[Paper Review] The Tadpole Conjecture in Asymptotic Limits

Mariana Graña, Thomas W. Grimm+3 more|arXiv (Cornell University)|Apr 11, 2022

Black Holes and Theoretical Physics41 references31 citations

TL;DR

This paper provides the first conceptual explanation for the Tadpole Conjecture in Type IIB and F-theory compactifications by leveraging asymptotic Hodge theory. It demonstrates that in asymptotic limits of Calabi-Yau four-fold moduli spaces, the number of stabilized complex structure moduli scales linearly with the number of sl(2)-representations supported by self-dual fluxes, and each such representation contributes a positive-definite term to the tadpole, leading to a linear scaling of the tadpole charge with the number of stabilized moduli—providing strong evidence for the refined Tadpole Conjecture.

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.

Motivation & Objective

• To provide a conceptual explanation for the linear scaling of the tadpole charge with the number of stabilized complex structure moduli in F-theory compactifications.
• To clarify why the tadpole conjecture's linear scaling only becomes relevant in the regime of large numbers of stabilized moduli.
• To establish that the tadpole contribution arises from independent, positive-definite contributions of sl(2)-representations in asymptotic regimes.
• To demonstrate that the minimal tadpole contribution per stabilized modulus is bounded from below, supporting the refined Tadpole Conjecture.

Proposed method

• Utilizes asymptotic Hodge theory to analyze the Hodge star operator in strict asymptotic regimes of Calabi-Yau four-fold moduli spaces.
• Identifies orthogonal sl(2)-block structures in the Hodge decomposition, allowing fluxes to be grouped into sl(2)-representations.
• Applies the self-duality condition for fluxes to derive polynomial equations in moduli, with degrees determined by sl(2)-weights.
• Projects fluxes onto individual sl(2)-eigenspaces and computes boundary norms using Cholesky decomposition to derive lower bounds on flux norms.
• Demonstrates that each sl(2)-representation fixes exactly one modulus and contributes a positive-definite term to the tadpole.
• Shows that K3-type representations (maximal weight 2) dominate the scaling behavior and that their fluxes contribute bounded, moduli-independent lower bounds to the tadpole.

Experimental results

Research questions

• RQ1 Why does the tadpole charge scale linearly with the number of stabilized complex structure moduli in F-theory compactifications?
• RQ2 What is the origin of the linear scaling behavior in asymptotic limits, and why does it only emerge for large numbers of moduli?
• RQ3 How do sl(2)-representations in the Hodge decomposition relate to moduli stabilization and tadpole contributions?
• RQ4 Can the tadpole contribution from each flux representation be bounded from below, and what does this imply for the refined Tadpole Conjecture?
• RQ5 Why do only certain sl(2)-representations—particularly those of K3-type—dominate the scaling behavior in Calabi-Yau four-folds?

Key findings

• The number of stabilized moduli is exactly equal to the number of sl(2)-representations supported by the flux, with each representation fixing one modulus.
• Each sl(2)-representation contributes a positive-definite, moduli-independent lower bound to the tadpole, ensuring linear scaling with the number of stabilized moduli.
• For Calabi-Yau four-folds, all but one sl(2)-representation are of K3-type (maximal weight 2), matching representations found in K3 surfaces.
• The minimal contribution to the tadpole from any K3-type flux representation is bounded from below by 7/6, as demonstrated in explicit examples.
• Fluxes in K3-type representations are constrained by self-duality to pair positive and negative weights, and their norm contributions scale with large moduli parameters, ensuring non-vanishing lower bounds.
• The analysis confirms that the tadpole grows linearly with the number of stabilized moduli, providing strong evidence for the refined Tadpole Conjecture in asymptotic regimes.