Mysterious zero in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ supergravity

It is shown that all the states in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ supergravity have zero eigenvalue for all Casimir eigenvalues of its symmetry group SU(2,2|4). To compute this zero in supergravity we refine the oscillator methods for studying the lowest weight unitary representations of $\mathrm{SU}(N,M|R+S).\mathrm{}$ We solve the reduction problem when one multiplies an arbitrary number of super-doubletons. This enters in the computation of the Casimir eigenvalues of the lowest weight representations. We apply the results to SU(2,2|4) that classifies the Kaluza-Klein towers of ten-dimensional type IIB supergravity compactified on ${\mathrm{AdS}}_5\times {\mathrm{S}}^5.$ We show that the vanishing of the SU(2,2|4) Casimir eigenvalues for all the states is indeed a group-theoretical fact in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ supergravity. By the AdS-CFT correspondence, it is also a fact for gauge invariant states of super-Yang-Mills theory with four supersymmetries in four dimensions. This nontrivial and mysterious zero is very interesting because it is predicted as a straightforward consequence of the fundamental local Sp(2) symmetry in 2T-physics. Via the 2T-physics explanation of this zero we find a global indication that these special supergravity and super-Yang-Mills theories hide a twelve-dimensional structure with (10,2) signature.