Instanton contributions to the ABJM free energy from quantum M2 branes

Abstract

We present a quantum M2 brane computation of the instanton prefactor in the leading non-perturbative contribution to the ABJM 3-sphere free energy at large N and fixed level k. Using supersymmetric localization, such instanton contribution was found earlier to take the form $$ {F}^{inst}\left(N,k\right)=-{\left({\sin}^2\frac{2\pi }{k}\right)}^{-1}\exp \left(-2\pi \sqrt{\frac{2N}{k}}\right)+.\dots $$ F inst N k = − sin 2 2 π k − 1 exp − 2 π 2 N k + . … The exponent comes from the action of an M2 brane instanton wrapped on S3/ℤk, which represents the M-theory uplift of the ℂP1 instanton in type IIA string theory on AdS4 × ℂP3. The IIA string computation of the leading large k term in the instanton prefactor was recently performed in arXiv:2304.12340. Here we find that the exact value of the prefactor $$ {\left({\sin}^2\frac{2\pi }{k}\right)}^{-1} $$ sin 2 2 π k − 1 is reproduced by the 1-loop term in the M2 brane partition function expanded near the S3/ℤk instanton configuration. As in the Wilson loop example in arXiv:2303.15207, the quantum M2 brane computation is well defined and produces a finite result in exact agreement with localization.

S-matrix on effective string and compactified membrane

“…We will also compute the 1-loop correction to the scattering amplitude of 4 massless particles with all massless and massive modes running in the loop. We will find that as in the examples considered in [13,14] this 1-loop amplitude is UV finite in analytic regularization as appropriate for a 3d theory. It is also IR finite as all massless modes have only derivative couplings like in the NG string case.…”

“…M-theory in AdS 4 × S 7 /Z k should be dual to the N = 6 supersymmetric 3d U k (N) × U −k (N) Chern-Simons matter (ABJM) theory [11,12]. Recent work [13,14] provided a remarkable evidence that direct semiclassical quantization of the M2 brane in AdS 4 × S 7 /Z k background reproduces the results of large N localization computations [15][16][17][18] of the 1 2 -BPS Wilson loop and instanton contributions to free energy in the ABJM gauge theory.…”

“…While the M2 brane action is highly non-linear, expanded near a classical solution with nondegenerate induced 3d metric it can be quantized in a static gauge with the 1-loop result being UV finite (containing no logarithmic divergences) [2,13,14,[19][20][21][22]. As the 1/N expansion of the localization results on the gauge theory side have the form of an expansion in the inverse of the effective M2 brane tension T 2 = √ 2k π √ N, this suggests that the matching with the 1-loop M2 brane computations [13,14] should, in fact, extend also to 2-loop and higher orders.…”

Quantum holographic surface anomalies

“…Our work also serves to enlarge the scope of recent progress on the semiclassical quantisation of M2-branes. This has been done in several contexts like Wilson loops and instanton corrections to the free energy of ABJM theory [58][59][60][61][62] and reproduced results from supersymmetric localisation. In most of those cases the classical M2-brane solutions are BPS and their spectrum can be solved exactly and summed up using ζ function regularisation.…”

“…In this appendix we therefore abandon the near boundary metric and use one near the classical brane solution. Indeed, the brane does not need to have a boundary and can be a bulk instanton, as those studied in [59,61].…”