Branes as solutions of gauge theories in gravitational field

The idea of the Gauss map is unified with the concept of branes as hypersurfaces embedded into $D$-dimensional Minkowski space. The map introduces new generalized coordinates of branes alternative to their world vectors $\mathbf{x}$ 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

Therefore, the geometric approach seems to be a powerfull tool for the investigation of integrability of nonlinear brane equations. A reformulation of the geometric approach to strings in terms of gauge field theories based on the use of the Cartan's moving coordinate frames [21] and ideas [22], [23] was developed in [24]. This gauge treatment allowed to reveal isomorphism between relativistic string and a closed sector of states of the exactly integrable two-dimensional SO(1,1)×SO(D-2) invariant gauge model. As a result, the classical string description in the geometric approach was mapped onto a chain of the two-dimensional exactly integrable equations.
So, it is interesting to generalize the mentioned gauge approach to the case of p-branes. This implies construction of a (p+1)-dimensional SO(1,p)×SO(Dp-1) invariant gauge model possessing the solution associated with p-brane. The problem finds its positive solution in the present paper.

Hypesurfaces in the Minkowski space
A time-like (p + 1)-dimensional hypersurface Σ p+1 embedded into the Ddimensional Minkowski spacetime with the signature η mn = (+, −, . . . , −) is described by its radius vector x(ξ µ ) parametrized by the coordinates ξ µ = (τ, σ r ), (r = 1, 2, .., p). Using the local orthonormal frame n A (ξ µ ) = (n i , n a ) with A = (i, a), attached to Σ p+1 , one can expand the infinitesimal displacements dx(ξ µ ) and dn A (ξ µ ) in the basis n A (ξ µ ) with the frame vectors n i , (i, k = 0, 1, ..., p) tangent and n a , (a, b = p + 1, p + 2, ..., D − p − 1) -normal to the hypersurface. The fixation ω a = 0 of the normal displacement of x breaks down the local Lorentz group SO(1, D − 1) of the moving frame to its local subgroup SO(1, p) × SO(D − p − 1). As a result, the antisymmetric matrix generators where A µi k and B µa b may be treated as SO(1, p) and SO(D − p − 1) gauge fields with their strengths F µνi k and H µνa b given by and W µi b is a charged vector multiplet with the covariant derivative including the SO(1, p) × SO(D − p − 1) gauge fields. The integrability conditions for the PDEs (1) and (2) generates the well-known Maurer-Cartan structure equations prescribing for the torsion and curvature forms of the Minkowski space to be equal to zero. After the break-up of the matrix indices in the tangent and normal components to Σ p+1 Eqs. (8) take the form of the field constraints where D || µ is the covariant derivative associated with the local Lorentz group SO(1, p) acting in the local planes tangent to Σ p+1 The object ω i µ has a geometric sense as the (p+1)-bein of Σ p+1 which connects the orthonormal moving frame n i with the natural frame e µ . Then the metric tensor G µν (ξ ρ ) of the hypersurface is presented as The solution of the constraints (13) introduces the second fundamental form l µν a of the hypersurface Σ p+1 The presented gauge reformulation [24] of the Regge-Lund geometric approach for strings revealed their new description by the closed exactly solvable sector of states of the two-dimensional SO(1,1)×SO(D-2) gauge invariant model. The model includes a massless scalar multiplet originating from the independent components of l µν a of the string worldsheet Σ 2 in a fixed gauge.

SO(1,p)× SO(D-p-1) gauge invariant model
Further let us consider a SO(1, p) × SO(D − p − 1) invariant gauge model in a curved (p + 1)-dimensional space with the space-time coordinates ξ µ and a given pseudo-Riemannian metric g µν whereŴ {µŴν} ≡Ŵ µŴν +Ŵ νŴµ , and the potential term V describes generally and gauge invariant nonlinear (self)interactions of the vector multiplet W µ ia . The generally and gauge covariant derivative∇ µ is defined bŷ and it differs from the usual general covariant derivative which contains only the metric compatible with the Levi-Chivita connection Γ ρ µν = Γ ρ νµ = 1 2 g ργ (∂ µ g νγ + ∂ ν g µγ − ∂ γ g µν ). Our objective is to study the equations of motion of this model and their exact analytical solutions.
The variation of S (17) in the gauge and vector fields results in the generalized Maxwell and Euler-Lagrange equations for W µiâ Using the shifted gauge field strengths F µν ik and H µν ab one can present Eqs. (21)(22)(23) in the compact form The next step is to apply the generalized first Bianchi identity where the Riemann-Cristoffel tensor action on a vector V γ is defined as Then one can present the EOM (28) for W ia µ in the form where R νλ := R µ ν µλ is the Ricci tensor. From the relation 1 4 including the commutators ofŴ µ one can introduce the shifted potential V where the trace (26), (27) and (31) of the model can be rewritten aŝ It is easily seen that the first-order PDEs form a particular solution of Eqs. (34-36), provided that the condition is satisfied. Using arbitrariness of V, and consequently of V , one can choose the latter in the form which solves Eq. (38) if R µν does not depend on W νia . Thus, we find that the model (17) with the Lagrangian density produces the following nonlinear Euler-Lagrange equationŝ for the gauge A µi k , B µa b and the vector W µia fields in an external gravitational field g µν (ξ ρ ). These equations have the particular solution (37) which coincides with the Maurer-Cartan Eqs. (9), (10) and (11) called the Gauss-Codazzi (G-C) equations in the classical differential geometry of surfaces.
One can weaken the demand for the gravitational field to be external and to consider it in the equal rights with the dynamical gauge and vector fields. Then EOM (41-43) have to be completed by the variational eqs. with respect to g µν . These equations will connect the Ricci tensor R µν with the gauge strengths, W µia and its covariant derivatives. However, solution of this equation even on the Gauss-Codazzi shell needs in an additional investigation and is not studied in this paper. It is also interesting that the model (17) with the Lagrangian (40) and a dynamical gravitational field g µν seems to be a natural generalization of the Dirac scale-invariant gravity theory with the four-dimensional action W [25] (see also [26]) where β is the dilaton field. Below we shall study another interesting possibility to treat the proposed action (17) as a gauge-invariant action associated with some hypersurfaces and branes embedded into the higher dimensional Minkowski space.

Branes as solutions of the gauge model
The observation that the Gauss-Codazzi equations give a particular solution of the gauge model in an external gravitational field points to the possibility to consider branes as solutions of gauge theories. To develop this conjecture one can treat the considered (p + 1)-dimensional pseudo-Riemannian spacetime of the gauge model (17)(18) as a (p+1)-dimensional world hypersurface swept by a p-brane in D-dim. Minkowski space.
To realize this conjecture one has to take into account the remaining M-C equations (12)(13). Here we observe that Eqs. (12) have the solution that generalizes the well-known tetrade postulate of the general relativity to higher dimensions and identifies the metric connection Γ ρ µν with A µi k by means of the gauge transformation The transformation function ω i µ in (46) coincides with the (p+1)-bein (15) of the brane hypersurface equipped with the metric G µν = g µν .
Using the expressions (30) and (4) for the Riemann tensor R µν γ λ and the strength F µνi k we find their connection Then, with the help of the Gauss-Codazzi condition for F µν i k we obtain which represents the Ricci tensor of the hypersurface on the G-C shell. As a result, the term R νλ W ia λ in Eq. (38) becomes the function ofŴ λ 1 2 and the potential V is to be found as the solution of this equation. Moreover, Eq. (49) has to take into account still remaining M-C equations (13) that show that not all of the components of W ia λ are independent DOF. This follows from the relation resulting from the symmetry l µν a = l νµ a for the second fundamental form components. Taking into account the linear dependence (50) gives where Sp(l a ) ≡ g µν l µν a = −ω j µ W µ ja , (a = p + 1, ..., D − p − 1). Substitution of (51) into Eq. (49) results in the equation which has the solution The constraint Sp(l c ) = 0 is the well-known minimality condition for the (p + 1)-dim. worldvolume equivalent to the brane EOM Beltrami operator on Σ p+1 [20]. Eq. (55) follows from the Dirac action where G is the determinant of the induced metric G αβ := ∂ α x∂ β x. So, EOM of gauge model (17) are compatible with the generalized tetrade anzats and have the solution describing the Dirac p-brane.

Reduction to SO(D-p-1) invariant gauge model
Having revealed the brane solutions one can use the tetrade postulate (45) directly in the gauge action (17) together with the transition from W µi a to l µν a = −ω i ν W µi a owing to solution (16) of the first Maurer-Cartan equations. Then the G-C eqs. (9-11) take the form where ∇ ⊥ µ l νρ a := ∂ µ l νρ a −Γ λ µν l λρ a −Γ λ µρ l νλ a +B ab µ l νρb . As is seen the relation (57) is the generalization of the Gauss theorema egregium to a (p+1)-dimensional hypersurface embedded into the D-dimensional Minkowski space.
The corresponding action which solutions are the G-C eqs. (57-59) is To prove this we consider the action (17) with l µν a substituted for W µi Variation of S (61) in the dynamical fields l µν a , B µ ab gives their EOM their particular solution provided the condition Using the G-C constraints (57-59) and generalized Bianchi identity one can transform Eq. (65) to the following form Eq. (67) fixes the potential term V and the trace of l µνa to be equal to We see that Dirac p-brane (56) is the solution of Eqs. (62-63) for S (60) which is presented by the M-C conditions (7)(8). The substitution of an arbitrary function f (R, ∇R) of the background Riemannian tensor for the R 2 term in S (60) preserves the above conclusion.
Variation of S (60) and its f(R) extension with respect to g µν will produce generalized G-E equations for the dynamical gravitational field associated with p-branes, or/and the Σ p+1 diffeomorphism constraints if f = const.

Summary
A new gauge reformulation of the geometric Regge-Lund approach for the relativistic string is generalized to the case of relativistic p-branes imbedded into D-dimensional Minkowski space. There is constructed a set of SO(1,p)× SO(D-p-1) invariant gauge models including massless vector multiplets in gravitational field and possessing exact solutions. The latter are presented by the first-order Gauss-Codazzi PDEs rewritten as gauge constraints.
It is shown that the generalized tetrade postulate allows to treat the exact solutions as associated with hypersurfaces, or branes embedded into higher dimensional Minkowski spaces. The treatment is accompanied with the transition from the vector multiplets to the massless tensor fields.
It has been surprising that the SO(1,p)× SO(D-p-1) gauge-invariant model (17) turns out to be a generalization of the Dirac scale invariant gravity theory [25] with the massless vector multiplet W µia or the tensor l µν a substituted for the Dirac dilaton field β. This observation needs in further study.