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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∫ ℳ g¯c g−1 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→ℳ 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)∫ ℳ g¯c g−1 3=B g2g(2g−2)B g−1(2g−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 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 2g−2F 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(1−q 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

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