nLab
M2-M5 brane bound state

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

Traditional discussion

Concretely, if

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

denotes the representation containing

• $N^{(M2)}_i$ direct summands

of the

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

(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

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

then:

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

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

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

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

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

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

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

   3. 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.

Formalization via weight systems on chord diagrams

To make this precise, let

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

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 $\mathfrak{sl}(2,C)$) and write

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

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

Finally, write

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

for the linear map which sends a formal linear combination or representations to the weight system on Sullivan chord diagrams with $deg \in \mathbb{N}$ 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

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

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

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

graphics from Sati-Schreiber 19c

Normalization and large $N$ 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 $N^{(M2)}$).

Related entries

brane intersections/bound states/wrapped branes

• D-branes and anti D-branes form bound states by tachyon condensation, thought to imply the classification of D-brane charge by K-theory

• intersecting D-branes/fuzzy funnels:

  • Dp-D(p+2)-brane bound state and Yang-Mills monopoles

    • D0-D2 brane bound state

    • D1-D3 brane bound state

    • D2-D4 brane bound state

    • D3-D5 brane bound state

    • D4-D6 brane bound state

    • D6-D8-brane bound state

  • Dp-D(p+4)-brane bound state and Yang-Mills instantons

    • D0-D4 brane bound state

    • D1-D5 brane bound state

    • D2-D6 brane bound state

    • D3-D7 brane bound state

    • D4-D8 brane bound state

    • D5-D9 brane bound state

      small instanton

• Dp-D(p+6) brane bound state

  • D0-D6 brane bound state

  • D1-D7 brane bound state

  • D2-D8 brane bound state

• NS5-D4-D2 bound states

S-duality$\,$bound states:

• (p,q)-string

• (p,q)5-branes

• string network

• 5-brane web

intersecting$\,$M-branes:

• M5-MO9 brane bound state

• M2-M5 brane bound states

• E-strings

• membrane triple junction

• M-theory on 8-manifolds

• M-brane

• wrapped brane

References

General

• J.M. Izquierdo, Neil Lambert, George Papadopoulos, Paul Townsend, Dyonic Membranes, Nucl. Phys. B460:560-578, 1996 (arXiv:hep-th/9508177)

• Michael Green, Neil Lambert, George Papadopoulos, Paul Townsend, Dyonic $p$-branes from self-dual $(p+1)$-branes, Phys.Lett.B384:86-92, 1996 (arXiv:hep-th/9605146)

• Troels Harmark, Section 3.1 of Open Branes in Space-Time Non-Commutative Little String Theory, Nucl.Phys. B593 (2001) 76-98 (arXiv:hep-th/0007147)

• Troels Harmark, N.A. Obers, Section 5.1 of Phase Structure of Non-Commutative Field Theories and Spinning Brane Bound States, JHEP 0003 (2000) 024 (arXiv:hep-th/9911169)

• George Papadopoulos, Dimitrios Tsimpis, The holonomy of the supercovariant connection and Killing spinors, JHEP 0307:018, 2003 (arXiv:hep-th/0306117)

• Giuseppe Dibitetto, Nicolò Petri, BPS objects in $D=7$ supergravity and their M-theory origin, JHEP 12 (2017) 041 (arxiv:1707.06152)

• Nicolò Petri, slide 14 of Surface defects in massive IIA, talk at Recent Trends in String Theory and Related Topics 2018 (pdf)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 4 of Twisted Cohomotopy implies M-theory anomaly cancellation (arXiv:1904.10207)

• Jay Armas, Vasilis Niarchos, Niels A. Obers, Section 2 of Thermal transitions of metastable M-branes (arXiv:1904.13283)

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 D$p$-D$(p+2)$-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):

• Anirban Basu, Jeffrey Harvey, The M2-M5 Brane System and a Generalized Nahm’s Equation, Nucl.Phys. B713 (2005) 136-150 (arXiv:hep-th/0412310)

• Jonathan Bagger, Neil Lambert, Sunil Mukhi, Constantinos Papageorgakis, Section 2.2.1 of Multiple Membranes in M-theory, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 (arXiv:1203.3546, doi:10.1016/j.physrep.2013.01.006)

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:

• Horatiu Nastase, Constantinos Papageorgakis, Sanjaye Ramgoolam, The fuzzy $S^2$ structure of M2-M5 systems in ABJM membrane theories, JHEP 0905:123, 2009 (arXiv:0903.3966)

• Sanjaye Ramgoolam, Fuzzy geometry of membranes in M-theory, 2009 (pdf)

• Horatiu Nastase, Constantinos Papageorgakis, Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors, SIGMA 6:058, 2010 (arXiv:1003.5590)

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, $AdS_2$ Holography: Mind the Cap, JHEP 1812 (2018) 028 (arXiv:1806.02834)