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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2020
    • (edited Jan 16th 2020)

    In order to formalize some physics, I am looking for a suitable mathematical concept. It looks like the putative concept ought to be something like “pro-finite 𝔰𝔲(2)\mathfrak{su}(2)-representations”, but I am not sure yet. And once I am sure, I’ll be wondering if there is any decent established theory for such things.

    The following is the motivation (taken from here):


    There is the remarkable observation (MSJVR 02, checked in AIST 17) that in the BMN matrix model supersymmetric M2-M5-brane bound states are identified with “limit sequences” of isomorphism classes of finite-dimensional complex Lie algebra representations of su(2).

    Concretely, if

    N({N i (M2),N i (M5)} i)iN i (M2)ρ N i (M5)𝔰𝔲(2)Rep fin \mathbf{N}( \{N^{(M2)}_i,N^{(M5)}_i\}_i ) \;\;\coloneqq\;\; \underset{ i }{\oplus} N^{(M2)}_i \cdot \rho_{N^{(M5)}_i} \;\;\in\;\; \mathfrak{su}(2)Rep^{fin}

    denotes the representation containing

    of the

    • N i (M5)N^{(M5)}_i-dimensional irrep

    (for {N i (M2),N i (M5)} i(×) I\{N^{(M2)}_i, N^{(M5)}_i\}_{i} \in (\mathbb{N} \times \mathbb{N})^I some finitely indexed set of pairs of natural numbers)

    with total dimension

    Ndim(N({N i (M2),N i (M5)} i)) N \;\coloneqq\; dim\big( \mathbf{N}( \{N^{(M2)}_i,N^{(M5)}_i\}_i ) \big)


    1. an M5-brane configuration corresponds to a sequence of such representations for which

      N i (M2)N^{(M2)}_i \to \infty

      for fixed N i (M5)N^{(M5)}_i

      and fixed ratios N i (M2)/NN^{(M2)}_i/N

    2. an M2-brane configuration corresponds to a sequence of such representations for which

      N i (M5)N^{(M5)}_i \to \infty

      for fixed N i (M2)N^{(M2)}_i

      and fixed ratios N i (M5)/NN^{(M5)}_i/N

    for all iIi \in I.

    Hence, by extension, any other sequence of finite-dimensional 𝔰𝔲(2)\mathfrak{su}(2)-representations is a kind of mixture of these two cases, interpreted as an M2-M5 brane bound state of sorts.


    Question. I’d like to extract a precise definition of “M2-M5 brane bound state” from the above. It must subsume suitable limits of finite-dimensional 𝔰𝔲(2)\mathfrak{su}(2)-representation as above. But taken where? And identified how?

    Is “profinite 𝔰𝔲(2)\mathfrak{su}(2)-reps” a thing in representation theory? (i.e. pro-objects in the category of finite-dimensional representations.) Or maybe ind-objects instead? Or something else?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2020

    Ah, never mind, I see the answer now. It was right in front of me all along.

    The space of interest here is not 𝔰𝔲(2)Rep\mathfrak{su}(2)Rep but the image of Span(𝔰𝔲(2)Rep)Span(\mathfrak{su}(2)Rep) in weight systems. Now taking the linear span is what makes sense of expressions like 1Nρ N \frac{1}{N} \cdot \rho_{N}. And their limit with NN\to \infty gotta be taken in the space of weight systems, because that’s precisely what the multi-trace observables of the BMN model observe.

    Okay, case closed. Thanks for listening.