On the Fermionic Frequencies of Circular Strings

TL;DR

The paper resolves longstanding mismatches between worldsheet and algebraic-curve computations of fermionic fluctuation frequencies for circular strings in $AdS_5\times S^5$ and $AdS_4\times \mathbb{CP}^3$ by carefully incorporating spin-bundle transition matrices, i.e. the target-space spin structure, into the semiclassical analysis. Through explicit treatment of patches around coordinate singularities and appropriate boundary conditions, the fermionic frequencies computed on the worldsheet align with the algebraic-curve predictions in both the $su(2)$ and $sl(2)$ sectors across the two backgrounds. It also discusses how a corrected classical solution and coset-based formalisms avoid these issues and ensure consistent periodicity. Overall, the work clarifies how global topological data of the target space influence semiclassical spectra in AdS/CFT contexts and provides a robust method for reconciling different computational approaches.

Abstract

We revisit the semiclassical computation of the fluctuation spectrum around different circular string solutions in AdS_5xS^5 and AdS_4xCP^3, starting from the Green-Schwarz action. It has been known that the results for these frequencies obtained from the algebraic curve and from the worldsheet computations sometimes do not agree. In particular, different methods give different results for the half-integer shifts in the mode numbers of the frequencies. We find that these discrepancies can be removed if one carefully takes into account the transition matrices in the spin bundle over the target space.