The n-Category Café

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.)

Make sure you ask Marco to whistle for you!

Maybe I am beginning to see how the Riemannian metric comes in from just the arrows.

Check section 6.1 in Gualtieri’s math.DG/0401221 and in particular proposition 6.6, the upshot of which is that with $$b$$ any 2-form on $X$ and with $$g$$ any Riemannian metric on $X$, we may regard $$g \pm b : T X \to T^* X$$ as a metric (that “open string metric”) in the context of generalized complex geometry.

Consider first the case that the Riemannian metric is absent $$g = 0 \,.$$ And notice that the 2-form $b$ has to be interpreted as a connection on our trivial $\Sigma U(1)$-bundle, defining a 2-functor $$\mathrm{tra}_b : P_2(X) \to (P_2(X)\times \Sigma^2 U(1)) \,.$$ Let’s abbreviate our Atiyah 2-groupoid of the trivial $\Sigma U(1)$-2-bundle on $X$ as $$\mathrm{At} := (P_2(X)\times \Sigma^2 U(1)) \,.$$ Then notice that, with our interpretation of $T X \oplus T^* X$ as the space of smooth 1-parameter families of inner automorphisms of the Atiyah 2-groupoid as above (now I do work in the picture where I have pulled back along $P_2(X) \to X \times X$) we can interpret the map $$b : T X \to T^* X$$ in terms of arrows simply as mapping flows on $P_2(X)$ to flows on the Atiyah 2-groupoid according to $$\array{ & {}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} \\ P_2(X) &\Downarrow& P_2(X) \\ & {}_{\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\exp(v)}} \\ && \downarrow^{\mathrm{tra}_b} \\ && At } \;\;\;\;\;\; = \;\;\;\;\;\; \array{ P_2(X) \\ \downarrow^{\mathrm{tra}_b} \\ & {}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} \\ At &\Downarrow& At \\ & {}_{\;\;\;}\searrow \nearrow_{} } \,.$$

So this does map flows on the base to flows on the total Atiyah 2-groupoid, and by unravelling what all these arrows mean one finds that this does indeed precisely correspond to the map $$b : T X \to T^* X \,.$$

(Compare this to the diagram in What is a Lie derivative really? to get a better idea for what’s going on.)

So among all maps from flows on the base to flows on the Atiyah 2-groupoid, those induced by a transport 2-functor as above do consistently correspond to maps induced from the corresponding connection 2-form.

But there are more morphisms from base flows to flows on the Atiyah 2-groupoid than that. Indeed, each $g \in S^2 T^* X$ does procide one. The underlying formula for the components is pretty much as above, only that now there is no 2-functor $P_2(X) \to At$ defined by this anymore.

So, I am thinking that a generalized metric in the sense of generalized complex geometry, hence an “open string metric” is – while in general no longer a transport 2-functor with values in the line 2-bundle, instead a functor from flows on the base to flows on the Atiyah Lie 2-groupoid of the line 2-bundle.

I’ll provide you with the precise diagrammatics of this next week.

I feel like a layman staring at awe at an abstract piece of art and commenting on how beautiful it is only to be told the painting is upside down.

I have no idea what you just said Urs, but it is beautiful to look at :)

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…

Umh… not quite.

The open string metric is $$G = g - B\cdot g^{-1} \cdot B$$ Its inverse is closer to the thing you’re considering $$G^{-1} = {\left[(g+B)^{-1}\right]}_{\text{sym}}$$ The bivector, $${\left[(g+B)^{-1}\right]}_{\text{anti-sym}}$$ is also significant.