Musings

Strings 2004: a Day Late and a Dollar Short

Wednesday morning was Cosmic Strings Morning. Rob Myers gave a very pretty review of his work with Copeland and Polchinski. Nick Jones talked about the evolution of cosmic F- and D-string networks. They compute the reconnection probability for colliding F- and D-strings (which get folded into existing computations of the evolution of string networks). They claim that the reconnection probablility can be much smaller ($P\sim 10^{-3}$ for F-F and $P\sim .1$ for D-D strings) than the standard Nielsen-Olesen strings ($P\sim 1$). I’m not sure why these results differ from the gauge theory answer (presumably because the quantum corrections to the latter have never been properly taken into account) but, in any case, the most interesting features of these string networks is that the collision of general $(p,q)$ strings produces 3-string junctions (they also compute the probability for this process) and the existence of “baryons” (D3-branes wrapped on cycles with $n$-units of $F_{(3)}$ flux) on which $n$ fundamental strings can end. These change the evolution of these string networks in ways that cannot be captured by simply taking existing simulations of string networks and changing the reconnection probablility, $P$.

Probably the most exciting talks so far have been about a cluster of work by Vafa and collaborators on topological string theory and its relationship with other aspects of string theory.

Strominger talked about the surprising relation between black hole entropy in $N=2$ supergravity and the topological A-model. If you take Type IIA string theory compactified on a Calabi-Yau manifold, and look for supersymmetric blackhole solutions, you find the well-known attractor mechanism where, whatever the values of the Kähler moduli, $X^\Lambda$, of the Calabi-Yau at spatial infinity, as you approach the horizon, they are attracted to the locus

where $(q_\Lambda,p^\Lambda)$ are the electric and magnetic charges of the blackhole, $F_{0\Lambda}= \partial F_0/\partial X^\Lambda$ and $F_0$ is the prepotential (related to the genus-zero topological string vacuum amplitude). The Kahler form is

and the attractor equations fix $C$ and the moduli, $X^\Lambda$ up to a Kähler transformation. Lopes, Cardoso, de Wit and Mohaupt, found a beautiful formula for the corrections to the area law expression for entropy of the blackhole. Define

where $F_h$ is proportional to the genus-$h$ topological string amplitude, and $T_{\mu\nu}$ is the (anti-self dual part of the) graviphoton field strength. At the horizon, the exact attractor equation is

and the blackhole entropy

where the first term is, essentially, the area law. This formula can be recast in terms of a mixed canonical/microcanonical partition function

The stunning result is

where

and $\log Z_{\text{top}}= \sum_{h=0}^\infty g_{\text{top}}^{2h-2} F_{h(\text{top})}(t^A)$ is the topological A-model partition function.

Robert Dijkgraaf talked about a 7-dimensional field theory based on work of Hitchin on $G_2$ structures, which might be called (the spacetime theory of) topological M-theory. When defined on a manifold of the form $M_{\text{CY}}\times S^1$, it provides a derivation of the proposed S-duality between the topological A-model and the topological B-model (whose spacetime theories are Kodaira-Spencer theory and Kähler gravity, respectively). There’s also an 8-dimensional theory, based on $Spin(7)$ structures.

Both Seiberg and Rastelli gave beautiful talks about $c\lt 1$ noncritical string theory, and progress in understanding them from the point of view of D-branes (whose collective field theory is none other than the Matrix model).