Friday, April 13, 2007

Gopakumar-Vafa invariants for the quintic at genus 51

Yesterday, Kirill Saraikin was talking about their work focusing on a generalization of the OSV conjecture to non-supersymmetric black holes. It was surprising - i.e. either unlikely or revolutionary - to many people that such a generalization should exist because these non-supersymmetric objects are not protected, according to the state-of-the-art understanding of protection.

The argument needed Nikita Nekrasov's unpublished extension of the topological string by one or two additional parameters corresponding to components of the graviphoton field strength in the 01 and 23 directions in the full string-theoretical spacetime. The statement that an additional subset of protected but non-supersymmetric quantities should exist was controversial and, as far as I can say, remains unsettled.

Duality seminar

Right now, Min-Xin Huang from Wisconsin talked about their methods to evaluate the topological A-model partition sum for various compact Calabi-Yau manifolds such as the quintic hypersurface in CP4 and other complete intersection algebraic varieties in weighted complex projective spaces. They start with the BCOV paper and the holomorphic anomaly equation and attempt to determine the integration constant - an ambiguity known as the holomorphic ambiguity - by imposing the right boundary conditions at special points of the moduli space such as the conifold point, Gepner point, or infinite complex structure limit.

It turns out that for the quintic, these considerations allow you to determine all the unknown coefficient and, indirectly, all the Gopakumar-Vafa invariants up to genus 51 diagrams although they have explicitly quantified the results using their laptop up to genus 26 only which is nevertheless sufficient to fill the whole hard disk and occupy your laptop's CPU unit for days. ;-)

Recall that the topological A-model partition sum is controlled by the worldsheet instantons and it thus counts the number of holomorphic curves of different genera and different degrees that can be embedded within a given Calabi-Yau manifold. These integers associated with the A-model are known as the Gopakumar-Vafa invariants and they are closely related to the Gromov-Witten invariants and, via a transformation, to Donaldson-Thomas invariants.

Note that 16 years ago, it was difficult to calculate the number of twisted rational cubics in the quintic. In fact, mathematicians got a wrong number, 2,682,549,425, while the physicists i.e. string theorists obtained the correct result of 317,206,375 by mirror symmetry. Today it's possible to calculate these invariants for curves whose degree is comparable to one hundred. Of course, the typical magnitude of these invariants is exponentially growing with a power of the genus.

1 comment:

  1. Naturally, generalization of the OSV conjecture to non-supersymmetric black holes should exist


    but too much work remain ;-D

    Otherwise, GW/GV/DT invariants in topological vertex are integrals of Chern classes on stable map moduli spaces


    Salute planckeon