3d dualities from 4d dualities (1305.3924v2)

Abstract: Many examples of low-energy dualities have been found in supersymmetric gauge theories with four supercharges, both in four and in three space-time dimensions. In these dualities, two theories that are different at high energies have the same low-energy limit. In this paper we clarify the relation between the dualities in four and in three dimensions. We show that every four dimensional duality gives rise to a three dimensional duality between theories that are similar, but not identical, to the dimensional reductions of the four dimensional dual gauge theories to three dimensions. From these specific three dimensional dualities one can flow to many other low-energy dualities, including known three dimensional dualities and many new ones. We discuss in detail the case of three dimensional SU(N_c) supersymmetric QCD theories, showing how to derive new duals for these theories from the four dimensional duality.

• The paper demonstrates that 4d dualities systematically reduce to 3d dualities while preserving equivalent IR physics.
• It employs circle compactification to analyze changes in the Coulomb branch and effects of monopole-induced superpotentials.
• The findings offer a unified framework for understanding supersymmetric gauge theories and guide further exploration into higher-dimensional quantum field theories.

Analyzing Low-Energy Dualities: A Transition From Four to Three Dimensions

This paper explores the intriguing interplay between supersymmetric gauge theories defined in four space-time dimensions (4d) and their counterparts in three dimensions (3d). The central theme revolves around dualities within these frameworks which, despite arising from seemingly distinct high-energy theories, yield equivalent infra-red (IR) physics. The manuscript makes notable contributions by demonstrating that dualities in four dimensions systematically reduce to dualities in three dimensions, thereby unifying these theoretical observations.

Dualities and Supersymmetry

Supersymmetric gauge theories are an active area of research due to their similarity to non-supersymmetric counterparts in phenomena such as confinement and chiral symmetry breaking, while simultaneously allowing precise control over strong coupling dynamics through supersymmetry. This paper explores dualities in theories with four supercharges, a class that exhibits low-energy equivalences across different high-energy configurations. Notably, the authors focus on supersymmetric QCD (sQCD) and USp(2N) theories, showcasing how their IR behavior is preserved under dimensional reduction from 4d to 3d.

Building the Bridge: Dimensional Reduction

The authors expound on the mechanistic transition from 4d dualities to their 3d equivalents. By compactifying 4d theories on a circle and analyzing the resulting effective field theories, they articulate how compactification affects the theory's Coulomb branch and moduli space. They address the transformations of gauge theories through careful consideration of monopole contributions and non-perturbative superpotentials induced by instantons on the compactified dimensions.

Key Results and Their Implications

Numerical Equivalence of Dualities

Numerous 4d dual pairs, such as the SU(N) and USp(2N) sQCD theories, are shown to yield valid 3d dualities. Notably, the dimensionally reduced 3d theories occasionally possess deformed or additional superpotentials, derived from the mother 4d theories’ non-perturbative dynamics. In particular, dualities in supersymmetric QCD are preserved when supplemented by specific monopole-induced superpotentials in the 3d setup.

Theoretical Continuity

The theoretical implications of these results are profound. By affirming that every 4d duality generates a consistent 3d duality, the paper enhances the understanding of dualities within the supersymmetric paradigm. The considerations of Euclidean partition functions on the torus further substantiate these observations, presenting a unifying thread that relates dualities across dimensions through consistency conditions on partition functions.

Future Prospects and Broader Context

The insights garnered from this analysis could pave the way for advancements in understanding the landscape of quantum field theories, particularly regarding their potential classification and interrelations in higher dimensions, such as 6d theories. Further, the exploration into Chern-Simons levels in 3d duals and the paper's combinatorial techniques could inspire fresh approaches in topological quantum field theories, potentially influencing fields such as condensed matter physics.

By methodically breaking down these complex duality structures, this paper provides a framework not only for unifying known dualities but also posits a hopeful conjecture that all known dualities might stem from higher-dimensional ancestor theories. This perspective could renew efforts towards deriving a comprehensive understanding of quantum field theories' IR landscapes, potentially unlocking novel insights into both gauge and string theory paradigms.