A Polynomial Weyl Invariant Spinning Membrane Action

A review of the construction of a Weyl-invariant spinning-membrane action that is polynomial in the fields, without a cosmological constant term, comprised of quadratic and quartic-derivative terms, and where supersymmetry is linearly realized, is presented. The action is invariant under a modified supersymmetry transformation law which is derived from a new Q+K + S sum-rule based on the 3D-superconformal algebra . A satisfactory spinning membrane Lagrangian has not been constructed yet, as far as we know. Satisfactory in the sense that a suitable action must be one which is polynomial in the fields, without (curvature) R terms which interfere with the algebraic elimination of the three-metric, and also where supersymmetry is linearly realized in the space of physical fields. Lindstrom and Rocek [1] were the first ones to construct a Weyl invariant spinning membrane action. However, such action was highly non− polynomial complicating the quantization process . The suitable action to supersymmetrize is the one of Dolan and Tchrakian (DT) [2] without a cosmological constant and with quadratic and quartic-derivative terms. Such membrane action is basically a Skyrmion like action. We shall write down the supersymmetrization of the polynomial DT action that is devoid of R and kinetic gravitino terms; where supersymmetry is linearly realized and the gauge algebra closes [3]. The crux of the work [3] reviewed here lies in the necessity to Weyl-covariantize the Dolan–Tchrakian action through the introduction of extra fields. These are the gauge field of dilations , bμ, and the scalar coupling , A0 of dimension (length) , that must appear in front of the quartic derivative terms of the DT action. Our coventions are: Greek indices stand for three-dimensional ones; Latin indices for spacetime ones : i, j = 0, 1, 2....D. The signature of the 3D volume is (−,+,+). The Dolan–Tchrakian Action for the bosonic p-brane ( extendon) with vanishing cosmological constant in the case that p = odd; p+ 1 = 2n is : L2n = √−ggμ1ν1 ....gμν∂[μ1X i1 .........∂μn]X i∂[ν1Xj .........∂νn]Xjηi1j1 ....ηinjn .. (1) ηij is the spacetime metric and g μν is the world volume metric of the 2n hypersurface spanned by the motion of the p − brane. Antisymmetrization of indices is also required. Upon the algebraic elimination of the world volume