Holomorphic Currents and Duality in N=1 Supersymmetric Theories

TL;DR

This paper explores holomorphic currents in twisted N=1 supersymmetric theories on Riemann surfaces, demonstrating their invariance under RG flow and their role in electric-magnetic duality, with detailed analysis of SU(N_c) SQCD.

Contribution

It introduces the algebra of holomorphic currents in twisted N=1 theories and shows their invariance under RG flow and duality, providing detailed checks in SU(N_c) SQCD.

Findings

Holomorphic current algebra is invariant under RG flow.

Duality maps preserve the algebra of holomorphic currents.

Agreement with Kutasov, Schwimmer, and Seiberg duality in SU(N_c) SQCD.

Abstract

Twisted supersymmetric theories on a product of two Riemann surfaces possess non-local holomorphic currents in a BRST cohomology. The holomorphic currents act as vector fields on the chiral ring. The OPE's of these currents are invariant under the renormalization group flow up to BRST-exact terms. In the context of electric-magnetic duality, the algebra generated by the holomorphic currents in the electric theory is isomorphic to the one on the magnetic side. For the currents corresponding to global symmetries this isomorphism follows from 't Hooft anomaly matching conditions. The isomorphism between OPE's of the currents corresponding to non-linear transformations of fields of matter imposes non-trivial conditions on the duality map of chiral ring. We consider in detail the $SU(N_c)$ SQCD with matter in fundamental and adjoint representations, and find agreement with the duality map proposed by Kutasov, Schwimmer and Seiberg.