Finite landscape of 6d N=(1,0) supergravity

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≤ 193 T ≤ 193 , and the rank of the gauge group is restricted to r(V)≤ 480 r ( V ) ≤ 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})≤ 491 , h^{1,1}({\rm Base})≤ 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≠0 T ≠ 0 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 a bound on the number of massless fields in the supergravity for d≥ 6 d ≥ 6 .