  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.
