Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification

TL;DR

The paper demonstrates that non-invertible symmetries in four-dimensional ${\cal N}=4$ SYM, arising from Montonen-Olive duality and one-form gauging, can be harnessed through twisted compactification to generate entirely new three-dimensional ${\cal N}=6$ theories. It systematically catalogs non-invertible defects across all gauge algebras (including exceptional and non-simply-laced cases) and analyzes how twisting by these defects yields novel RG flows and 3d SCFTs with moduli spaces described by complex reflection groups. In particular, the resulting 3d theories are often ABJ-type, and the moduli spaces match ${\mathbb C}^{4n}/\Gamma$ for appropriate complex reflection groups $\Gamma$, extending the landscape beyond invertible twists. This work broadens the role of non-invertible symmetries as a constructive tool for exploring new QFTs and their IR dynamics, with implications for dualities, RG flows, and string/M-theory realizations of 3d ${\cal N}=6$ theories.

Abstract

Non-invertible symmetries have recently been understood to provide interesting contraints on RG flows of QFTs. In this work, we show how non-invertible symmetries can also be used to generate entirely new RG flows, by means of so-called "non-invertible twisted compactification". We illustrate the idea in the example of twisted compactifications of 4d $\mathcal{N}=4$ super-Yang-Mills (SYM) to three dimensions. After giving a catalogue of non-invertible symmetries descending from Montonen-Olive duality transformations of 4d $\mathcal{N}=4$ SYM, we show that twisted compactification by non-invertible symmetries can be used to obtain 3d $\mathcal{N}=6$ theories which appear otherwise unreachable if one restricts to twists by invertible symmetries.

Figures (12)

• Figure 1: At $\tau_{\mathrm{YM}}=i$, the $SU(2)$ theory has a non-invertible defect ${\cal N}$, which can be understood as the composition of a defect $\sigma$ implementing gauging of the ${\mathbb Z}_2^{(1)}$ one-form symmetry, together with an invertible ${\mathsf S}$ defect.
• Figure 2: Left: an insertion of ${\cal N}^2$ at a point on $S^1$ and wrapping $M_d$. Right: after having used the fusion rules, we may replace this with an insertion of the condensate (blue) on all two-cycles of $M_d$. This shows that twisted compactification by the square ${\cal N}^2$ of a duality defect is equivalent to normal compactification together with gauging in the remaining three-dimensions (up to charge conjugation).
• Figure 3: Twisted compactification of $SU(2)$ SYM and $SO(3)_+$ SYM by the non-invertible defect ${\cal N}$ gives the same theory in three dimensions. This can be seen by splitting and moving the topological defects in the manner shown above.
• Figure 4: Modular transformations for theories with gauge algebra $\mathfrak{su}(2)$, with background gauge fields for the one-form symmetry turned off.
• Figure 5: Web of transformations for theories with gauge algebra $\mathfrak{su}(2)$. The transformations in green are the Montonen-Olive duality transformations, generating $SL(2, {\mathbb Z})$. The transformations in red are topological manipulations generating $SL(2, {\mathbb Z}_2)$.