Topological wave functions and the 4D-5D lift

TL;DR

This paper explores the relationship between topological string amplitudes and 4D-5D black hole physics in supergravity, revealing how canonical transformations relate different charge configurations and constraining the topological wave function.

Contribution

It provides a new interpretation of the Gopakumar-Vafa relation as a canonical transformation and extends the analysis to a generalized topological string amplitude within duality groups.

Findings

Interpreted Gopakumar-Vafa relation as a canonical transformation.

Used black hole entropy to constrain topological wave function behavior.

Identified Weyl reflection as a canonical transformation in the duality group.

Abstract

We revisit the holomorphic anomaly equations satisfied by the topological string amplitude from the perspective of the 4D-5D lift, in the context of ''magic'' N=2 supergravity theories. In particular, we interpret the Gopakumar-Vafa relation between 5D black hole degeneracies and the topological string amplitude as the result of a canonical transformation from 4D to 5D charges. Moreover we use the known Bekenstein-Hawking entropy of 5D black holes to constrain the asymptotic behavior of the topological wave function at finite topological coupling but large K\"ahler classes. In the process, some subtleties in the relation between 5D black hole degeneracies and the topological string amplitude are uncovered, but not resolved. Finally we extend these considerations to the putative one-parameter generalization of the topological string amplitude, and identify the canonical transformation as a Weyl reflection inside the 3D duality group.