Emanuel SCHEIDEGGER
(Institute For Theoretical Physics, Vienna)
Higher genus topological string amplitudes
We will show how to compute the higher genus instanton numbers in the topological string with a
compact Calabi-Yau target space. We will explain the different methods involved and some of their
relations to mathematics. In particular, we show that for K3 fibrations there exist generating
functions for the Gopakumar-Vafa invariants which are automorphic forms. The knowledge of these
automorphic forms at genus zero yields severe constraints for higher genus. However, additional
information coming from the embedding of the Picard lattice of K3 into the one of the Calabi-Yau
threefold plays a role.
References:
- M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa,
"Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes",
Commun.Math.Phys. 165 (1994) 311-428,
hep-th/9309140.
- R. Gopakumar, C. Vafa, "M-Theory and
Topological Strings - II",
hep-th/9812127.
- M. Marino, G. Moore, "Counting higher genus
curves in a Calabi-Yau manifold", Nucl.Phys. B543 (1999) 592-614,
hep-th/9808131.
- S. Katz,
A. Klemm, C. Vafa, "M-Theory, Topological Strings and Spinning Black Holes",
Adv.Theor.Math.Phys. 3 (1999) 1445-1537,
hep-th/9910181.
- A. Klemm, M. Kreuzer, E. Riegler,
E. Scheidegger, to appear