The String Coffee Table

Let $$V = \oplus_{n \in \mathbb{Z}} V_n$$ be any $\mathbb{Z}$-graded vector space with $$V_n = 0 \,\,\,\,\,\, \forall n \lt -1,\, n \geq p$$ for some $p \in \mathbb{Z}$, and with a nilpotent linear operator $$\delta : V_n \to V_{n-1}$$ of grade $-1$ such that we have a complex $$0 \to V_{p-1} \overset{\delta}{\to}\cdots \to V_3 \overset{\delta}{\to} V_2 \overset{\delta}{\to} V_1 \overset{\delta}{\to} V_0 \overset{\delta}{\to} V_{-1} \to 0 \,.$$

Then the following structures all encode precisely the same information:

1. a $p$-algebroid structure on $V$
2. a graded differential algebra on $\Lambda V^*$
3. an $L_{\infinity}$-algebra on $V$ (aka a strong(ly) homotopy Lie algebra structure, aka an sh Lie algebra structure on $V$).
4. a semistrict Lie $p$-algebra whose space of $n$-morphisms is $V_n$ (and which would be called a Lie p-algebroid when $V_{-1} \neq 0$).

Here is how and why these are all the same:

The equivalence of 1) and 4) for $p \leq 1$ was the content of the first row of the table that I mentioned in part I, which was taken from HDA6.

In order to see this translate a 1-algebroid $$\array{ A = & \{ & E \to M \,, & \\ & \,& E \overset{\rho}{\to} TM \,, & \\ & \,& [\cdot,\cdot] : \Gamma(E) \times \Gamma(E) \to \Gamma(E) & \} }$$ into an $L_{\infty}$ algebra as follows:

• set $V_0 = \Gamma(E)$
  - set $V_{-1} = \Gamma(TM)$
• set $l_1 = \rho$
• set $$\array{ l_2 : V_0 \times V_0 &\to& V_0 \\ (e_1,e_2) &\mapsto& [e_1,e_2]}$$ where the bracket on the right is that on $\Gamma(E)$, and $$\array{ l_2 : V_0 \times V_{-1} &\to& V_{-1} \\ (e,t) &\mapsto& [\rho(e),t]}\,,$$ where now the bracket on the right is the Lie bracket of vector fields in $\Gamma(TM)$. (Thanks to Thomas and Melchior for pointing this out to me. It is really very obvious, but I was confused about this point for a while.)

The equivalence of $L_\infty$-algebras with semistrict Lie $p$-algebras (which are $p$-fold categorified Lie algebras) is a result of general abstract nonsense using $L_\infty$-operads which is discussed at the beginning of section 4.3 in HDA6. For the special case $p=2$ and $V_{-1} = 0$ the detailed proof for the equivalence of the categories semistrict Lie 2-algebras and $L_{\infinity}$ algebas on $V$ is that of theorem 36 in that paper.

The equivalence between $L_\infty$ algebras on $V$ and differential graded algebras on $\Lambda V^*$ is a consequence of theorem 2.3 in

Tom Lada & Martin Markl
Strongly Homotopy Lie Algebras
hep-th/9406095

which makes use of the theorem on the top of p. 8 in

Tom Lada & Jim Stasheff
Introduction to sh Lie Algebras for Physicists
hep-th/9209099

and which I stated here in the form as for instance given in the remark below definition 5 on p.7 of

G. Barnich, R. Fulp, T. Lada & J. Stasheff
The sh Lie Structure of Poisson Brackets in Field Theory
hep-th/9702176

The idea here is beautifully simple, all the trouble comes from keeping track of the signs: An ordinary Lie algebra has a single bracket $$l_2(\cdot,\cdot) \equiv [\cdot,\cdot] : V \times V \to V \,.$$ For an $L_\infty$-algebra this is generalized to an infinite family of $n$-ary ‘brackets’ $$l_n(\cdots) : \otimes^n V \to V$$ which are of grade $n-2$. These can be extended to coderivations $\hat l_n$ of the algebra $\Lambda V$ regarded as a coalgebra (see the above two papers for the details) such that the defining property of the $L_\infty$-algebra $(V,\{l_n\}_n)$ simply reads $$\sum_{i,j} \hat l_i \circ \hat l_j = 0 \,.$$ This of course means that we have a nilpotent operator $D = \sum_i \hat l_i$ which again defines a nilpotent operator $Q$ on $\Lambda V^*$ (I am glossing over sign issues related to redefining the grading by a shift here and there) given by $$Q \omega(v_1,v_2,\dots,v_n) = \sum_i \omega(\hat l_i(v_1,v_2,\dots,v_n)) \,.$$ For the algebroid case where $V_{-1} \neq 0$ one has to interpret $\omega(l_1(v_1\in V_0),\cdots))$ as $\rho(v)(\omega(\cdots))$. Then one reobtains for $p=1$ the usual dual formulation of the definition of a (1-)algebroid (which I mentioned in part I)

It is noteworthy that what is called the Courant algebroid is a special case of a Lie 2-algebroid $$0 \to V_1 \overset{\delta}{\to} V_0 \overset{\delta}{\to} V_{-1} \to 0 \,.$$

This is discussed in section 2.4 of the thesis

Dmitry Roytenberg
Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds
math.DG/9910078

which I mentioned last time.

As Roytenberg points out on the top of p. 17 of the above thesis, the family of Courant algebroids over a point, i.e. of those with $V_{-1} = 0$, is the same as the family of semistrict Lie $2$-algebras called $\mathfrak{g}_k$ as defined by Baez and Crans in HDA6. As I had mentioned these could recently been shown to be equivalent to infinite dimensional Lie $2$-algebras which are, for the case that $\mathfrak{g} = \mathrm{Lie}(\mathrm{Spin}(n))$ related to the group $\mathrm{String}(n)$. See for instance John Baez’s Irvine Algebra Seminar talk on this.

Thomas Strobl emphasizes the fact that the description of Lie $p$-algebroids in terms of differential graded algebras on $\Lambda V^*$ with differential $Q$ is from many points of views the most elegant and convenient one. It leaves us with just a complex $$0 \to \Lambda^0 V^* \overset{Q}{\to} \Lambda^1 V^* \overset{Q}{\to} \Lambda^2 V^* \overset{Q}{\to} \cdots$$ and all the rather dodgy relations of an $L_{\infty}$ are encoded in $$Q^2 = 0 \,.$$ Also, $p$-algebroid morphisms in this language simply become chain maps between these complexes and 2-morphisms become chain homotopies between these. This gives an extremely elegant language to talk about equations of motion and gauge transformation of the Poisson Sigma Model (PSM) and related systems. This is described in detail in

Martin Bojowald, Alexeij Kotov & Thomas Strobl
Lie Algebroid Morphisms, Poisson Sigma Models, and Off-Shell Closed Gauge Symmetries
math.DG/0406445

It turns out that in general the equations of motion of these $\sigma$-models specify morphisms between Lie ($p$-)algebroids. For the Courant algebroid ($p=2$) this is discussed in

Alexei Kotov, Peter Schaller & Thomas Strobl
Dirac Sigma Models
hep-th/0411112

This formalism suggests a way how to formulate Yang-Mills-like theories using algebroids instead of algebras, which is discussed in

Thomas Strobl
Algebroid Yang-Mills Theories
hep-th/0406215

For $2$-algebroids these contain (possibly nonabelian) $2$-form fields. One very attractive aspect of this approach is that using the language of differential graded algebras the treatment of gauge transformations and invariances of action functionals for such theories, which has been a source of trouble before, becomes much more transparent.

When taking algebroids over a point these $p$-algebroid YM theories reduce to Lie $p$-algebra YM-like theories. It seems to me that the algebroid case would correspond to what should be obtained by replacing the structure 2-group in 2-bundles with 2-connection by a (weak) 2-groupoid, but that needs more thinking.

What is however more or less clear already is the relation of algebroids to abelian $(p-1)$-gerbes and hence to abelian $p$-bundles. This is discussed in section 3.8 of

Marco Gualtieri
Generalized Complex Geometry
math.DG0401221

Given any base manifold $M$ there are Lie $p$-algebroids coming from the bundle $$TM \oplus \Lambda^{(p-1)} T^*M$$ and these describe abelian $p$-gerbes.

For $p=1$ we have an ordinary Lie (1-)algebroid $TM\oplus 1 \overset{\rho}{\to} M$ which characterizes a (possibly twisted) $U(1)$ principal (1-)bundle (= possibly twisted 0-gerbe) over $M$.

For $p=2$ we have the Courant algebroid (a 2-algebroid) given by $TM\oplus T^*M \overset{\rho}{\to} M$ which characterizes a (possibly twisted) $U(1)$-2-bundles ($U(1)$-(1-)gerbe) over $M$.

This algebroid is at the heart of Hitchin’s generalized geometry and it knows all about the Kalb-Ramond $B$-field in string theory. For instance in

Anton Alekseev & Thomas Strobl
Current Algebras and Differential Geometry
hep-th/0410183

it is shown how a Courant algebroid can be reconstructed from knowledge of the current algebra of $2D-\sigma$-models with $2$-form backgrounds.

Abelian 3-bundles/2-gerbes (which should describe the supergravity 3-form like abelian 1-gerbes describe the KR 2-form) are related to 3-algebroids $$TM \oplus \Lambda^2 T^*M \to M \,,$$ and so on. (Which answers Luboš’s remark at the very end of his entry on generalized geometry.)