nLab
M-brane 3-algebra

Contents

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

String theory

Contents

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 VV

VVV[,,]3-bracketV 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.

{M2-brane 3-algebras}{faithful orthogonal representationsof metric Lie algebras} \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.

(…)

Definition

Definition

(M2-brane 3-algebra)

An M2-brane 3-algebra is

  1. a real finite-dimensional vector space VV;

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

  3. a trilinear function, the 3-bracket

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

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

  1. (unitarity)

    [x,y,z],w=z,[x,y,w] \big\langle [x,y,z],w \big\rangle \;=\; -\big\langle z, [x,y,w]\big\rangle
  2. (symmetry)

    [x,y,z],w=[z,w,x],y \big\langle [x,y,z],w \big\rangle \;=\; -\big\langle [z,w,x], y \big\rangle
  3. (fundamental identity)

    [x,y,[v,w,z]] =+[[x,y,v],w,z] =+[v,[x,y,w],z] =+[v,w,[x,y,z]] \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

    ((𝔤,[,]),g) \big((\mathfrak{g},[-,-]), g \big)
  2. a metric vector space

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

    hence a non-degenerate real inner product space;

  3. an orthogonal Lie algebra representation

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

of 𝔤\mathfrak{g} on (V,k)(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 gg on 𝔤\mathfrak{g} and kk on VV 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

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

equivalently as linear maps of the form

Wf¯ZV *, W \overset{\overline{f}}{\longrightarrow} Z \otimes V^\ast \,,

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

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

(1)k ijk(v i,v j) 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{1,,dim(𝔤)}\{t_a\}_{a \in \{1,\cdots, dim(\mathfrak{g})\}} for the Lie algebra, and the induced components

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

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

(3)ρ(t a,v i)ρ a i jv i \rho(t_a, v_i) \;\coloneqq\; \rho_a{}^{i}{}_j v_i

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

(4)VVρ˜𝔤 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)(ρ˜ a ij) a,i,j=g ljk abρ b l i (\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 representationFaulkner construction)

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

  1. a bilinear map

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

    given by

    (6)(X,D(x,y))=ρ(X,x),yfor allX𝔤,x,yV (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)VVV[,,] ρρ(D(,),)V, V \otimes V \otimes V \overset{ [-,-,-]_{\rho} \coloneqq \rho(D(-,-),-) }{\longrightarrow} V \,,

    hence

    [x,y,z] ρρ(D(x,y),z) [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 “ρ(X,x)\rho(X,x)” for their “XxX \cdot x”). The construction itself, up to one dualization of VV, 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 X𝔤X \in \mathfrak{g} are expanded as XX at aX \coloneqq X^a t_a and representation vectors xVx \in V are expanded as xx iv ix \coloneqq x^i v_i, the defining equation (6)

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

reads

k abX aD ij bx iy j=g ljρ a l iX ax iy jfor all components 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 abD ij b=g ljρ a l ifor all indices k_{a b} D_{i j}{}^b \;=\; g_{l j} \rho_a{}^l{}_i \;\;\;\; \text{for all indices}

hence equivalently reads

D ij a=g ljk abρ b l ifor all indices D_{i j}{}^a \;=\; g_{l j} k^{a b} \, \rho_b{}^l{}_i \;\;\;\; \text{for all indices}

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

D ij a=ρ a ijfor all indices. D_{i j}{}^a \;=\; \rho^a{}_{i j} \;\;\;\; \text{for all indices} \,.

With this, the induced trilinear bracket (7)

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

is given in components as

[v i,v j,v k] ρ =ρ(D(v i,v j),v k) =ρ(D ij at a,v k) =ρ a m kD ij a=g ljk abρ b l iv m =ρ a m kg ljk abρ b l iv m =(ρ a m kρ a j i)v m. \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 𝔤VρV\mathfrak{g} \otimes V \overset{\rho}{\to} V of a metric Lie algebra as in Def. , its induced trilinear map [,,] ρ[-,-,-]_\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

{orthogonal representationsmetric Lie algebras} /ρ[,,] ρ{M2-brane3-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:

{faithfulorthogonal representationsmetric Lie algebras} /ρ[,,] ρ{M2-brane3-algebras} /. \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

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

further highlighted as such in

(whence the “BLG” of the BLG model)

and further explored in

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

(…)

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

The full identification of M2-brane 3-algebras with dualizable Lie algebra representations over metric Lie algebras is due to

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

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

See also:

Last revised on January 29, 2020 at 03:58:22. See the history of this page for a list of all contributions to it.