/nlab/revision/PU%28%E2%84%8B%29/1">v1</a>, <a href="https://ncatlab.org/nlab/show/PU%28%E2%84%8B%29">current</a>

   for completeness, to go with U(ℋ), for the moment mainly in order to record references, such as:

   • David John Simms, Topological aspects of the projective unitary group, Math. Proc. Camb. Phil. Soc. 68 1 (1970) 57-60 (doi:10.1017/S0305004100001043)

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 19th 2021
   • (edited Sep 19th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the statement that $PU(\mathcal{H})$ is well-pointed, which is made explicit on p. 23 of * [[Fabian Hebestreit]], [[Steffen Sagave]], *Homotopical and operator algebraic twisted K-theory*, Mathematische Annalen **378** (2020) 1021-1059 ([arXiv:1904.01872](https://arxiv.org/abs/1904.01872), [doi:10.1007/s00208-020-02066-6](https://doi.org/10.1007/s00208-020-02066-6)) but follows by a more general result due to Dardalat and Pennig. <a href="https://ncatlab.org/nlab/revision/diff/PU%28%E2%84%8B%29/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/PU%28%E2%84%8B%29/3">v3</a>, <a href="https://ncatlab.org/nlab/show/PU%28%E2%84%8B%29">current</a>

   added the statement that $PU(\mathcal{H})$ is well-pointed, which is made explicit on p. 23 of

   • Fabian Hebestreit, Steffen Sagave, Homotopical and operator algebraic twisted K-theory, Mathematische Annalen 378 (2020) 1021-1059 (arXiv:1904.01872, doi:10.1007/s00208-020-02066-6)

   but follows by a more general result due to Dardalat and Pennig.

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 19th 2021
   • (edited Sep 19th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement that $S^1 \to \mathrm{U}(\mathcal{H}) \to PU(\mathcal{H})$ is a fiber bundle. Am copying this statement also to [[U(H)]] and to *[[coset space coprojection admitting local sections]]* <a href="https://ncatlab.org/nlab/revision/diff/PU%28%E2%84%8B%29/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/PU%28%E2%84%8B%29/5">v5</a>, <a href="https://ncatlab.org/nlab/show/PU%28%E2%84%8B%29">current</a>

   added statement that $S^1 \to \mathrm{U}(\mathcal{H}) \to PU(\mathcal{H})$ is a fiber bundle. Am copying this statement also to U(H) and to coset space coprojection admitting local sections

   diff, v5, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 30th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Ellen Maycock Parker]], *The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras*, Transactions of the American Mathematical Society **308** 1 (1988) ([jstor:2000953](https://www.jstor.org/stable/2000953)) <a href="https://ncatlab.org/nlab/revision/diff/PU%28%E2%84%8B%29/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/PU%28%E2%84%8B%29/6">v6</a>, <a href="https://ncatlab.org/nlab/show/PU%28%E2%84%8B%29">current</a>

   added pointer to:

   • Ellen Maycock Parker, The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras, Transactions of the American Mathematical Society 308 1 (1988) (jstor:2000953)

   diff, v6, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 13th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor the graded version of $PU(\mathcal{H})$ I have added pointer also to * [[Alan Carey]], [[Bai-Ling Wang]], p. 5 of: *Thom isomorphism and Push-forward map in twisted K-theory*, Journal of K-Theory **1** 2 (2008) 357-393 ([arXiv:math/0507414](https://arxiv.org/abs/math/0507414), [doi:10.1017/is007011015jkt011](https://doi.org/10.1017/is007011015jkt011)) <a href="https://ncatlab.org/nlab/revision/diff/PU%28%E2%84%8B%29/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/PU%28%E2%84%8B%29/7">v7</a>, <a href="https://ncatlab.org/nlab/show/PU%28%E2%84%8B%29">current</a>

   for the graded version of $PU(\mathcal{H})$ I have added pointer also to

   • Alan Carey, Bai-Ling Wang, p. 5 of: Thom isomorphism and Push-forward map in twisted K-theory, Journal of K-Theory 1 2 (2008) 357-393 (arXiv:math/0507414, doi:10.1017/is007011015jkt011)

   diff, v7, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 13th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexThese authors also point (in addition to the reference in #4) to * E. Parker, *Graded continuous trace $C^\ast$-algebras and duality*, Operator algebras and topology (Craiova, 1989), 130–145, Pitman Res. Notes Math. Ser., 270. but I haven't yet found any online trace of this article/book (I find the 1983 version of the series, but not the one from 1989).

   These authors also point (in addition to the reference in #4) to

   • E. Parker, Graded continuous trace $C^\ast$-algebras and duality, Operator algebras and topology (Craiova, 1989), 130–145, Pitman Res. Notes Math. Ser., 270.

   but I haven’t yet found any online trace of this article/book (I find the 1983 version of the series, but not the one from 1989).

7. • CommentRowNumber7.
   • CommentAuthorperezl.alonso
   • CommentTimeApr 16th 2024
   • (edited Apr 16th 2024)
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexThe setting of $N\to \infty$ D-branes giving rise to an infinite-dimensional gauge bundle reminds me of the difference in presentations of the string group, where $PU(H)$ itself also appears in the construction by Stolz. In [p.28](https://arxiv.org/pdf/1201.5277.pdf#page=28) of 1201.5277 it is reviewed that the map of topological spaces $dd:BPU(H)\to B^2 U(1)$ is actually the identity morphism, but surely its differential refinement **dd** of smooth stacks is not. But one can still have a fibration $\mathbf{B}^2 U(1)\to \mathbf{B} PU(H)\to \mathbf{B} G$ for $BG=*$, i.e. a nontrivial extension of 2-groups only visible at the smooth setting but not at the topological level, so that the $N\to \infty$ branes actually carry a nonabelian gerbe (with hopefully a finite-dimensional presentation). Do the arguments for quantization in differential K-theory rule out this possibility, or are they really just based on the need to incorporate the differential picture into the original hypothesis for quantization? The fact that those large $N$ matrix models describe some aspects of string and M-theory, which does feature these non-abelian gerbes, might suggest these already appear on D-branes, no?

   The setting of $N\to \infty$ D-branes giving rise to an infinite-dimensional gauge bundle reminds me of the difference in presentations of the string group, where $PU(H)$ itself also appears in the construction by Stolz. In p.28 of 1201.5277 it is reviewed that the map of topological spaces $dd:BPU(H)\to B^2 U(1)$ is actually the identity morphism, but surely its differential refinement dd of smooth stacks is not. But one can still have a fibration $\mathbf{B}^2 U(1)\to \mathbf{B} PU(H)\to \mathbf{B} G$ for $BG=*$, i.e. a nontrivial extension of 2-groups only visible at the smooth setting but not at the topological level, so that the $N\to \infty$ branes actually carry a nonabelian gerbe (with hopefully a finite-dimensional presentation). Do the arguments for quantization in differential K-theory rule out this possibility, or are they really just based on the need to incorporate the differential picture into the original hypothesis for quantization? The fact that those large $N$ matrix models describe some aspects of string and M-theory, which does feature these non-abelian gerbes, might suggest these already appear on D-branes, no?