From Large to Small $\mathcal{N}=(4,4)$ Superconformal Surface Defects in Holographic 6d SCFTs

Abstract

Two-dimensional (2d) $\mathcal{N}=(4,4)$ Lie superalgebras can be either "small" or "large", meaning their R-symmetry is either $\mathfrak{so}(4)$ or $\mathfrak{so}(4)\oplus \mathfrak{so}(4)$, respectively. Both cases admit a superconformal extension and fit into the one-parameter family $\mathfrak{d}\left(2,1;{\hat{I}}^3\right)\oplus \mathfrak{d}\left(2,1;{\hat{I}}^3\right)$, with parameter ${\hat{I}}^3\in (-\infty ,\infty )$. The large algebra corresponds to generic values of ${\hat{I}}^3$, while the small case corresponds to a degeneration limit with ${\hat{I}}^3\to -\infty$. In 11d supergravity, we study known solutions with superisometry algebra $\mathfrak{d}\left(2,1;{\hat{I}}^3\right)\oplus \mathfrak{d}\left(2,1;{\hat{I}}^3\right)$ that are asymptotically locally AdS${}_7\times S^4$. These solutions are holographically dual to the 6d maximally superconformal field theory with 2d superconformal defects invariant under $\mathfrak{d}\left(2,1;{\hat{I}}^3\right)\oplus \mathfrak{d}\left(2,1;{\hat{I}}^3\right)$. We show that a limit of these solutions, in which ${\hat{I}}^3\to -\infty$, reproduces another known class of solutions, holographically dual to small $\mathcal{N}=(4,4)$ superconformal defects. We then use this limit to generate new small $\mathcal{N}=(4,4)$ solutions with finite Ricci scalar, in contrast to the known small $\mathcal{N}=(4,4)$ solutions. We then use holography to compute the entanglement entropy of a spherical region centered on these small $\mathcal{N}=(4,4)$ defects, which provides a linear combination of defect Weyl anomaly coefficients that characterizes the number of defect-localized degrees of freedom. We also comment on the generalization of our results to include $\mathcal{N}=(0,4)$ surface defects through orbifolding.

1+35 pages, 4 figures, v2: added (5.13)