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We generalize the construction of M. Lieblich for the compactification of themoduli stack of $\PGL_r$-bundles on algebraic spaces to the moduli stack ofTanaka-Thomas $\PGL_r$-Higgs bundles on algebraic schemes. The method we use isthe moduli stack of Higgs version of Azumaya algebras. In the case of smoothsurfaces, we obtain a virtual fundamental class on the moduli stack of$\PGL_r$-Higgs bundles. An application to the Vafa-Witten invariants isdiscussed.

On the construction of moduli stack of projective Higgs bundles over surfaces