Supercurrents and (partial) supersymmetry in adjoint QCD2 and its generalizations

A bstract 1 + 1-dimensional SU( N ) gauge theory coupled to an adjoint Majorana fermion, also known as adjoint QCD 2 , has the surprising feature that at fermion mass $$ \sqrt{\frac{g^2N}{2\pi }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>g</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:mfrac> </mml:msqrt> </mml:math> it exhibits supersymmetry. In this paper, we obtain a deeper insight into how the supersymmetry works by constructing the gauge invariant, Lorentz covariant supercurrent j μA . Its conservation relies crucially on the presence of a quantum anomaly. We generalize this construction to a class of models where, in addition to an adjoint Majorana fermion of an appropriate mass, the gauge theory is coupled to some collection of massless fermions (SU( N ) may be replaced by a more general gauge group). In general, these models have a supersymmetric massive sector and a non-supersymmetric CFT sector [1], but there are cases in which both sectors are supersymmetric. An example of such a gapless, fully supersymmetric model is SU( N ) gauge theory coupled to three adjoint Majorana fermions, of which two are massless and the third has mass $$ \sqrt{\frac{3{g}^2N}{2\pi }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> <mml:msup> <mml:mi>g</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:mfrac> </mml:msqrt> </mml:math> .