The Tadpole Conjecture in Asymptotic Limits

TL;DR

This paper uses asymptotic Hodge theory to analyze the tadpole conjecture, revealing why stabilizing many complex structure moduli in flux compactifications requires fluxes with a tadpole that scales linearly with the number of stabilized fields.

Contribution

It provides the first conceptual argument explaining the linear scaling in the tadpole conjecture using asymptotic Hodge theory and sl(2)-representation structures.

Findings

Number of stabilized moduli scales with the number of sl(2)-representations supported by fluxes.

Each sl(2)-representation fixes a single modulus.

For Calabi-Yau four-folds, most representations relate to two-folds, enabling explicit stabilization analysis.

Abstract

The tadpole conjecture suggests that the complete stabilization of complex structure deformations in Type IIB and F-theory flux compactifications is severely obstructed by the tadpole bound on the fluxes. More precisely, it states that the stabilization of a large number of moduli requires a flux background with a tadpole that scales linearly in the number of stabilized fields. Restricting to the asymptotic regions of the complex structure moduli space, we give the first conceptual argument that explains this linear scaling setting and clarifies why it sets in only for a large number of stabilized moduli. Our approach relies on the use of asymptotic Hodge theory. In particular, we use the fact that in each asymptotic regime an orthogonal sl(2)-block structure emerges that allows us to group fluxes into sl(2)-representations and decouple complex structure directions. We show that the number of stabilized moduli scales with the number of sl(2)-representations supported by fluxes, and that each representation fixes a single modulus. Furthermore, we find that for Calabi-Yau four-folds all but one representation can be identified with representations occurring on two-folds. This allows us to discuss moduli stabilization explicitly and establish the relevant scaling constraints for the tadpole.