Remarks on the geometrical properties of semiclassically quantized strings

We discuss some geometrical aspects of the semiclassical quantization of string solutions in type IIB Green–Schwarz action on ${{\rm{AdS}}}_{5}\times {{\rm{S}}}^{5}.$ We concentrate on quadratic fluctuations around classical configurations, expressing the relevant differential operators in terms of (intrinsic and extrinsic) invariants of the background geometry. The aim of our exercise is to present some compact expressions encoding the spectral properties of bosonic and fermionic fluctuations. The appearing of non-trivial structures on the relevant bundles and their role in concrete computations are also considered. We corroborate the presentation of general formulas by working out explicitly a couple of relevant examples, namely the spinning string and the latitude BPS Wilson loop.