nLab embedding tensor

Idea

In the context of gauging of U-duality-symmetry groups of supergravity-theories to gauged supergravity, the embedding tensor (Nicolai-Samtleben 00) is the datum that specifies which subgroup of the global U-duality is promoted to a gauge group. The requirement of supersymmetry and of consistency then implies conditions on this choice, called the “linear constraint” and the “quadratic constraint”.

Formalized in terms of Lie theory (Lavau 17) these conditions say that an embedding tensor is a homomorphism of Leibniz algebras from a Lie module to the underlying Lie algebra (the “quadratic constaint”) where the Leibniz-product on the module is given by the Lie action induced by that homomorphism itself (the “linear constraint”).

Any choice of embedding tensor for a gauged supergravity is supposed to induce a tensor hierarchy (de Wit-Samtleben 05, 08) of higher degree differential form-fields which jointly serve as ever higher order corrections to the resulting gauge-covariance of the field strengths. This tensor hierarchy may be understood as a dg-Lie algebra/L-∞ algebra-structure which lifts the Leibniz algebra-structure implied/induced by the embedding tensor (Lavau 17, Lavau-Palmkvist 19, Lavau-Stasheff 19).

Definition

Embedding tensors and tensor hierarchy in super/dg-Lie theory

We spell out aspects of the formalization of the concept of embedding tensors and their tensor hierarchies in terms of super Lie algebras/dg-Lie algebras.

The super Lie algebra of multi-endomorphisms

The algebra of embedding tensors and their tensor hierarchies turns out to be neatly captured by structure found in or induced from the following super Lie algebra.

The following construction is briefly highlighted in Palmkvist 09, 2.3 Palmkvist 13, 3.1 (reviewed more clearly in Lavau-Palmkvist 19, 2.4) where it is attributed to Kantor 70:

Definition

(super Lie algebra of multi-endomorphisms)

Let $V$ be a finite-dimensional vector space over some ground field $k$.

Define a $\mathbb{Z}$-graded vector space

$$MultEnd(V) \;\in \; Vect_k^{\mathbb{Z}} \,,$$

concentrated in degrees $\leq 1$, recursively as follows:

For $n =1$ we set

$$MultEnd(V)_{1} \;\coloneqq\; V \,.$$

For $n \leq 0 \in \mathbb{Z}$, the component space in degree $n-1$ is taken to be the vector space of linear maps from $V$ to the component space in degree $n$:

$$MultEnd(V)_{n-1} \;\coloneqq\; Hom_k( V, MultEnd(V)_n ) \,.$$

Hence:

(1)$$\begin{aligned} MultEnd(V)_1 & = V \\ MultEnd(V)_0 & = Hom_k(V,V) = \mathfrak{gl}(V) \\ MultEnd(V)_{-1} & = Hom_k(V, Hom_k(V,V)) \simeq Hom_k(V \otimes V, V) \\ MultEnd(V)_{-2} & = Hom_k(V, Hom_k(V, Hom_k(V,V))) \simeq Hom_k(V \otimes V \otimes V, V) \\ \vdots \end{aligned}$$

Consider then the direct sum of these component spaces as a super vector space with the even number/odd number-degrees being in super-even/super-odd degree, respectively.

On this super vector space consider a super Lie bracket defined recusively as follows:

For all $v_1, v_2 \in MultEnd(V)_1 = V$ we set

$$[v_1, v_2] \;=\; 0 \,.$$

For $f \in MultEnd(V)_{n \leq 0}$ and $v \in MultEnd(V)_1 = V$ we set

(2)$$[f, v] \;\coloneqq\; f(v)$$

Finally, for $f\in MultEnd(V)_{ deg(f) \leq 0 }$ and $g\in MultEnd(V)_{deg(g) \leq 0}$ we set

(3)$$\begin{aligned} [f, g] & \colon\; v \;\mapsto\; [f, g(v)] - (-1)^{ deg(f) deg(g) } [ g, f(v) ] \\ \end{aligned}$$

Remark

By (2) the definition (3) is equivalent to

$$[ [f,g],v ] \;=\; [f, [g,v] ] - (-1)^{ deg(f) deg(g) } [ g, [f,v] ]$$

Hence (3) is already implied by (2) if the bracket is to satisfy the super Jacobi identity

It remains to show that:

Proposition

Def. indeed gives a super Lie algebra in that the bracket (3) satisfies the super Jacobi identity.

Proof

We proceed by induction:

By Remark we have that the super Jacobi identity holds for all triples $f_1, f_2, f_3 \in MultEnd(V)$ with $deg(f_3) \geq 0$.

Now assume that the super Jacobi identity has been shown for triples $(f_1, f_2, f_3(v))$ and $(f_1, f_3, f_2(v))$, for any $v \in V$. The following computation shows that then it holds for $(f_1, f_2, f_3)$:

$$\begin{aligned} [f_1, [f_2, f_3] ] (v) & = [ f_1, [f_2, f_3](v) ] - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} [ [ f_2, f_3 ], f_1(v) ] \\ & = [ f_1, [ f_2, f_3(v) ] ] \\ & \phantom{=} - (-1)^{deg(f_2)deg(f_3)} [ f_1, [ f_3, f_2(v) ] ] \\ & \phantom{=} - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} [ f_2, [ f_3, f_1(v) ] ] \\ & \phantom{=} + (-1)^{deg(f_1)(deg(f_2) + deg(f_3)) + deg(f_2)deg(f_3)} [ f_3, [ f_2, f_1(v) ] ] \\ & = [ f_1, [ f_2, f_3(v) ] ] - (-1)^{deg(f_1) deg(f_2)} { \color{green} [ f_2, [ f_1, f_3(v) ] ] } \\ & \phantom{=} - (-1)^{deg(f_2) deg(f_3)} \big( [ f_1, [ f_3, f_2(v) ] ] - (-1)^{deg(f_1) deg(f_3)} { \color{orange} [ f_3, [ f_1, f_2(v) ] ] } \big) \\ & \phantom{=} - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} \big( [ f_2, [ f_3, f_1(v) ] ] - (-1)^{deg(f_1)deg(f_3)} { \color{blue} [ f_2, [ f_1, f_3(v) ] ] } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_2 ) + deg(f_1) deg(f_3) + deg(f_2) deg(f_3)} \big( [ f_3, [ f_2, f_1(v) ] ] - (-1)^{deg(f_1) deg(f_2)} { \color{cyan} [ f_3, [ f_1, f_2(c) ] ] } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_2)} \big( \underset{ = 0 }{ \underbrace{ + { \color{green} [ f_2, [ f_1, f_3(v) ] ] } - { \color{blue} [ f_2, [ f_1, f_3(v) ] ] } } } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_3) + deg(f_2) deg(f_3)} \big( \underset{ = 0 }{ \underbrace{ { \color{orange} [ f_3, [ f_1, f_2(v) ] ] } - { \color{cyan} [ f_3, [ f_1, f_2(v) ] ] } } } \big) \\ & = \big[ [f_1, f_2], f_3(v) \big] \\ & \phantom{=} - (-1)^{ deg(f_2) deg(f_3) } \big[ [f_1, f_3], f_2(c) \big] \\ & \phantom{=} + (-1)^{ deg(f_1) deg(f_2) } \big[ f_2, [f_1, f_3](v) \big] \\ & \phantom{=} - (-1)^{ deg(f_3)( deg(f_1) + deg(f_2) ) } \big[ f_3, [f_1, f_2](v) \big] \\ & = \big[ [f_1, f_2], f_3 \big](v) + (-1)^{deg(f_1)deg(f_2))} \big[ f_2, [f_1, f_3] \big](v) \end{aligned}$$

(Fine, but is this sufficient to induct over the full range of all three degrees?)

Example

For $f,g \in MultEnd(V)_0 = Hom_k(V,V)$ (1) we have that the bracket on $MultEnd(V)$ in Def. restricts to

$$[f,g](v) \;=\; [f,g(v)] - [g,f(v)] \;=\; f(g(v)) - g(f(v))$$

(by combining (3) with (2)).

This is the Lie bracket of the general linear Lie algebra $\mathfrak{gl}(V)$, as indicated on the right in (1).

Embedding tensors

Definition

(embedding tensor)

Given

• $\mathfrak{g}$ a Lie algebra,

• $\mathfrak{g} \otimes V \overset{\rho}{\longrightarrow} V$ a Lie algebra representation of $\mathfrak{g}$

  (typically required to be a faithful representation, see Remark below),

then an embedding tensor is a linear map

$$\Theta \;\colon\; V \longrightarrow \mathfrak{g}$$

such that for all $v_i \in V$ the following condition (“quadratic constraint”) is satisfied:

(4)$$[\Theta(v_1), \Theta(v_2)] \;=\; \Theta \big( \rho_{\Theta(v_1)}(v_2) \big) \,,$$

where on the left we have the Lie bracket of $\mathfrak{g}$.

The idea of this definition goes back to Nicolai-Samtleben 00, with many followups in the literature on tensor hierarchies in gauged supergravity. The above mathematical formulation is due to Lavau 17.

Remark

(Leibniz algebra-structure)

The “quadratic constraint” (4) implies (see this Prop.) that the product

(5)$$\array{ V \otimes V &\overset{ }{\longrightarrow}& V \\ (v_1, v_2) &\mapsto& v_1 \cdot v_2 \mathrlap{ \;\coloneqq\; \rho_{\Theta(v_1)}(v_2) } }$$

makes (the underlying vector space of) $V$ a Leibniz algebra. Conversely, if a Leibniz algebra structure “$\cdot$” on $V$ is already given, we may ask that it coincides with this one induced from the embedding tensor, a condition then called the linear constraint:

(6)$$v_1 \cdot v_2 \;=\; \rho_{\Theta(v_1)}(v_2) \,.$$

With respect to this induced Leibniz algebra structure, hence equivalently with the “linear constraint” (6) understood, the “quadratic constraint” (4) equivalently says that the embedding tensor is a homomorphism of Leibniz algebras (using that Lie algebras are special cases of a Leibniz algebras):

$$[\Theta(v_1), \Theta(v_2)] \;=\; \Theta(v_1 \cdot v_2) \,.$$

Proposition

(embedding tensors are square-0 elements in $MultEnd(V)$)

Let $k$ be a ground field of characteristic zero.

An element in degree -1 of the super Lie algebra $MultEnd(V)$ from Def. ,

$$\Theta \in MultEnd(V)_{-1} \simeq Hom_{k}(V, \mathfrak{gl}(V)) \,,$$

which by Example is identified with a linear map

$$\Theta \;\colon\; V \longrightarrow \mathfrak{g} \coloneqq \mathfrak{gl}(V)$$

from $V$ to the general linear Lie algebra on $V$, is square-0 precisely if it is an embedding tensor (Def. ), in that:

$$[\Theta, \Theta] \;=\; 0 \phantom{AAA} \Leftrightarrow \phantom{AAA} [\Theta(v_1), \Theta(v_2) ] \;=\; \Theta( \rho_{\Theta(v_1)}(v_2) ) \,.$$

Here on the right, $[-,-]$ denotes the Lie bracket in $\mathfrak{gl}(V)$, while $\rho$ denotes the canonical Lie algebra action of $\mathfrak{gl}(V)$ on $V$.

Proof

By unwinding of the definition (2) and (3) and using again Example we compute as follows:

$$\begin{aligned} \big( \tfrac{1}{2} [\Theta,\Theta](v_1) \big)(v_2) & = [\Theta, \Theta(v_1)](v_2) \\ & = [\Theta, \underset{ \mathclap{ = \rho_{\Theta(v_1)}(v_2) } } { \underbrace{ (\Theta(v_1))(v_2) } } ] - [\Theta(v_1), \Theta(v_2)] \\ & = \Theta( \rho_{\Theta(v_1)}(v_2) ) - [ \Theta(v_1), \Theta(v_2) ] \end{aligned}$$

Tensor hierarchies

Remark

(embedding tensors induce tensor hierarchies)

In view of the relation between super Lie algebras and dg-Lie algebras (above), Prop. says that every choice of an embedding tensor for a faithful representation on a vector space $V$ induces a dg-Lie algebra $(MultEnd(V), [-,-], \partial \coloneqq [\Theta, -])$.

According to Palmkvist 13, 3.1, Lavau-Palmkvist 19, 2.4 this dg-Lie algebra (or some extension of some sub-algebra of it) is the tensor hierarchy associated with the embedding tensor.

Relation with double and exceptional field theory

See Hohm-Samtleben 19 for a review.

Related concepts

• gauged supergravity

• tensor hierarchy

• double field theory, exceptional field theory

References

The concept in gauged supergravity originates with

• Hermann Nicolai, Henning Samtleben, Maximal gauged supergravity in three dimensions, Phys. Rev. Lett. 86 (2001) 1686-1689 (arXiv:hep-th/0010076)

For further references see at tensor hierarchy.

The mathematical formulation above is due to

• Sylvain Lavau, Tensor hierarchies and Leibniz algebras, J. Geom. Phys. 144:147-189 (2019) (arXiv:1708.07068)

with further discussion in:

• Sylvain Lavau, Jakob Palmkvist, Infinity-enhancing of Leibniz algebras (arXiv:1907.05752)

• Sylvain Lavau, Jim Stasheff, $L_\infty$-algebra extensions of Leibniz algebras (arXiv:2003.07838)

Review of the relation with double field theory and exceptional field theory:

• Edvard Musaev, U-dualities in Type II string theories and M-theory, Symmetry 2019, 11(8), 993 (arXiv:1311.3331, doi:10.3390/sym11080993)

• Henning Samtleben, Olaf Hohm, Higher Gauge Structures in Double and Exceptional Field Theory, Contribution to Proceedings of Higher Structures in M-Theory 2018 (arXiv:1903.02821, doi:10.1002/prop.201910008)