The String Coffee Table

There was a talk on topological membranes by Allesandro Tanzini

Giulio Bonelli, Alessandro Tanzini, Maxim Zabzine
On topological M-theory
hep-th/0509175.

As you all know, the idea of topological M-theory is to realize topological string theories as limits of topological membrane theories, in analogy to how phyiscal superstrings are realized as limits of physical membranes.

Instead of saying anything about this particular work, I’ll restate a question which was raised, but not fully answered after the talk:

The topological membrane couples to the supergravity 3-form just like the topological string couples to the Kalb-Ramond 2-form. We know that this 3-form is best thought of as the Chern-Simons 3-form of an $E_8$ connection on spacetime. Hence, if we forget about the other couplings of the toplogical membrane for a moment, its coupling to the SUGRA 3-form makes it look a loot like $E_8$ Chern-Simons theory on its 3-dimensional worldvolume.

Not quite of course, because what we vary is the membrane “embedding” into spacetime, not the $E_8$ connection, which is merely pulled back from spacetime to the membrane worldvolume using this embedding. But anyway, it seems suggestive that, (according to CMP 121, 3 (1989), 351-399 ) on the boundary of an $E_8$ Chern-Simons theory there lives a 2D conformal field theory with $E_8$ current algebra.

Hence it looks like the boundary theory of topological membranes coupled to just the SUGRA 3-form should be closely related to $E_8$ WZW models. But a similar picture is of course implied by Hořava-Witten theory, where the boundary theory of the membranes is the heterotic string carrying $E_8$ current algebras.

Ok, so the question is: Does anyone know if there is a precise way in which the $E_8$ current algebra of the heterotic string can be understood as the boundary theory of an $E_8$-Chern-Simons theory living on a membrane, induced from its coupling to the SUGRA 3-form?

A few notes from another talk: Jarah Evslin sketched work in progress which is supposed to illuminate the true nature of RR-fields, clarifying in particular the behaviour under S-duality ($\to$). In the available 45 minutes he could only sketch the most basic setup, which, even further condensed, is the following:

Let $G_p$ be the $p$-form field strength of an RR-$(p-1)$-form in either IIA or IIB. Let $H$ be the 3-form field-strength of the Kalb-Ramond 3-form. (This induces the “twist” in the twisted K-theory classification of the RR-fields.) The equations of motion say that

Jarah’s idea is to use this equation in order to regard the operator of left wedge multiplication with $H$ as a “BRST operator” in the following sense.

The $G_p$ are the “physical fields”.

Gauge transformation parameters $\Lambda$ of these fields are one type of “ghosts” and D-branes are another type of “ghosts”.

This makes sense in that $d G_p$ is proportional to magnetic “brane density”, so that $H\wedge$ can be regarded as mapping “fields” ($G_p$) to “ghosts” (brane current densities). Hence we have a complex looking something like this:

Think of this as $\mathbb{Z}_2$-graded with the central term living in grade 0.

Now, physical fields, namely gauge orbits of field strenghts, do live in the ghost 0 cohomology of $Q = H\wedge$, which corresponds to the K-theory $K^0_H(M)$, with $M$ being spacetime and $H$ indicating twisted K-theory.

Similarly, the D-branes themselves live in ghost-number 1 cohomology of $Q = H\wedge$, corresponding to $K^1_H(M)$.

I can sort of see how this should work, but if you find any of this suspicious we will have to ask Jarah to help us with the details…

Jarah said this way of looking at things should eventually be useful for understanding S-duality covariant K-theory. Namely, in order to obtain something of this sort, he expects it to be sufficient to enlarge the above described $H\wedge$-cohomology by similar cohomologies which include also the gauge transformations of the KR-field itself.

He would probably have needed 450 instead of 45 minutes to get the details across. But a pdf with more notes is supposed to appear in the online proceedings of the conference.

Finally, I should mention that Brano Jurco gave nice overview talk about the latest on 1- and 2-gerbes with structure group the string group for $\mathrm{Spin}(n)$ and for $E_8$, and how this is related to the Green-Schwarz anomaly and possibly to M5-brane anomalies. If I find the time, I’ll report on more details of this talk. But right now I have absolutely no time left.