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Abstract: We observe that the Hamiltonian , where is the flat 4d Dirac operator in a self-dual gauge background, is supersymmetric, admitting 4 different real supercharges. A generalization of this model to the motion on a curved conformally flat 4d manifold exists. For an Abelian self-dual background, the corresponding Lagrangian can be derived from known harmonic superspace expressions. [Copyright &y& Elsevier]

Abstract: We study the duality of the supersymmetric self-dual and Maxwell–Chern–Simons theories coupled to a fermionic matter superfield, using a master action. This approach evades the difficulties inherent to the quartic couplings that appear when matter is represented by a scalar superfield. The price is that the spinorial matter superfield represents a unusual supersymmetric multiplet, whose main physical properties we also discuss. [Copyright &y& Elsevier]

We formulate noncommutative self-dual N=4 supersymmetric Yang–Mills theory in D=2+2 dimensions. As in the corresponding commutative case, this theory can serve as the possible master theory of all the noncommutative supersymmetric integrable models in lower dimensions. As a by-product, noncommutative self-dual N=2 supersymmetric Yang–Mills theory is obtained in D=2+2. We also perform a dimensional reduction of the N=2 theory further into N=(2,2) in D=1+1, as a basis for more general future applications. As a typical example, we show how noncommutative integrable matrix N=(1,0) supersymmetric KdV equations in D=1+1 arise from this theory, via the Yang–Mills gauge groups GL(n,R) or SL(2n,R). [Copyright &y& Elsevier]

We present self-dual N=2 supergravity in superspace for Euclidean seven dimensions with the reduced holonomy G2⊂SO(7), including all higher-order terms. As its foundation, we first establish N=2 supergravity without self-duality in Euclidean seven dimensions. We next show how the generalized self-duality in terms of octonion structure constants can be consistently imposed on the superspace constraints. We found two self-dual N=2 supergravity theories possible in 7D, depending on the representations of the two spinor charges of N=2. The first formulation has both of the two spinor charges in the 7 of G2 with 24+24 on-shell degrees of freedom. The second formulation has both charges in the 1 of G2 with 16+16 on-shell degrees of freedom. [Copyright &y& Elsevier]

We construct chiral N=(1,0) self-dual supergravity in Euclidean eight dimensions with reduced holonomy Spin(7), including all the higher-order interactions in a closed form. We first establish the non-chiral N=(1,1) superspace supergravity in eight dimensions with SO(8) holonomy without self-duality, as the foundation of the formulation. In order to make the whole computation simple, and the generalized self-duality compatible with supersymmetry, we adopt a particular set of superspace constraints similar to the one originally developed in ten-dimensional superspace. The intrinsic properties of octonionic structure constants make local supersymmetry, generalized self-duality condition, and reduced holonomy Spin(7) all consistent with each other. [Copyright &y& Elsevier]