Abstract
We investigate holomorphic anomalies of partition functions underlying string compactifications on Calabi-Yau fourfolds with background fluxes. For elliptic fourfolds the partition functions have an alternative interpretation as elliptic genera of N = 1 supersymmetric string theories in four dimensions, or as generating functions for relative, genus zero Gromov-Witten invariants of fourfolds with fluxes. We derive the holomorphic anomaly equations by starting from the BCOV formalism of topological strings, and translating them into geometrical terms. The result can be recast into modular and elliptic anomaly equations. As a new feature, as compared to threefolds, we find an extra contribution which is given by a gravitational descendant invariant. This leads to linear terms in the anomaly equations, which support an algebra of derivatives mapping between partition functions of the various flux sectors. These geometric features are mirrored by certain properties of quasi-Jacobi forms. We also offer an interpretation of the physics from the viewpoint of the worldsheet theory, and comment on holomorphic anomalies at genus one.
Highlights
1.1 Overview and summaryThe computation of non-perturbatively exact partition functions of supersymmetric string theories, such as elliptic genera and various pre- and superpotentials, has attracted a lot of attention over the years
[23–27], the role of nearly tensionless critical strings at infinite distance points has been clarified in the context of quantum gravity conjectures such as the Weak Gravity Conjecture [28] or the Swampland Distance Conjecture [29]; the modularity of the partition function of these strings lies at the heart of the proof of the Weak Gravity Conjecture in such theories [23, 30–32]
Before we delve into the intricate mathematical details of the holomorphic anomaly equations for Calabi-Yau fourfolds, we briefly review the original work [50, 51] of BCOV, which was primarily aimed at threefolds, and outline how it extends to fourfolds at genus zero
Summary
The computation of non-perturbatively exact partition functions of supersymmetric string theories, such as elliptic genera and various pre- and superpotentials, has attracted a lot of attention over the years. It makes use of the fact that modular anomalies are equivalent to holomorphic ones, and the latter naturally arise from contact terms in the CFT that underlies topological strings This essentially boils down to the question of how to generalize the celebrated work of BCOV [50, 51] on threefolds to fourfolds with fluxes. Such geometries are dual to perturbative or non-perturbative heterotic strings For these we evaluate the modular and elliptic anomaly equations, and notably the descendant invariant, to put them in a concise form directly in terms of partition functions. We work out a detailed example, for which we explicitly determine the various flux-induced partition functions in terms of quasi-Jacobi forms These are shown to satisfy the modular and elliptic anomaly equations that we derived from geometry. It turns out that the situation is a straightforward generalization of the one of threefolds, in that the relevant partition functions are independent of the flux and the anomaly equations do not receive a contribution from a gravitational descendant invariant
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
