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?

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).

]]>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)

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)

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*

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.

]]>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)