In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to
oriented $\mathbb C^*$-equivariant cohomology theories. Here we study the
K-theoretic refinement. It gives rational functions in $t^{1/2}$ invariant
under $t^{1/2}\leftrightarrow t^{-1/2}$ which specialise to numerical
Vafa-Witten invariants at $t=1$.
On the "instanton branch" the invariants give the virtual
$\chi_{-t}^{}$-genus refinement of G\"ottsche-Kool. Applying modularity to
their calculations gives predictions for the contribution of the "monopole
branch". We calculate some cases and find perfect agreement. We also do
calculations on K3 surfaces, finding Jacobi forms refining the usual modular
forms, proving a conjecture of G\"ottsche-Kool.
We determine the K-theoretic virtual classes of degeneracy loci using
Eagon-Northcott complexes, and show they calculate refined Vafa-Witten
invariants. Using this Laarakker [Laa] proves universality results for the
invariants.