### I’m *Melting* …

Okounkov, Reshetikhin and Vafa have a paper out today in which they relate the Gopakumar-Vafa conjecture for the Partition Function of the Topological A-Model to the statitstical mechanics of melting crystals.

In the limit of large-volume Calabi-Yau, with Euler characteristic, $\chi$, the genus $g$ vacuum amplitude of the A-Model is

where $c_i$ is the $i^{\mathrm{th}}$ Chern Class of the Hodge bundle, $H\to \overline{\mathcal{M}_g}$ (the bundle whose fiber over $\Sigma_g$ is spanned by the $g$ holomorphic 1-forms on $\Sigma_g$).

The integral is given by

where the $B_i$ are Bernoulli numbers. Gopakumar and Vafa argued that the all-genus result could be evaluated by a 1-loop computation in M-theory. Summing over BPS states, they obtained

where

The present paper notes that this is the partition function for the statistical mechanics of corner-melting of a 3D cubical crystal. They argue for a rather literalist interpretation, where the 3D space is the base of Calabi-Yau, written as a $T^3$ fibration. And they suggest various generalizations, relating the recently-constructed Topological-Vertex of the A-Model to a dimer problem.

Wild stuff!

Posted by distler at September 23, 2003 8:50 AM