Six Easy Pieces: interplays among dualities in 4d, 3d and 2d

In this paper we consider 4d $\mathrm{SU}(N)$ gauge theories with $N+1$ fundamentals, five antifundamentals and a conjugate two index antisymmetric tensor. The model has been shown to be in a mixed phase in the IR, splitting in an interacting non-Abelian Coulomb phase and a free magnetic phase. Through tensor deconfinement, we show that baryonic deformations lead to a non-Abelian free magnetic phase. Along the analysis we obtain a duality with symplectic SQCD that can be further reduced to 3d and 2d. In the 3d case the analysis of the three sphere partition function allows one to obtain dualities between $\mathrm{SU}(N)$ with a two index symmetric tensor and $\mathrm{SO}(N)$ theories. On the other hand, in 2d we recover dualities already known in the literature and propose new ones between special unitary and symplectic gauge theories.

💡 Research Summary

In this work the authors investigate a class of four‑dimensional supersymmetric gauge theories with gauge group SU(N) that contain a conjugate antisymmetric tensor, N + 1 fundamentals and five antifundamentals. Earlier studies have shown that, in the infrared, these models reside in a mixed phase: an interacting non‑Abelian Coulomb sector coexists with a free magnetic sector. The present paper introduces baryonic superpotential deformations that are classically irrelevant in the UV but become “dangerously irrelevant’’ along the RG flow. Three families of deformations are considered: W = Āⁿ, W = Āⁿ⁻¹ Q̃₁ Q̃₂ and W = Āⁿ⁻² Q̃₁ Q̃₂ Q̃₃ Q̃₄.

Using the technique of tensor deconfinement, the antisymmetric tensor is replaced by a strongly coupled symplectic gauge node USp(2 m) (with m = n − 2, n − 1 or n depending on the deformation). The resulting quiver contains a spectator SU(2n) node that is flipped and forced to s‑confine. After s‑confinement and a subsequent Seiberg duality on the symplectic node, the magnetic description is a USp(2 m) SQCD with 2 m + 6 fundamentals together with singlet mesons and baryons coupled through a superpotential of the form
Wₘₐg = det(M₁|M₂) + B M₁ B̃₁ + M₂ B̃₂,
with additional σ R R terms when the deformation involves Q̃’s. The analysis of the superconformal index confirms the equality of the electric and magnetic partition functions.

Crucially, the baryonic deformations trigger an RG flow that eliminates the interacting Coulomb sector, leaving a purely free magnetic phase described either by an IR‑free USp(2) gauge theory or by an IR‑free USp(2 M) theory (M depends on the specific deformation). The latter theory is essential for the dimensional reductions that follow.

The authors then compactify the four‑dimensional duality on a circle to three dimensions, applying the ARSW prescription to translate the equality of superconformal indices into an identity between squashed‑three‑sphere partition functions. By freezing appropriate mass parameters and employing the duplication formula for the one‑loop determinant, the antisymmetric tensor of SU(N) is turned into a symmetric tensor, while the symplectic gauge group is mapped to an orthogonal group. This manipulation yields new three‑dimensional dualities between SU(N) gauge theories with a symmetric tensor and SO(K) gauge theories with vectors, together with the expected IR‑free USp(2) phase. The monopole superpotential generated by the compactification imposes constraints on the mass parameters, ensuring consistency of the duality.

Finally, the paper studies a twisted compactification on S² to obtain two‑dimensional (2,2) theories. The baryonic deformations dictate integer R‑charges that are used to define the twist. The authors perform a systematic analysis for the SU(2n) case, enumerating all admissible charge assignments. They recover several known 2d dualities (e.g. between SU and USp GLSMs) and, more importantly, discover new dualities that only arise when the four‑dimensional parent theory includes the baryonic superpotentials. In these new dualities, the IR description involves a special unitary gauge group on one side and a symplectic gauge group on the other, with matching chiral rings and elliptic genera.

Overall, the paper demonstrates that tensor deconfinement combined with carefully chosen baryonic deformations provides a powerful framework to resolve mixed phases into free magnetic phases, to generate a web of dualities across dimensions, and to uncover previously unknown relationships between SU, USp and SO gauge theories. The work opens several avenues for future research, such as extending the construction to other tensor representations, exploring the role of discrete anomalies in the reduction process, and applying the methodology to non‑Lagrangian sectors.