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for completeness, to go with U(ℋ), for the moment mainly in order to record references, such as:
added the statement that $PU(\mathcal{H})$ is well-pointed, which is made explicit on p. 23 of
but follows by a more general result due to Dardalat and Pennig.
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 pointer to:
for the graded version of $PU(\mathcal{H})$ I have added pointer also to
These authors also point (in addition to the reference in #4) to
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).
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?
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