Semiclassical quantisation of finite-gap strings

We perform a first principle semiclassical quantisation of the general finite-gap solution to the equations of a string moving on × S³. The derivation is only formal as we do not regularise divergent sums over stability angles. Moreover, with regards to the AdS/CFT correspondence the result is incomplete as the fluctuations orthogonal to this subspace in AdS₅ × S⁵ are not taken into account. Nevertheless, the calculation serves the purpose of understanding how the moduli of the algebraic curve gets quantised semiclassically, purely from the point of view of finite-gap integration and with no input from the gauge theory side. Our result is expressed in a very compact and simple formula which encodes the infinite sum over stability angles in a succinct way and reproduces exactly what one expects from knowledge of the dual gauge theory. Namely, at tree level the filling fractions of the algebraic curve get quantised in large integer multiples of ℏ = 1/(λ)^1/2. At 1-loop order the filling fractions receive Maslov index corrections of ½ℏ and all the singular points of the spectral curve become filled with small half-integer multiples of ℏ. For the subsector in question this is in agreement with the previously obtained results for the semiclassical energy spectrum of the string using the method proposed in hep-th/0703191. Along the way we derive the complete hierarchy of commuting flows for the string in the × S³ subsector which are generated by the Taylor coefficients of the quasi-momentum p(x) through Hamilton's equation. Moreover, we also derive a very general and simple formula for the stability angles around a generic finite-gap solution which may be used in the study of stability properties of solutions in the × S³ subsector. We also stress the issue of quantum operator orderings and whether or not a given ordering preserves integrability since this problem already crops up at 1-loop in the form of the subprincipal symbol.