nLab tensor hierarchy

Idea

In gauged supergravity the gauging of a subgroup of a global U-duality symmetry group (via a choice of embedding tensor) in general requires the existence of a hierarchy of differential p-form-fields to ensure the gauge covariance of the resulting field strengths. This is known as the tensor hierarchy (de Wit-Samtleben 05, deWit-Samtleben 08)

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.

Related concepts

• gauged supergravity

• embedding tensor

• $E L_\infty$-algebra

• adjusted Weil algebra

References

The concept originates with

• Bernard de Wit, Henning Samtleben, Gauged maximal supergravities and hierarchies of nonabelian vector-tensor systems, Fortsch. Phys. 53 (2005) 442-449 (arXiv:hep-th/0501243)

• Bernard de Wit, Henning Samtleben, The end of the $p$-form hierarchy, JHEP 0808:015, 2008 (arXiv:0805.4767)

See also

• Jesper Greitz, Paul Howe, Jakob Palmkvist, The tensor hierarchy simplified, Class. Quantum Grav. 31 (2014) 087001 (arXiv:1308.4972)

Possible relation to U-duality and M-theory:

• Bernard de Wit, Hermann Nicolai, Henning Samtleben, Gauged Supergravities, Tensor Hierarchies, and M-Theory, JHEP 0802:044, 2008 (arXiv:0801.1294)

Relation to Borcherds algebras:

• Jakob Palmkvist, The tensor hierarchy algebra, J. Math. Phys. 55, 011701 (2014) (arXiv:1305.0018)

Relation to Leibniz algebras:

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

A proposal for understanding tensor hierarchies as dg-Lie algebra/L-infinity algebra-refinenents of Leibniz algebras:

• 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)

A proposal for understanding tensor hierarchies via higher gauge theory with adjusted Weil algebras:

• Leron Borsten, Hyungrok Kim, Christian Saemann. $E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies [arXiv:2106.00108]

survey and review:

• Christian Saemann, $E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies, talk at SFT@Cloud 2021 (2021) [pdf, pdf]