Scalar fields in curved backgrounds are assumed to be composite objects. As an example realizing such a possibility we consider a model of the massless tensor field l_μν(x) in a 4-dim. background g_μν(x) with spontaneously broken Weyl and scale symmetries. It is shown that the potential of l_μν, represented by a scalar quartic polynomial, has the degenerate extremal described by the composite Nambu–Goldstone scalar boson ϕ(x):=g^μνl_μν. Removal of the degeneracy shows that ϕ acquires a non-zero vev ⟨ϕ⟩0=μ which, together with the free parameters of the potential, defines the cosmological constant. The latter is zero for a certain choice of the parameters.

The braneworld scenario is studied and the effective action of 3-brane living in 5-dim. Minkowski space is constructed. This action is proved to be invariant under spontaneously broken scale and Weyl symmetries and to encode a model of quadratic gravity generalizing the Starobinsky model. The symmetry breaking generates the Hilbert–Einstein term with the Newton constant GN∼1/μ2, where μ is a mass scale equal to the mean curvature of the vacuum hyper-worldsheet of 3-brane. This result proposes a brane modification of the mechanism of spontaneously generated gravity.

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 R1,3 this extremum is shown to be a saddle 3-dim. hypersurface which defines classically instable vacuum.

We discuss the gauge theory approach to consideration of the Nambu Goldstone bosons as gauge and vector fields represented by the Cartan forms of spontaneously broken symmetries. The approach is generalized to describe the fundamental branes in terms of (p + 1)-dimensional worldvolume gauge and massless tensor fields consisting of the Nambu-Goldstone bosons associated with the spontaneously broken Poincare symmetry of the D-dimensional Minkowski space.

Mapping of fundamental branes to their worldsheet (ws) multiplets originating from spontaneous breaking of the Poincare symmetry is studied. The interaction Lagrangian for fields of the Nambu-Goldstone multiplet is shown to encode R-2 gravity on the ws. The power law k(p) similar to T-p(3-p/2(p+1)) for the SO(D - p - 1) gauge coupling k(p) as the function of the p-brane tension T-p is assumed. It points to the presence of asymptotic freedom and confinement phases in brane matter. Their connection with collapse and inflation of the branes is discussed.

Phenomenological Lagrangians for physical systems with spontaneously broken symmetries are reformulated in terms of gauge field theory. Description of the Dirac p-branes in terms of the Yang-Mills-Cartan gauge multiplets interacting with gravity, is proved to be equivalent to their description as a closed dynamical system with the symmetry ISO(1, D - 1) spontaneously broken to ISO(1,p) x SO(D - p - 1).

The idea of the Gauss map is unified with the concept of branes as hypersurfaces embedded into D-dimensional Minkowski space. Themap introduces new generalized coordinates of branes alternative to their world vectors x and identified with the gauge and other massless fields. In these coordinates the Dirac p-branes realize extremals of the Euler-Lagrange equations of motion of a (p + 1)dimensional SO(D - p - 1) gauge-invariant action in a gravitational background.

Proposed is a new approach to finding exact solutions of nonlinear p-brane equations in D-dimensional Minkowski space based on the use of various initial value constraints. It is shown that the constraints Delta((p))(x) over right arrow =0 and Delta((p))(x) over right arrow = -Lambda(t, sigma(r))(x) over right arrow give two sets of exact solutions.

Nonlinear equations of p-branes in D = (2p + 1)-dimensional Minkowski space are discussed. Presented are new exact solutions for a set of spinning p-branes with the Abelian symmetries U(1) x U(1) x ... x U(1) of their shapes.

Exact solvability of brane equations is studied, and a new U(1) x U(I) x ... x U(1) invariant anzats for the solution of p-brane equations in D = (2p + 1)-dimensional Minkowski space is proposed. The reduction of the p-brane Hamiltonian to the Hamiltonian of p-dimensional relativistic anharmonic oscillator with the monomial potential of the degree equal to 2p is revealed. For the case of degenerate p-torus with equal radii it is shown that the p-brane equations are integrable and their solutions are expressed in terms of elliptic (p = 2) or hyperelliptic (p > 2) functions. The solution describes contracting p-brane with the contraction time depending on p and the brane energy density. The toroidal brane elasticity is found to break down linear Hooke law as it takes place for the anharmonic elasticity of smectic liquid crystals.