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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 $\mathfrak{su}(2)$-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):

   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

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

   denotes the representation containing

   • $N^{(M2)}_i$ 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 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 $N^{(M2)}_i/N$

   2. 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?

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 $\mathfrak{su}(2)Rep$ 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.