  The Virasoro constraints for moduli spaces of stable torsion free sheaves on
a surface with only $(p,p)$-cohomology were recently proved by
Bojko-Moreira-Lim. The rank 1 case, which is not restricted to surfaces with
only $(p,p)$-cohomology, was established by Moreira. We prove Virasoro
constraints for K3 surfaces using Markman monodromy operators, which allow us
to reduce to the rank 1 case. We also prove new Virasoro constraints in rank 0.
Finally, for K3 surfaces, we introduce new Virasoro operators in negative
degree which, together with the previous Virasoro operators, give a
representation of Virasoro algebra with central charge $24$.
