4d/3d reduction of dualities with O6

TL;DR

This work studies the circle reduction of 4d Seiberg-like dualities for $\mathrm{U}(N)$ gauge theories with two-index tensor matter in the presence of $\mathrm{O6}$ orientifolds, using brane constructions and T-duality to guide the 3d dynamics. It compares the brane-derived dualities with field-theory analyses and localization results, identifying when the ARSW prescription suffices and when a double scaling limit on the 4d index is required to obtain convergent, consistent $3d$ partition functions. The authors obtain explicit 3d dual pairs for three tensor configurations (conjugate antisymmetric, conjugate symmetric, and mixed symmetric/antisymmetric) and provide detailed $S^3_b$ partition-function identities, including the emergence of monopole singlets and AHW-like couplings in the reduced theories. Their results unify the 4d/3d correspondence in these orientifolded setups, offering a robust framework to derive pure 3d dualities from higher-dimensional parents and to explore rich monopole operator structures. The work also outlines potential generalizations, such as adjoint deformations, alternative orientifold planes, and CS-term flows, highlighting the broader impact on the landscape of 3d dualities and their brane realizations.

Abstract

We consider $\mathrm{U}(N)$ gauge theories with a pair of two-index tensors interacting through a quartic superpotential, in addition to fundamentals and antifundamentals. The models have a brane engineering in terms of NS, D4, D6 branes and an O6 plane. Depending on the representation of the tensorial matter we have either an O6$^{+}$ plane, an O6$^{-}$ plane or a combined state of O6$^{+}$ and O6$^{-}$, with the addition of 8 semi-infinite half-D6 branes, where the last case realizes a chiral theory. The 4d IR duality is realized through an HW transition in the brane description. Here we study the circle reduction of these dualities from the brane perspective by T-dualizing along the compact direction. We then compare the results against the one obtained from field theoretical considerations and from localization, finding a precise agreement. When we consider the reduction of the 4d superconformal index to the 3d squashed three sphere partition function we observe that it is not always possible to obtain convergent 3d result with the standard reduction prescription, and that the double scaling limit is necessary.

Figures (5)

• Figure 1: In this figure we give a pictorial representation of the fivebranes and of the D4 branes considered in this paper, by representing their extension in directions $x_{45689}$.
• Figure 2: In this figure we consider an O6 plane extended along the directions $x_{0123789}$, an NS brane extended along $x_{012345}$ and an NS$'$ brane $x_{012389}$. If we consider an O6 plane with an NS$'$ brane the orientifold changes its RR charge at $x_7=0$ and RR charge conservation imposes the addition of eight semi-infinite (along $x_7$) half-D6 branes.
• Figure 3: In these pictures we represent the consistent ways to add Euclidean D1 branes (depicted in grey) to the brane setup in presence of an O5 plane on the central NS$'$ brane. The three configurations are obtained by moving some of the D3 branes (extended along $x_6$ in the figures) along the direction $x_3$. The first two configurations correspond to monopoles with opposite charges under the topological global symmetry $\mathop{\mathrm{U}}\nolimits(1)_J$ arising from the $\mathop{\mathrm{U}}\nolimits(1) \subset \mathop{\mathrm{U}}\nolimits(N)$ gauge symmetry. The last configurations is less common and refers to a monopole arising from $\mathop{\mathrm{SU}}\nolimits(N) \subset \mathop{\mathrm{U}}\nolimits(N)$. It is uncharged under the topological symmetry.
• Figure 4: This is a pictorial representation of the compact direction $x_3$, where $\frac{1}{r}$ represents is the periodicity of the compact scalar $\sigma$ on $S^1$. The red circles at the origin and at the point $\frac{1}{2r}$, often denoted in the body of the paper as the mirror point, represent the positions of the orientifolds in the geometric setup.
• Figure 5: A pictorial representation of the $\mathop{\mathrm{U}}\nolimits(3) \times \mathop{\mathrm{U}}\nolimits(3)$ studied in this appendix.