We review Vafa-Witten theory in the more general setting, where the underlying moduli space is not the moduli space of instantons, but the full Vafa-Witten equation. (i) the new Vafa-Witten 4-manifold invariant associated with this moduli space, (ii) the relation with the Gromov-Witten invariant, (iii) the new Vafa-Witten Floer homology assigned to the 3-manifold boundary, (iv) New Vafa-Witten Atiyah-Floer support (v) Abouzaid-Manolescu proof and generalization of the conjecture [1] For the hypercohomology of perverse sheaves of vanishing cycles, (vi) the Langlands duality of these invariants, the Floer homology and the hypercohomology, and (vii) with purely imaginary parameters dedicated to the classical correspondences in zero coupling Quantum geometric Langlands correspondences (ii), (iv), (vi), and (vii) restrict the Higgs bundle features. We also describe how these invariants and homologies are classified in the process and discuss their higher classification. Thus, we relate differential and enumerative geometry, topology and geometric representation theory in mathematics via maximum supersymmetric topological quantum field theory with electromagnetic duality in physics.
