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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
of the
(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:
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 $N^{(M2)}_i/N$
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?
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.
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