### The G and the B

#### Posted by Urs Schreiber

We have a pretty good understanding of what the “$B$-field” in string theory really is, in terms of arrow-theory.

This nicely explains a bunch of things. I used to be very annoyed with myself, though, for not understanding, on this nice abstract level, one of the more intriguing aspects of the $B$-field:

beyond its mere nature as being the 2-categorical version of a line bundle with connection, it turns out that the connection 2-form $B$ here unifies in an intriguing way with a *Riemannian metric* (and with a dilaton field, in fact).

Both these rank-2 tensors $B$ and $g$ sum up to an object $g + B$ which is known as the “open string metric” to string theorists, and which happens to have a surprisingly nice and natural geometrical interpretation in the context of what is called generalized complex geometry.

In this approach, due to Nigel Hitchin, one studies the geometry of manifolds $X$ all in terms of the *sum* of their tangent and cotangent bundle
$T^* X \oplus T X
\,,$
making use of various kinds of useful natural structures present on this bundle, like its canonical bilinear pairing as well as the Courant bracket.

As is indicated to some extent in section 3.8 of

Marco Gualtieri
*Generalized Complex Geometry*

math.DG/0401221

this bundle $T X \oplus T^* X$ is to be thought of as the *Atiyah Lie 2-algebroid* of an abelian gerbe on $X$, hence as the infinitesimal version of something like the 2-groupoid of automorphisms of the corresponding $\Sigma U(1)$ 2-bundle.

The more or less obvious $n$-algebroid structures on $T X \oplus \wedge^{n-1} T^* X$ should corespond to the Atiyah Lie $n$-algebroids of $\Sigma^{n-1}$-$n$-bundles.

While I do follow all the algebra underlying generalized complex geometry, I had always said to myself that at some point I should better understand what all this *really* means. Given the available nice categorical picture of line 2-bundles, there should be a nice arrow-theoretic integrated version of understanding all this, showing how a line 2-bundle with connection knows something about Riemannian structures.

Unfortunately, I had never really taken the time to think this through.

Now, today Marco Gualtieri approached me with a question. Since he is already happy with having unified the 2-form connection with the Riemannian metric in his formalism, he was trying to get hints for how all the other string background fields would fit into this mathematical picture, like the RR-forms and in particular the dilaton.

This made me really angry with myself. I wished I had figured out at least some of the answers to these obviously open and crucial questions *before* he asked me! I knew these needed to be answered at some point.

As it goes, this kind of frustration sometimes is the best motivatin to get going. While S. Merkulov was teaching us this morning about operads, Poisson structures and graph complexes, I believe I made some progress with understanding what’s going on.

The key is, it seems, to first figure out what the Atiyah $n$-groupoid of an $n$-bundle *really* is. I’ll discuss this using tangent $n$-categories and flows on $n$-categories, as described in Arrow-theoretic differential geometry.

I shall try to indicate what I mean by that, and how it gives rise to the appearance of the structures we see in generalized complex geometry.

So let $X$ be a smooth manifold. Let $X \times X$ be its pair Lie groupoid.

Notice that we obtain the corresponding Lie algebroid as the one whose sections are the $\mathbb{R}$-flows on $X \times X$ $T X \ni v : \mathbb{R} = \to \mathrm{INN}(X \times X) \,,$ using the reasoning described here, and using the notion of inner automorphisms $(n+1)$-groups as described in The inner automorphism 3-group of a strict 2-group.

Now let $P \to X$ a principal $G$-bundle on $X$ and consider its integrated Atiyah sequence $\mathrm{Ad} P \to P \times_G P \to X \times X \,.$

Here $P \times_G P$ is the Atiyah groupoid of $P$: its objects are the fibers of $P$, its morphism are all fiber morphisms (necessarily isomorphisms).

The sections of the corresponding Lie algebroid are again the $\mathbb{R}$-flows on this groupoid, hence the smooth group homomorphisms $\mathbb{R} \to \mathrm{INN}(P \times_G P) \,.$

Locally that’s clearly sections of $T X \oplus \mathrm{Lie}(G)$. I think globally it correctly $T P / G$, as it should be.

When passing to 2-bundles now, I will simply my and your lifes by only working locally, for the time being.

So for $G_2 :=\Sigma U(1)$ the Lie 2-group shifted $U(1)$, consider a trvial principal $G_2$-2bundle $X \times G_{2} \to X$ with the corresponding Atiyah Lie 2-groupoid, which is simply the cartesian product $(X \times X) \times (\Sigma^2 U(1) ) \,.$

A smooth $\mathbb{R}$-flow on this 2-groupoid $\mathbb{R} \to \mathrm{INN}( (X \times X) \times (\Sigma^2 U(1) ))$ is the following:

a vector field $v \in \Gamma(T X)$ as before, encoding the flow on $X \times X$ itself, together with a functorial smooth assignment of rectangles $\array{ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet \\ \downarrow^{\mathrm{Id}} &\Downarrow^{\exp(i \int_\gamma A_{v,t})}& \downarrow^{\mathrm{Id}} \\ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet }$ in $\Sigma^2 U(1)$ to paths $(x \stackrel{\gamma}{\to} y) \,,$ coming from a 1-form $A \in \Omega^1(X)$ smoothly parameterized by $t$, by the usual logic.

So indeed, a smooth $\mathbb{R}$-flow on the Atiyah 2-groupoid of a trivial $\Sigma U(1)$-2-bundle is a section of $T X \oplus T^* X \,.$

By precisely the same kind of argument we’d get $T X \oplus \wedge^{(n-1)} T^* X$ for $n$-bundles with structure $n$-group $\Sigma^{n-1} U(1)$. (To see that in detail, you need to either work with pseudofunctors or pull back the Atiyah groupoid to $n$-paths along $P_n(X) \to X \times X$ first.)

That sets the scene. Next I should try to reformulate all the stuff done in generalized complex geometry using arrow-theoretic differential theory.

## Re: The G and the B

That method of flows on categories is really useful, it seems. For instance consider the strict version of the String 2-group. Smooth morphisms $\mathbb{R} \to \mathrm{INN}(\mathrm{String}_k(G))$ (notice: not $\mathrm{INN}_0(\mathrm{String}_k(G))$!, we are really mapping into all automorphisms which are connected by a pseudonatural transformation to the identity) come, for each element $p$ of the path group of $G$, from a smooth family of squares $\array{ \bullet &\stackrel{p}{\to}& \bullet \\ \downarrow^{p_g(t)} &\Downarrow^{\omega(t)}& \downarrow^{p_g(t)} \\ \bullet &\to& \bullet }$ with $t \mapsto p_g(t)$ denoting a smooth 1-parameter family of based paths in $G$, starting, for $t=0$ at the constant path on the identity, with the endpoint evolving as $t \mapsto g(t)$.

Then the element $\omega(t)$ of the centrally extended loop group is already essentially fixed up to the central part. In the limit $t \to 0$ only the endpoint of the path contributes (all wiggling of the path is of higher than linear order in $t$), hence we get at once that $\mathbb{R}$-flows on the String 2-group live in the vector space $\mathrm{Lie}(G) \oplus \mathbb{R} \,.$ And that’s just the vector space on which, indeed the skeletal weak version of the Lie 2-algebra of the string group is built.

(And the grading is clearly that induced by the the degree $k$ of $k$-morphisms: the $\mathrm{Lie}(G)$-part comes from 1-morphisms and is degree 1, while the $\mathbb{R}$-part comes from the 2-morphisms.)