nLab BLG model

Idea

The BLG model is a 3-dimensional SCFT involving a Chern-Simons theory coupled to matter. It is argued to be the worldvolume theory of 2 coincident black M2-branes with 16 manifest supersymmetries. The generalization to an arbitrary number of M2-branes is supposed to be given by the ABJM model.

$N$ Killing spinors on spherical space form $S^7/\widehat{G}$	$\phantom{AA}\widehat{G} =$	spin-lift of subgroup of isometry group of 7-sphere	3d superconformal gauge field theory on back M2-branes with near horizon geometry $AdS_4 \times S^7/\widehat{G}$
$\phantom{AA}N = 8\phantom{AA}$	$\phantom{AA}\mathbb{Z}_2$	cyclic group of order 2	BLG model
$\phantom{AA}N = 7\phantom{AA}$	—	—	—
$\phantom{AA}N = 6\phantom{AA}$	$\phantom{AA}\mathbb{Z}_{k\gt 2}$	cyclic group	ABJM model
$\phantom{AA}N = 5\phantom{AA}$	$\phantom{AA}2 D_{k+2}$ $2 T$, $2 O$, $2 I$	binary dihedral group, binary tetrahedral group, binary octahedral group, binary icosahedral group	(HLLLP 08a, BHRSS 08)
$\phantom{AA}N = 4\phantom{AA}$	$\phantom{A}2 D_{k+2}$ $2 O$, $2 I$	binary dihedral group, binary octahedral group, binary icosahedral group	(HLLLP 08b, Chen-Wu 10)

• José Figueroa-O'Farrill et al. 2009 (arXiv:0909.0163, pdf slides)

Properties

The Lagrangian

(…)

Boundary conditions

Discussion of boundary conditions of the BLG model, leading to brane intersection with M-wave, M5-brane and MO9-brane is in (Chu-Smith 09, BPST 09).

The “$3$-algebra” structure

The BLG model Lagrangian involves a trilinear operation on the scalar fields $\phi \in V$

$$[-,-,-] : V^{\otimes 3} \to V \,,$$

the M2-brane 3-algebra.

Moreover, the supersymmetry of the Lagrangian hinges on the fact that this map satisfies a condition that has some similarity to a Jacobi identity for the binary operation on a Lie algebra.

Therefore, superficially, it looks like this might be the trinary bracket on an L-∞ algebra structure on the space $V$.

On the one hand, indeed, by the discussion at supergravity C-field , the M2-brane is charged under a circle 3-bundle with connection whose higher gauge theory is controlled by Lie 3-algebras in direct analogy to how the higher gauge theory of the string is controled by gerbes/principal 2-bundles and their Lie 2-algebras and that of charged particles by ordinary Lie algebras.

Motivated by attempts in Basu-Harvey 04 to generalize Nahm's equations for fuzzy funnels of D2-D4-brane intersections to M2-M5-brane intersections, Bagger-Lambert 06 introduced the algebraic structure that came to be known the M2-brane 3-algebra, further highlighted by Gustavsson 07. This terminology was picked up by many authors In the process, it transmuted sometimes to “3-Lie algebra” and sometimes even to “Lie 3-algebra”.

Beware that the Bagger-Lambert “M2-brane 3-algebra” is not- aLie 3-algebra_ in the established sense of an L-∞ algebra structure on a graded vector space $V$ concentrated in the lowest three degrees.

The reason is that for the notion of an L-∞ algebra (as discussed there) it is crucial that $V$ is a $\mathbb{N}$-graded (or $\mathbb{Z}$-graded) vector space and that the $n$-ary brackets respect the degree in a certain way. But in the BaggerLambert-proposal, $V$ is all concentrated in a single degree (is regarded as ungraded). One immediately finds that in this case the $L_\infty$-respect of $[-,-,-]$ for the grading would imply that $V$ is taken to be in degree $1/2$. Since this is not in $\mathbb{N}$, it does not yield an $L_\infty$-algebra.

Notice that the $\mathbb{N}$-grading (or $\mathbb{Z}$-grading) of $L_\infty$-algebras is crucial for the homotopy theoretic interpretation of L-∞ algebras as higher Lie algebras. None of the good theory of $L_\infty$-algebras survives when this grading is dropped. This grading has its origin in the Dold-Kan correspondence, which establishes integral graded homological structures as models for structures in higher category theory. Notably, a higher Lie algebra is supposed to have a Lie integration to a smooth $n$-groupoid. Under this process, the elements in degree $k$ of the higher Lie algebra become tangents to the space of k-morphisms of this smooth $n$-groupoid. Clearly, here only integer $k$ do make sense.

On the other hand, it is of course possible to consider the structure on “$L_\infty$-algebras without grading”, even if these will not have a good theory. This notion has once been introduced by Filippov (Sib. Math. Zh. No 6 126–140 (195)) under the term n-Lie algebra .

Beware, therefore, that the innocent-looking difference between the terms

• Lie n-algebra

• n-Lie algebra

corresponds, unfortunately, to a major difference in the behaviour of the concepts behind these terms.

In conclusion, it is clear that 2-brane physics is governed by Lie 3-algebraic structures, but it is not yet clear how the trinary operation highlighted by BaggerLambert would be an example.

In view of this it might be noteworthy that the equivalent reformulation and generalization of the BLG model by the ABJM model does not involve any “3-algebras” at all. In fact at least most of the “3-algebras” appearing in the membrane literature may be understood as being data of a plain Lie algebra with an invariant product and a representation (MFMR 08). These authors summarize the state of affiars on p. 3 as

All this prompts one to question whether the 3-algebras appearing in the constructions [13,10,11] play a fundamental role in M-theory or, at least insofar as the effective field theory is concerned, are largely superfluous. The equivalence of [10] and [6] and the abundance of new theories (dual to known M-theory backgrounds) which seem not to involve a 3-algebra might suggest the latter. Nonetheless, given our lack of understanding of how to incorporate in Lie-algebraic terms the expected properties of M-theoretic degrees of freedom, like the entropy scaling laws for M2- and M5-brane condensates, it may be useful to understand the precise relation between the 3-algebras appearing in the recent literature on superconformal Chern–Simons theory and Lie algebras.

The article (MFMR 08) provides this relation and under this relation the “3-algebraic” BLG model has then been understood as a special case of the ordinary Lie algebraic ABJM theory. A review is in (Bagger-Lambert 12).

It has also been suggested that “3-algebras” are to be interpreted in 2-plectic structure (Saemann-Szabo).

Related concepts

• M2-brane 3-algebra

• ABJM theory

• membrane matrix model

$d$	$N$	superconformal super Lie algebra	R-symmetry	black brane worldvolume superconformal field theory via AdS-CFT
$\phantom{A}3\phantom{A}$	$\phantom{A}2k+1\phantom{A}$	$\phantom{A}B(k,2) \simeq$ osp$(2k+1 \vert 4)\phantom{A}$	$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$	$\phantom{A}2k\phantom{A}$	$\phantom{A}D(k,2)\simeq$ osp$(2k \vert 4)\phantom{A}$	$\phantom{A}SO(2k)\phantom{A}$	M2-brane D=3 SYM BLG model ABJM model
$\phantom{A}4\phantom{A}$	$\phantom{A}k+1\phantom{A}$	$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$	$\phantom{A}U(k+1)\phantom{A}$	D3-brane D=4 N=4 SYM D=4 N=2 SYM D=4 N=1 SYM
$\phantom{A}5\phantom{A}$	$\phantom{A}1\phantom{A}$	$\phantom{A}F(4)\phantom{A}$	$\phantom{A}SO(3)\phantom{A}$	D4-brane D=5 SYM
$\phantom{A}6\phantom{A}$	$\phantom{A}k\phantom{A}$	$\phantom{A}D(4,k) \simeq$ osp$(8 \vert 2k)\phantom{A}$	$\phantom{A}Sp(k)\phantom{A}$	M5-brane D=6 N=(2,0) SCFT D=6 N=(1,0) SCFT

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

References

Precursors

• John Schwarz, Superconformal Chern-Simons Theories (arXiv:arXiv:hep-th/0411077

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 (this eventually motivated 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)

General

The original articles are

• Jonathan Bagger, Neil Lambert, Modeling Multiple M2’s, Phys. Rev. D75, 045020 (2007). (hep-th/0611108).

• Jonathan Bagger, Neil Lambert, Gauge Symmetry and Supersymmetry of Multiple M2-Branes, Phys. Rev. D77, 065008 (2008). (arXiv:0711.0955).

and with special emphasis on the “M2-brane 3-algebra”-structure also:

• Andreas Gustavsson, Algebraic structures on parallel M2-branes Nucl. Phys. B811, 66-76 (2009) (arXiv:0709.1260)
• Andreas Gustavsson, One-loop corrections to Bagger-Lambert theory, Nuclear Physics B 807, n. 1-2 (2009) 315–333 doi

The interpretation in terms of branes at a $\mathbb{Z}/2$-ADE-singularities with discrete torsion in the supergravity C-field is due to

• Neil Lambert, David Tong, Membranes on an Orbifold, Phys.Rev.Lett.101:041602, 2008 (arXiv:0804.1114)

• Jacques Distler, Sunil Mukhi, Constantinos Papageorgakis, Mark Van Raamsdonk, M2-branes on M-folds, JHEP 0805:038,2008 (arXiv:0804.1256)

Review:

• Jonathan Bagger, Neil Lambert, Sunil Mukhi, Constantinos Papageorgakis, 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)

• Neil Lambert, M-Theory and Maximally Supersymmetric Gauge Theories, Annual Review of Nuclear and Particle Science, Vol. 62:285-313 (arXiv:1203.4244, doi:10.1146/annurev-nucl-102010-130248)

• Neil Lambert, Lessons from M2’s and Hopes for M5’s, Proceedings of the LMS-EPSRC Durham Symposium: Higher Structures in M-Theory 2018 Fortschritte der Physik, 2019 (arXiv:1903.02825, slides pdf, video recording)

Discussion of boundary conditions leading to brane intersection with M-wave, M5-brane and MO9-brane is in

• Chong-Sun Chu, Douglas J. Smith, Multiple Self-Dual Strings on M5-Branes, JHEP 1001:001, 2010 (arXiv:0909.2333)

• David Berman, Malcolm J. Perry, Ergin Sezgin, Daniel C. Thompson, Boundary Conditions for Interacting Membranes, JHEP 1004:025, 2010 (arXiv:0912.3504)

Discussion in Horava-Witten theory reducing M2-branes to heterotic strings is in

• Neil Lambert, Heterotic M2-branes, Physics Letters B

  Volume 749, 7 October 2015, Pages 363-367 (arXiv:1507.07931)

The interpretation of at least most of the “M2-brane 3-algebra” appearing in the membrane literature in terms of plain metric Lie algebras is due to

• Paul de Medeiros, José Figueroa-O'Farrill, Elena Méndez-Escobar, Patricia Ritter, On the Lie-algebraic origin of metric 3-algebras, Commun. Math. Phys. 290: 871-902, 2009 (arXiv:0809.1086)

See also

• José Figueroa-O'Farrill, Triple systems and Lie superalgebras in M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf, pdf)

The suggestion that BGL “3-algebras” are to be interpreted in 2-plectic geometry appears in

• Christian Saemann, Richard Szabo, Quantization of 2-Plectic Manifolds (arXiv:1106.1890)