BPS surface operators and calibrations

TL;DR

This paper studies the holographic duals of BPS surface operators in 6d ${ m N}=(2,0)$ theory, revealing how calibration forms relate to M2-brane actions and introducing new expectation values at large N.

Contribution

It classifies calibration forms for various surface operators and links their geometry to M2-brane actions, including special cases involving Lagrangian submanifolds and Calabi-Yau structures.

Findings

Calibration forms are closed for most classes, simplifying M2-brane action calculations.

Surface anomalies relate to ratios of surfaces with identical anomalies, leading to new expectation values.

Special Lagrangian submanifolds correspond to a rich structure for certain surface operators.

Abstract

We present here a careful study of the holographic duals of BPS surface operators in the 6d ${\cal N}=(2,0)$ theory. Several different classes of surface operators have been recently identified and each class has a specific calibration form - a 3-form in $AdS_7\times S^4$ whose pullback to the M2-brane world-volume is equal to the volume form. In all but one class, the appropriate forms are closed, so the action of the M2-brane is easily expressed in terms of boundary data, which is the geometry of the surface. Specifically, for surfaces of vanishing anomaly, it is proportional to the integral of the square of the extrinsic curvature. This can be extended to the case of surfaces with anomalies, by taking the ratio of two surfaces with the same anomaly. This gives a slew of new expectation values at large $N$ in this theory. For one specific class of surface operators, which are Lagrangian submanifolds of ${\mathbb R}^4\subset {\mathbb R}^6$, the structure is far richer and we find that the M2-branes are special Lagrangian submanifold of an appropriate six-dimensional almost Calabi-Yau submanifold of $AdS_7\times S^4$. This allows for an elegant treatment of many such examples.