The String Coffee Table

I am currently staying at the Erwin-Schrödinger institute in Vienna, attending a workshop on mirror symmetry ($\to$).

In parallel, there is a program here on things related to free differential algebras and gerbes. I am not sure what the precise title of that parallel program is, actually.

But in any case, today I heard an interesting talk by Leonardo Castellani, which is what I more or less talk about in the following.

Originally, I intended to produce a transcript of what Kontsevich and Fukaya talked about this morning. But first I need to find somebody willing and able to decode for me the notes I have taken in these talks. I surely cannot…

1) survey of FDAs, Lie $n$-algebras and $n$-connections and $n$-bundles

Free differential graded algebra (we should really say “free differential graded commutative algebras”, as Ezra Getzler kindly emphasized to me today), FDA for short, are essentially the same ($\to$) as

$\bullet$ semistrict Lie $n$-algebras and Lie $n$-algebroids ($\to$, $\to$)

$\bullet$ $n$-term $L_\infty$ algebras .

More precisely, from $L_\infty$-algebras and Lie $n$-algebras, which come with complexes of vector spaces with lots of graded brackets on them, we obtain free differential graded algebras simply by passing to the dual vector spaces and equipping them with a differential whose nilpotency is equivalent to the (intricate) system of higher Jacobi almost-identities defining the original structure.

This is nice, for two reasons:

1) FDAs are much easier to compute with than Lie $n$-algebras.

2) Lie $n$-algebras have a much clearer conceptual interpretation in higher gauge theory ($\to$) than their mere FDA structure suggests.

So we can pass between the two pictures as convenient. In particular, if we encounter considerations in just one picture, we know how to translate it to the other.

The conceptual understanding of Lie $n$-algebras allows us to easily understand their role in higher gauge theory.

An (integrable) connection on an $n$-bundle ($\to$) is, by definition, something that allows us, consistently, to perform parallel transport ($\to$) over $n$-dimensional volumes.

In other words, it is a morphism

from the $n$-groupoid of $n$-paths in the base space $X$ to the transport $n$-groupoid of the $n$-bundle $T\to X$ with connection ($\to$).

But this setup is easily differentiated. Passing to infinitesimals, $n$-groupoids become $n$-algebroids. Hence, infinitesimally, an (integrable) $n$-connection on an $n$ bundle is a morphism

of the corresponding algebroids ($\to$).

Knowing this, we may pass to the dual FDA description of this situation, and study connections on $n$-bundles in terms of morphisms of FDA algebras (differential graded algebras).

Motivated by the Poisson $\sigma$-model, Thomas Strobl and collaborators have looked at such morphisms ($\to$) from the point of view of gauge theory.

One finds a couple of nice, unifying structures in this context.

i) First of all, one should note that a morphism $d\mathrm{tra} : A \to B$ of $n$-algebroids corresponds to a chain map $(d_B, B^\bullet) \to (d_A,A^\bullet)$ of the corresponding dg-algebras.

ii) Naturally, then, 1-morphisms of $n$-algebroid morphisms correspond to chain homotopies, 2-morphisms to homotopies of homotopies, and so on.

iii) If we look at the double complex $(Q := d_A \pm d_B , A^\bullet \oplus B^\bullet)$, these conditions read as follows:

- a map $\Phi$ of dg-algebras (=FDAs) has to be $Q$-closed $[Q,\Phi] = 0$.

- a map of complexes $\epsilon : B^\bullet \to A^{\bullet-1}$ is a 1-morphisms of maps of dg-algebras, with $\phi' = \phi + [Q,\epsilon]$ (where the bracket is graded, hence now an anticommutator).

- similarly, a map $\epsilon_p : B^\bullet \to A^{\bullet-p}$ is a $p$-morphisms of (the underlying) Lie $n$-algebras, relating two $(p-1)$-morphisms that differ by $[Q,\epsilon_\p]$ ($\to$).

iv) the failure of a morphism $\Phi$ to be a chain map in degree $p$ is, when this map is interpreted as a connection on an $n$-bundle, precisely the $p$-form curvature (for $p \lt n$ also known as “fake curvatures”)

v) Bianchi identites in the gauge theory sense are nothing but $Q^2\Phi = 0$.

vi) Infinitesimal gauge transformations in the gauge theory sense are nothing but exact morphisms $[Q,\epsilon]$.

vii) more generally, infinitesimal transformations are generated by generalized Lie derivatives $\{Q,i_t\}$. These are symmetries of $\Phi$ iff $[L_t,\Phi] := [\{Q,i_t\},\Phi] = 0$.

2) translating Castellani’s paper into Lie $n$-algebra language

We can now, step by step, go through Castellani’s paper hep-th/0508213 and interpret the FDA constructions there in the context of $n$-connections on $n$-bundles.

$\bullet$ equations (2.1) and (2.2) are the dual formulation of a certain semistrict Lie $n$-algebra, which plays the role of the $n$-algebra of the gauge $n$-group. The number $n$ is detrmined, in this paper, by the highest $p$-form degree appearing, as $n = p$.

For $p=1$ we get only 1-forms and the formalism described charged points (section 13.5).

For $p=2$ we get 1- and 2-forms. If the 1-forms are trivial and the 2-form is abelian this described the Kalb-Ramond gerbe connection that the fundamental string couples to (section 13.6)

For $p=3$ we get 1-, 2, and 3-forms. The 3-form of 11D supergravity should be a realization of this (compare section 13.8).

Indeed, that’s the case the Castellani studies in section 3 of his paper.

$\bullet$ We may interpret all the constants appearing there intrinsically. In particular, the coefficients $C^i{}_{A_1 A_2A_2}$, which relate the $p$-forms to the connection 1-form encode nonvanishing Jacobiators (measuring the failure of the Jacobi identity to hold).

$\bullet$ The concept referred to as soft group manifolds in the last paragraph of page 2 is secretly precisely the concept of a map from the dg-algebra characterizing a group to that of the ordinary deRham complex, i.e. a morphisms from $n$-paths to an $n$-group characterizing an $n$-connection.

$\bullet$ Equations (2.6) and (2.7) give the curvatures, which encode the failure of this map to be a chain map.

$\bullet$ Equation (2.16) is a realization of the statement that Lie derivates split into a pure gauge part and a contraction of the curvature

$\bullet$ The closure of the algebra of generalized Lie derivatives, given in the particular example in equations (2.28)-(2.30), is guaranteed by the general structure

3) Castellani’s result

The crucial new result of the paper is given in section 3. With hindsight, given the above considerations, I think can rephrase this main result as follows.

The author notes that there is semistrict Lie $3$-algebra ($\simeq$ 3-term $L_\infty$-algebra $\simeq$ a certain dg-algebra) whose Lie algebra generated by the $[Q,i_v]$ is precisely the super-Poincaré Lie algebra in eleven dimensions, centrally extended by the central charge $Z^{ab}$ corresponding to membrane (“M-branes”).

Moreover, a 3-connection with values in that Lie 3-algebra encodes, locally, precisely the field content of 11D supergravity.

There is one more nice fact, which builds on an older, well known, result, as discussed for instance in

L. Castellani, R. D’Auria & P. Fré
Supergravity and superstrings: a geometric perspective
World Scientific, Singapore (1991),

namely that imposing the condition that the curvature $[Q,\Phi]$ of this 3-connection is horizontal, meaning that it takes values only in the algebra of objects of the gauge $3$-algebra, is equilvalent to the equations of motion of the graviton, the gravitino and the vielbein.

In closing, I just make the following remark:

If the connection $\Phi$ were to come from a strict transport $3$-functor, then all but its highest curvature component would have to vanish, since then it would have to be a chain map.

Apparently, what is going on here is a weakening of this strict condition, where we only require those parts of the fake curvatures to vanish, which lie in the image of certain maps.

This reminds me that I still need to solve the exercise that Larry Breen has given me last month:

Exercise: Redo the study of 2-transport ($\to$), now allowing the transport 2-functors to be weak, and see if that allows the fake curvature to be nonvanishing.

Since I already said so much, I can just as well say the following, too:

When weakening the transport 2-functor, we find all kinds of new freedoms, namely that encoded in some compositor, in the unitor, in the associator and what not.

Using this indeed goes beyond the known fake-flat 2-connections – but in different directions. We get something weaker, but it’s both more general and still more restrictive than the data Breen & Messing found.

So I am inclined to worry about the converse

Exercise: Check if the local connection data found by Breen and Messing can, or cannot, be integrated to finite surface transport.

I am saying this here, because observations along the above lines might help to see what is really going on. It seems that Castellani has, in terms of FDAs, a veryprecise weakening of the morphism condition.

All I’d need to do is to translate the horizonatllity of $[Q,\Phi]$ back to statements about $n$-functors between $n$-groupoids…