form gauge symmetry. These zero modes were understood from the 11D perspective [@Bhattacharyya:2012ye] and the resulting logarithmic correction agrees with the one expected from the solution of the dual ABJM theory via localization [@Marino:2009jd; @Fuji:2011km; @Marino:2011eh]. We hope to report on these elaborations elsewhere.

Boundary Modes in AdS$_4$
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The evanescent part of the $2$-form — the quantum contribution of an antisymmetric tensor that is above and beyond that of its dual scalar field — is naturally interpreted as a boundary mode, at least in the context of AdS$_4$. A boundary mode is formally a pure gauge field configuration but it is physical because the putative gauge parameter is non normalizable and so the field configuration cannot be gauged away by any element of the symmetry group. This mechanism is unimportant in classical field theory but it matters in the quantum theory, as expected for a feature related to quantum inequivalence.

The boundary modes reside in the kernel of the classical duality transformation (\[eq:Bphiduality\]) between an antisymmetric tensor and a scalar. Their $3$-form field strength vanishes identically in bulk, since they are formally pure gauge, and so they are not assigned a scalar dual $$H^{\textrm{(bndy~mode)}}_{\mu\nu\sigma} = 0 = \epsilon_{\mu\nu\sigma\lambda}\nabla^\lambda \phi~,$$ 