Abstract
We classify orbifold geometries which can be interpreted as moduli spaces of four-dimensional \mathcal{N}\geq 3𝒩≥3 superconformal field theories up to rank 2 (complex dimension 6). The large majority of the geometries we find correspond to moduli spaces of known theories or discretely gauged version of them. Remarkably, we find 6 geometries which are not realized by any known theory, of which 3 have an \mathcal{N}=2𝒩=2 Coulomb branch slice with a non-freely generated coordinate ring, suggesting the existence of new, exotic \mathcal{N}=3𝒩=3 theories.
Highlights
Theoretical physicists’ wild dream of mapping the space of quantum field theories, even when restricted to unitary, local, and Poincaré-invariant ones, is probably unattainable
In this manuscript we have carried out the analysis of the rank 2 geometries which can be interpreted as moduli spaces of N = 3 theories
A crucial assumption that we make is that all such geometries are orbifolds of 3r though, as explained in [7], it remains an open question whether this is the case
Summary
Theoretical physicists’ wild dream of mapping the space of quantum field theories, even when restricted to unitary, local, and Poincaré-invariant ones, is probably unattainable. It follows in these cases that as a complex space CΓ is an algebraic variety not isomorphic to r. It is worth remarking that we don’t perform our classification by directly studying the finite subgroups of Sp(4, ) which would give rise to consistent rank-2 TSK geometries, but instead by using a related property of MΓ that follows from N = 3 supersymmetry This property is that, for any rank r, the matrix τi j of EM couplings on MΓ , which by standard arguments is an element of the fundamental domain of the Siegel upper half space, Hr , is fixed by the action of Γ ⊂ Sp(2r, ). We will show how to do this when r ≤ 2 in section 3, and turn to justifying assertions (1)–(3)
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