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M2-M5 brane bound state

Contents

Contents

Idea

bound states of M2-branes with M5-branes;

in particular dyonic\,black M2-branes, i.e. M5-branes wrapped on a 3-manifold (ILPT 95)

Properties

Image in type II string theory

Under suitable duality between M-theory and type IIA string theory the M2/M5brane bound states become Dp-D(p+2)-brane bound states (Basu-Harvey 05).

Relation to giant gravitons

for the relation to giant gravitons: (Camino-Ramallo 01)


M2-M5 brane bound states in the BMN matrix model

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×NN \times N-matrix-fields constituting finite-dimensional complex Lie algebra representations of su(2).

Traditional discussion

Concretely, if

i(N i (M2)N i (M5))𝔰𝔲(2) Rep fin \underset{ i }{\oplus} \big( N^{(M2)}_i \cdot \mathbf{N}^{(M5)}_i \big) \;\;\in\;\; \mathfrak{su}(2)_{\mathbb{C}}Rep^{fin}

denotes the representation containing

of the

  • N i (M5)N^{(M5)}_i-dimensional irrep N i (M5)𝔰𝔲(2) Rep\mathbf{N}^{(M5)}_i \in \mathfrak{su}(2)_{\mathbb{C}}Rep

(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(i(N i (M2)N i (M5))) N \;\coloneqq\; dim \big( \underset{ i }{\oplus} \big( N^{(M2)}_i \cdot \mathbf{N}^{(M5)}_i \big) \big)

then:

  1. a configuration of a finite number of stacks of coincident M5-branes corresponds to a sequence of such representations for which

    1. N i (M2)N^{(M2)}_i \to \infty (this being the relevant large N limit)

    2. for fixed N i (M5)N^{(M5)}_i (being the number of M5-branes in the iith stack)

    3. and fixed ratios N i (M2)/NN^{(M2)}_i/N (being the charge/light-cone momentum carried by the iith stack);

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

    1. N i (M5)N^{(M5)}_i \to \infty (this being the relevant large N limit)

    2. for fixed N i (M2)N^{(M2)}_i (being the number of M2-brane in the iith stack)

    3. and fixed ratios N i (M5)/NN^{(M5)}_i/N (being the charge/light-cone momentum carried by the iith stack)

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.

Formalization via weight systems on chord diagrams

To make this precise, let

𝔰𝔲(2) MetMod /Set \mathfrak{su}(2)_{\mathbb{C}} MetMod_{/\sim} \;\in\; Set

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 𝔰𝔩(2,C)\mathfrak{sl}(2,C)) and write

Span(𝔰𝔲(2) MetMod /)Vect Span \big( \mathfrak{su}(2)_{\mathbb{C}} MetMod_{/\sim} \big) \;\in\; Vect_{\mathbb{C}}

for its linear span (the complex vector space of formal linear combinations of isomorphism classes of metric Lie representations).

Finally, write

Span(𝔰𝔲(2) MetMod /) (𝒲 s) deg c 1V 1+c 2V 2 c 1w V 1+c 2w V 2 \array{ Span \big( \mathfrak{su}(2)_{\mathbb{C}}MetMod_{/\sim} \big) & \longrightarrow & \big( \mathcal{W}^{{}^{s}} \big)^{deg} \\ c_1 \cdot V_1 + c_2 \cdot V_2 &\mapsto& c_1 \cdot w_{V_1} + c_2 \cdot w_{V_2} }

for the linear map which sends a formal linear combination or representations to the weight system on Sullivan chord diagrams with degdeg \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

Ψ {N i (M2),N i (M5)} ideg(𝒲 s) deg \Psi_{ \left\{ N^{(M2)}_i, N^{(M5)}_i \right\}_{i} } \;\in\; \underset{ deg \in \mathbb{N} }{\prod} \big( \mathcal{W}^{{}^{\mathrm{s}}} \big)^{deg}

hence a weight system horizontal chord diagrams closed to Sullivan chord diagrams, these now being the multi-trace observables on these these) is

(1)Ψ {N i (M2),N i (M5)} i4π2 2deg iN i (M2)i(N i (M2)1((N i (M5)) 21) 1+2degN i (M5)) \Psi_{ \left\{ N^{(M2)}_i, N^{(M5)}_i \right\}_{i} } \;\coloneqq\; \tfrac{ 4 \pi \, 2^{2\,deg} }{ \sum_i N^{(M2)}_i } \underset{i}{\sum} \left( N^{(M2)}_i \cdot \tfrac{1}{ \left( \sqrt{ \left( N^{(M5)}_i \right)^2 - 1 } \right)^{1 + 2 deg} } \mathbf{N}^{(M5)}_i \right)

graphics from Sati-Schreiber 19c

Normalization and large NN 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 2deg2\,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 Ψ {N i (M2),N i (M5)} i \Psi_{ \left\{ N^{(M2)}_i, N^{(M5)}_i \right\}_{i} } for which N i (M2)=δ i i 0N (M2)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)N^{(M2)} \to \infty and N (M5)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 i (M2)=δ i i 0N (M2)N^{(M2)}_i = \delta_i^{i_0} N^{(M2)} then the N (M2)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)N^{(M2)}).


brane intersections/bound states/wrapped branes

S-duality\,bound states:

intersecting\,M-branes:

References

General

Further bound states of M2-branes/M5-branes to giant gravitons:

  • J. M. Camino, A. V. Ramallo, M-Theory Giant Gravitons with C field, Phys.Lett.B525:337-346,2002 (arXiv:hep-th/0110096)

Reduction to Dpp-D(p+2)(p+2)-brane bound states

The lift of Dp-D(p+2)-brane bound states in string theory to M2-M5-brane bound states/E-strings in M-theory, under duality between M-theory and type IIA string theory+T-duality, via generalization of Nahm's equation (eventually motivating the BLG model/ABJM model):

Argument, using the ABJM model, that the apparent fuzzy 3-sphere geometry of M2-M5 brane bound states effectively collapses to a fuzzy 2-sphere geometry of Dp-D(p+2)-brane intersection fuzzy funnels:

KK-reduction to JT gravity

Realization of JT-gravity as Kaluza-Klein reduction of D=6 supergravity on the worldvolume of D1-D5 brane bound states or M2-M5 brane bound states:

  • Yue-Zhou Li, Shou-Long Li, H. Lu, Exact Embeddings of JT Gravity in Strings and M-theory, Eur. Phys. J. C (2018) 78: 791 (arXiv:1804.09742)

  • Iosif Bena, Pierre Heidmann, David Turton, AdS 2AdS_2 Holography: Mind the Cap, JHEP 1812 (2018) 028 (arXiv:1806.02834)

Last revised on February 1, 2020 at 11:16:43. See the history of this page for a list of all contributions to it.