Membrane instantons and non-perturbative effects in AdS4/CFT3

A

bstract

We study Euclidean M2-brane instantons in Freund-Rubin backgrounds AdS ₄ ×Y ₇ . For a seven-dimensional weak G ₂ manifold Y ₇ , we show that the BPS condition for an M2-brane wrapping a three-cycle Σ ⊂ Y ₇ is equivalent to the associativity condition with respect to the nearly parallel G ₂ -structure. When 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 / ℤ _k , recovering the known result for the S ³ / 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.