Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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, χ\chi, the genus gg vacuum amplitude of the A-Model is

(1)F g=12χ g¯c g1 3+𝒪(e A) F_g = \textstyle{\frac{1}{2}}\chi \int_{\overline{\mathcal{M}_g}} c_{g - 1}^3 + \mathcal{O}\left(e^{ - A}\right)

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

The integral is given by

(2) g¯c g1 3=B g2g(2g2)B g1(2g2)! \int_{\overline{\mathcal{M}_g}} c_{g - 1}^3= \frac{B_g}{2g(2g - 2)} \frac{B_{g - 1}}{(2g - 2)!}

where the B iB_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 gg s 2g2F g=f(q) χ/2 Z(q)= exp \sum_g g_s^{2g - 2} F_g = f(q)^{\chi/2}

where

(4)f(q)= n=1 1(1q n) n,q=e g s 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 3T^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

0 Comments & 0 Trackbacks

Post a New Comment