Supersymmetric exact sequence, heat kernel and super KdV hierarchy

TL;DR

This paper develops an algebraic and analytic framework for the N=1 super KdV hierarchy, establishing an exact supersymmetric sequence and analyzing the super heat operator's Green's function to generate hierarchy members.

Contribution

It introduces a supersymmetric derivation ring, proves an exact sequence, and links the super heat kernel's asymptotics to the super KdV hierarchy.

Findings

Established an exact sequence of supersymmetric rings.

Derived the Green's function for the super heat operator.

Connected heat kernel asymptotics to super KdV hierarchy.

Abstract

We introduce the free N=1 supersymmetric derivation ring and prove the existence of an exact sequence of supersymmetric rings and linear transformations. We apply necessary and sufficient conditions arising from this exact supersymmetric sequence to obtain the essential relations between conserved quantities, gradients and the N=1 super KdV hierarchy. We combine this algebraic approach with an analytic analysis of the super heat operator.We obtain the explicit expression for the Green's function of the super heat operator in terms of a series expansion and discuss its properties. The expansion is convergent under the assumption of bounded bosonic and fermionic potentials. We show that the asymptotic expansion when $t\to0^+$ of the Green's function for the super heat operator evaluated over its diagonal generates all the members of the N=1 super KdV hierarchy.