There is the suggestion (MSJVR 02, checked in AIST 17a, AIST 17b) that, in the BMN matrix model, supersymmetric M2-M5-brane bound states are identified with isomorphism classes of certain “limit sequences” of longitudinal-light cone-constant $N \times N$-matrix-fields constituting finite-dimensional complex Lie algebra representations of su(2).
Concretely, if
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
then:
a configuration of a finite number of stacks of coincident M5-branes corresponds to a sequence of such representations for which
$N^{(M2)}_i \to \infty$ (this being the relevant large N limit)
for fixed $N^{(M5)}_i$ (being the number of M5-branes in the $i$th stack)
and fixed ratios $N^{(M2)}_i/N$ (being the charge/light-cone momentum carried by the $i$th stack);
an M2-brane configuration corresponds to a sequence of such representations for which
$N^{(M5)}_i \to \infty$ (this being the relevant large N limit)
for fixed $N^{(M2)}_i$ (being the number of M2-brane in the $i$th stack)
and fixed ratios $N^{(M5)}_i/N$ (being the charge/light-cone momentum carried by the $i$th stack)
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.
To make this precise, let
be the set of isomorphism classes of complex metric Lie representations (hence finite-dimensional representations) of su(2) (hence of the special linear Lie algebra $\mathfrak{sl}(2,C)$) and write
for its linear span (the complex vector space of formal linear combinations of isomorphism classes of metric Lie representations).
Finally, write
for the linear map which sends a formal linear combination or representations to the weight system on Sullivan chord diagrams with $deg \in \mathbb{N}$ chords which is given by tracing in the given representation.
Then a M2-M5-brane bound state as in the traditional discussion above, but now formalized as an su(2)-weight system
hence a weight system horizontal chord diagrams closed to Sullivan chord diagrams, these now being the multi-trace observables on these) is
graphics from Sati-Schreiber 19c
Normalization and large $N$ limit. The first power of the square root in (1) reflects the volume measure on the fuzzy 2-sphere (by the formula here), while the power of $2\,deg$ (which is the number of operators in the multi-trace observable evaluating the weight system) gives the normalization (here) of the functions on the fuzzy 2-sphere.
Hence this normalization is such that the single-trace observables among the multi-trace observables, hence those which come from round chord diagrams, coincide on those M2-M5 brane bound states $\Psi_{ \left\{ N^{(M2)}_i, N^{(M5)}_i \right\}_{i} }$ for which $N^{(M2)}_i = \delta_i^{i_0} N^{(M2)}$, hence those which have a single constitutent fuzzy 2-sphere, with the shape observables on single fuzzy 2-spheres discussed here:
graphics from Sati-Schreiber 19c
Therefore, with this normalization, the limits $N^{(M2)} \to \infty$ and $N^{(M5)} \to \infty$ of (1) should exist in weight systems. The former trivially so, the latter by the usual convergence ofthe fuzzy 2-sphere to the round 2-sphere in the large N limit.
Notice that the multi trace observables on these states only see the relative radii of the constitutent fuzzy 2-spheres: If $N^{(M2)}_i = \delta_i^{i_0} N^{(M2)}$ then the $N^{(M2)}$-dependence of (1) cancels out, reflecting the fact that then there is only a single constituent 2-sphere of which the observable sees only the radius fluctuations, not the absolute radius (proportional to $N^{(M2)}$).
Last revised on May 6, 2021 at 09:59:38. See the history of this page for a list of all contributions to it.