Modularity from Monodromy

Abstract

In this note we describe a method to calculate the action of a particular Fourier-Mukai transformation on a basis of brane charges on elliptically fibered Calabi-Yau threefolds with and without a section. The Fourier-Mukai kernel is the ideal sheaf of the relative diagonal and for fibrations that admit a section this is essentially the Poincaré sheaf. We find that in this case it induces an action of the modular group on the charges of 2-branes.

19 pages

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• Elliptic Blowup Equations for 6d SCFTs. IV: Matters
• The discrete Green-Schwarz mechanism in 6D F-Theory and Elliptic Genera of Non-Critical Strings
• Quasi-Jacobi Forms, Elliptic Genera and Strings in Four Dimensions
• Twisted 6d $(2,0)$ SCFTs on a Circle
• Heterotic strings on $(K3\times T^2)/{\mathbb{Z}}_3$ and their dual Calabi-Yau threefolds
• Refined BPS numbers on compact Calabi-Yau threefolds from Wilson loops
• State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture