Static gauge and energy spectrum of single-mode strings in $\mathrm{AdS_5\times S^5}$

Motivated by the notorious difficulties in determining the first quantum corrections to the spectrum of short strings in $\mathrm{AdS_5\times S^5}$ from first principles, we study closed bosonic strings in this background employing a static gauge. In this gauge the world-sheet Hamiltonian density is constant along the extension of the string and directly proportional to the square of the spacetime energy. We quantize this system in a minisuperspace approach, in which we consider only a single $\mathrm{AdS_5}$ string-mode excitation next to an arbitrary particle like zero-mode contribution in the full $\mathrm{AdS_5\times S^5}$ background. We determine the quantum spectrum using this method to the next-to-next-to-leading order in the large 't Hooft coupling expansion. We argue for an ordering prescription which should arise from supersymmetrization and indeed recover the integrability based predictions for the spectrum of the lightest excitation, dual to the Konishi field scaling dimensions. The higher excitations fail to agree, but this is shown to be a consequence of the string-mode truncation employed. Despite this simple setup, our system reveals intriguing features, such as a close connection to particles in $\mathrm{AdS_6}$, classical integrability and preservation of the isometries of $\mathrm{AdS_5\times S^5}$ at the quantum level.