The n-Category Café

Does this work tell us anything about all those dualities (S, T, U, etc.) between string theories?

In the context of the above, I’d like to look again at the question we were discussing here in the thread “On BV Quantization, Part VIII”, about whether and how bundles with connection can be classified by maps of “spaces”.

To some degree, the developments in Part B indicate an answer to that.

To see that, one should notice that graded-commutative differential algebras (dg-algebras for short, in the following) are pretty close to being spaces. I’ll say something about that now. With that in mind, everything we say about bundles with connections in terms of dg-algebras can be translated into a description in terms of spaces.

It’s essentially the idea of rational homotopy theory: whenever you see a graded-commutative dg-algebra, you can think of it as being the dg-algebra of differential forms on some space.

But I’ll slightly vary the theme of rational homotopy theory (or rather, of what I know about it) here and instead of relating simplicial spaces to dg-algebras, relate them to smooth spaces in the sense of sheaves on manifolds.

So for me now, a smooth space is a sheaf on a site $S$, which might be open (convex) subsets of $\mathbb{R} \cup \mathbb{R}^2 \cup \mathbb{R}^3 \cup \cdots$, or might be the site of all manifolds, or some other site on whose objects we naturally have smooth differential forms.

I write $$S^\infty := \mathrm{Sh}(S)$$ for the category of smooth spaces in this sense.

There is a natural notion of differental forms on such a smooth space:

differential forms themeselves form a smooth space, given by the sheaf that sends each manifold $U$ to the set of differential forms on it: $$\Omega^\bullet : U \mapsto \Omega^\bullet(U) \,.$$

Given any other sheaf $X \in \mathrm{Sh}(S)$, a differential form $\omega$ on $X$ is a morphisms (of sheaves)

$$\omega : X \to \Omega^\bullet \,.$$

Here you should think for each object $U$ of our site of each element in the set $X(U)$ as a smooth map $f$ from $U$ into $X$, and of the image $\omega_U(f)$ of that element under $\omega_U$ as the differential form $f^* \omega$ on $U$ obtained by pullback along that map $f$ of a differential form $\omega$ on $X$.

The set of all differential forms on $X$ is hence

$$\Omega^\bullet(X) := \mathrm{Hom}_{S^\infty}(X, \Omega^\bullet) \,.$$

One checks easily that this inherits the structure of a graded-commutative dg-algebra from the fact that $\Omega^\bullet(U)$ is a commutative dg-algebra for all $U$ and that exterior derivative and wedge product commute with pullbacks.

This gives us a contravariant functor from smooth spaces to graded-commutative dg-algebras:

$$\Omega^\bullet : S^\infty \to DGCA \,.$$

There is another functor going the other way:

given any differential graded-commutative algebra, which I’ll write $\mathrm{CE}(g,V)$, we obtain a smooth space $X_{g,V}$ by setting

$$X_{g,V} : U \mapsto \mathrm{Hom}_{dg-Alg}(\mathrm{CE}(g,V), \Omega^\bullet(U)) \,.$$

(This is the basic idea underlying Sullivan models in rational homotopy theory.)

This yields a contravariant functor $$S^\infty \leftarrow DGCA : Hom_{dg-Alg}(-- , \Omega^\bullet(--)) \,.$$

I am not sure right now in full generality to which extent these two functors are mutual inverses or adjoints. But they should be.

One thing that is directly clear is that every graded-commutative algebra $\mathrm{CE}(g,V)$ sits at least inside the algebra of differential forms $\Omega^\bullet(X_{g,V})$ on the space $X_{g,V}$ it defines:

$$\mathrm{CE}(g,V) \hookrightarrow \Omega^\bullet(X_{g,V}) \,.$$

This is a tautology following from the way $X_{g,V}$ and $\Omega^\bullet(X)$ were defined.

But for instance I think one easily checks that $$\Omega^\bullet(X) = \Omega^\bullet(X_{\Omega^\bullet(X)}) \,,$$ which is reassuring.

Now, take our article and hit everything in sight with the functor $$S^\infty \leftarrow DGCA : Hom_{dg-Alg}(-- , \Omega^\bullet(--))$$ thus sending our discussion in terms of DGCAs entirely to a discussion in terms of smooth spaces.

This then yields, in particular, the following statements:

for every Lie $\infty$-algebra $g$ there is a smooth space $X_{\mathrm{inn}(g)}$ such that $g$-valued forms $A$ on any smooth space $Y$ correspond to smooth maps $$f_A : Y \to X_{\mathrm{inn}(g)} \,.$$

The “$g$-connection descent objects” on a space $X$ which we describe, and which encode $n$-bundles with connection with values in $g$ are characterized by maps of smooth spaces $$X \to X_{\mathrm{inv}(g)}$$ with the property that we can complete the two squares in $$\array{ F &\to & X_g \\ \downarrow && \downarrow \\ Y &\to & X_{\mathrm{inn}(g)} \\ \downarrow && \downarrow \\ X &\to& X_{\mathrm{inv}(g)} } \,.$$

The horizontal morphism at the bottom here plays the role of the classifying map as familiar from ordinary bundles. Indeed, for $g$ an ordinary Lie algebra, $X_{\mathrm{inv}(g)}$ is a smooth space with the same deRham cohomology as (any smooth model of) $B G$.

Then the fact that a surjective submersion $$\array{ Y \\ \downarrow \\ X }$$ appears is the phenomenon that John likes to refer to in terms of anafunctors. The smooth map $$Y \to X_{\mathrm{inn}(g)}$$ which covers our classifying map $$X \to X_\mathrm{inv}(g)$$ is the one that actually encodes the connection in detail, whereas $Y \to X_{\mathrm{inv}(g)}$ only knows about the characteristic forms of the conneciton form(s).

So, we don’t see here, directly, a single smooth space such that smooth maps into it classify bundles with connection.

Rather, we see a smooth space $X_{\mathrm{inv}(g)}$ characterizing bundles and being covered by another smooth space, $X_{\mathrm{inn}(g)}$, maps into which characterize connections on a trivial bundle. Fit together into a diagram as above, these maps characterize (nontrivial) bundles with connections.

Despite of this, we can after all speak of a universal connection in this context, and that’s probably all we actually want.

I talked about that before (also at the end of the slide show), but let me say it again, with the above picture now more clearly present:

from what I just said, if follows that the space $$X_{\mathrm{inv}(g)}$$ (which is essentially $B G$) supports not only a canonical $G$-bundle $$\array{ X_{\mathrm{inn}(g)} \\ \downarrow \\ X_{\mathrm{inv}(g)} }$$

but also that this universal bundle carries canonically a $g$-connection: this is because we can canonically complete the identity morphism $$\mathrm{Id} : X_{\mathrm{inv}(g)} X_{\mathrm{inv}(g)}$$ to a $g$-connection descent object, simply by having identities on all horizontal edges: $$\array{ X_g &\stackrel{\mathrm{Id}}{\to} & X_g \\ \downarrow && \downarrow \\ X_{\mathrm{inn}(g)} &\stackrel{\mathrm{Id}}{\to}& X_{\mathrm{inn}(g)} \\ \downarrow && \downarrow \\ X_{\mathrm{inv}(g)} &\stackrel{\mathrm{Id}}{\to}& X_{\mathrm{inv}(g)} } \,.$$

This is the universal $g$-connection (descent object) on the unversal $g$-bundle (descent object).

(But notice the remark from our introduction: right now we haven’t shown that the classification of these things by deRham classes lifts to integral classes. And also notice that what I am saying here would be fully satisfactory only when we know to which extent the two contravariant functors $\Omega^\bullet : S^\infty \to DGCAs$ and $Hom(--,\Omega^\bullet(--)) : DGCAs \to S^\infty$ are inverses of each other, at least for the DDCAs we are encountering here.)

And clearly, any other $g$-connection (descent object) is the pullback of this universal one, simply in the tautological sense that every morphism is the pullback of the identity on its target along itself.

A somewhat more polished version is now available:

Part A, part B.

We now have an arXiv number:

arXiv:0801.3480.

But I’ll keep the pdf here with corrections and improvements.

Currently it just contains precisely one typo less than the arXiv version. ;-)

As you may have noticed, many of the concepts I used to discuss here do appear in our article in their Lie $\infty$-algebraic incarnation: $n$-transport, its $n$-curvature with values in $INN(G)$, the charged $n$-particle, etc.

One concept which I liked to discuss a lot, however, does not appear at the moment: sections of $n$-bundles and their covariant derivatives.

One reason is that it turned out to be not quite straightforward to move the definition of that which I am so fond of ($n$-sections and their covariant derivative as morphisms into the $n$-curvature, as described here) to the Lie $\infty$-algebraic world.

There is a good reason for why that’s non-straightforward: this description of sections makes crucial use of non-invertible morphisms in the $n$-category of $n$-vector spaces. This means it falls out of the realm of $\infty$-groupoids. So our map from Lie $\infty$-groupoids to Lie $\infty$-algebroids fails and hence this concept does not internalize properly in the differential realm.

I was pretty upset about that. quantization of the $n$-particle is supposed to be all about taking $n$-spaces of sections of the background field $n$-bundle. And the Lie $\infty$-algebraic formulation is supposed to be the powerful tool to handle this $n$-bundle. So it’s too bad that this tool doesn’t admit taking sections.

I thought for a while that it just means that before taking sections I simply need to send everything Lie $\infty$-algebraic back to the integral world by hitting everything in sight with $\Pi_\infty(Hom(--,\Omega^\bullet(--)))$ and then proceed there.

While that might be quite an interesting thing to do, it seems comparatively cumbersome for just taking $n$-sections, compared to how nicely everything else goes throu on the Lie $\infty$-algebraic level.

But now I think I understood how to get the best of both worlds.

There is a reformulation of the concept of a morphism into the $n$-curvature of a $G_{(n)}$-bundle with connection in terms of a $V//G_{(n)}$-$n$-groupoid bundle, where $V$ is an $n$-representation and $V//G_{(n)}$ the corresponding action $n$-groupoid. And that reformulation does fit nicely into the Lie $\infty$-algebraic world.

There it looks like this:

as we describe in the article, for $g$ an $L_\infty$-algebra a corresponding bundle with connection can be represented by a diagram which involves, among other things, a morphism of the kind

$$\Omega^\bullet(Y) \stackrel{(A,F_A)}{\leftarrow} \mathrm{W}(g) \,,$$

where $Y \to X$ is some surjective submersion over base space $X$ and $\mathrm{W}(g)$ is the Weil algebra of the $L_\infty$-algebra $g$.

Now, pick a representation $V$ of $g$ and form the corresponding action Lie $\infty$-algebroid which comes with its Chevalley-Eilenberg algebra $CE(g,V)$ and Weil algebra $\mathrm{W}(g,V)$.

We have a canonical injection $$\array{ \mathrm{W}(g,V) \\ \uparrow \\ \mathrm{W}(g) } \,.$$

A section $\sigma$ of the given $g$-bundle is then a completion of $$\array{ &&\mathrm{W}(g,V) \\ &&\uparrow \\ \Omega^\bullet(Y)&\stackrel{(A,F_A)}{\leftarrow}&\mathrm{W}(g) }$$

to

$$\array{ &&&\mathrm{W}(g,V) \\ &{}^{(\sigma,\nabla_A \sigma,A,F_A)}\swarrow&&\uparrow \\ &\Omega^\bullet(Y)&\stackrel{(A,F_A)}{\leftarrow}&\mathrm{W}(g) } \,.$$

The “curvature” part of that is, automatically, $\nabla_A \sigma$, the covariant derivative of the section $\sigma$.

As a reaction to a question the table on p. 7 has now been replaced by the following more detailed table: