We recently conjectured a set of dualities relating two-dimensional orthogonal gauge theories with $mathcal{N}=(4,4)$ supersymmetry, analogous to Horis dualities with $mathcal{N}=(2,2)$ supersymmetry. Here we provide a quantitative test of this conje
cture by computing the elliptic genera of the dual pairs and showing that they agree. The elliptic genus of orthogonal gauge theories has multiple topological sectors that depend on the global structure of the group and on the value of a discrete $theta$ parameter. We derive the dependence on the $theta$ parameter by determining whether a given sector has $(S)Pin$ structure or not.
We refine a previous proposal for obtaining the multi-instanton partition function from the supersymmetric index of the 1d supersymmetric gauge theory on the worldline of D0-branes. We provide examples where the refinements are crucial for obtaining the correct result.
We construct two-dimensional N=(2,2) supersymmetric gauge theories with orthogonal and symplectic groups using branes and orientifold planes in Type IIA string theory. A number of puzzles regarding the construction, including the effect of exchanging
NS5-branes on an orientifold 2-plane, are resolved by lifting the configurations to M theory. The low energy properties and conjectured dualities of these theories are reproduced in the M-brane description. A similar construction of N=(4,4) theories with orthogonal and symplectic groups leads to new duality conjectures for these theories.
