## September 23, 2003

### 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

(1)$F_g = \textstyle{\frac{1}{2}}\chi \int_{\overline{\mathcal{M}_g}} c_{g - 1}^3 + \mathcal{O}\left(e^{ - A}\right)$

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

(2)$\int_{\overline{\mathcal{M}_g}} c_{g - 1}^3= \frac{B_g}{2g(2g - 2)} \frac{B_{g - 1}}{(2g - 2)!}$

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

(3)$Z(q)= exp \sum_g g_s^{2g - 2} F_g = f(q)^{\chi/2}$

where

(4)$f(q)=\prod_{n=1}^\infty \frac{1}{(1 - q^n)^n},\quad q=e^{ - g_s}$

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

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/224