Abstract
We study the compactification of the 6d ${\cal N}=(2,0)$ SCFT on the product of a Riemann surface with flux and a circle. On the one hand, this can be understood by first reducing on the Riemann surface, giving rise to 4d ${\cal N}=1$ and ${\cal N}=2$ class ${\cal S}$ theories, which we then reduce on $S^1$ to get 3d ${\cal N}=2$ and ${\cal N}=4$ class ${\cal S}$ theories. On the other hand, we may first compactify on $S^1$ to get the 5d ${\cal N}=2$ Yang-Mills theory. By studying its reduction on a Riemann surface, we obtain a mirror dual description of 3d class ${\cal S}$ theories, generalizing the star-shaped quiver theories of Benini, Tachikawa, and Xie. We comment on some global properties of the gauge group in these reductions, and test the dualities by computing various supersymmetric partition functions.
Highlights
Studying quantum field theories (QFTs) on compact spaces often leads to insights into complicated dynamics of lower dimensional theories
Another example is the many insights derived in recent years, following the seminal paper [6], about strong coupling dynamics of 4D N ≥ 1 by understanding them as compactifications of 6D superconformal field theories (SCFTs) on Riemann surfaces
We may compactify with a flux, n, for the Spð1ÞF 6D flavor symmetry on the Riemann surface, which leads to an N 1⁄4 2 model in 3D, and here we find simple dual descriptions with a number of adjoint chiral multiplets for the central node which is linear in the flux, n
Summary
Studying quantum field theories (QFTs) on compact spaces often leads to insights into complicated dynamics of lower dimensional theories. A particular example is understanding N 1⁄4 2 dualities in three-dimensional (3D) starting from 4D N 1⁄4 1 dual theories compactified on a circle [1,2,3,4,5] Another example is the many insights derived in recent years, following the seminal paper [6], about strong coupling dynamics of 4D N ≥ 1 by understanding them as compactifications of 6D SCFTs on Riemann surfaces. We can either first try to understand the reduction on a circle down to 5D and a subsequent compactification to 3D, or first compactify to 4D and to 3D The former way has the advantage that, the 6D theories are not given in terms of Lagrangians, often when compactified on a circle (possibly with holonomies for various symmetries), they possess an effective 5D description in terms of fields. We focus on the case of the 5D N 1⁄4 2 SYM theory, which will be our main example
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