Hidden Symmetries of 4D N=2 Gauge Theories

TL;DR

This paper uncovers hidden symmetries in 4D N=2 gauge theories using Lie algebroid structures and Drinfeld twists, revealing non-associative algebraic features after marginal deformations.

Contribution

It introduces a novel Lie algebroid framework and Drinfeld twist approach to analyze symmetries in N=2 gauge theories beyond traditional Lie algebra methods.

Findings

Hidden symmetries are recovered via Lie algebroids.

Marginal deformations induce non-associative algebraic structures.

Planar Lagrangian invariance under twisted symmetries is demonstrated.

Abstract

We study the global symmetries of the $\mathbb{Z}_2$-orbifold of N=4 Super-Yang-Mills theory and its marginal deformations. The process of orbifolding to obtain an N=2 theory would appear to break the $\mathrm{SU}(4)$ R-symmetry down to $\mathrm{SU}(2)\times \mathrm{SU}(2)\times \mathrm{U}(1)$. We show that the broken generators can be recovered by moving beyond the Lie algebraic setting to that of a Lie algebroid. This remains true when marginally deforming away from the orbifold point by allowing the couplings of the $ \mathrm{SU}(N)\times \mathrm{SU}(N)$ gauge groups to vary independently. The information about the marginal deformation is captured by a Drinfeld-type twist of this $\mathrm{SU}(4)$ Lie algebroid. The twist is read off from the F- and D- terms, and thus directly from the Lagrangian. Even though at the orbifold point the algebraic structure is associative, it becomes non-associative after the marginal deformation. We explicitly check that the planar Lagrangian of the theory is invariant under this twisted version of the $\mathrm{SU}(4)$ algebroid and we discuss implications of this hidden symmetry for the spectrum of the N=2 theory.