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Probe Branes Dynamics in Nonconstant Background Fields

We consider probe p-branes dynamics in string theory backgrounds of general type. We use an action, which interpolates between Nambu-Goto and Polyakov type actions. This allows us to give a unified description for the tensile and tensionless branes. Firstly, we perform our analysis in the frequently used static gauge. Then, we obtain exact brane solutions in more general gauges. The same approach is used to study the Dirichlet p-brane dynamics and here exact solutions are also found. As an illustration, we apply our results to the brane world scenario in the framework of the mirage cosmology approach.


Introduction
The classical p-brane is a p-dimensional relativistic object, which evolving in the space-time, describes a (p + 1)-dimensional worldvolume. In this terminology, p = 0 corresponds to a point particle, p = 1 corresponds to a string, p = 2 corresponds to a membrane and so on.
Every p-brane characterizes by its tension T p with dimension of (mass) p+1 . When the tension T p = 0, the p-brane is called tensionless one. This relationship between the tensionless branes and the tensile ones generalizes the correspondence between massless and massive particles for the case of extended objects. Thus, the tensionless branes may be viewed as a high-energy limit of the tensile ones.
The point particle (0-brane) couples to 1-form gauge field (the electro-magnetic potential). Generalizing to arbitrary p, one obtains that the p-branes couple to (p + 1)-form gauge potentials.
As is known, there exist five consistent superstring theories in ten dimensions: Type IIA with N = 2 non-chiral supersymmetry, type IIB with N = 2 chiral supersymmetry, type I with N = 1 supersymmetry and gauge symmetry SO(32) and heterotic strings with N = 1 supersymmetry and SO(32) or E 8 × E 8 gauge symmetry.
The superstring dynamics unify all fundamental interactions between the elementary particles, including gravity, at super high energies. The p-branes arise naturally in the superstring theory, because there exist exact brane solutions of the superstring effective equations of motion. The 2-branes and the 5-branes are the fundamental dynamical objects in eleven dimensional M -theory, which is the strong coupling limit of the five superstring theories in ten dimensions, and which low energy field theory limit is the eleven dimensional supergravity. Particular type of 3-branes arise in the Randall-Sundrum brane world scenario.
In this talk, we consider the probe p-branes dynamics in string theory backgrounds of generic type. In the probe brane limit, one neglects the back-reaction of the brane on the background fields. We use an action, which interpolates between Nambu-Goto and Polyakov type actions. Thus, we give a unified description for the tensile and tensionless branes. Our aim is to obtain the conditions on the background metric and the (p+1)-form gauge field, under which the nonlinear brane equations of motion and constraints can be solved exactly. Firstly, we perform our analysis in the frequently used static gauge. Then, we obtain exact brane solutions in a more general gauge. The same approach is used to study the Dirichlet p-brane dynamics and here exact solutions are also found. As an illustration, we apply our results to the brane world scenario in the framework of the mirage cosmology approach.

Probe p-Branes Dynamics
The Nambu-Goto type action for bosonic p-brane in a D-dimensional curved space-time with metric tensor g MN (x), interacting with a background (p+1)-form gauge field B p+1 via Wess-Zumino term, can be written as The corresponding Polyakov type action is given by where γ is the determinant of the auxiliary worldvolume metric γ mn , and γ mn is its inverse. None of these actions is appropriate for description of the tensionless branes. To have the possibility for unified description of the tensile and tensionless branes, we will use the action in which the limit T p → 0 may be taken. It can be shown that this action is classically equivalent to the above two actions. The Lagrange multipliers λ 0 and λ i are connected to the lapse function N and the shift vector N i as follows: Varying the action S p with respect to λ m , one obtains the constraints We will work in the gauge λ m = constants, in which the equations of motion take the form:

Static Gauge
In the commonly used static gauge, we have the following identification: X m (ξ) = ξ m . The other part of the coordinates X a are supposed to be functions only of ξ 0 ≡ τ . The typical string theory backgrounds do not depend on x m , so that Under these conditions, the action S p reduces to To have finite action, we require the fraction V p /λ 0 to be finite. In the string case (p = 1) and in conformal gauge, for example, this corresponds to have the fraction V 1 /α ′ = 2πV 1 T 1 finite. Now, the constraints are: The Lagrangian L SG does not depend on τ explicitly, so the energy E is conserved: With the help of the constraints, we can replace this equality by the following one It can be shown that the equations of motion and all these constraints can be reduced to the equalities It turns out that for background fields depending on only one coordinate x a , we can always integrate these equations, and the solution is Otherwise, supposing the metric g ab is a diagonal one, we can rewrite the equations of motion in the form To find solutions of the above equations without choosing particular background, we fix all coordinates X a except one. Then the exact probe brane solution of the equations of motion and constraints is given again by the same expression for τ (X a ).
To find solutions depending on more than one coordinate, we have to impose further conditions on the background fields. An example of such sufficient conditions, which allow us to reduce the order of the equations of motion by one, is given below (we split the index a in such a way that X r is one of the coordinates X a , and X α are the others): The result of integrations is the following where D α , E α and E r are arbitrary functions of their arguments, and n α is the number of the coordinates X α . Further progress is possible, when working with particular background configurations.

Rotated Gauge
Now we will repeat our analysis of the probe p-brane dynamics in a more general gauge than the static one. Namely, our ansatz for the coordinates X m (ξ) is the following We call this ansatz rotated gauge, because Λ m n ξ n look like rotations in the space described by the coordinates ξ n . However, there are no restrictions on the parameters Λ m n . They are arbitrary constants. For Λ m n = δ m n , we come back to static gauge. In the same way as before, one obtains the Lagrangian the constraints and finally, the equations of motion and the effective constraint We see that these equalities have the same form as in static gauge, but with Therefore, the corresponding exact solution, depending on one of the coordinates X a will be

Probe Dp-Branes Dynamics
The Dirichlet p-branes are extended objects on which the open strings can end. On the other hand, they are solutions of the superstring effective equations of motion, carrying Ramond-Ramond charges. The Dirac-Born-Infeld type action for the bosonic part of the super-Dp-brane in a D-dimensional curved space-time with metric tensor g MN (x), interacting with a background (p+1)-form Ramond-Ramond gauge field C p+1 via Wess-Zumino term, can be written as T D =(2π) −(p−1)/2 g −1 s T p is the D-brane tension, g s = exp Φ is the string coupling expressed by the dilaton vacuum expectation value Φ and 2πα ′ is the inverse string tension. G mn = ∂ m X M ∂ n X N g MN (X), B mn = ∂ m X M ∂ n X N b MN (X) and Φ(X) are the pullbacks of the background metric, antisymmetric tensor and dilaton to the Dp-brane worldvolume, while F mn (ξ) is the field strength of the worldvolume U (1) gauge field A m (ξ). The parameter a(p, D) depends on the brane and space-time dimensions p and D respectively.
To be able to take the limit T D → 0, we will work with the classically equivalent action Here additional Lagrange multipliers κ i are introduced, in order to linearize the quadratic term For simplicity, we restrict our considerations to constant dilaton Φ = Φ 0 and constant electromagnetic field F mn on the Dp-brane worldvolume.

Static Gauge
In static gauge, the reduced Lagrangian is given by Using the same approach as before, we obtain the equations of motion and the effective constraint, which have the previous form The corresponding exact solution, depending on one of the coordinates X a , is

Rotated Gauge
In the same way, one obtains the exact probe Dp-brane solution in this gauge. It is given by the above equality for τ (X a ), where instead of V SG D we have to use the potential

Application to Mirage Cosmology
The idea of the mirage cosmology approach [1] - [16] to the brane world model is the following [2]. The motion of the brane universe in a curved higher dimensional bulk space induces a cosmological evolution on the universe brane that is indistinguishable from a similar one induced by matter density on the brane. It can be shown that the motion of the probe brane in ambient space induces cosmological expansion or contraction on our universe simulating various kinds of "matter" or a cosmological constant (inflation). There are two steps in the procedure: 1. Determine the probe brane motion by solving the worldvolume field equations.
2. Determine the induced metric on the brane which now becomes an implicit function of time. This gives a cosmological evolution in the induced brane metric. This cosmological evolution can be reinterpreted in terms of cosmological "mirage" energy densities on the brane via a Friedman-like equation for the scale factor a: