D=10 CHIRAL TENSIONLESS SUPER p-BRANES
^{1}^{1}1Work supported in part by the National Science Foundation
of Bulgaria under contract

P. Bozhilov
^{2}^{2}2E-mail: ; permanent address:
Dept.of Theoretical Physics,”Konstantin Preslavsky” Univ. of
Shoumen, 9700 Shoumen, Bulgaria

Bogoliubov Laboratory of Theoretical Physics,

JINR, 141980 Dubna, Russia

We consider a model for tensionless (null) super -branes with chiral supersymmetries in ten dimensional flat space-time. After establishing the symmetries of the action, we give the general solution of the classical equations of motion in a particular gauge. In the case of a null superstring (=1) we find the general solution in an arbitrary gauge. Then, using a harmonic superspace approach, the initial algebra of first and second class constraints is converted into an algebra of Lorentz-covariant, BFV-irreducible, first class constraints only. The corresponding BRST charge is as for a first rank dynamical system.

## 1 Introduction

The tensionless (null) -branes correspond to usual -branes with their tension taken to be zero. This relationship between null -branes and the tensionful ones may be regarded as a generalization of the massless-massive particles correspondence. On the other hand, the limit corresponds to the energetic scale . In other words, the null -brane is the high energy limit of the tensionful one. There exist also an interpretation of the null and free -branes as theories, corresponding to different vacuum states of a -brane, interacting with a scalar field background [1]. So, one can consider the possibility of tension generation for null -branes (see [2] and references therein). Another viewpoint on the connection between null and tensionful -branes is that the null one may be interpreted as a ”free” theory opposed to the tensionful ”interacting” theory [3]. All this explains the interest in considering null -branes and their supersymmetric extensions.

Models for tensionless -branes with manifest supersymmetry are proposed in [4]. In [1] a twistor-like action is suggested, for null super--branes with -extended global supersymmetry in four dimensional space-time. In the recent work [5], the quantum constraint algebras of the usual and conformal tensionless spinning -branes are considered.

In a previous paper [6], we announced for a null super -brane model, and here we are going to formulate it, and to consider its classical properties. After establishing the symmetries of the action, we give the general solution of the classical equations of motion in a particular gauge. In the case of a null superstring, (=1), we find the general solution in an arbitrary gauge. Then, in the framework of a harmonic superspace approach, the initial algebra of first and second class constraints is converted into an algebra of Lorentz-covariant, BFV-irreducible, first class constraints only. The corresponding BRST charge is as for a first rank dynamical system.

## 2 Lagrangian formulation

We define our model for -extended chiral tensionless super -branes by the action:

(1) | |||

Here are the superspace coordinates, are left Majorana-Weyl space-time spinors ( , being the number of the supersymmetries) and are the 10-dimensional Pauli matrices (our spinor conventions are given in the Appendix). Actions of this type are first given in [7] for the case of tensionless superstring () and in [8] for the bosonic case ().

The action (1) has an obvious global Poincar invariance. Under global infinitesimal supersymmetry transformations, the fields , and transform as follows:

As a consequence and hence also.

To prove the invariance of the action under infinitesimal diffeomorphisms, we first write down the corresponding transformation law for the (r,s)-type tensor density of weight

where is the Lie derivative along the vector field . Using (2), one verifies that if , are world-volume scalars () and is a world-volume (1,0)-type tensor density of weight , then is a (0,1)-type tensor, is a (0,2)-type tensor and is a scalar density of weight . Therefore,

and this variation vanishes under suitable boundary conditions.

Let us now check the kappa-invariance of the action. We define the -variations of , and as follows:

(3) | |||

Therefore, are left Majorana-Weyl space-time spinors and world-volume scalar densities of weight .

The algebra of kappa-transformations closes only on the equations of motion, which can be written in the form:

(4) |

As usual, an additional local bosonic world-volume symmetry is needed for its closure. In our case, the Lagrangian, and therefore the action, are invariant under the following transformations of the fields:

Now, checking the commutator of two kappa-transformations, we find:

Here , and are given by the expressions:

We note that in (3) has the following property on the equations of motion

This means, that the local kappa-invariance of the action indeed eliminates half of the components of as is needed.

For transition to Hamiltonian picture, it is convenient to rewrite the Lagrangian density (1) in the form ():

(5) |

where

The equations of motion for the Lagrange multipliers and which follow from (5) give the constraints ( and are the momenta conjugated to and ):

(6) |

The remaining constraints follow from the definition of the momenta :

(7) |

## 3 Hamiltonian formulation

The Hamiltonian which corresponds to the Lagrangian density (5) is a linear combination of the constraint (6) and (7):

(8) |

It is a generalization of the Hamiltonians for the bosonic null -brane and for the -extended Green-Schwarz superparticle.

The equations of motion which follow from the Hamiltonian (8) are:

(9) |

In (9), one can consider , and as depending only on , but not on as a consequence from their equations of motion.

In the gauge when , and are constants, the general solution of (9) is

(10) | |||||

where , , and are arbitrary functions of their arguments

In the case of null strings (), one can write explicitly the general solution of the equations of motion in an arbitrary gauge: , , . This solution is given by

(11) | |||||

Here , , and are arbitrary functions of the variable

The solution (10) at differs from (11) by the choice of the particular solutions of the inhomogenious equations. As for and , one can write for example (, , are now constants)

and analogously for the other arbitrary functions in the general solution of the equations of motion.

Let us now consider the properties of the constraints (6), (7). They satisfy the following (equal ) Poisson bracket algebra

From the condition that the constraints must be maintained in time, i.e. [9]

(12) |

it follows that in the Hamiltonian one has to include the constraints

instead of . This is because the Hamiltonian has to be first class quantity, but are a mixture of first and second class constraints. has the following non-zero Poisson brackets

In this form, our constraints are first class and the Dirac consistency conditions (12) (with replaced by ) are satisfied identically. However, one now encounters a new problem. The constraints , and are not BFV-irreducible, i.e. they are functionally dependent:

It is known, that in this case after BRST-BFV quantization an infinite number of ghosts for ghosts appear, if one wants to preserve the manifest Lorentz invariance. The way out consists in the introduction of auxiliary variables, so that the mixture of first and second class constraints can be appropriately covariantly decomposed into first class constraints and second class ones. To this end, here we will use the auxiliary harmonic variables introduced in [10] and [11]. These are and , where superscripts and transform under the ’internal’ groups and respectively. The just introduced variables are constrained by the following orthogonality conditions

where

is the invariant metric tensor in the relevant representation space of and as a consequence of the Fierz identity for the 10-dimensional -matrices. We note that and do not depend on .

Now we have to ensure that our dynamical system does not depend on arbitrary rotations of the auxiliary variables , . It can be done by introduction of first class constraints, which generate these transformations

(13) | |||||

In the above equalities, and are the momenta canonically conjugated to and .

The newly introduced constraints (13) obey the following Poisson bracket algebra

This algebra is isomorphic to the algebra: generate rotations, is the generator of the subgroup and generate the transformations from the coset .

Now we are ready to separate into first and second class constraints in a Lorentz-covariant form. This separation is given by the equalities [12]:

(14) | |||||

Here are first class constraints and are second class ones:

It is convenient to pass from second class constraints to first class constraints , without changing the actual degrees of freedom [12], [13] :

where are fermionic ghosts which abelianize our second class constraints as a consequence of the Poisson bracket relation

It turns out, that the constraint algebra is much more simple, if we work not with and but with given by

After the introduction of the auxiliary fermionic variables , we have to modify some of the constraints, to preserve their first class property. Namely , and change as follows

As a consequence, we can write down the Hamiltonian for the considered model in the form:

The constraints entering are all first class, irreducible and Lorentz- covariant. Their algebra reads (only the non-zero Poisson brackets are written):