Duflo--Serganova functor for chiral differential operators on classical invariant rings

We study the Duflo--Serganova functor for vertex algebras of chiral differential operators on classical invariant rings, introduced in \cite{ADS}. Our main result is a general cancellation theorem showing that a common bosonic and fermionic summand can be removed by Duflo--Serganova reduction. We apply this theorem to obtain Duflo--Serganova reduction isomorphisms for two families of simple orthosymplectic affine vertex superalgebras at levels $-2n$ and $n$, which are not covered by the existing results of Gorelik and Serganova. As a consequence, we obtain supercharacter identities, including the conjecture of Adamović and Milas as a special case.