Super-Liouville Theory from Superstring Theory in the Presence of Gauge Worldsheet Fields

Abstract

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 𝐵-fields 𝐵=𝑒^𝜑+(𝑖/2)𝜓𝜓. We discuss the integrability of super-Liouville theory through the Lax formulation, and we establish also the superfields formulation of super-Liouville equations.

1. Introduction

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 [6].

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 𝑆₁=−𝑑²𝜎14𝜋𝛼^′𝜕_𝑎𝑋^𝜇𝜕^𝑎𝑋_𝜇+14𝑔²𝐹_𝑎𝑏𝐹^𝑎𝑏, (1) where 𝑔 is the gauge coupling constant, 𝐹_𝑎𝑏=𝜕_𝑎𝐴_𝑏−𝜕_𝑏𝐴_𝑎 are the fields strength associated with the worldsheet gauge fields, 𝐴_𝑎(𝜎,𝜏), {𝜎,𝜏} are the worldsheet coordinates, and 𝑋^𝜇 are the string coordinates. This action has the gauge symmetry. Therefore, we have the condition 𝜕_𝑎𝐴^𝑎=0. (2)

2.2. Supersymmetric Action

To get the supersymmetric action the bosonic fields 𝑋^𝜇 and 𝐴^𝑎 should be replaced by the superfields 𝑌^𝜇𝜎,𝜏;𝜃¹,𝜃²=𝑋^𝜇(𝜎,𝜏)+𝜃𝜓^𝜇()+1𝜎,𝜏2𝜃𝜃𝐵^𝜇(),A𝜎,𝜏_𝑎𝜎,𝜏;𝜃¹,𝜃²=𝐴^𝑎(𝜎,𝜏)+𝜃𝜌^𝑎𝜒()+1𝜎,𝜏2𝜃𝜃𝑊^𝑎(𝜎,𝜏), (3) where 𝜓^𝜇=(𝜓^𝜇_𝛼) are dynamical fields on the worldsheets, 𝐵^𝜇 and 𝑊^𝑎 are auxiliary fields, the Majorana spinor 𝜒 is the superpartner of 𝐴_𝑎, and the Grassmannian coordinates 𝜃¹ and 𝜃² form a Majorana spinor 𝜃=^𝜃1_𝜃2.

We introduce also the following superspace covariant derivative: 𝒟^𝑎=𝑘𝜀^𝑎𝑏𝜌_𝑏𝜕𝐷,𝐷=𝜕𝜃−𝑖𝜌^𝑎𝜃𝜕_𝑎, (4) where 𝜀⁰¹=−𝜀¹⁰=1 and 𝑘 is a constant which is finding such that 𝒟^𝑎𝑌_𝜇𝒟_𝑎𝑌^𝜇=𝐷𝑌_𝜇𝐷𝑌^𝜇; this gives 𝑘∈{±1/√2,±𝑖/√2}.

Reformulating the action by introducing (3) and (4) we obtain the following supersymmetric action: 𝑆=𝑑²𝜎𝑑²𝜃𝑖8𝜋𝛼^′𝒟^𝑎𝑌_𝜇𝒟_𝑎𝑌^𝜇+𝑖4𝑔²ℱ_𝑎𝑏ℱ^𝑎𝑏. (5) The superfield strength ℱ_𝑎𝑏 is defined as follows: ℱ_𝑎𝑏=𝒟_𝑎𝒜_𝑏−𝒟_𝑏𝒜_𝑎. (6) After making integration over the Grassmannian coordinates 𝜃¹ and 𝜃², this action takes the form 𝑆=𝑑²𝜎−14𝜋𝛼^′𝜕_𝑎𝑋^𝜇𝜕^𝑎𝑋_𝜇−𝑖𝜓^𝜇𝜌^𝑎𝜕_𝑎𝜓_𝜇−𝐵^𝜇𝐵_𝜇−12𝑔²𝜕_𝑎𝐴^𝑏𝜕^𝑎𝐴_𝑏−𝑊_𝑎𝑊^𝑎. (7) As we see, the gaugino field 𝜒 from the two-dimensional action disappeared.

2.3. Extracoordinates in Gauged Superstring Action

In the action (7) the kinetic terms of the fields 𝑋^𝜇 and 𝐴^𝑏 have the same feature. In other words, 𝐴⁰ and 𝐴¹ have the roles of the time and space coordinates. Let {𝑋^𝑎}={𝐴⁰,𝐴¹} denote the coordinates of this (1+1) dimensional spacetime. Thus, we have the field redefinition 𝑋^𝑎=2𝜋𝛼^′𝑔𝐴^𝑎,𝐵^𝑎=2𝜋𝛼^′𝑔𝑊^𝑎. (8) According to these definitions, the action (7) can be written as 1𝑆=−4𝜋𝛼^′𝑑²𝜎𝜕_𝑎𝑋^𝑀𝜕^𝑎𝑋_𝑀−𝑖𝜓^𝜇𝜌^𝑎𝜕_𝑎𝜓_𝜇−𝐵^𝑀𝐵_𝑀, (9) where 𝑀∈{𝜇,𝑎}, and we will use the convention 𝑎∈{0,1} and 𝜇∈{0^′,1^′,…,9^′}, 𝑋^𝑀=𝑋^𝜇∪𝑋^𝑎. (10) Since both 𝑋^𝑎 and 𝜎^𝑎 carry the worldsheet index, the partial derivative 𝜕_𝑎 always shows derivative with respect to 𝜎^𝑎. The bosonic part of this action apparently describes a 12-dimensional spacetime with the signature 10+2 and the coordinates. However, in the superstring theory the dimension of the spacetime is always 9+1. Therefore, they are called the extradimensions or the fictitious coordinates [6].

The fermionic term of the action (9) also can be written with the 12-dimensional indices. For this, the Majorana spinor 𝜓^𝑎 is defined by 𝜓⁰=2𝜋𝛼^′𝑔𝜒₂𝜒₁,𝜓¹=2𝜋𝛼^′𝑔𝜒₂−𝜒₁. (11) The spinors 𝜓⁰ and 𝜓¹ satisfy the identities 𝜓^𝑏𝜌^𝑎𝜕_𝑎𝜓_𝑏=0,𝜓^𝑎𝜓_𝑎=4𝜋𝛼^′𝑔²𝜒𝜒, (12) where 𝜒=^𝜒1_𝜒2. Introducing the identity (12) in the action (9) leads to the covariant form of this action 1𝐼=−4𝜋𝛼^′𝑑²𝜎𝜕_𝑎𝑋^𝑀𝜕^𝑎𝑋_𝑀−𝑖𝜓^𝑀𝜌^𝑎𝜕_𝑎𝜓_𝑀−𝐵^𝑀𝐵_𝑀. (13) The metric of the extended manifold is 𝜂_𝑀𝑁=diag𝜂_𝜇𝜈,𝜂_𝑎𝑏, (14) where 𝜂_𝜇𝜈 belongs to the 9+1 actual spacetime and 𝜂_𝑎𝑏 to the fictitious coordinates.

The equations of motion, extracted from the action (13), are 𝜕_𝑎𝜕^𝑎𝑋^𝑀𝜌=0,_𝑎𝜕_𝑎𝜓^𝑀𝐵=0,_𝑀=0. (15) In addition, we should also consider the gauge condition 𝜕_𝑎𝑋^𝑎=0. (16) This condition and the equation of motion of 𝑋^𝑎 can be written as 𝑋^𝑎=𝜀^𝑎𝑏𝜕_𝑏𝜕𝜙,_𝑎𝜕^𝑎𝜙=𝑐, (17) respectively. The constant 𝑐 is independent of 𝜎 and 𝜏.

2.4. Symmetries of the Model

2.4.1. Worldsheet Supersymmetry

Using the superfield (3) we obtain the supersymmetry transformations of 𝐴^𝑎 and 𝜒 as in the following: 𝛿𝐴^𝑎=𝜖𝜌^𝑎𝜒,𝛿𝑊^𝑎=−𝑖𝜖𝜌^𝑏𝜌^𝑎𝜕_𝑏𝑖𝜒,𝛿𝜒=−4𝜌^𝑎𝑏𝐹_𝑎𝑏1𝜖−2𝜌_𝑎𝑊^𝑎𝜖, (18) where 𝜌^𝑎𝑏=(1/2)[𝜌^𝑎,𝜌^𝑏]. The supersymmetry parameter 𝜀 is an anticommuting infinitesimal constant spinor. In terms of the fields {𝑋^𝑎,𝜓^𝑎,𝐵^𝑎} these transformations take the form 𝛿𝑋^𝑎=𝑖𝜀^𝑎𝑏𝜖𝜓_𝑏,𝛿𝐵^𝑎=𝜀^𝑎𝑏𝜖𝜌^𝑐𝜕_𝑐𝜓_𝑏,𝛿𝜓^𝑎1=−2𝜌^𝑎𝜀^𝑏𝑐𝜕_𝑏𝑋_𝑐−𝑖𝜀^𝑎𝑏𝜌_𝑏𝜌_𝑐𝐵^𝑐𝜖. (19) The transformations (19) form a closed algebra. The supercurrent associated with the supersymmetry transformations (19), accompanied by 𝛿𝑋^𝜇=𝛿𝜓^𝜇=0, is 𝑘_𝑎=𝑖4𝜌^𝑎′𝜌_𝑎𝜓_𝑎′𝜀^𝑏𝑐𝜕_𝑏𝑋_𝑐. (20) According to the identity 𝜌^𝑎𝜌^𝑎′𝜌_𝑎=0 there is 𝜌^𝑎𝑘_𝑎=0 then 𝑘_𝑎 is a conserved current 𝜕^𝑎𝑘_𝑎=0.

2.4.2. The Poincare Symmetry

The action (13), with 𝐵^𝑀=0, under the Poincare transformations 𝛿𝑋^𝑀=𝑎^𝑀_𝑁𝑋^𝑁+𝑏^𝑀,𝛿𝜓^𝑀=𝑎^𝑀_𝑁𝜓^𝑁, (21) is symmetric. The matrix 𝑎_𝑀𝑁 is a constant antisymmetric, and 𝑏^𝑀 is a constant vector. The associated currents to these transformations are 𝑃^𝑀_𝑎=12𝜋𝛼^′𝜕_𝑎𝑋^𝑀,𝐽_𝑎^𝑀𝑁=12𝜋𝛼^′𝑋^𝑀𝜕_𝑎𝑋^𝑁−𝑋^𝑁𝜕_𝑎𝑋^𝑀+𝑖𝜓^𝑀𝜌_𝑎𝜓^𝑁. (22) There are a conserved currents, that is, 𝜕^𝑎𝑃^𝑀_𝑎=𝜕^𝑎𝐽_𝑎^𝑀𝑁=0.

3. Liouville Equations from Gauged Superstring Action

Let us consider the action (13) of the superstring in the presence of the worldsheet gauge fields 1𝑆=−4𝜋𝛼^′𝑑²𝜎𝜕_𝑎𝑋^𝑀𝜕^𝑎𝑋_𝑀−𝑖𝜓^𝑀𝜌^𝑎𝜕_𝑎𝜓_𝑀−𝐵^𝑀𝐵_𝑀, (23) where 𝑋^𝑀,𝜓^𝑀 and 𝐵^𝑀 are the fields of conformal weights 1, 3/2, and 2, respectively.

We assume that these fields can be processed as follows: 𝑋^𝑀=𝑘^𝑀𝜓⋅𝜑,_𝑀=𝑘^𝑀𝐵⋅𝜓,_𝑀=𝑘^𝑀⋅𝐵, (24) where 𝑘^𝑀 is the Lorentz field of conformal weight 1 which does not depend on worldsheet variables, and the new fields 𝜑, 𝜓, and 𝐵 are, respectively, of conformal weights 0, 1/2 and 1.

By introducing the Ansatz (24), the action (23) becomes 𝑆₁𝑘=−²4𝜋𝛼^′𝑑²𝜎𝜕_𝑎𝜑𝜕^𝑎𝜑−𝑖𝜓𝜌^𝑎𝜕_𝑎𝜓−𝐵⋅𝐵. (25) The equations of motion relative to the fields 𝜑, 𝜓, 𝜓, and 𝐵 are given by 𝜕_𝑎𝜕^𝑎𝜌𝜑=0,_𝑎𝜕_𝑎𝜕𝜓=0,_𝑎𝜓𝜌^𝑎=0,𝐵=0. (26) To obtain the Liouville equations we need to take the following value for the field 𝐵: 𝐵=𝑒^𝜑+𝑖2𝜓𝜓. (27) In that case the action becomes 𝑆₂𝑘=−²4𝜋𝛼^′𝑑²𝜎𝜕_𝑎𝜑𝜕^𝑎𝜑−𝑖𝜓𝜌^𝑎𝜕_𝑎𝜓−𝑒^2𝜑+𝑖𝜓𝜓𝑒^𝜑, (28) where (𝜓𝜓)²=0.

This last action (28) has the same form as that found in our paper [7]; by consequence we find the same equations of motion: 𝜕_𝑎𝜕^𝑎𝜑=𝑒^2𝜑+𝑖2𝜓𝜓𝑒^𝜑,𝜌^𝑎𝜕_𝑎𝜓=𝜓𝑒^𝜑,𝜕_𝑎𝜓𝜌^𝑎=−𝜓𝑒^𝜑. (29)

4. Superfield Analysis

In terms of a superfield formulation associated with an 𝑁=1 supersymmetry we can set [8] Φ=𝜑+𝜃𝜓+𝜃𝜓+𝜃𝜃𝐵. (30) We can then easily show that the system of super-Liouville equations (29) can be expressed in terms of superderivative of the superfield Φ as follows: 𝐷𝐷Φ=𝑒^Φ, (31) where 𝐷=𝜕_𝜃+𝜃𝜕 and 𝐷=𝜕_𝜃+𝜃𝜕. Indeed, straightforward computations lead to 𝐷𝐷Φ=−𝐵−𝜃𝜕𝜓+𝜃𝜕𝜓+𝜃𝜃𝜕𝜕𝜑. (32) By virtue of (30) and expanding the exponential of the superfield Φ, we find 𝑒^Φ=𝑒^𝜑1−𝜃𝜓+𝜃𝜓+𝜃𝜃−𝑒^𝜑+𝜓𝜓. (33) Identifying (31) with (32) one finds easily super-Liouville equations of motions (29).

Furthermore, using the complex transformations 𝑧=𝜎+𝑖𝜏 and 𝑧=𝜎−𝑖𝜏 one can easily rewrite the previous super-Liouville equations to become 𝜕𝜕𝜑=−𝑒^2𝜑+𝜓𝜓𝑒^𝜑,𝜕𝜓=𝜓𝑒^𝜑,𝜕𝜓=−𝜓𝑒^𝜑 (34) with 𝜕_𝑧=𝜕 and 𝜕_𝑧=𝜕. With the equation of motion (34) and the previous discussion, we write the superstring-Liouville action in terms of 𝑁=1 superfield Φ (30) as follows: 𝑆=𝑑𝜃𝑑𝜃𝑑²𝜎12𝐷Φ𝐷Φ+exp(Φ). (35) Forgetting about the fermionic fields, the super-Liouville equations are reduced simply to the Liouville equation 𝜕𝜕𝜑=−𝑒^2𝜑, (36) while the scalar superfield Φ is reduced to the scalar real-field 𝜑. The associated Liouville action is 1𝑆=2𝑑²𝜎𝜕𝜑𝜕𝜑+exp2𝜑, (37) whose single constant of motion is the stress energy momentum tensor 𝑇 of weight 2 such that 𝑇₂(𝑧)=𝜕²𝜑−𝜕𝜑²,𝜕𝑇₂(𝑧)=0. (38)

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 𝐷𝐴_𝜃+𝐷𝐴_𝜃+𝐴_𝜃,𝐴_𝜃=0, (39) where the Lax pair (𝐴_𝜃,𝐴_𝜃) is defined as functions of the osp(1∣2) Lie superalgebra generators. One possible realization of this Lax pair is given by 𝐴_𝜃=𝐷Φℎ+𝑓₊,𝐴_𝜃=−2𝑖exp(Φ)𝑓₋. (40) Indeed, by virtue of the zero curvature condition and the commutations relations of the osp(1∣2) Lie superalgebra, we recover easily the super-Liouville equation of motion (34). Indeed 𝐷𝐴_𝜃=−2𝑖𝐷Φexp(Φ)𝑓₋,𝐷𝐴_𝜃=𝐴𝐷𝐷Φℎ,_𝜃,𝐴_𝜃=−2𝑖𝐷Φexp(Φ)ℎ,𝑓₋−2𝑖exp(Φ)𝑓₊,𝑓₋=2𝑖𝐷Φexp(Φ)𝑓₋−exp(Φ)ℎ. (41) Then with respect to the zero curvature condition 𝐷𝐴_𝜃+𝐷𝐴_𝜃+{𝐴_𝜃,𝐴_𝜃}=0, and as suspected, were cover the super-Liouville equation of motion 𝐷𝐷Φ=expΦ. (42) The 𝑁=1 super-Liouville conserved current can be written as 𝐺_3/2(𝑧,𝜃)=𝐽_3/2(𝑧)+𝜃𝑇₂(𝑧),𝐷𝐺_3/2=0, (43) where 𝑇₂(𝑧) is the Virasoro conformal current of weight 2 and 𝐽_3/2(𝑧) is its supersymmetric partner of conformal spin 3/2. The explicit form of this 𝑁=1 supercurrent in terms of the superfield Φ is given by 𝐺_3/2(𝑧,𝜃)=𝐷²Φ𝐷Φ−𝐷³Φ,𝐷𝐺_3/2=0. (44)

6. Conclusions

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.