Compactifying 5d superconformal field theories to 3d

Abstract

Building on recent progress in the study of compactifications of 6d (1, 0) superconformal field theories (SCFTs) on Riemann surfaces to 4d$$ \mathcal{N} $$ N = 1 theories, we initiate a systematic study of compactifications of 5d$$ \mathcal{N} $$ N = 1 SCFTs on Riemann surfaces to 3d$$ \mathcal{N} $$ N = 2 theories. Specifically, we consider the compactification of the so-called rank 1 Seiberg $$ {E}_{N_f+1} $$ E N f + 1 SCFTs on tori and tubes with flux in their global symmetry, and put the resulting 3d theories to various consistency checks. These include matching the (usually enhanced) IR symmetry of the 3d theories with the one expected from the compactification, given by the commutant of the flux in the global symmetry of the corresponding 5d SCFT, and identifying the spectrum of operators and conformal manifolds predicted by the 5d picture. As the models we examine are in three dimensions, we encounter novel elements that are not present in compactifications to four dimensions, notably Chern-Simons terms and monopole superpotentials, that play an important role in our construction. The methods used in this paper can also be used for the compactification of any other 5d SCFT that has a deformation leading to a 5d gauge theory.

On the 3d compactifications of 5d SCFTs associated with SU(N+1) gauge theories

“…In this paper we extend this analysis to the study of compactifications of other known 5d SCFTs, which can be considered as a higher rank generalization of the findings of [2]. The rank 1 E N f +1 SCFTs for N f ≤ 7 can be obtained by starting from the 6d rank 1 E-string theory [5,6], compactifying it on a circle to obtain a 5d SCFT with E 8 global symmetry and then performing mass deformations that lead to lower values of N f .…”

“…Given the tremendous success in the study of dualities and symmetry enhancements of three-dimensional supersymmetric quantum fields theories, it is natural to wonder whether there is some general organizing principle behind them. One possible approach to this type of questions has been recently proposed in [2] following a similar idea that has proven to be very successful in the context of 4d N = 1 theories. Specifically, we can try to construct 3d N = 2 theories by compactifying 5d N = 1 SCFTs on Riemann surfaces with fluxes for their global symmetry through the surface.…”

“…Hence, the two different looking Lagrangians flow to this same SCFT in the IR, that is they are dual. This approach has been successfully applied to the rank 1 E N f +1 Seiberg SCFTs [3] with N f ≤ 6 2 in [2], where many three-dimensional models with symmetry enhancements or related by dualities have been found. Among these, it was in particular possible to recover one instance of Aharony duality [4].…”

“…One of these is the possibility of a Chern-Simons interaction for the gauge fields in three dimensions. In [2] it was observed that, even if the gauge theory description of the 5d SCFT doesn't have any CS interaction, it is in some cases necessary to turn on a non-trivial CS level for the gauge nodes of the resulting 3d quiver gauge theory in order for it to pass all the consistency checks. This is also required in order for the theory to be gauge invariant at the quantum level.…”

Holographic description of SCFT5 compactifications

“…There should be a way of relating our family of solutions with that of [53]. The formalism developed in [86] should apply for our backgrounds in section 5.2. There may be similar field theory elaborations for our backgrounds in sections 5.3 and 5.4.…”