Minimal surfaces with mixed three-form flux

Abstract

We study minimal area world sheets ending on two concentric circumferences on the boundary of Euclidean AdS 3 with mixed Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz three-form fluxes. We solve the problem by reducing the system to a one-dimensional integrable model. We find that the Neveu-Schwarz-Neveu-Schwarz flux term either brings the surface near to the boundary or separates the circumferences. In the limit of pure Neveu-Schwarz-Neveu-Schwarz flux, the solution adheres to the boundary in the former case and the outer radius diverges in the latter. We further construct the underlying elliptic spectral curve, which allows us to analyze the deformation of other related minimal surfaces. We show that in the regime of pure Neveu-Schwarz-Neveu-Schwarz flux the elliptic curve degenerates.

Quantum corrections to minimal surfaces with mixed three-form flux

“…We will show that the problem trivializes due to the adhesion of the classical world-sheet to the boundary. We will also argue that a similar phenomenon extends to the remaining surfaces in the class of solutions of [26] that remain stuck at the boundary in this limit, even if the factorization of the NS-NS flux into the conformal anomaly breaks down. In section 6 we will summarize our results and comment on some possible future developments.…”

“…where the flux mixing parameter is restricted to lie within 0 ≤ q ≤ 1. The solution is [26] θ(τ, σ) = τ , u(τ, σ) = 1 − q 2 sinh σ , v(τ, σ) = arctanh (q tanh σ) , (2.4) where τ ∈ [0, 2π) and σ ∈ [0, ∞) are the Euclidean time and space world-sheet coordinates, respectively. We must emphasize that (2.4) constitutes a particular limit of the general solution to the underlying integrable mechanical system, a feature that will justify the special properties that it presents as an effective background in the semiclassical quantization scheme.…”

“…Then, we impose asymptotic Dirichlet boundary conditions at the endpoints of the spatial interval. 9 In fact, we may consider a finite interval for the annulus surface, thereby extending the computation to every solution that belongs to the general class of world-sheets with vanishing dilatation charge in the limit of pure NS-NS flux [26] (for simplicity we set the winding number to k = 1 in this discussion).…”

“…Indeed, the extension under the presence of NS-NS flux of the minimal surface termed "quark-antiquark potential" in the context of the AdS 5 /CFT 4 correspondence, which subtends two parallel lines at the boundary of Euclidean AdS 3 , can be shown to adhere to the boundary in one of the two limits in which the R-R flux vanishes (that is, q = 1 or q = −1, depending on the conventions). 10 Similarly, the class of minimal surfaces subtending two concentric circumferences at the boundary of Euclidean AdS 3 considered in [26] displays a range of parameters for which the confinement of the world-sheet to the boundary in the limit of pure NS-NS flux again occurs. We may proceed analogously in these generalized cases, although the study of quadratic perturbation around those solutions and their associated functional determinants is considerably more involved.…”

“…The relevance of minimal area surfaces in the AdS 5 /CFT 4 correspondence has motivated their study in lower-dimensional avatars of the duality. One of the contributions along these lines, within the framework of type IIB string theory on AdS 3 × S 3 × T 4 , was the construction in [26] of minimal surfaces in Euclidean AdS 3 with mixed Ramond-Ramond (R-R) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) three-form fluxes on the basis of the underlying integrable mechanical model (see [27] for other findings of minimal surfaces under the presence of mixed fluxes). The authors employed the periodic ansatz of [3] to study the extension of the class of classical world-sheets subtending two concentric circumferences at the boundary of Euclidean anti-de Sitter space by the introduction of NS-NS flux.…”