Wess-Zumino actions and Dirichlet Boundary Conditions for Super p-branes with Exotic Fractions of Supersymmetry

The general solutions in the models of closed and open superstring and super p-branes with exotic fractions of the N=1 supersymmetry are considered and the spontaneously broken character of the $OSp(1,2M)$ symmetry of the models is established. It is shown that extending these models by Wess-Zumino terms generates the Dirichlet boundary conditions for superstring and super p-branes. Using the generalized Wess-Zumino terms new $OSp(1,2M)$ invariant super p-brane and Dp-brane-like actions preserving $\frac{M-1}{M}$ fraction of supersymmetry are proposed. For M=32 these models suggest new superbrane vacua of M-theory preserving 31 from 32 global supersymmetries.

1 Introduction necessary to include the fermionic degrees of freedom and to permit exotic vacuum states preserving 31 from 32 supersymmetries [8], [9].
The string/brane description of the vacuum state with the so high supersymmetry was given by the model [10] of tensionless superstring and super p-brane. A connection of this model with the description [9] (see also [11]) of the BPS states in M-theory was discussed in [12]. The model [10] develops the approach [13], [14] and [15] to the description of string/brane dynamics in superspaces extended by the addition of tensor central charge (TCC) coordinates. The centrally extended superspaces are characterized by the orthosymplectic symmetries and are closely connected with gauge theory of massless fields with higher spins [16], [17] which already appear in the quantized superparticle models with exotic supersymmetries [18], [19].
It was observed in [20] that OSp(1, 64) symmetry is spontaneously broken in D = 11 supergravity which is the low energy phase of M/string-theory containing massive higher spin states. This observation gave a reason to suppose that the superbrane microscopic structure also may be described in terms of the spontaneously broken orthosymplectic symmetries [17]. Taking into account the connection of tensionless strings and branes with higher spin field theory [21], [22] it is important to understand whether the OSp(1, 2M) symmetry of the model [10] is spontaneously broken.
Here we study this question for the case of closed and open tensionless superstring and super p-brane and find that the OSp(1, 2M) symmetry of the model is spontaneously broken by the general static solutions of the brane equations of motion. This effect is similar to the partial supersymmetry breaking by the super four-brane [23] and the generalized coordinates of the model [10] are the Goldstone fields of the OSp(1, 2M) symmetry. These Goldstone fields may be associated with effective long wave description of the vacua in microscopic higher spin theories. Also we construct new topological Wess-Zumino like superstring and super p-brane actions generating the Dirichlet boundary conditions and spontaneously breaking supersymmetry and OSp(1, 2M) symmetry. In addition we propose a new set of the OSp(1, 2M) invariant super p-brane and Dp-brane like actions preserving M −1 M fraction of the N = 1 supersymmetry.
2 A simple super p-brane model with extra supersymmetry: the general solution and symmetries.
The exactly solvable supersymmetric model of closed tensionless super p-brane (p = 1, 2, 3, ...) with extra κ-symmetry has been studied in [10]. This model includes the Cartan differential one-form W ab invariant under the N = 1 global supersymmetry transformations of the generalized superspace composed by the spin-tensor Y ab , the Grassmannian Majorana spinor θ a and an auxiliary commuting Majorana spinor U a [24] parametrizing the light-like density of the brane momentum. The world-volume density ρ µ = (ρ τ , ρ) [25], invariant under the N = 1 supersymmetry (3), provides reparametrization invariance of S p (1). The real symmetric spin-tensor Y ab unifies the space-time coordinates x m and the TCC coordinates z kl..m of the D-dimensional Minkowski space-time with D = 2, 3, 4 mod(8). The spin-tensor Y ab is a realization of the symmetric matrix of generalized symplectic coordinates previously considered in [16], [17]. The action S p is invariant under the transformations of the enhanced κ−symmetry with the parameter κ restricted by one real condition U a κ a = 0 (7) and the super p-brane model (1) where the Majorana spinorỸ a is defined by the relation and is a new effective variable substituted for Y ab andη is the Lorentz invariant Grassmannian field describing the Goldstone fermion of the model. The action (8) is the component representation of the OSp(1, 2M) invariant action where Y Λ = (iU a ,Ỹ a ,η) is a real OSp(1, 2M) supertwistor and G ΛΞ = (−1) ΛΞ+1 G ΞΛ is the invariant supersymplectic metric previously considered in superparticle dynamics [18]. The equations of motion following from S p (11) are invariant under the linearly realized OSp(1, 2M) symmetry, worldvolume reparametrizations and the Weyl gauge symmetry [26] In the partially fixed reparametrization gauge [10] ρ i (τ, σ) = 0, (i = 1, 2, ..., p), removing p of (p + 1) components of the worldvolume density ρ µ (τ, σ) without breaking of the Weyl and OSp(1, 2M) symmetries, the general solution of Eqs.(12) is given by The static fields Y Λ ( σ) in (15) are restricted by the p initial data constraints which are the invariants of the Weyl and OSp(1, 2M) symmetries. In the case of closed super p-brane the components of Y Λ ( σ) and ρ τ (τ, σ) are periodic functions of σ i The components of the arbitrary supertwistor Y Λ ( σ) in the general solution (15) are the invariants of the Weyl gauge symmetry (13) due to the presence of the ρ τ (τ, σ) factor. However, they form the linear representation of the OSp(1, 2M) group, because ρ τ is the invariant of this group. The ρ τ (τ, σ)-factor in (15) concentrates all dependence of the general solution on the evolution parameter τ and it may be removed by the additional to (14) gauge fixing The gauge condition (18) breaks the Weyl symmetry, but preserves the OSp(1, 2M) symmetry and simplifies the general solution (15) to the pure static form where ρ τ (τ, σ) = ρ τ 0 (σ) was moved in Y Λ 0 (σ i ). One remarkss that the solutions (15) and (19) are equivalent on the classical level, because of a correlation between the Weyl and space-time conformal symmetries on the quantum level of the tensionless string treatment [27].
The components of the static field Y Λ 0 (σ i ) describe the shape of the super p-brane. The superbrane has a freedom to choose any shape restricted by the initial data constraints (16) and this shape will remains frozen during the evolution. Any other shape obtained from the initially randomly chosen by any transformation belonging to the OSp(1, 2M) group will have the same rights. However, a fixing of the brane shape by any fixed initial data for Y Λ 0 (σ i ) will break the OSp(1, 2M) symmetry. From the point of view of the general theory of system with broken global symmetry [28] fixing of the form of Y Λ 0 (σ i ) may be interpreated as a choice of the vacuum state of the underlying field system with the spontaneously broken global OSp(1, 2M) symmetry. As a result, the static fields Y Λ 0 (σ i ) are interpreted similarly to [23] as the Goldstone fields associated with the spontaneously broken OSp(1, 2M) symmetry and the action (11) is an effective long wave action for the Goldstone fields associated with the super p-brane. It proves the spontaneously broken character of the OSp(1, 2M) symmetry as the symmetry of the brane action (11).
Using the supersymmetry laws (3) and the definitions (9),(10) of the components of the supertwistor Y Λ = (iU a ,Ỹ a ,η) we find the transformation properties of the Goldstone fields under the supersymmetry transformations from OSp(1, 2M) The N = 1 supersymmetry transformations (20) are nonlinear, as it have to be for the spontaneously broken symmetries [28], because the original Goldstone fields θ a are presented in (20) (7) U a κ a = 0.
The non-zero projection (U a ε a ) of the supersymmetry parameter ε a U a ε a = 0 (21) defines the direction of the spontaneously broken ( 1 M ) fraction of the N = 1 supersymmetry which can not be compensated any of the (M − 1) κ-symmetry transformations. So, the condition (21) is antipodal to the condition (7) in the correspondence with the aforesaid and the Goldstone fermionη has a non-zero shift (21).
It is easy to check that the action S p (8), and respectively (11), are invariant under the N = 1 global supersymmetry transformations (20), because of the cancellation between the contributions given byỸ a and the fermionic Goldstone fieldη An interesting and open question is to clarify the effect of the boundary terms for the dynamics of the open super p-branes and we turn to this question below.

Boundary conditions for the open super p-brane
Here we study the case of open super p-brane (11). The contribution of the boundary terms in the variation of S p (11) is given by where ds ν = 1 p! ε νµ 1 µ 2 ...µp dS µ 1 µ 2 ...µp . Here, we consider the variational problem with the fix initial (τ = τ i ) and final (τ = τ f ) data, so the integral along the super p-brane profile for As a result, the variation δS p | Γ (23) is filled out by the integrals along the p-dimensional boundaries of the brane worldvolume containing the τ -direction In the case of variational problem with free ends, i.e. when the field variations on the p-brane boundaries are arbitrary, the vanishing of these hypersurface terms in δS p | Γ (25) gives the open super p-brane boundary conditions One of the solutions of (26) is The second possibility to satisfy the boundary conditions (26) implies the zero boundary conditions for the supertwistor Y Λ = (iU a ,Ỹ a ,η) values on the boundaries or, equivalently, in terms of the supertwistor components The boundary conditions (27)  As a result, the general solution (15) for the closed super p-brane in this gauge gives also the general solution of the boundary problem (12), (27) for the open super p-brane.
Concerning the boundary conditions (28), one can note that the zero boundary values U a | σ i =0,π = 0 (29) result in some problem in the geometric interpretation of the auxiliary spinor field U a as a basic constituent of the local spinor repere attached to the super p-brane worldvolume. For example, in the case of the 4-dimensional Minkowski space, where U a is treated [24] as one of the components of the Newman-Penrose dyads [29], these boundary conditions result in the condition which breaks the basis relation U a V a = 1 defining the dyads U a , V a [29]. To preserve this condition the spinor field V a should be singular on the brane/string boundaries and it signals on some instabilities on the brane boundaries. Therefore, the solution (27) have to be choosen for the considerd simple model (11) and in this case the open and closed super p-brane are described by the same general solution (15) for the static Goldstone fields. This result is based on use of the gauge condition (14) for the auxiliary field ρ µ .
To overcome the problem of the singular character of the boundary conditions (28) we need to extend the simple action (11) and to this end we may generalize the topological actions studied in [11], [27]. An example of that generalization will be done in the next section, where we will present of a topological action which yields the Dirichlet boundary conditions for open superstring, resulting to the spontaneous breakdown of the OSp(1, 2M) symmetry and N = 1 supersymmetry.

A topological action generating the Dirichlet boundary conditions for the superstring
The superstring action with enhanced supersymmetry given by contributes only on the superstring ends and yields the Dirichlet boundary conditions similar to those for the Nambu strings [30]. The integrand in the integral S W Z (31) is a total derivative and is presented in the form of the integral along the one-dimensional boundary of the superstring worldsheet The integral (31) is similar to the curvature integral for the open string where R/2 is the Gauss curvature of the string worldsheet. It was shown in [31] that taking into account of the nonlinear boundary conditions generated by S R reveals a topological structure of the string action extrema. To find the effect resulted in by S W Z (31) one notes that the integrand of S W Z (32) coincides with the differential form (U a W ab µ U b ) in (1) and therefore S W Z is invariant of the original symmetries of the action (11) besides of the Weyl gauge symmetry (13). The latter restriction follows from the absence of the ρ µ density in the integral (32) which results in it change under the Weyl transformation (11). It means that the Weyl symmetry is explicitly broken by the boundary terms, already on the classical level unlike the Green-Schwarz superstring, where the breakdown appears only on the quantum level. The variation of the Wess-Zumino term (32) gives where the initial and final variational conditions δY Ξ (τ i , σ) = 0, δY Ξ (τ f , σ) = 0 have been used. Next, taking into account the freedom in the variations δY Λ (τ i , σ)| σ=0,π on the string ends we obtain the following boundary conditions presented in the component form as The boundary conditions (36) which contains the integration constants A Λ and B Λ , defined by the initial data The solution (40) will partially preserve the supersymmetry if the conditions for the projection (U a ε a ) on the superstring ends are satisfied, as it follows from the transformation rules (20). The conditions (41) impose two real conditons for the supersymmetry parameters ε a resulting to the breaking of ( 2 M ) fraction of N = 1 supersymmetry or, in the special case to the breaking only ( 1 M ) fraction of N = 1 supersymmetry.

The superstring model with the Wess-Zumino term
Here we show that the addition of the Wess-Zumino term (31) in the original action removes the problem of the singular character of the second solution (28) of the boundary conditions (26). The extended action modifies the boundary conditions (26) to the conditions The conditions (44) are invariant under the OSp(1, 2M) symmetry similarly to (27) and (28) and their general solution if the gauge (14) (for p = 1) is choosen. The substitution of (46) in the boundary conditions (44) with ρ σ = 0 results in the boundary conditions ∂ τ ρ τ (τ, σ)| σ=0,π = 0 (47) which are satisfied by the additional gauge fixing (18) ∂ τ ρ τ (τ, σ) = 0.
In this gauge the general solution (46) coincides with the static solution (19) describing the above studied closed and opened superstrings but has the Dirichlet boundary conditions (38). One notes that the initial data Y Λ 0 (σ) (48) are restricted by the constraint (16) The matching (45) and (48) confirms that the integration constants A Λ , B Λ coincide with the Y Λ 0 (σ) values taken on the string ends σ = 0, π We conclude that the superstring action (43) with the Dirichlet boundary conditions (45) describes a static BPS state with the spontaneously broken OSp(1, 2M) symmetry.

Wess-Zumino actions of higher orders
Using the OSp(1, 2M) invariant character of the differential one-form Y Λ G ΛΞ dY Ξ and two-form dY Λ G ΛΞ dY Ξ one can construct more general OSp(1, 2M) invariant super pbrane actions with enhanced supersymmetry. At first, we note that the closed 2n- which is not equal to zero, because of the symplectic character of the supertwistor metric G ΛΞ , can be used to generate the Dirichlet boundary terms for the open super p-brane (p = 2n − 1) described by the generalized action (43) Similarly to the open superstring case (32), the Wess-Zumino integral in (52) is transformed to the integral along the (2n−1)-dimensional boundary M 2n−1 of the super (2n−1)brane worldvolume The sufficient conditions for the vanishing of the variations of the integral (53) with the fix initial and final data are the conditions generalizing the Dirichlet boundary condition (36). Therefore, in the gauge (14) and (18) this open super p-brane is described by the pure static solution generalizing the superstring static solution (48). On the other hand the integrals (53) can be considered as the OSp(1, 2M) invariant actions for the new models of super pbranes (p = 2n − 2) with enhanced supersymmetry. For n = 1 we get the known action [18] for superparticles, but for n = 2, 3 we find the new actions for the supermembrane or a domain wall in the symplectic superspace, and for the super four-brane We shall analyse these models in another place.

7
The Weyl symmetry restoration for the Wess-Zumino actions A characteristic feature of the proposed Wess-Zumino actions is the explicit breaking of the Weyl gauge symmetry (13). When the Wess-Zumino terms are considered as the boundary terms generating the Dirichlet boundary conditions for the superstring (36) and super p-branes (54) the breaking of the Weyl symmetry is localized at the boundaries. It shows that the spontaneous breaking of the OSp(1, 2M) symmetry on the boundaries is accompanied by the explicit breakdown of the Weyl gauge symmetry on the boundaries. Because the Dirichlet boundary conditions are associated with the Dp-branes attached on their boundaries [30], a question on the action of Dp-branes in the symplectic superspaces considered here appears. It implies the correspondent generalization of the proposed Wess-Zumino actions. One of the posssible generalizations is rather natural and is based on the observation that the Weyl invariance of the considered Wess-Zumino actions may be restored by the minimal lengthening of the differentials d → D = (d − A), where the worldvolume one-form A is the gauge field associated with the Weyl symmetry. The covariant differentials DY Σ are homogeneously transformed under the Weyl symmetry transformations (13) Then the generalized OSp(1, 2M) invariant two and one-forms become the invariants of the Weyl symmetry also, where the compensating scalar field φ, with the transformation low was introduced. Then the closed 2n-differential form Ω 2n = (G ΛΞ dY Λ ∧ dY Ξ ) n may be changed by the Weyl invariant 2n-differential formΩ 2n and Ω 2n−1 byΩ 2n−1Ω As a result, the actions (53) is transformed to the new super (2n − 1)-brane actioñ invariant under the OSp(1, 2M) and Weyl symmetries. Respectively, the actioñ will describe a new OSp(1, 2M) and Weyl invariant super (2n − 2)-brane. These actions may be presented in the Dp-brane like form, e.g.

Conclusion
We considered the general solutions of the equations of motion in the simple model of closed and open tensionless superstring and super p-branes and found that these static solutions spontaneouly break the OSp(1, 2M) symmetry and N = 1 supersymmetry. Next, we generalized this model to the higher orders in the derivatives of the Goldstone fields and constructed the new Wess-Zumino like actions supposed to describe tensile super p-branes. These actions generate the Dirichlet boundary conditions which, in particular, break the Weyl gauge symmetry. The introduction of additional vector and scalar fields restores the Weyl symmetry and results in the Weyl and OSp(1, 2M) invariant Dp-brane like actions. The open problem is to find supersymmetric Y-M field theories having the considered superbranes as vacuum states spontaneously breaking the OSp(1, 2M) symmetry. One can conjecture that these branes appear as supersymmetric solutions of D = 11 supergravity [1], [2], where the OSp(1, 64) symmetry is also spontaneously broken [20]. Then a connection between the R 31 holonomy and space-time symmetries [1], [2] with the local Abelian shifts of the space-time and TCC brane coordinates by the null multivectors [26] may appear. We will study these problems in another place.