Abstract
We construct gauge theories with a vector multiplet and hypermultiplets of (1,0)(1,0) supersymmetry on the six-sphere. The gauge coupling on the sphere depends on the polar angle. This has a natural explanation in terms of the tensor branch of (1,0)(1,0) theories on the six-sphere. For the vector multiplet we give an off-shell formulation for all supersymmetries. For hypermultiplets we give an off-shell formulation for one supersymmetry. We show that the path integral for the vector multiplet localizes to solutions of the Hermitian-Yang-Mills equation, which is a generalization of the (anti-)self duality condition to higher dimensions. For the hypermultiplet, the path integral localizes to configurations where the field strengths of two complex scalars are related by an almost complex structure.
Highlights
Introduction and summaryThe study of supersymmetric theories on curved spaces has led to insights into the dynamics of strongly coupled theories
We show that the path integral for the vector multiplet localizes onto solutions of the Hermitian Yang-Mills (HYM) equation on 6 which is a generalization of theself-duality condition on gauge field in 4D
We wrote down explicit Lagrangians for vector and hypermultiplets with off-shell supersymmetry
Summary
The study of supersymmetric theories on curved spaces has led to insights into the dynamics of strongly coupled theories. This can be done by adding terms in the Lagrangian that depend on the position on the sphere Such constructions, though somewhat exotic, are not unfamiliar in supergravity and field theory. It is conceivable that the VEV on 6 gives a supersymmetric theory with non-constant coupling Let us demonstrate this point in a little more detail for the case of 4D N = 4 SYM. Broken supersymmetries correspond to special conformal supersymmetries in the flat space while the Poincaré supersymmetries are preserved This analysis is closely related to the existence of half-BPS scalar field configurations of maximally supersymmetric theories on AdS spacetime [22]. In this paper we construct theories on 6 with features described above, i.e., a non-constant coupling and (1, 0) supersymmetry. Appendices contain our conventions and technical details of numerous computations
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