The n-Category Café

Andrew Stacey rightly writes in to point out that I have stolen the idea of an alpha-release of mathematical thought from the first line of the abstract of his notes on Comparative Smootheology.

I think I have now figured out how the adjoint representation of an $L_\infty$-algebra on itself works. (currently def. 2).

(When I say this, I am saying this in the context that I am looking at here. Actions of $L_\infty$-algebras on complexes and on themselves have of course been considered before, noticably at the end of Lada-Markl. I think what I am doing here is the same, but from a different point of view.)

So this now allows me to say in particular what an adjoint String 2-bundle is. And what a section of that is: locally such a section is a $g$-valued 0-form together with an ordinary 1-form.

I can compute the covariant derivative of that section, with respect to the given String 2-connection, but need to better understand what the result is telling me.

Maybe nothing much. What we really want to understand are sections of Chern-Simons 3-bundles and their relation to WZW theory.

So on with that…

I wrote:

What we really want to understand are sections of Chern-Simons 3-bundles

The states of Chern-Simons theory should be obtained like this:

- start with the Chern-Simons 3-bundle (2-gerbe) with connection on $B G$;

- pick a 2-dimensional manifold $\Sigma$, form the space of maps $\Sigma \to BG$;

- transgress the Chern-Simons 3-bundle to that space of maps;

- form the vector space of sections of the resulting line bundle.

That should be the space of states.

Using the technology we’ve been setting up, in fact all of these steps now have a well-defined meaning:

- the Chern-Simons 3-bundle with connection on $B G$ is represented by the $b^2 u(1)$-connection descent object in prop. 35, p. 64;

- the space of maps is the presheaf $\mathrm{maps}(\mathrm{inv}(g),\Omega^\bullet(\Sigma))$ from def. 4, p. 17;

- the transgression of the Chern-Simons bundle to the space of maps is the image of the original $b^2 u(1)$-connection descent object under the functor $\Omega^\bullet(\mathrm{maps}(--,\Omega^\bullet(\Sigma)))$ from def. 5, p. 18 as described for the case we are interested in on p. 87;

- the vector space of sections, finally, we form as described here in this entry, (after quotienting out the irrelevant symmetries, thereby performing the integration without integration of the Chern-Simons action functional, as in section 4);

So this gives a well-defined, effectively computable construction of the space of states of Chern-Simons theory.

And so I sat down and effectively computed the well-defined result.

The result which I found tonight is this:

after unwrapping the definitions here, a state is a $\mathbb{C}$-valued function $$\psi \in \Omega^0(\Omega^1(\Sigma,g))$$ on the space $\Omega^1(\Sigma,g)$ of $g$-valued 1-forms on $\Sigma$ which, when pulled back to the sub-space of flat $g$-valued 1-forms satisfies the descent condition

$$d \psi + (tg_\Sigma \mu)\wedge \psi = 0$$

where $tg_\Sigma \mu$ is the 1-form on the space of $g$-valued 1-forms which we obtain from transgressing the connection 3-form $\mu$ of the Chern-Simons 3-bundle over $B G$.

To compare this expression with the existing physics literature one can rephrase it in terms of the kind of component-based functional derivative equations used there (which actually receive a rigorous interpretation thereby). Doing so, I found that the above is equivalently expressed as

$$C^c{}_{ab} \left( A^a_\mu(x) \frac{\delta}{\delta A_{b\nu}(x)} + A^a_\mu(x) A^b_\nu(x) \right) \psi = 0 \,,$$

for all $c$ ranging over a chosen basis $\{t_a\}$ of $g$, for all $x \in \Sigma$ and for all $\mu,\nu$ ranging over a basis of $T_x \Sigma$.

Here $A^a_\mu(x) \in \Omega^0(\Omega^1(\Sigma,g))$ is the function (the “field”) on the space of $g$-valued forms, as in section 9.3.1, whose value on a given $g$-valued connection $A \in \Omega^1(\Sigma,g)$ is the $a,\mu$-component of that 1-form at $x \in \Sigma$ in the chosen basis, and $\{C^c_{ab}\}$ are the structure constant of the Lie algebra $g$ in that basis.

A description of this computation is now in section 4, p. 8.

I am hoping I didn’t make a mistake. It is getting late here. Take this with a grain of salt for the moment.

What I need to do next is to figure out how to bring a complex structure into the game. The sections we really want are holomorphic sections with respect to some complex structure.

Following up on the discussion here on how transgression of String 2-bundles and the corresponding obstructing Chern-Simons 3-bundles yields Kac-Moody centrally extended loop bundles and the corresponding obstructing line 2-bundle (lifting gerbes) I have started writing out the theory of the loop Lie $\infty$-algebroids in more detail.

In the new section 2.3 I do the following:

a) quickly review Simon Willerton’s neat observation about the loop groupoid of a finite group, which can be found summarized in The baby version of Freed-Hopkins-Teleman;

b) then generalize the discussion from finite groups to Lie groups using the result of Parallel transport and functors (arXiv, blog);

c) then use that to guide the study of the corresponding loop Lie $\infty$-algebroids using the transgression operation from section 9 of $L_\infty$-connections and applications (pdf, blog).

I show that

d) the loop Lie algebra of an ordinary Lie algebra $g$ is, as you’d expect, just the standard loop Lie algebra $\Omega g$

e) that the loop Lie algebra of the (skeletal version of the) String Lie 2-algebra $g_\mu$ is the Kac-Moody central extension $$\hat \Omega g$$ of $\Omega g$.

This is to be read as a reply to my remark here.

Clearly, this is closely related to the step From loop groups to Lie 2-groups (arXiv, blog).

But the precise relation I won’t start spelling out before I had some sleep and took care of other things I am supposed to be doing.