### Location, Location, Location

Here’s my office at the KITP (on a less foggy day and minus the construction, which is currently causing the building to shake).

Ron Donagi gave a great talk this morning about a version of Dijkgraaf-Vafa and large-N duality for compact Calabi-Yau’s. DV’s setup can be thought of as a noncompact Calabi-Yau

which, generically, has $n$ isolated $A_1$ singularities at $u=v=y=W'(x)=0$. If $W(x) = \sum_i^0^{n+1} a_i x^i$, these can be smoothed by adding $f_\mu(x)=\sum_{i=0}^{n-1}\mu_i x^i$

The salient feature, according to Ron is that this degenerates to a curve (the plane $\{ u=v=y=0\}$) when the parameters $a=\mu=0$.

Their compact example is the intersection of a quadric and a quartic in $\mathbb{C}P^5$. When the quadric has rank 3, one has a genus-3 curve of $A_1$ singularities, while the generic quadric gives a smooth Calabi-Yau. To first order in the deformation, they compute the corresponding integrable system and solve for the open-string invariants.

It’s quite a *tour de force*, made all the more remarkable by the fact that it’s not clear what they are calculating. Since the Calabi-Yau is compact, the total D-brane charge must vanish. So they need to put in both $N$ and $N$ antibranes (on the open string side). In a supersymmetric theory, the quantity they calculate would be the related to the glueball superpotential. In this non-supersymmetric gauge theory, I’m not sure what the interpretation should be. But it’s pretty exciting if there’s something *exact* that you can calculate in this nonsupersymmetric theory.

Greg Moore also gave a great talk about the C-field in M-theory. Yup, it’s an element of differential cohomology, $\check{H}^4(X)$. Mike Hopkins gave a series of lectures last week touching on this, which went *way* over my head. But these guys actually have a concrete model one can wrap one’s head around.

The idea is that, up to 15-skeletons, the classifying space of $E_8$ bundles is an Eilenberg-MacLane space

So an isomorphism class of $E_8$ bundle on an 11-manifold is equivalent to a choice of cohomology class $a\in H^4(X,\mathbb{Z})$. Their concrete model for the C-field is a pair $(A,c)$, consisting of connection on an $E_8$ principal bundle, $P$, and an honest-to-God, globally-defined, 3-form $c$. They can write explicit formulas for the bosonic part of the 11D supergravity action in terms of $(A,c)$ and the field strength,

which are invariant under gauge transformations

where $(\alpha,\omega) \in \Omega^1( ad P)\times \Omega^3_{\mathbb{Z}}(X)$. Naively, the gauge group is $\Omega^1( ad P)\ltimes \Omega^3_{\mathbb{Z}}(X)$, with the composition law

Now, Greg says this is too naive, and one should really take the gauge group to be $\Omega^1( ad P)\ltimes \check{H}^3(X)$ which is a nontrivial extension

That means I still have to understand differential $H^3$, understand how to compose two such guys, *etc*.

Rats!

Posted by distler at July 25, 2003 3:04 AM