Geometric shape optimization via gravity duality in specific condensed matter systems

Since the advent of the Anti-deSitter/Conformal Field Theory (AdS/CFT) correspondence of string theory, the implementation of gravity duality theories has seen a plethora of use for applications ranging from black hole thermodynamics in quantum gravity to phase transitions and their critical temperatures in condensed matter. In this presentation we detail two new examples of such dualities for specific constrained condensed matter systems. First, we outline the duality between the extraordinary magnetoresistance (EMR), observed in semiconductor-metal hybrids, and non-symmetric gravity coupled to a diffusive $U(1)$ gauge field. The corresponding gravity theory may be interpreted as the generalized complex geometry of the semi-direct product of the symmetric metric and the antisymmetric Kalb-Ramond field: ($g_{\mu\nu}+\beta_{\mu\nu}$). We construct the four dimensional covariant field theory and compute the resulting equations of motion. The equations encode the most general form of EMR within a well defined variational principle, for specific lower dimensional embedded geometric scenarios. Second, We provide a unique and novel extension of da Costa's calculation of a quantum mechanically constrained particle. This is achieved by analyzing the perturbative back reaction of the quantum confined particle's eigenstates and spectra upon the geometry of the curved surface itself, thereby addressing the problem of shape optimization in this scenario. We do this by first formulating a two dimensional action principle of the quantum constrained particle, which upon wave function variation reproduces Schr\"odinger's equation including da Costa's surface curvature induced potentials. We further demonstrate that our derived action principle is dual to a two dimensional dilation gravity theory. We vary this dual gravity theory with respect to the embedded two dimensional inverse-metric to obtain the respective geometrodynamical Einstein equation, providing a full pathway for shape optimization in this scenario.