As was shown in arXiv:2303.15207, the leading large $N$, fixed $k$ correction in the localization result for the expectation value of the $\frac{ 1}{ 2}$-BPS circular Wilson loop in $U(N)_{ k}\times U(N)_{ -k}$ ABJM theory given by the $(\sin\frac{ 2\pi}{ k})^{ -1}$ factor can be reproduced on the dual M-theory side as the one-loop correction in the partition function of an M2 brane in AdS$_{ 4}\times S^{ 7}/\mathbb{ Z}_{ k}$ with AdS$_{ 2}\times S^{ 1}$ world volume. Here we prove, following the suggestion in arXiv:2408.10070, that the analogous fact is true also for the corresponding correction $B_1=-\frac{ 1}{ 2\pi k}\cot\frac{ 2\pi}{ k}$ in the localization result for the Bremsstrahlung function associated with the Wilson line with a small cusp in either AdS$_4$ or $\rm CP^3$. The corresponding M2 brane is wrapped on the 11d circle and generalizes the type IIA string solution in AdS$_{ 4}\times \rm CP^3$ ending on the cusped line. We show that the one-loop term in the M2 brane partition function reproduces the localization expression for $B_1$ as the coefficient of the leading term in its small cusp expansion.
Categories: hep-th