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

Branes with constant mean curvature of their hyper-worldsheets of codim 1 are treated as the Nambu-Goldstone fields of the broken Poincare symmetry. Mapping of their action into quadratic curvature gravity action in hyper-ws is shown. Equation for the brane potential extremals and its solution describing hyper-ws of constant curvature are found. For membranes in $\mathbf{R}^{1,3}$ this extremum is shown to be a saddle 3-dim. hypersurface which defines classically unstable vacuum.


Introduction
Differential geometric description of hyper-worldsheets (h-ws) [1 -4] of the Dirac (or fundamental) branes [5] as hypersurfaces embedded into Riemannian spaces revealed new connections between the theories of strings, gauge and gravitational fields [6 -9]. Embedding of a h-ws Σ min p+1 of fundamental p-brane into D-dim. Minkowski space results in the spontaneous breaking of the Poincare group ISO(1, D − 1) to its subgroup ISO(1, p) × SO(D − p − 1) [10 -13]. The corresponding Nambu-Goldstone (N-G) fields (see [14] and refs. there) were used for construction of the invariant action of fundamental p-branes in the gauge theory formalism on the curved h-ws [15]. The latter introduced the SO(D − p − 1) Y-M multiplet B ab µ , the N-G tensor multiplet l a µν identified with the second fundamental form of Σ min p+1 equipped with the metric tensor g µν . As shown in [16] for codim 1, the action [15] reailizes a (p + 1)-dim. h-ws version of quadratic curvature gravity investigated in [17 -21]. The pure R 2 term and cosmological constant, represented in the action S Dir [16] by an integration constant, form the quartic scalar potential of interaction V Dir (l, g) for the fields l µν and g µν . For zero cosmological constant the action S Dir contains only one coupling constant κ p ∼ T 3−p 2(p+1) p depending on the p-brane tension T p . The power law provides three different regimes of the behavior of k p as a function of the tension and, consequently, energy corresponding to the cases p < 3, p = 3 and p > 3. These regimes show the presence of the asymptotic freedom and confinement phases and their connection with inflation and collapse of the branes. Exact solutions for p-brane collapse were earlier obtained in [22], [23]. The dimensionless coupling k 3 associated with three-branes characterizes a scale invariant model belonging to those considered in [24 -29]. The model [16] does not contain invariants linear in components of the h-ws Riemannian tensor. This necessitates the model extension which could encode the Hilbert-Einstein(H-E) term [30]. To this objective we note that h-ws of fundamental branes are minimal hypersurfaces with zero mean curvature, i.e. Spl ≡ g νµ l νµ = 0. A natural generalization of the latter condition is Spl = µ, where µ is a constant with the dimension of mass [µ] = [L −1 ]. This condition selects hypersurfaces with the constant mean curvature (cmc) equal to µ. The string model with cmc ws embedded into R 1,2 was studied in [31] by the extension of the Nambu-Goto action. In the gauge formalism µ is treated as a physical constant fixing a characteristic scale of the symmetry breaking. A desired extension is supposed to have two fundamental couplings T p and µ defining the brane dynamics in the presence of gravitational and elastic forces. A similar dynamics is observed for string in curved space [32]. Here we solve the problem in question and build the potential V cmc (l, g, µ) which encodes the H-E and cosmological terms (50) in addition to the pure R 2 term. The proposed potential extends the Starobinsky model [18] and its generalization [33] including a massless scalar field φ. In particular, the potential contains the cosmological and interaction terms. The latter term is proportional to µR νλ l λν . It should be also noted that the field φ from [33] can be identified with Spl treated as the dynamical mean curvature field constrained to be constant in our approach.
Then we derive the equation for extremals of the functional V cmc (l, g, µ) and find its general solution for the most interesting case R = 0. Consistency of the solution with the Gauss and Peterson-Codazzi (P-C) embedding conditions results in the matrix equation for null vectors of the (p + 1)-dim. Einstein tensor (matrix) G µν . Its solution reveals the presence of the extremal l oµν (ξ) = µ p+1 g µν (ξ). This critical point corresponds to an AdS p+1 -like hyper-ws Σ o p+1 of constant negative curvature. Next we analyse the behavior of V cmc in the vicinity of l oµν (ξ). The second partial derivatives of V cmc with respect to l at l o (ξ) have different signs depending, in particular, on the brane dimension p. So, for string the extremum is the minimum that shows stability of the string potential in R 1,2 . But for membrane (p = 2) the critical point l o (ξ) is proved to be the saddle 3-dim. hypersurface. It points to the classical unstability of the cmc membrane potential in 4-dim. Minkowski space-time. The classical unstability of the fundamental (super)membrane [36], described by the potential V (x(ξ)) [37], is well known. Since fundamental membrane sweeps minimal h-ws, its change by cmc h-ws embedded into R 1,3 does not remove the unstability. The obtained results can be used while studying the braneworld cosmological models [34], [35], as well as for investigation of inflation and reheating mechanisms in R 2 gravity models.

Branes with minimal hyper-worldsheets
In the geometrical approach Dirac p-branes embedded into R 1,D−1 with the signature (+ − ...−) are described as dynamical systems with the Poincare symmetry ISO(1, D −1) spontaneously broken to ISO(1, p)×SO(D −p−1). The p-branes sweep minimal h-ws Σ min p+1 with zero mean curvature where g µν is the induced metric in Σ min p+1 . The symmetric tensor l a µν (ξ) is the second fundamental form of the h-ws treated as a constrained multiplet of the local group SO(D − p − 1). The indexes a, b = p + 1, p + 2, ..., D − 1 enumerate the orts n a (ξ ρ ) of an orthonormal moving frame attached to Σ min p+1 and orthogonal to it. The internal coordinates ξ µ = (τ, σ r ), r = 1, 2, .., p parametrize the h-ws world vector x(ξ ρ ). The map of the Dirac p-brane action to the diff and SO(D − p − 1) invariant h-ws action is realized by the field action [15] where the brackets {µν}, [µν] imply the µ, ν symmetrization and antisymmetrization, respectively. The potential energy term V (l, g) is equal to The kinetic term for the SO( The metric and Y-M covariant derivative ∇ ⊥ µ l νρ a is defined by the expression The h-ws metric g µν (ξ) in (2) is treated as a background field since its equations are encoded by the Gauss's Theorema Egregium used under construction of action (2). The discussed gauge approach can be generalized for the construction of h-ws with non-zero mean curvature. Solution of this problem is reduced to such a deformation of V Dir (3) which makes it consistent with the cmc condition. Below we solve this problem.

Brane h-ws with constant mean curvature
For h-ws of codimension 1 the gauge field B ab µ ≡ 0 since a = b = p + 1 and (2) encodes description of quadratic curvature gravity on the string/brane-ws with zero mean curvature by means of the reduced potential (3) as shown in [16]. To find a deformed potential V cmc corresponding to a cmc hypersurface Σ cmc p+1 of codim 1 we consider the h-ws action and obtain the following EOM for the reduced N-G field l µν := −l (p+1) µν The (P-C) conditions for the covariant derivatives (5) used in (9) nullify their l.h.s. and yield the consistency condition for V(l, g) Using the Bianchi identities (BI) for the commutator in (11) and the Gauss eqs. (6) we present the r.h.s. of Eqs. (11) in the form where Sp(l 2 ) = l µρ l ρ ν g µν . Substitution of (13) into (11) yields the PDE's 1 2 for V (l, g). Solution of Eqs. (14) corresponding to the cmc µ(p) is where c p is an integration constant and the cmc condition is treated as the inverse Higgs effect. So, the action (8) with V cmc (15) is consistent with the Gauss (6) and P-C (10) conditions. This action describes diff invariant self-interaction of the N-G field l µν . From the viewpoint of the variational principle the h-ws metric g µν works as a background metric since its evolution is defined by Eqs. (6). The latter permits to present the Euler-Lagrange eqs. produced by (16) as Using the identity (17) into the equation Contracting (18) with g ρν and using the cmc Spl = µ we obtain the covariant conservation law for the divergence ∇ ρ l ρ Now it is suitable to rewrite EOM (18) in the form Using the relation for the (ν, ρ)-antisymmetric part of the commutator given by the Bianchi identities and Gauss conditions, shows vanishing of (20) for codim 1. As a result, EOM (17) are represented in the form Eqs. (21) are consistent with the P-C eqs. (10) in view of the identity following from (10) after contraction of µ and ρ and using the cmc condition. Consideration of the local conservation law (19) together with (22) hints that the latter can be treated as initial data constraints (IDC) preserved by EOM (21). Moreover, we intend to check a possibility to interpret all the P-C eqs.
Next we prove that the IDC (23) together with the local conservation laws (31) provide fulfillment of the P-C conditions (10) at any τ To this objective consider ν = r and ρ = τ in (31) and obtain the relations which in combination with the IDC (24) show that The substitution of (34) into the expansion (26) proves that The choice of both indices ν, ρ in (31) as space ones ν = r, ρ = s gives Eqs. (36) and (24) show that Eqs. (37) and (35) The remaining subset of the P-C eqs. to be proved contains only space-like indices r, s, q. Therefore, we consider the expansion where the relation ∇ τ ∇ [r l s]q | τ =0 = g τ τ ∇ τ ∇ [r l s]q | τ =0 together with (23), (24) are used. The proof of (40) reduces to the that of the relation Using the BI (12) we present the first term in the r.h.s. of (42) as The conditions (37) permit to rewrite the second term in the form The sum of (43) and (44) transformss (42) into after using (38) and (24). Further, we apply the Gauss theorem (6) and obtain the following expressions for the terms in (45) Here we observe mutual cancellation of the coorresponding terms in the sum of (47) and (48). This proves Eqs. (40) and shows validity of Eqs. (32) . Then the cmc condition Spl = µ emerges as a consequence of Eqs. (30) and (32). So, we see that EOM (17) together with the Gauss eqs. (6) provide conservation of the IDC (30) and (32). The latter select a closed sector of the solutions descrbing cmc h-ws of codim 1.

Extremals of the brane potential
The Gauss's theorem permits to express V cmc (15) associated with the h-ws Σ cmc p+1 in terms of the Riemann tensor. Actually, using Eqs. (6) we obtain where R νρ is the Ricci tensor. Using (49) we obtain the R-reps. for V cmc which reduces to V Dir when µ = 0. Reps. (50) can be rewritten as where Λ(p) := ( µ 2 (p) 4 − 3cp µ 2 (p) ) is the h-ws cosmological constant. We see emergence of the H-E term together with the interaction term which extend the pure R 2 gravity encoded by V Dir . The potential V cmc reduces to the constant 2 3 µ 2 Λ(p) when R νρ = 0. Then (49) yields the relations (l n ) νρ = µ n−1 l νρ , (n = 2, 3, ...) =⇒ l νρ = µg νρ , detl ν α = 0 (53) and Spl = (p + 1)µ, respectively. The latter is compatible with the cmc condition only if p = 0 that describes a worldline. So, the case when R νρ = 0, but µ = 0, detl νρ = 0 describes point-like 0-branes. Further we study extremals of V cmc defined by solutions of the equation In the R-reps. this equation takes the form after using relations (49). The general solution of (55) is The substitution of this extremal in the Gauss equation yields the relation The extremal (56) should obey the P-C eqs. (10) and Eq. (22) which follows from (10) and has the form after substitution of (56) in (22). The use of the relation which follows from the Bianchi identities transforms Eq. (59) into the equation The latter means that ∂ µ R should be a null vector for the Einstein matrix G µν := R µν − 1 2 g µν R associated with Σ extr p+1 . In the R-reps. this h-ws is fixed by Eqs. (57), (62). Their solution for the case when detG µν = 0 is This solution defines a h-ws Σ o p+1 of constant curvature (cc) characterized by the Riemann tensor and the Gauss curvature K o R oµνγλ = K o (g µγ g νλ − g νγ g µλ ). (64) The substitution of (64) in the Gauss eqs. (57) results in the relation which shows proportionality of R oµν to g µν and defines K o The latter completely fixes the Riemann tensor of Σ o and the function Λ o = pK o , respectively. Then we obtain As shows the substitution of (68) in (56), this extremal acquires the form Solutions (69) are proportional to g µν (ξ), and therefore they are solutions of the P-C eqs. (10) in the l-reps. and eqs. (58) in the R-reps., respectively. Coming back to Eq. (54) in the l-reps. we rewrite it in the form l ν α (µl αρ − g αρ Sp(l 2 )) = 0 which shows that there exist two sets of the solutions. For the first of them detl µν = 0, for the other it is nonzero. In the latter case (54) reduces to l αρ = Sp(l 2 ) µ g αρ , detl µν = 0 (70) and corresponds to the above-considered solution (69). Indeed, in this case contraction of (70) with l ρα and g ρα , respectively, yields the conditions Then the substitution of (71) in (70) shows that Σ o p+1 is a (p + 1)-dim. Einstein space with the negative cosmological constant Λ o which can be treated as the anti-de Sitter space AdS p+1 .
The vacuum value of V cmc (15) is the constant given by The action (16) at the extremum l o (ξ) is also constant equal to due to vanishing of the kinetic term of l o (ξ) because l oµν ∼ g µν . The differential of Eq. (14) yields the second partial derivative of V cmc The substition of l µν = l oµν (69) in Eq. (74) gives its value since g τ g σ < 0 and g τ g η < 0 due to the time-like character of Σ o 3 . However, the mixed derivative with respect to l σ and l η is negative since g σ g η > 0. It means that the extremal of the potential V cmc (l, g, µ) is a 3-dim. saddle hypersurface that shows classical unstability of the membrane vacuum. So, the potential of cmc 3-brane with R µν = 0 in 4-dim. Minkowski space turns out to be unstable. This result based on the sign indefiniteness of the second derivatives in (75) hints that extremals of V cmc (l, g, µ) for branes with codim 1 and higher p might also be saddle hypersurfaces.

Conclusion
We considered the extension of the gauge theory approach [15] to branes sweeping hyper-worldsheets of codimension 1 with constant mean curvature. Constructed were invariant action and interaction potential for the h-ws Nambu-Goldstone bosons l µν of the broken Poincare symmetry. The potential was shown to encode the generalized h-ws action of quadratic curvature gravity. The equation for the brane potential extremals and its solution describing h-ws of negative constant curvature, were revealed. This set of maximally symmetric spaces also includes anti-de Sitter space AdS p+1 which is the exact solution of the H-E equations for an empty universe with negative cosmological constant. The found extremum for p = 2 was proved to be realized by a saddle 3-dim. hypersurface. This points to classical unstability of the constant mean curvature membrane potential in R 1,3 . It is interesting to connect the saddle-like unstability with the inflation and reheating mechanisms studied in quadratic curvature models of gravity.