p-Branes with $$AdS_{p+1}$$ vacuum as models of $$R^2$$ gravity

Abstract

Branes with constant mean curvature of their hyper-worldsheets of codimension 1 are treated as the Nambu-Goldstone fields of the broken Poincare symmetry. Mapping of their action into quadratic curvature gravity action with spontaneously generated gravity, is shown. Equation for the brane potential extremals and its solution describing hyper-ws of constant curvature are found. For membranes in R 1,3 this extremum is shown to be a saddle 3-dim. hypersurface which defines classically instable vacuum.633 Page 2 of 11 Eur. Phys. J. C (2019) 79 :633 p = 3, D = 6, 8; p = 4, D = 9 and p = 5, D = 10 corresponding to the embeddings of membranes and 3-, 4-, 5-branes into the mentioned higher-dimensional Minkowski spaces. The pair p = 5, D = 10 describes the heterotic 5-brane associated with the soliton solution of the heterotic string equations found in [10]. In M-theory the supermembrane and super 5-brane are called M2 and M5 branes. Mbranes together with superstrings are treated as fundamental constituents of supersymmetric 11-dimensional M-theory unifying gravity with other fundamental forces. Thus, both branes and strings play key roles in theoretical understanding of fundamental interactions, phenomenology of elementary particle physics, black holes, dark matter and AdS/CFT correspondence. A comprehensive review of these and other problems of contemporary physics in which branes play a vital role may be found in [11]. The breakthrough role of branes in attempts to reveal new physics sharpens the problem of their quantum consistency. Solution of this problem has to shed a new light on quantization of gravity. Until now quantization of extended objects remains an open problem even for the case of (super)membranes (p = 2) [12]. This uncertainty in the quantum status of the brane paradigma requires new tools for analyzing its viability. A weak progress in this matter is explained by a complicated non-linear character of brane equations and constraints. The non-linearity problem is typical of the theories unifying the Yang-Mills, gravitational and other fields covariant under diffeomorphisms, gauge and internal symmetries. To solve the problem, we need to deepen understanding of the structure of brane non-linearities.Here we assume that the non-linearities can be converted into the geometric structures known from gravity and gauge theories. For branes, the proof of this assumption allows to implement the BRST-BFV quantization and others methods used for general non-linear systems. We expect that the nonlinearities of ordinary p-branes can be described by a combination of scalars built from the curvature tensor of ( p + 1)dimensional hyper-worldsheets (h-ws). 1 It implies a map of non-linear terms into those known from ( p + 1)-dim. f (R) gravity which generalizes the Starobinsky 4-dim. model [13]. Construction of such a map will also give an additional motivation for the hypothesis that our world is a 3-brane embedded into higher-dimensional spaces [14][15][16][17].The inherent non-linearities prevent the use of string har...