Remarks on the topological open membrane

Abstract

Just as non-commutative gauge theories arise from quantising open strings in a large magnetic field, non-Abelian two-form gauge theories may conceivably be constructed by quantising open membranes in a large three-form magnetic background. We make some observations that arise in following this strategy, with an emphasis on the relation to the quantisation of volume-preserving diffeomorphisms (vpd). In particular, we construct consistent non-Abelian interactions of a two-form in 3+1 dimensions, based on gauge invariance under vpd.PACS numbers: 11.10. Kk, 11.10.Lm, 47.37.+q Keywords: Non-commutative geometry, Nambu bracket, Non-Abelian tensor, Superfluid HeliumOf the many mysteries surrounding the theory formerly known as string theory, the dynamics of the five-brane is perhaps the most intriguing. While a single type IIA or M5 brane supports a free self-dual two-form gauge field on its world-volume, together with five transverse scalar degrees of freedom and their fermionic partners [1], the case of N five-branes at small separation is much less understood. Analogy with the case of coinciding D-branes and duality with ALE singularities in type IIB string theory suggest that the chiral multiplet now transforms in the adjoint representation of a spontaneously broken U (N ), with the open membranes stretched between two five-branes playing the rôle of stringy W bosons on the five-brane worldvolume [2]. This picture is at best suggestive however, since the quantization of the membrane, if at all sensible, is still beyond reach (see [4] for a review of recent attempts however). Disregarding the fundamental origin of these low-energy degrees of freedom, no consistent local theory of interacting non-Abelian selfdual two-forms has been found to date, in line with no go theorems [5]. The nature of the U (N ) non-Abelian symmetry itself is unclear, since anomaly and entropy considerations hint at N 3 degrees of freedom rather than N 2 as expected on a naive perturbative basis [6] -an admittedly unwarranted expectation due to the absence of a tunable coupling.On the other hand, recent progress in the understanding of D-branes in background fields has taught us that non-Abelian dynamics arise even at N = 1, in the presence of a strong background magnetic or electric field [7]. Indeed, the effects of the higher-derivative Born-Infeld couplings can be resummed by going to a non-commutative description, where the U (1) gauge invariance δa µ = ∂ µ λ is replaced by a transformation δa µ = ∂ µ λ + a µ * λ − λ * a µ formally identical to the usual Yang-Mills gauge invariance [8]. The Moyal deformation of the ordinary commutative product can in fact be derived without detailed knowledge of the dynamics of open strings in a magnetic background B = dA: in the limit of large B, one may retain only the topological coupling Σ B from the action of the string with worldsheet Σ (we omit the pull-back from target space). This in turn reduces to a coupling γ A on the boundary, hence to a topological quantum mechanical model. For constant back...

A toy model of open membrane field theory in constant 3-form flux

“…It has a long history of the struggle to achieve this goal, for example, see [10,11,14,[17][18][19]21,28,29]. One of the difficulty of the problem is that there seems to be no natural principle and therefore we have a variety of proposals with both merits and demerits.…”

Classical and quantum Nambu mechanics

“…There are in the literature several persuasive but inconclusive arguments that Nambu brackets are a natural language for describing extended objects, for example: [2,6,7,14,18,27,31,34,36,40,46].…”

BV Quantization of Topological Open Membranes

“…This model is a toy model for string theory in a background 3-form field, and was studied in [3,4,5,6], but rather from the point of view of the somewhat ill-defined boundary string theory. In [3] it was shown that in a particular decoupling limit a stack of M5-branes in a 3-form field the open membrane action of the M2-brane reduces to this particularly simple topological membrane model.…”