We derive a 2 + 1 dimensional model with unconventional supersymmetry at the boundary of an AdS 4 N -extended supergravity, generalizing previous results. The (unconventional) extended supersymmetry of the boundary model is instrumental in describing, within a topdown approach, the electronic properties of graphene-like 2D materials at the two Dirac points, K and K . The two valleys correspond to the two independent sectors of the OSp(p|2)×OSp(q|2) boundary model in the p = q case, which are related by a parity transformation. The Semenoff and Haldane-type masses entering the corresponding Dirac equations are identified with the torsion parameters of the substrate in the model.

We develop in detail the holographic framework for an $$ \mathcal{N} $$ N = 2 pure AdS supergravity model in four dimensions, including all the contributions from the fermionic fields and adopting the Fefferman-Graham parametrization. We work in the first order formalism, where the full superconformal structure can be kept manifest in principle, even if only a part of it is realized as a symmetry on the boundary, while the remainder has a non-linear realization. Our study generalizes the results presented in antecedent literature and includes a general discussion of the gauge-fixing conditions on the bulk fields which yield the asymptotic symmetries at the boundary. We construct the corresponding super- conformal currents and show that they satisfy the related Ward identities when the bulk equations of motion are imposed. Consistency of the holographic setup requires the super- AdS curvatures to vanish at the boundary. This determines, in particular, the expression of the super-Schouten tensor of the boundary theory, which generalizes the purely bosonic Schouten tensor of standard gravity by including gravitini bilinears. The same applies to the superpartner of the super-Schouten tensor, the conformino. Furthermore, the vanishing of the supertorsion poses general constraints on the sources of the three-dimensional boundary conformal field theory and requires that the super-Schouten tensor is endowed with an antisymmetric part proportional to a gravitino-squared term.

Motivated by the appearance of associativity anomalies in the context of superstring field theory, we give a generalized solution built from boundary condition changing operators which can be associated to a generic tachyon vacuum in the KBc subalgebra of the Okawa form. We articulate sufficient conditions on the choice of tachyon vacuum to ensure that ambiguous products do not appear in the equations of motion.