Advances in Theoretical and Mathematical Physics

Volume 20 (2016)

Number 1

Some details on the Gopakumar–Vafa and Ooguri–Vafa formulas

Pages: 1 – 133



Mykola Dedushenko (Joseph Henry Laboratories, Princeton University, Princeton, New Jersey, U.S.A.)

Edward Witten (School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A.; and Department of Physics, University of Washington, Seattle, Wash., U.S.A.)


The Gopakumar–Vafa (GV) formula expresses certain couplings that arise in Type IIA compactification to four dimensions on a Calabi–Yau manifold in terms of a counting of BPS states in M-theory. The couplings in question have applications to topological strings and supersymmetric black holes. In this paper, we reconsider the GV formula, taking a close look at the Schwinger-like computation that was suggested in the original GV work. The goal is to understand the background that must be used in this computation, the role played by the extended supersymmetry of this background, and how the computation gives a holomorphic result though superficially depending only on particle masses. We also examine in a similar way the Ooguri–Vafa (OV) formula, which is an extension of the GV formula to include D4-branes.

Published 25 July 2016