| Summary: | We use conformal embeddings involving exceptional affine Kac-Moody algebras
to derive new dualities of three-dimensional topological field theories. These
generalize the familiar level-rank duality of Chern-Simons theories based on
classical gauge groups to the setting of exceptional gauge groups. For
instance, one duality sequence we discuss is $(E_{N})_{1}\leftrightarrow
SU(9-N)_{-1}$. Others such as $SO(3)_{8}\leftrightarrow PSU(3)_{-6},$ are
dualities among theories with classical gauge groups that arise due to their
embedding into an exceptional chiral algebra. We apply these equivalences
between topological field theories to conjecture new boson-boson Chern-Simons
matter dualities. We also use them to determine candidate phase diagrams of
time-reversal invariant $G_{2}$ gauge theory coupled to either an adjoint
fermion, or two fundamental fermions.
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