From N=2 strings to M-Theory

Abstract

Taking the N=2 strings as the starting point, we discuss the equivalent self-dual field theories and analyse their symmetry structure in 2 + 2 dimensions. Restoring the full 'Lorentz' invariance in the target space necessarily leads to an extension of the N=2 string theory to a theory of 2 + 2 dimensional supermembranes propagating in 2 + 10 dimensional target space. The supermembrane requires maximal conformal supersymmetry in 2 + 2 dimensions, in the way advocated by Siegel. The corresponding self-dual N=4 Yang-Mills theory and the self-dual N=8 (gauged) supergravity in 2+2 dimensions thus appear to be naturally associated to the membrane theory, not a string. Since the same theory of membranes seems to represent the M-theory which is apparently underlying the all known N=1 string theories, the N=2 strings now appear on equal footing with the other string models as particular limits of the unique fundamental theory. Unlike the standard 10-dimensional superstrings, the N=2 strings seem to be much closer to a membrane description of the F & M theory.

geometry from Fedosov's deformation quantization

“…We apologize for excluding many relevant references. The relevance of Self Dual Gravity in connection with N = 2 strings was initiated by [20] and pursued by many others [30,31]. Important remarks about M (atrix) models and N = 2 strings have appeared in [29] .…”

Towards an Ashtekar formalism in eight dimensions

“…Thus, a natural step forward is to consider the N = (2, 1) heterotic string [4]. In this scenario, it was observed [5] that a consistent N = (2, 1) string should consider right movers 'living' in 2 + 2 dimensions and left movers in 2 + 10 dimensions, rather than in 1 + 9.…”

“…One of the interesting aspects of 2 + 2-branes is that their dynamic lead to self-dual self-gravity coupled to self-dual supermatter in 2 + 10 dimensions [5]. Meanwhile, a realistic 1 + 3-dimensional gravitational theory should arise from the 'compactification' of 1 + 7 dimensions out of the full 2 + 10-dimensional spacetime background.…”

Superstrings from theories with N > 1 world-sheet supersymmetry

“…The (4,4) world-sheet theory therefore has eight bosonic generators for both left-movers and rightmovers. Just as the (2, 2) string super world-sheet can be thought of as a [2,2] world-volume when the Kac-Moody parameters are included as bosonic dimensions, the (4, 4) world-sheet can be interpreted as an [8,8] dimensional world-volume. This is suggestive of an underlying connection with octonions [7].…”

“…(b) The quantization of the heterotic (2, 1) theories was shown in [5] to also give theories with either [1,1] or [2,1] target spaces. Those theories with [1,1] target spaces have precisely the field content needed to describe the world-sheets of the critical superstring theories [6] (see also [8]).…”