Membrane instantons and non-perturbative effects in ${\mathrm{AdS}}_4/{\mathrm{CFT}}_3$

Abstract

We study Euclidean M2-brane instantons in Freund-Rubin backgrounds ${\mathrm{AdS}}_4\times {\mathrm{Y}}_7$. For a seven-dimensional weak $G_2$ manifold ${\mathrm{Y}}_7$, we show that the BPS condition for an M2-brane wrapping a three-cycle $\hat{I}\pounds \subset {\mathrm{Y}}_7$ is equivalent to the associativity condition with respect to the nearly parallel $G_2$-structure. When ${\mathrm{Y}}_7$ is Sasaki-Einstein, we identify a special class of BPS M2-branes that preserve both real internal Killing spinors and correspond to invariant three-dimensional submanifolds inheriting a Sasakian structure. We analyse the quadratic fluctuations around BPS M2-brane instantons in these backgrounds. For the special class of M2-branes in Sasaki-Einstein manifolds, the fluctuation problem reduces to transversely elliptic complexes, and the one-loop partition function can be expressed in terms of the corresponding equivariant indices. We then apply the index formula for the one-loop partition function to invariant M2-branes in $S^7/{\mathbb{Z}}_k$, recovering the known result for the $S^3/{\mathbb{Z}}_k$ instantons and discussing more general invariant BPS cycles. As a further application, we consider M2-brane instantons with $S^3$-quotient worldvolumes in the $(p,q)$-model geometry.

45 pages. v2: references added, minor improvements