Nilpotent local fermionic symmetry with generalized self-duality in D = n + n dimensions embedding supersymmetric integrable models

Abstract

We present local nilpotent fermionic symmetry with a vector-spinor as its gauge field in D=2+2 space-time dimensions. Our field-content is (AμI,ψμI,χI), where I=1,2,⋯,g≡dim G is the adjoint-index of a gauge group G, while ψμI and χI are Majorana spinors. The field strengths FμνI of AμI and RμνI of ψμI satisfy the self-duality conditions FμνI=+(1/2)ϵμνρσFρσI and RμνI=+(1/2)ϵμνρσRρσI. The lowest flow of super-KdV equations and super-KP equations are embedded into this system in ways more natural than the conventional supersymmetric self-dual theories in D=2+2. This indicates that our theory even without supersymmetry in D=2+2 can serve as the “Master Theory” of all supersymmetric integrable systems in lower dimensions. Our D=2+2 result is further generalized to D=n+n(n=3,4,⋯) with the field content (AμI,ψμI,χI,Bμ1⋯μn−3), with the (n−3)-rank auxiliary Abelian tensor Bμ1⋯μn−3. The generalized self-dual conditions are ▪ and ▪. We conjecture that our theory in D=n+n is the “Grand-Master Theory” of all supersymmetric integrable systems in lower dimensions.