nLab M-brane 3-algebra

Idea

In the construction of the Lagrangian density for the supersymmetric worldvolume gauge quantum field theory on coincident M2-branes (the BLG model/ABJM model, a conformal super-Chern-Simons theory coupled to matter-fields) a certain algebraic structure appears – and at least in the BLG model it appears at a pivotal stage in the proof of supersymmetry – which is given by a trilinear map on an inner product space $V$

$$V \otimes V \otimes V \overset{ \overset{ \color{blue} \;\; { \text{3-bracket} \atop \phantom{-} } \;\; }{ [-,-,-] } }{\longrightarrow} V$$

that is subject to some axioms which are different from but clearly reminiscent to the axioms on the Lie bracket of a Lie algebra, (which is of course a bilinear map).

In special cases, but not generally, this algebraic structure is a special case of a structure introduced by Filippov, which could be called a Filippov algebra, but which mostly came to alternatively be called a 3-Lie algebra or a Lie 3-algebra or often just a 3-algebra.

Unfortunately, in homotopy theory and higher algebra, these terms refer to something different, namely to L-infinity algebras or categorifications of associative algebras to n-algebras – but the structure seen in the BLG model on M2-branes is none of these.

Hence to be on the safe side, one may want to speak of M2-brane 3-algebras or maybe M-brane algebras, for definiteness

Moreover, while this “M2-brane 3-algebra” appeared to be a central ingredient in the BLG model, it makes no appearance in the ABJM model, which however generalizes and hence subsumes the BLG model, up to slight re-identifications of terms.

Indeed, inspection reveals that “M2-brane 3-algebras” give rise to and may be re-constructed from data in ordinary Lie theory, namely from pairs consisting of a metric Lie algebra and a dualizable Lie algebra representation.

$$\big\{ \text{M2-brane 3-algebras} \big\} \;\;\overset{\simeq}{\leftrightarrow} \left\{ { \text{faithful orthogonal representations} \atop \text{of metric Lie algebras} } \right\}$$

This is due to dMFFMER 08, Theorem 11, recalled as Prop. below.

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Definition

Definition

(M2-brane 3-algebra)

An M2-brane 3-algebra is

1. a real finite-dimensional vector space $V$;

2. a metric $\langle -,-\rangle$ on $V$, hence a non-degenerate inner product space (not required to be positive definite);

3. a trilinear function, the 3-bracket

   $$V \otimes V \otimes V \overset{[-,-,-]}{\longrightarrow} V$$

such that the following three axioms hold, for all elements $w,x,y,z \in V$:

1. (unitarity)

   $$\big\langle [x,y,z],w \big\rangle \;=\; -\big\langle z, [x,y,w]\big\rangle$$

2. (symmetry)

   $$\big\langle [x,y,z],w \big\rangle \;=\; -\big\langle [z,w,x], y \big\rangle$$

3. (fundamental identity)

   $$\begin{aligned} \big[x,y, [v,w,z] \big] & = \phantom{+}\; \big[ [x,y,v], w, z \big] \\ & \phantom{=\;} + \big[ v, [x,y, w] , z \big] \\ & \phantom{=\;} + \big[ v, w , [x,y, z ] \big] \end{aligned}$$

In this generality, and under the name “generalized metric Lie 3-algebras”, this is due to Cherkis-Saemann 08, 41, see dMFFMER 08, Def. 1

Properties

Equivalence to metric Lie representations

For the following, let the ground field be the real numbers and take all vector spaces involved to be real and finite-dimensional.

For definiteness, we make the following explicit:

Definition

(metric Lie representation)

An orthogonal representation of a metric Lie algebra is

1. a metric Lie algebra

   $$\big((\mathfrak{g},[-,-]), g \big)$$

2. a metric vector space

   $$(V,k) \,,$$

   hence a non-degenerate real inner product space;

3. an orthogonal Lie algebra representation

   $$\mathfrak{g} \otimes V \overset{\rho}{\to} V$$

of $\mathfrak{g}$ on $(V,k)$.

Remark

(linear duals and adjuncts/transposition)

Given an orthogonal representation of a metric Lie algebra as in Def. , the non-degenerate inner products $g$ on $\mathfrak{g}$ and $k$ on $V$ exhibit these as self-dual objects in the monoidal category of finite-dimensional vector spaces (with respect to the tensor product of vector spaces).

This means that by passsing to adjuncts we may regard linear maps of the form

$$V \otimes W \overset{f}{\longrightarrow} Z$$

equivalently as linear maps of the form

$$W \overset{\overline{f}}{\longrightarrow} Z \otimes V^\ast \,,$$

where $V^\ast$ is the dual vector space; and similarly for $\mathfrak{g}$.

Once we choose a linear basis $\{v_i\}_{i \in \{1, \cdots, dim(V)\}}$ of $V$, with respect to which the metric tensor $k$ has components

(1)$$k_{i j} \;\coloneqq\; k(v_i, v_j)$$

this forming of adjuncts is equivalently the yoga of “raising and lowering of indices via the metric”. Similarly for a choice of linear basis $\{t_a\}_{a \in \{1,\cdots, dim(\mathfrak{g})\}}$ for the Lie algebra, and the induced components

(2)$$g_{a b} \;\coloneqq\; g(t_a, t_b) \,.$$

Specifically, given the basis component $(\rho_a{}^{i}{}_j)_{a,i,j}$ of the Lie action $\mathfrak{g} \otimes V \overset{\rho}{\longrightarrow} V$, defined by

(3)$$\rho(t_a, v_i) \;\coloneqq\; \rho_a{}^{i}{}_j v_i$$

we get the components $(\tilde \rho^a{}_{i j})_{a,i,j}$ of the adjunct

(4)$$V \otimes V \overset{ \tilde \rho }{\longrightarrow} \mathfrak{g}$$

by contracting the original component (3) with the components (1) (2) of the metric tensors:

(5)$$(\tilde \rho^a{}_{i j})_{a,i,j} \;=\; g_{l j} k^{a b} \, \rho_b{}^l{}_i$$

Definition

(trilinear map induced from metric Lie representation – Faulkner construction)

Given an orthogonal representation $\mathfrak{g} \otimes V \overset{\rho}{\to} V$ of a metric Lie algebra as in Def. , define

1. a bilinear map

   $$V \otimes V \overset{ \;\;D\;\; }{\longrightarrow} \mathfrak{g}$$

   given by

   (6)$$(X, D(x,y)) \;=\; \langle \rho(X,x), y \rangle \;\;\;\; \mathrlap{ \text{for all}\; X \in \mathfrak{g}\,, x, y \in V }$$

2. a trilinear map

   (7)$$V \otimes V \otimes V \overset{ [-,-,-]_{\rho} \coloneqq \rho(D(-,-),-) }{\longrightarrow} V \,,$$

   hence

   $$[x, y, z]_\rho \;\coloneqq\; \rho\big( D(x, y), z \big)$$

In this form and with this notation this is dMFFMER 08, equation (9) and over Prop. 10 (except that we write “$\rho(X,x)$” for their “$X \cdot x$”). The construction itself, up to one dualization of $V$, was introduced in Faulkner 73.

Remark

(component expression of trilinear bracket)

After a choice of linear bases as in Remark , in terms of which Lie algebra elements $X \in \mathfrak{g}$ are expanded as $X \coloneqq X^a t_a$ and representation vectors $x \in V$ are expanded as $x \coloneqq x^i v_i$, the defining equation (6)

$$(X, D(x,y)) \;=\; \langle \rho(X,x), y \rangle \;\;\;\; \text{for all arguments}$$

reads

$$k_{a b} X^a \, D_{i j}{}^b \, x^i y^j \;=\; g_{l j} \, \rho_a{}^l{}_i \, X^a x^i y^j \;\;\;\; \text{for all components}$$

(where here and in the following we use the Einstein summation convention), hence reads

$$k_{a b} D_{i j}{}^b \;=\; g_{l j} \rho_a{}^l{}_i \;\;\;\; \text{for all indices}$$

hence equivalently reads

$$D_{i j}{}^a \;=\; g_{l j} k^{a b} \, \rho_b{}^l{}_i \;\;\;\; \text{for all indices}$$

hence says that the tensor $D$ is equal to the adjunct (4) of the Lie action tensor $\rho$, given in components by the evident “raising and lowering of indices via the metrics” as in (5):

$$D_{i j}{}^a \;=\; \rho^a{}_{i j} \;\;\;\; \text{for all indices} \,.$$

With this, the induced trilinear bracket (7)

$$V \otimes V \otimes V \overset{ [-,-,-]_{\rho} \coloneqq \rho(D(-,-),-) }{\longrightarrow} V$$

is given in components as

$$\begin{aligned} [v_i, v_j, v_k]_\rho & =\; \rho(D(v_i, v_j), v_k) \\ & =\; \rho( D_{i j}{}^a t_a , v_k ) \\ & = \rho_{a}{}^m{}_k \underset{ = g_{l j} k^{a b} \, \rho_b{}^l{}_i }{ \underbrace{D_{i j}{}^a} } v_m \\ & = \rho_{a}{}^m{}_k g_{l j} k^{a b} \, \rho_b{}^l{}_i v_m \\ & = \big( \rho_{a}{}^m{}_k \rho^a{}_j{}_i \big) v_m \end{aligned} \,.$$

Proposition

(M2-brane 3-algebras are equivalent to metric Lie representations)

Given an orthogonal representation $\mathfrak{g} \otimes V \overset{\rho}{\to} V$ of a metric Lie algebra as in Def. , its induced trilinear map $[-,-,-]_\rho$ (Def. ) satisfies the axioms of an M2-brane 3-algebra (Def. ).

Hence this assignment defines a function from isomorphism classes of orthogonal representations of metric Lie algebras to isomorphism classes of M2-brane 3-algebras

$$\left\{ { {\text{orthogonal representations}} \atop {\text{metric Lie algebras}} } \right\}_{/\sim} \underoverset {} { \;\;\;\; \rho \;\mapsto\; [-,-,-]_\rho \;\;\;\; }{\longrightarrow} \left\{ { {\text{M2-brane}} \atop {\text{3-algebras}} } \right\}_{/\sim} \,.$$

Moreover, restricted to faithful representations, this function is a bijection:

$$\left\{ { { {\text{faithful}} \atop {\text{orthogonal representations}} } \atop {\text{metric Lie algebras}} } \right\}_{/\sim} \underoverset {\simeq} { \;\;\;\; \rho \;\mapsto\; [-,-,-]_\rho \;\;\;\; }{\longrightarrow} \left\{ { {\text{M2-brane}} \atop {\text{3-algebras}} } \right\}_{/\sim} \,.$$

This is dMFFMER 08, Prop. 10 and Theorem 11.

Relation to Lie algebra weight systems on chord diagrams

In Penrose notation (string diagram-notation), the trilinear bracket induced by a metric Lie representation according to Def. looks as follows:

With Prop. this shows that M2-brane 3-algebras are equivalently the datum that Lie algebra weight systems on (horizontal) chord diagrams assign to a single chord.

from Sati-Schreiber 19c

References

Appearance in M2-brane theory

The idea that some higher-arity version of the Lie bracket may be relevant for M2-M5 brane intersections originates with attempts of generalizing Nahm's equations for fuzzy funnels of D2-D4 brane intersections in

• 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)

Motivated by this, the M2-brane 3-algebra appears in the construction of the BLG model for the worldvolume quantum field theory on 2 coincident M2-branes in

• 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)

further highlighted as such in

• Andreas Gustavsson, Algebraic structures on parallel M2-branes Nucl. Phys. B811, 66-76 (2009) (arXiv:0709.1260)

(whence the “BLG” of the BLG model)

and further explored in

• Paul de Medeiros, José Figueroa-O'Farrill, Elena Méndez-Escobar, Metric Lie 3-algebras in Bagger-Lambert theory, JHEP 0808 : 045, 2008 (arXiv:0806.3242)

From here on a myriad of references followed up. Review includes:

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Equivalence to metric Lie representations

The full generalized axioms on the M2-brane 3-algebra and first insights into their relation to Lie algebra representations of metric Lie algebras is due to

• Sergey Cherkis, Christian Saemann, Section 4 of: Multiple M2-branes and Generalized 3-Lie algebras, Phys. Rev. D78:066019, 2008 (arXiv:0807.0808)

The full identification of M2-brane 3-algebras with dualizable Lie algebra representations over 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)

reviewed in

• José Figueroa-O'Farrill, slide 145 onwards in: Triple systems and Lie superalgebras in M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf, pdf)

further explored in

• José Figueroa-O'Farrill, Simplicity in the Faulkner construction, Journal of Physics A: Mathematical and Theoretical, Volume 42, Number 44 (arXiv:0905.4900)

and putting to use the Faulkner construction that was previously introduced in

• John Faulkner, On the geometry of inner ideals, Journal of Algebra, Volume 26, Issue 1, July 1973, Pages 1-9 (doi:10.1016/0021-8693(73)90032-X)

See also:

• Sam Palmer, Christian Saemann, section 2 of M-brane Models from Non-Abelian Gerbes, JHEP 1207:010, 2012 (arXiv:1203.5757)

• Patricia Ritter, Christian Saemann, section 2.5 of Lie 2-algebra models, JHEP 04 (2014) 066 (arXiv:1308.4892)

• Christian Saemann, appendix A of Lectures on Higher Structures in M-Theory (arXiv:1609.09815)

and

• Jakob Palmkvist, Three-algebras, triple systems and 3-graded Lie superalgebras, J. Phys. A43:015205, 2010 (arXiv:0905.2468)