Integrability and Conformal Blocks for Surface Defects in $\mathcal{N}=4$ SYM

TL;DR

This work analyzes half-BPS Gukov–Witten surface defects in $\mathcal{N}=4$ SYM, focusing on the defect CFT data and the interplay between symmetry constraints and integrability. By employing superconformal Ward identities in a harmonic superspace framework, it derives the structure of one-point and two-point functions and decomposes bulk correlators into defect conformal blocks, taking into account the unusual transverse rotation breaking induced by the defect. At weak coupling, leading-order perturbative checks confirm the defect-block organization and reveal integrability in restricted sectors, while generically the transverse breaking spoils integrability; a singular, rigid point ($\alpha=\beta=\gamma=0$) is identified where integrability may be restored via a mapping to integrable quenches. The paper also develops the semiclassical quantization around the defect by mapping to $AdS_3\times S^1$ and classifies the spectrum into easy/hard bosons and fermions, laying groundwork for perturbative calculations and potential localization bootstrap programs. Overall, the results illuminate how defect data, residual symmetry, and holography constrain correlators and provide a roadmap for extracting CFT data via bootstrap/analytic techniques in defect setups.

Abstract

We study various aspects of half-BPS surface defect operators in $\mathcal{N}=4$ SYM. For defects on generic points on the moduli space we use superconformal symmetry to fix the form of one-point and two-point functions of half-BPS operators and solve the superconformal Ward identities in terms of superconformal blocks, emphasizing the role of the broken rotational symmetry transverse to the defect in the superconformal block expansion. We verify this expansion by the leading-order perturbative calculation for the two-point functions. We also investigate the integrability of the defect CFT in the planar limit and argue that the integrability is broken at generic points of the defect moduli. The integrability is expected to be restored in the singular point of this moduli space where another "rigid" branch appears, and we provide evidence for this by showing that the defect one-point functions in this case can be mapped to a class of known integrable quenches.

Figures (7)

• Figure 1: An example of the leading-order diagrams for the two-point functions of the single trace operators. Two vertices at the centers represent each of the single trace operators, connected by a single edge corresponding to the scalar propagator. The black dots represent the one-point functions $\expval{\phi}$ for each Higgs scalar in the surface defect background.
• Figure 2: The decomposition structure of the long representation $\{m,n\}$ of $\mathfrak{sl}(2|2)$ to the representations of the bosonic subalgebra $\mathfrak g_0 = \mathfrak{sl}(2)\oplus \mathfrak{sl}(2)$. The boxed component is the top component. The sequence of the numbers labeling the arrows denote which odd operators are turned on, e.g. $234$ means the action with $z_4 z_3 z_2$.
• Figure 3: The decomposition structure of the massive short representations $\expval{j_1 , j_2}_{\text{I}\pm}$ with (a) $C = 2\left(j_1 - j_2\right)$ and (b) $C = -2\left(j_1 - j_2\right)$.
• Figure 4: Special cases of (a) $\expval{j_1 , 0}_{\text{I}+}$ or (b) $\expval{0 , j_2}_{\text{I}+}$ for the representations satisfying the shortening condition $C = 2 \left(j_1 - j_2\right)$.
• Figure 5: The multiplet structure of a massless representation $\expval{j, j}_{\text{I}\pm}$ with $j = j_1 = j_2$ satisfying both of the Type I conditions.