Spectrum and Quantum Symmetries of the ${\rm AdS}_5 \times {\rm S}^5$ Superstring

The the duality between ${\cal N}=4$ SYM and the AdS$_5\times\;$S$^5$ superstring appears to enjoy quantum integrability in the planar limit, which allowed to devise powerful methods ostensibly solving the spectral problem. However, quantization of the AdS$_5\times\;$S$^5$ superstring from first principles is still an open question and especially the spectrum of short string states has previously been derived only at leading order in large 't Hooft coupling. In this thesis we investigate possible routes to take the perturbative quantization of short string states beyond the leading order, where equally our aim is to gain better appreciation of the quantum symmetries at play. A prominent role is played by the lowest excited string states, dual to the Konishi supermultiplet, and we start by reviewing critically an asserted derivation of the Konishi anomalous dimension in the setup of pure spinor string theory. Next, we study the bosonic AdS$_5\times\;$S$^5$ string in static gauge, where we construct a so-called single-mode string solution, which shows classical integrability and invariance under the isometries SO(2,4)$\;\times\;$SO(6) at the quantum level. Arguing heuristically about the effects of supersymmetry, we indeed recover the first non-trivial quantum correction to the Konishi anomalous dimension. We continue by implementing static gauge for the full AdS$_5\times\;$S$^5$ superstring and find elegant expressions for the Lagrangian density and the supercharges. We then constrain our interest to the superparticle and, using two different methods, find canonical coordinates at quadratic order in fermions. As the single-mode string is just the SO(2,4)$\;\times\;$SO(6) orbit of the pulsating string, we conclude by applying orbit method quantization to the particle and spinning string solutions in bosonic AdS$_3\times\;$S$^3$ yielding consistent results for the spectra.