Defects in the supersymmetric mKdV hierarchy via Bäcklund transformations

Abstract

The integrability of the N = 1 supersymmetric modified Korteweg de-Vries (smKdV) hierarchy in the presence of defects is investigated through the construction of its super Bäcklund transformation. The construction of such transformation is performed by using essentially two methods: the Bäcklund-defect matrix approach and the superfield approach. Firstly, we employ the defect matrix associated to the hierarchy which turns out to be the same for the supersymmetric sinh-Gordon (sshG) model. The method is general for all flows and as an example we derive explicitly the Bäcklund equations in components for the first few flows of the hierarchy, namely t 3 and t 5 . Secondly, the supersymmetric extension of the Bäcklund transformation in the superspace formalism is constructed for those flows. Finally, this super Bäcklund transformation is employed to introduce type I defects for the supersymmetric mKdV hierarchy. Further integrability aspects by considering modified conserved quantities are derived from the defect matrix.

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy

“…Moreover, based on this fact it has been shown that the spatial component of the Bäcklund transformation, and consequently the associated defect matrix, are also common and henceforth universal within the entire bosonic hierarchy [10,11]. This agrees More recently, in [1], this result has been extended to the supersymmetric mKdV hierarchy with the following defect matrix…”

“…where m is the degree of the subspace G m according to Q, i.e., [Q, (1) and M(E) is its complement ( image). Now we can define the Lax pair as…”

“…In [1] this procedure have been aplied to obtain these transformations for the t 5 member of the smKdV hierarchy.…”

“…Now, using the x-part of the Bäcklund transformation (64)-(66) in these two equations we end up with the corresponding Bäcklund transformation for n = 3, that was obtained in [1].…”

Quantum anomalies in A(1) r Toda theories with defects

“…For theories defined on the whole line the quantum conservation laws have been studied in [17] where a suitable renormalization of the currents preserving conservation laws has been found. Classically it is possible to introduce boundary [58] and defects [59][60][61][62][63] but a systematic classification of possibly susy preserving boundaries and defects at quantum level is still lacking.…”