Finite Landscape of 6d N=(1,0) Supergravity

TL;DR

This paper proves that six-dimensional N=(1,0) supergravity theories with eight supercharges have a finite number of massless fields, establishing bounds on tensor multiplets and gauge group rank, and providing evidence for the string lamppost principle.

Contribution

It introduces a bottom-up approach to bound the number of fields in 6d supergravity, connecting supergravity, string theory, and geometric constraints, and demonstrates the finiteness of the landscape.

Findings

Bound on tensor multiplets: T ≤ 193

Bound on gauge group rank: r(V) ≤ 480

Hodge number bounds for elliptic CY 3-folds

Abstract

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 $d\geq 6$.