Hidden 12-dimensional structures in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ and $M^4\times R^6$ supergravities

It is shown that ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ supergravity has hitherto unnoticed supersymmetric properties that are related to a hidden 12-dimensional structure. The totality of the ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ supergravity Kaluza-Klein towers is given by a single superfield that describes the quantum states of a 12-dimensional supersymmetric particle. The particle has super phase space ${(X}^M{,P}^M,\Theta ),$ with a $\mathrm{}(10,2)\mathrm{}$ signature and 32 fermions. The worldline action is constructed as a generalization of the supersymmetric particle action in two-time physics. $\mathrm{SU}(2,2\mathrm{}|4)$ is a linearly realized global supersymmetry of the 2T action. The action is invariant under the gauge symmetries Sp $\mathrm{}(2,R),$ $\mathrm{SO}\mathrm{}(4,2),$ $\mathrm{SO}\mathrm{}\left(6\right),$ and fermionic kappa. These gauge symmetries ensure unitarity and causality while allowing the reduction of the 12-dimensional super phase space to the correct super phase space for ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ or $M^4\times R^6$ with 16 fermions and one time, or other dually related one time spaces. One of the predictions of this formulation is that all of the SU$(2,2\mathrm{}|4)$ representations that describe Kaluza-Klein towers in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ or $M^4\times R^6$ supergravity universally have vanishing eigenvalues for all the Casimir operators. This prediction has been partially verified directly in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ supergravity. This suggests that the supergravity spectrum supports a hidden $\mathrm{}(10,2)\mathrm{}$ structure. A possible duality between ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ and $M^4\times R^6$ supergravities is also indicated. Generalizations of the approach applicable 10-dimensional super Yang-Mills theory and 11-dimensional M theory are briefly discussed.