Abstract
Following a recent proposal, we delineate a general procedure to classify 5d SCFTs via compactifications of 6d SCFTs on a circle (possibly with a twist by a discrete global symmetry). The path from 6d SCFTs to 5d SCFTs can be divided into two steps. The first step involves computing the Coulomb branch data of the 5d KK theory obtained by compactifying a 6d SCFT on a circle of finite radius. The second step involves computing the limit of the KK theory when the inverse radius along with some other mass parameters is sent to infinity. Under this RG flow, the KK theory reduces to a 5d SCFT. We illustrate these ideas in the case of untwisted compactifications of rank one 6d SCFTs that can be constructed in F-theory without frozen singularities. The data of the corresponding KK theory can be packaged in the geometry of a Calabi-Yau threefold that we explicitly compute for every case. The RG flows correspond to flopping a collection of curves in the threefold and we formulate a concrete set of criteria which can be used to determine which collection of curves can induce the relevant RG flows, and, in principle, to determine the Calabi-Yau geometries describing the endpoints of these flows. We also comment on how to generalize our results to arbitrary rank.
Highlights
There is a long standing dream that it will be possible to obtain all lower dimensional quantum field theories via compactifications of higher dimensional quantum field theories
Under this RG flow, the KK theory reduces to a 5d SCFT
The data of the corresponding KK theory can be packaged in the geometry of a Calabi-Yau threefold that we explicitly compute for every case
Summary
There is a long standing dream that it will be possible to obtain all lower dimensional quantum field theories via compactifications of higher dimensional quantum field theories. The process of blowing down curves physically corresponds to performing RG flows where some BPS particles are integrated out of the theory Based on this observation, the following conjecture was proposed in [3], which we reiterate: Every rank n 5d SCFT can be obtained by a rank preserving RG flow starting from a 6d SCFT compactified on a circle with/without a discrete twist. We note that some special cases of our results were already obtained by [12] who studied the Calabi-Yau threefolds corresponding to very special rank one 6d SCFTs that are completely Higgsed in the sense that they cannot be Higgsed to obtain some other 6d theory Let us close this introduction with a justification for capturing the data of 5d KK theories in terms of Calabi-Yau geometries rather than proceeding field theoretically. In appendix B, we provide sample computations of the Calabi-Yau threefolds for some hand-picked KK theories that illustrate some key features of our results
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