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FRB 20210912A is a fast radio burst (FRB), detected and localised tosub-arcsecond precision by the Australian Square Kilometre Array Pathfinder. Nohost galaxy has been identified for this burst despite the high precision ofits localisation and deep optical and infrared follow-up, to 5-$\sigma$ limitsof $R=26.7$ mag and $K_\mathrm{s}=24.9$ mag with the Very Large Telescope. Thecombination of precise radio localisation and deep optical imaging has almostalways resulted in the secure identification of a host galaxy, and this is thefirst case in which the line-of-sight is not obscured by the Galactic disk. Thedispersion measure of this burst,$\mathrm{DM_{FRB}}=1233.696\pm0.006~\mathrm{pc}\ \mathrm{cm}^{-3}$, allows fora large source redshift of $z>1$ according to the Macquart relation. It couldthus be that the host galaxy is consistent with the known population of FRBhosts, but is too distant to detect in our observations ($z>0.7$ for a hostlike that of the first repeating FRB source, FRB 20121102A); that it is morenearby with a significant excess in $\mathrm{DM_{host}}$, and thus dimmer thanany known FRB host; or, least likely, that the FRB is truly hostless. Weconsider each possibility, making use of the population of known FRB hosts toframe each scenario. The fact of the missing host has ramifications for the FRBfield: even with high-precision localisation and deep follow-up, some FRB hostsmay be difficult to detect, with more distant hosts being the less likely to befound. This has implications for FRB cosmology, in which high-redshiftdetections are valuable.

The unseen host galaxy and high dispersion measure of a precisely-localised Fast Radio Burst suggests a high-redshift origin