Is every anomaly-free six-dimensional supergravity realized in string theory
In plain words
Six-dimensional supergravity theories (gravity plus a symmetry between bosons and fermions) can be listed by checking quantum consistency conditions called anomaly cancellation. Whether every theory that passes these checks actually comes from string theory is unknown.
Precise statement
For 6D $N=(1,0)$ supergravity with $T$ tensor multiplets, gauge group $G$ and hypermultiplet content $H$, Green-Schwarz anomaly cancellation restricts $(T, G, H)$. Determine whether every anomaly-free model that meets all known consistency conditions is realized by a string compactification (string universality, Kumar and Taylor 2009), or identify the additional conditions that remove the remainder.
What would settle it
A proof that consistency conditions (anomalies, brane-probe unitarity, black-hole arguments) reduce the anomaly-free list exactly to F-theory and heterotic models, or an anomaly-free model shown consistent with no string realization.
Status in the literature
In ten dimensions the unrealized anomaly-free theories were excluded (Adams, DeWolfe, Taylor 2010); in 6D brane-probe unitarity removed some anomaly-free models (Kim, Shiu, Vafa 2019), and anomaly-free models without known string realization remain.