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

Equivariant K-theory and refined Vafa-Witten invariants