The tadpole conjecture in asymptotic limits

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.

Topological constraints in the LARGE-volume scenario

“…It is difficult to estimate in general how much these fluxes contribute to the tadpole. However, recent results suggest that this contribution grows faster than Q 3 as the number of complex-structure/D7-brane moduli becomes large [48,49] (unless one considers singular compactifications [8]). Indeed, for h 1,1 + = 2, large Q 3 requires a large number of complex-structure/D7-brane moduli (see, e.g., section 5 in [31]).…”

Mass Hierarchies and Quantum Gravity Constraints in DKMM‐refined KKLT

“…This will have many complex structure moduli so that one very likely encounters the problem of the tadpole conjecture. [42][43][44][45] Namely, that it is not possible to freeze all of these many moduli using three-form fluxes, something that was silently assumed before we focused just on the final two moduli Z and S. Hence, we arrive at the conclusion that a working DKMM-refined KKLT Scenario 1 would require very large fluxes that are in conflict with the tadpole constraint. Scenario 2 is expected to be much more constrained in concrete cases, so that the flux tadpole could become dangerously large, likewise.…”

“…[ 36–40 ] One of the main results of [36] is that the uplift term could destabilize the complex structure modulus Z , that controls the size of the 3‐cycle that shrinks to zero size at the conifold locus $$Z=0$$. This provides a lower bound on the parameter $$g_s M^2 \gtrsim O(50)$$ pointing into the direction that KKLT is maybe not compatible with tadpole cancellation, [ 42–45 ] which in fact is thought to be a genuine quantum gravity effect.…”

“…Second, the above mentioned constraints lead to much larger tadpoles $$N\sim 10^{7-8}$$ than previously envisioned and are almost certainly in conflict with tadpole cancellation respectively the recent tadpole conjecture. [ 42–45 ]…”

“…Second, the above mentioned constraints lead to much larger tadpoles N ∼ 10 7−8 than previously envisioned and are almost certainly in conflict with tadpole cancellation respectively the recent tadpole conjecture. [42][43][44][45] For the less constrained Scenario 2 the hierarchy could be controlled by choosing still moderately large fluxes leading to a tadpole of the order N ∼ 10 2−3 . However, here certain parameters were treatad as freely tunable, while in practice they will also depend on flux quanta.…”