Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

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 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 nforum nlab noncommutative noncommutative-geometry number-theory 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 sheaves 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2021

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

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021
    • (edited Sep 19th 2021)

    added the statement that PU()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.

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021
    • (edited Sep 19th 2021)

    added statement that S 1U()PU()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

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2021

    added pointer to:

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

    diff, v6, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2021

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

    diff, v7, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2021

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

    • E. Parker, Graded continuous trace C *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).

    • CommentRowNumber7.
    • CommentAuthorperezl.alonso
    • CommentTimeApr 16th 2024
    • (edited Apr 16th 2024)

    The setting of NN\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)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)B 2U(1)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 B 2U(1)BPU(H)BG\mathbf{B}^2 U(1)\to \mathbf{B} PU(H)\to \mathbf{B} G for BG=*BG=*, i.e. a nontrivial extension of 2-groups only visible at the smooth setting but not at the topological level, so that the NN\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 NN 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?