Research Article | Open Access
Super-Liouville Theory from Superstring Theory in the Presence of Gauge Worldsheet Fields
We obtain in this work the super-Liouville theory induced from the action of the superstring theory in the presence of gauge worldsheet fields; such construction is based on a special ansatz that gives a special value of -fields . We discuss the integrability of super-Liouville theory through the Lax formulation, and we establish also the superfields formulation of super-Liouville equations.
Gauge superstring theories have been studied in different points of view [1, 2]. They are obtained by introducing the worldsheet abelian gauge fields in superstring action [3–5]. The action of the gauged superstring allows us to build two worldsheet fields from elements of the gauge field. They appear as the coordinates of space and time.
The super-Liouville field theory (SLFT) is a generalization supersymmetric of classical bosonic Liouville theory, which is known to be theory of matter-induced gravity in two dimensions. Similarly SLFT describes 2D supergravity, induced by supersymmetric matter .
Our goal in this paper is to find the super-Liouville fields equations from gauged superstring theory and study the superfields formulation and integrability of such equations.
This paper is organized as follows. In Section 2, we introduce some elements of the action of the superstring in the presence of worldsheet gauge fields and its symmetries. In Section 3, we present in detail the method to find the Liouville equations from gauged superstring action. In Section 4, we give the superfield formulation. In Section 5, the integrability of Liouville superstring theory is discussed. Section 6 is for concluding remarks.
2. Superstring Theory Coupled to Gauge Theory
In this section we recall some basic elements of superstring theory in the presence of gauge superfields, so that we have the following.
2.1. Bosonic Action
To obtain the action with gauge field, we use the superfields in the worldsheet superspace. The bosonic action is given by where is the gauge coupling constant, are the fields strength associated with the worldsheet gauge fields, , are the worldsheet coordinates, and are the string coordinates. This action has the gauge symmetry. Therefore, we have the condition
2.2. Supersymmetric Action
To get the supersymmetric action the bosonic fields and should be replaced by the superfields where are dynamical fields on the worldsheets, and are auxiliary fields, the Majorana spinor is the superpartner of , and the Grassmannian coordinates and form a Majorana spinor .
We introduce also the following superspace covariant derivative: where and is a constant which is finding such that ; this gives .
Reformulating the action by introducing (3) and (4) we obtain the following supersymmetric action: The superfield strength is defined as follows: After making integration over the Grassmannian coordinates and , this action takes the form As we see, the gaugino field from the two-dimensional action disappeared.
2.3. Extracoordinates in Gauged Superstring Action
In the action (7) the kinetic terms of the fields and have the same feature. In other words, and have the roles of the time and space coordinates. Let denote the coordinates of this () dimensional spacetime. Thus, we have the field redefinition According to these definitions, the action (7) can be written as where , and we will use the convention and , Since both and carry the worldsheet index, the partial derivative always shows derivative with respect to . The bosonic part of this action apparently describes a 12-dimensional spacetime with the signature and the coordinates. However, in the superstring theory the dimension of the spacetime is always . Therefore, they are called the extradimensions or the fictitious coordinates .
The fermionic term of the action (9) also can be written with the 12-dimensional indices. For this, the Majorana spinor is defined by The spinors and satisfy the identities where . Introducing the identity (12) in the action (9) leads to the covariant form of this action The metric of the extended manifold is where belongs to the actual spacetime and to the fictitious coordinates.
The equations of motion, extracted from the action (13), are In addition, we should also consider the gauge condition This condition and the equation of motion of can be written as respectively. The constant is independent of and .
2.4. Symmetries of the Model
2.4.1. Worldsheet Supersymmetry
Using the superfield (3) we obtain the supersymmetry transformations of and as in the following: where . The supersymmetry parameter is an anticommuting infinitesimal constant spinor. In terms of the fields these transformations take the form The transformations (19) form a closed algebra. The supercurrent associated with the supersymmetry transformations (19), accompanied by , is According to the identity there is then is a conserved current .
2.4.2. The Poincare Symmetry
The action (13), with , under the Poincare transformations is symmetric. The matrix is a constant antisymmetric, and is a constant vector. The associated currents to these transformations are There are a conserved currents, that is, .
3. Liouville Equations from Gauged Superstring Action
Let us consider the action (13) of the superstring in the presence of the worldsheet gauge fields where and are the fields of conformal weights 1, 3/2, and 2, respectively.
We assume that these fields can be processed as follows: where is the Lorentz field of conformal weight 1 which does not depend on worldsheet variables, and the new fields , , and are, respectively, of conformal weights 0, 1/2 and 1.
By introducing the Ansatz (24), the action (23) becomes The equations of motion relative to the fields , , , and are given by To obtain the Liouville equations we need to take the following value for the field : In that case the action becomes where .
4. Superfield Analysis
In terms of a superfield formulation associated with an supersymmetry we can set  We can then easily show that the system of super-Liouville equations (29) can be expressed in terms of superderivative of the superfield as follows: where and . Indeed, straightforward computations lead to By virtue of (30) and expanding the exponential of the superfield , we find Identifying (31) with (32) one finds easily super-Liouville equations of motions (29).
Furthermore, using the complex transformations and one can easily rewrite the previous super-Liouville equations to become with and . With the equation of motion (34) and the previous discussion, we write the superstring-Liouville action in terms of superfield (30) as follows: Forgetting about the fermionic fields, the super-Liouville equations are reduced simply to the Liouville equation while the scalar superfield is reduced to the scalar real-field . The associated Liouville action is whose single constant of motion is the stress energy momentum tensor of weight 2 such that
5. Integrability of Liouville Superstring Theory
The super-Liouville Lax pair one way to introduce the integrability of the Liouville superstring theory, we are discussing here, is through the Lax pair formulation [9, 10]. A key step towards establishing this integrability is through an explicit determination of the Lax pair generators. The zero curvature condition is given by where the Lax pair is defined as functions of the osp() Lie superalgebra generators. One possible realization of this Lax pair is given by Indeed, by virtue of the zero curvature condition and the commutations relations of the osp() Lie superalgebra, we recover easily the super-Liouville equation of motion (34). Indeed Then with respect to the zero curvature condition , and as suspected, were cover the super-Liouville equation of motion The super-Liouville conserved current can be written as where is the Virasoro conformal current of weight 2 and is its supersymmetric partner of conformal spin . The explicit form of this supercurrent in terms of the superfield is given by
We have studied the superstring theory in the presence of worldsheet gauge fields, and we have extracted the associated super-Liouville theory by a special ansatz. This work contains some connections between the string and gauge fields on one hand and super-Liouville theory on the other hand.
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