Lax operator and superspin chains from 4D CS gauge theory

Abstract

Abstract We study the properties of interacting line defects in the four-dimensional Chern Simons (CS) gauge theory with invariance given by the S L m | n super-group family. From this theory, we derive the oscillator realisation of the Lax operator for superspin chains with SL(m|n) symmetry. To this end, we investigate the holomorphic property of the bosonic Lax operator L and build a differential equation D L = 0 solved by the Costello–Gaioto–Yagi realisation of L in the framework of the CS theory. We generalize this construction to the case of gauge super-groups, and develop a Dynkin super-diagram algorithm to deal with the decomposition of the Lie superalgebras. We obtain the generalisation of the Lax operator describing the interaction between the electric Wilson super-lines and the magnetic ’t Hooft super-defects. This coupling is given in terms of a mixture of bosonic and fermionic oscillator degrees of freedom in the phase space of magnetically charged ’t Hooft super-lines. The purely fermionic realisation of the superspin chain Lax operator is also investigated and it is found to coincide exactly with the Z 2 -gradation of Lie superalgebras.