Hidden symmetries of 4D $$ \mathcal{N} $$ = 2 gauge theories

A bstract We study the global symmetries of the ℤ 2 -orbifold of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 Super-Yang-Mills theory and its marginal deformations. The process of orbifolding to obtain an $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 theory would appear to break the SU(4) R-symmetry down to SU(2) × SU(2) × 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 SU( N ) × SU( N ) gauge groups to vary independently. The information about the marginal deformation is captured by a Drinfeld-type twist of this 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 SU(4) algebroid and we discuss implications of this hidden symmetry for the spectrum of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 theory.