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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 16th 2020
   • (edited Jan 16th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn 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 $\mathfrak{su}(2)$-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](https://ncatlab.org/nlab/show/BMN+matrix+model#M2M5BraneBoundStatesInTheBMNMatrixModel)): **Motivation** There is the remarkable observation ([MSJVR 02](https://ncatlab.org/nlab/show/BMN+matrix+model#MSJVR02), checked in [AIST 17](https://ncatlab.org/nlab/show/BMN+matrix+model#AIST17)) that in the [[BMN matrix model]] supersymmetric [[M2-M5-brane bound states]] are identified with "limit sequences" of [[isomorphism classes]] of [[finite-dimensional vector space|finite-dimensional]] [[complex numbers|complex]] [[Lie algebra representations]] of [[su(2)]]. Concretely, if $$ \mathbf{N}( \{N^{(M2)}_i,N^{(M5)}_i\}_i ) \;\;\coloneqq\;\; \underset{ i }{\oplus} N^{(M2)}_i \cdot \rho_{N^{(M5)}_i} \;\;\in\;\; \mathfrak{su}(2)Rep^{fin} $$ denotes [[generalized the|the]] [[Lie algebra representation|representation]] containing * $N^{(M2)}_i$ [[direct sum|direct summands]] of the * $N^{(M5)}_i$-dimensional [[irrep]] (for $\{N^{(M2)}_i, N^{(M5)}_i\}_{i} \in (\mathbb{N} \times \mathbb{N})^I$ some [[finite set|finitely]] [[indexed set]] of [[pairs]] of [[natural numbers]]) with total [[dimension]] $$ N \;\coloneqq\; dim\big( \mathbf{N}( \{N^{(M2)}_i,N^{(M5)}_i\}_i ) \big) $$ then: 1. an [[M5-brane]] configuration corresponds to a sequence of such representations for which $N^{(M2)}_i \to \infty$ for fixed $N^{(M5)}_i$ and fixed [[ratios]] $N^{(M2)}_i/N$ 1. an [[M2-brane]] configuration corresponds to a sequence of such representations for which $N^{(M5)}_i \to \infty$ for fixed $N^{(M2)}_i$ and fixed [[ratios]] $N^{(M5)}_i/N$ for all $i \in I$. Hence, by extension, any other sequence of finite-dimensional $\mathfrak{su}(2)$-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 $\mathfrak{su}(2)$-representation as above. But taken where? And identified how? Is "profinite $\mathfrak{su}(2)$-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?

   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 -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):

   Motivation

   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

   denotes the representation containing

   • direct summands

   of the

   • -dimensional irrep

   (for some finitely indexed set of pairs of natural numbers)

   with total dimension

   then:

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

      for fixed

      and fixed ratios

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

      for fixed

      and fixed ratios

   for all .

   Hence, by extension, any other sequence of finite-dimensional -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 -representation as above. But taken where? And identified how?

   Is “profinite -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?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 16th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, never mind, I see the answer now. It was right in front of me all along. The space of interest here is not $\mathfrak{su}(2)Rep$ but the image of $Span(\mathfrak{su}(2)Rep)$ in weight systems. Now taking the linear span is what makes sense of expressions like $ \frac{1}{N} \cdot \rho_{N}$. And their limit with $N\to \infty$ 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.

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

   The space of interest here is not but the image of in weight systems. Now taking the linear span is what makes sense of expressions like . And their limit with 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.