We present a bottom-up argument showing that the number of massless fields in six-dimensional quantum gravitational theories with eight supercharges is uniformly bounded. Specifically, we show that the number of tensor multiplets is bounded by $T\leq 193$, and the rank of the gauge group is restricted to $r(V)\leq 480$. Given that F-theory compactifications on elliptic CY 3-folds are a subset, this provides a bound on the Hodge numbers of elliptic CY 3-folds: $h^{1,1}({\rm CY_3})\leq 491$, $h^{1,1}({\rm Base})\leq 194$ which are saturated by special elliptic CY 3-folds. This establishes that our bounds are sharp and also provides further evidence for the string lamppost principle. These results are derived by a comprehensive examination of the boundaries of the tensor moduli branch, showing that any consistent supergravity theory with $T\neq0$ must include a BPS string in its spectrum corresponding to a "little string theory" (LST) or a critical heterotic string. From this tensor branch analysis, we establish a containment relationship between SCFTs and LSTs embedded within a gravitational theory. Combined with the classification of 6d SCFTs and LSTs, this then leads to the above bounds. Together with previous works, this establishes the finiteness of the supergravity landscape for $d\geq 6$.
Author indications on fulfilling journal expectations
Provide a novel and synergetic link between different research areas.
Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
Detail a groundbreaking theoretical/experimental/computational discovery
Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
In refereeing
Reports on this Submission
Report #1 by
Lara Anderson
(Referee 1) on 2025-10-20
(Invited Report)
Strengths
This paper is an important exploration of finiteness questions in the context of string compactifications and 3 (complex) dimensional Calabi-Yau manifolds. As a result, its results strengthen important synergistic links between geometry and string theory.
In particular, past work (in both the mathematics and physics literature) has explored the question of finiteness of genus-one fibered Calabi-Yau manifolds. While past results established finiteness up to binational classes and certain bounds on the Hodge numbers of the Calabi-Yau threefold, this paper clarifies and solidifies these results and provides tighter bounds on the topological data.
In general, the question addressed by this work is a central one in string compactifications and the string swampland program and this paper makes substantial new progress.
Weaknesses
While not a weakness, the central finiteness arguments rely heavily on physical arguments derived from superconformal field theories and little string theories in 6-dimensions. It would be intriguing if some of these results could be independently derived in the context of differential/algebraic geometry in the future. However, this question is of course beyond the scope of the present work.
Report
This is an excellent and important paper. I highly recommend it for publication.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
