Abstract
We consider the vacuum geometry of supersymmetric theories with 4 supercharges, on a flat toroidal geometry. The 2 dimensional vacuum geometry is known to be captured by the tt * geometry. In the case of 3 dimensions, the parameter space is (T 2 × \( \mathbb{R} \))N and the vacuum geometry turns out to be a solution to a generalization of monopole equations in 3N dimensions where the relevant topological ring is that of line operators. We compute the generalization of the 2d cigar amplitudes, which lead to S 2 × S 1 or S 3 partition functions which are distinct from the supersymmetric partition functions on these spaces, but reduce to them in a certain limit. We show the sense in which these amplitudes generalize the structure of 3d Chern-Simons theories and 2d RCFT’s. In the case of 4 dimensions the parameter space is of the form X M,N = (T 3 × \( \mathbb{R} \))M × T 3N , and the vacuum geometry is a solution to a mixture of generalized monopole equations and generalized instanton equations (known as hyper-holomorphic connections). In this case the topological rings are associated to surface operators. We discuss the physical meaning of the generalized Nahm transforms which act on all of these geometries.
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Cecotti, S., Gaiotto, D. & Vafa, C. tt * geometry in 3 and 4 dimensions. J. High Energ. Phys. 2014, 55 (2014). https://doi.org/10.1007/JHEP05(2014)055
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DOI: https://doi.org/10.1007/JHEP05(2014)055