𝒩 = 8 superconformal gauge theories andM2 branes

Abstract

Based on recent developments, in this letter we find 2 + 1 dimensional gauge theories with scale invariance and N = 8 supersymmetry. The gauge theories are defined by a lagrangian and are based on an infinite set of 3-algebras, constructed as an extension of ordinary Lie algebras. Recent no-go theorems on the existence of 3-algebras are circumvented by relaxing the assumption that the invariant metric is positive definite. The gauge group is non compact, and its maximally compact subgroup can be chosen to be any ordinary Lie group, under which the matter fields are adjoints or singlets. The theories are parity invariant and do not admit any tunable coupling constant. In the case of SU(N) the moduli space of vacua contains a branch of the form (R 8 ) N /S N . These properties are expected for the field theory living on a stack of M2 branes.

Supersymmetric Yang-Mills theory from lorentzian three-algebras

“…As already mentioned in [13] [14] [15], the BF membrane model has ghosts in the classical theory, arising from the timelike direction in the "three-algebra". Despite the fact that ghosts cannot propagate in loops due to the special nature of the interactions of the BF membrane model, it remains an important issue to understand whether the theory has negative norm states.…”

“…5 A proposal for the worldvolume theory of coincident M2-branes in flat space has been made in [13] [14] [15], henceforth called the BF membrane model. This theory is based on a Lorentzian "three-algebra," 6 and its gauge structure is that of a three dimensional BF theory.…”

Superconformal M2-branes and generalized Jordan triple systems

“…As discussed above, if one wants to describe more general Lie algebras than so (4), this metric must be degenerate [10] or non-degenerate but indefinite [11][12][13]. Finally, to construct an action one also needs to introduce the basic gauge field A μab 3 which is related to the previously defined gauge field and structure constants as follows:…”

“…at the negative levels). These elements will satisfy 12) due to antisymmetry of the Lie bracket and the Jacobi identity, but also further conditions that amount to factoring out ideals from the free Lie algebra generated by T a and T a . Assume that g 0 is semisimple.…”

“…The drawback of using a degenerate metric as done in [10] is that it produces a set of field equations which cannot be integrated to a Lagrangian if the zero norm mode is not assumed constant. This subsequently led to a number of attempts to use a non-degenerate but Lorentzian metric [11][12][13]. Again there are problems; these theories make sense only provided the negative norm modes can be rendered harmless.…”