Remarks on the geometrical properties of semiclassically quantized strings

Abstract

We discuss some geometrical aspects of the semiclassical quantization of string solutions in Type IIB Green-Schwarz action on AdS 5 × 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 exercises 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.1

Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT

“…A brief account of the subject and references are in [37,38]. 3 Among the many reviews on the topic we suggest [28,[43][44][45][46][47] and the excellent textbook [48].…”

“…In chapter 3, which is largely based on [3], we present a pedagogical discussion about the general structure of the quadratic fluctuation Lagrangian around arbitrary classical configurations in AdS 5 × S 5 , expressing the relevant differential operators in terms of geometric invariants of the background geometry.…”

“…In addition to the Z 2 -grading implicit in the definition of superalgebra g -for whichg (0) ,g (2) are even andg (1) ,g (3) are odd subspaces -the automorphism endows g with the structure of a Z 4 -graded algebra [269].…”

“…and split it as A = A (0) + A (1) + A (2) + A (3) according to the Z 4 -decomposition. By construction the current is flat, namely it has vanishing two-form curvature F ≡ dA − A ∧ A = 0.…”

“…or AdS 4 × CP 3 . Note that quantum mechanical consistency guarantees the absence of conformal anomaly when the dimensionality of the spacetime is 10.…”