Non-planar corrections to ABJM Bremsstrahlung function from quantum M2 brane

As was shown in Giombi and Tseytlin (2023 Phys. Rev. Lett. 130 201601), the leading large N, fixed k correction in the localization result for the expectation value of the 1/2 BPS circular Wilson loop in U (N)k × U (N)−k ABJM theory given by the (sin ⁡ 2π/k)−1 factor can be reproduced on the dual M-theory side as the one-loop correction in the partition function of M2 brane in AdS4×S7/ℤk with AdS2×S1 world volume. Here we prove, following the suggestion in Giombi et al (2024 J. High Energy Phys. JHEP11(2024)056), that the analogous fact is true also for the corresponding correction B1 = − 1/2π k cot 2π/k in the localization result for the Bremsstrahlung function associated with the Wilson loop with a small cusp in either AdS4 or CP3. The corresponding M2 brane is wrapped on the 11d circle and generalizes the type IIA string solution in AdS4×CP3 ending on the cusped line. We show that the one-loop term in the M2 brane partition function reproduces the localization expression for B1 as the coefficient of the leading term in its small cusp expansion.