Topological String Theory: A Source for Conformal Field Theories

Abstract

In this paper, we first recall the approach to extract Liouville field theories from the action of worldsheets superstring in the standard case and also in the presence of gauge fields. In order to extend this formalism, we treat the case of Toda field equations based on a special ansatz of our superfields. Then, we study the integrability of the system by the Lax formulation based on the structure of the associated Lie superalgebra.

1. Introduction

It is commonly known that bosonic string theory has several problems. In particular, it is tachyonic, and it does not have fermionic states describing matter [1, 2]. It cannot be used to unify particle physics and gravity. To overcome such a problem, we should add fermions on the worldsheet ${\psi }^{\mu }$ [3, 4]. To make this realization, we can introduce fermionic fields into the Ramond–Neveu–Schwarz string theory [5, 6]. However, the action of this RNS model is obtained by extending the Polyakov action to a 2D worldsheet action with a local supersymmetry [7, 8, 9]. Consequently, the action obtained has in addition to Weyl symmetries a local superconformal symmetries [10, 11].

In 1967, Leonard Susskind is the first one to introduce the worldsheet surface as a generalization of the worldline describing the motion of a point particle in special and general relativity, and it is seen in topological string theory as a two-dimensional manifold that describes the motion of the string in spacetime [12, 13].

However, when we are only interested in the topology of the worldsheet, the resulting theory is called the topological string theory where only the variables of the worldsheet enter in the calculation of the associated partition function [14, 15]. Consequently, the topological string theory corresponds in the case of the conformal theory coupled to gravity to a nonlinear two-dimensional integrable model [16, 17].

In the present paper, we would like to introduce a new ansatz on the 2D superstring worldsheet action in order to give the super-Liouville and super Toda field theories. Such concept is based on a specific ansatz of our basic superfields on a well-defined subspace. We study the integrability criterion of such models by the Lax method.

This paper is structured as follows. In Section 2, we introduce an approach to deriving Liouville equations from the action of the superstring, and we present the supersymmetric formulation and the integrability criterion. We finish this section by studying the case of the gauge string theory. In Section 3, we present our basic notations of the super Toda theory from the 2D superstring worldsheet theory, and we also discuss the integrability of our system. Section 4 is devoted to conclusions and remarks.

2. Toward super-Liouville field theory

2.1 On RNS superstring action

Let us consider the worldsheet action for the superstring theory with a conformal gauge in two-dimensional worldsheet supersymmetry as follows [1, 8].

$$S_0=-\frac{1}{4\pi {\alpha }^{\prime }}\int d^2\sigma \left({\partial }^a{X}^{\mu }{\partial }_a{X}_{\mu }-i{\overline{\psi }}^{\mu }{\rho }^a{\partial }_a{\psi }_{\mu }\right),$$E1

with ${\sigma }^a=\left(\tau ,\sigma \right),$$a=0,1,$$X^{\mu }\left(z,\overline{z}\right),$$\mu =0,..,9$ are spacetime coordinates for a string, ${\psi }^{\mu }$ are a set of Majorana spinors, and ${\rho }^a$ are the Dirac matrices written in the Majorana basis as follows

$${\rho }^0=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right);{\rho }^1=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right).$$E2

In our paper, we consider two types of flat metrics, which are associated with two-dimensional worldsheet surface, and ${\eta }^{\mathit{\mu \nu }}=\mathit{diag}\left(-1,1,\dots ,1\right)$ is associated with the ten-dimensional spacetime. However, by using the variation of action (1), we have the following equations of motion

$${\partial }_a{\partial }^a{X}^{\mu }=0,{\rho }^a{\partial }_a{\psi }^{\mu }=0,{\partial }_a{\overline{\psi }}^{\mu }{\rho }^a=0.$$E3

The supersymmetric transformations consist of finding the transformations relative to the bosonic and fermionic variables; however, in our case, we have the following transformations

$$\delta X^{\mu }=\overline{\mathrm{ϵ}}{\psi }^{\mu },{\mathit{\delta \psi }}^{\mu }=-i{\rho }^a{\partial }_a{X}^{\mu }\mathrm{ϵ},$$E4

where $\mathrm{ϵ}$ are infinitesimally small parameters. Using the principle of least action $\mathit{\delta S}=0$, we obtain the following form of the conserved supercurrent

$$j_a=\frac{1}{2}{\rho }^b{\rho }_a{\psi }^{\mu }{\partial }_b{X}_{\mu },{\partial }^a{j}_a=0.$$E5

In what follows, we will use this formalism to describe the motion of the worldsheet in the form of nonlinear Liouville and Toda equations.

2.2 Liouville’s field equations

This section is based on one of our previous works concerning Liouville theory from superstring field theory [18], we adopt the ansatz proposed in such a paper concerning $X^{\mu }$ and ${\psi }^{\mu }$, namely

$$X^{\mu }≔\phi k^{\mu },{\psi }^{\mu }≔\psi k^{\mu },$$E6

where $\phi$ and $\psi$ are two Lorentz scalar fields and $k^{\mu }$ as a constant basis vector that satisfies $k^{\mu }{k}_{\mu }=k^2$, $\mu =0,1,\dots ,9$. In accordance with this hypothesis (6), the action of the superstring becomes

$$S_0=-\frac{k^2}{4\pi {\alpha }^{\prime }}\int d^2\sigma \left({\partial }^a\phi {\partial }_a\phi -i\overline{\psi }{\rho }^a{\partial }_a\psi \right).$$E7

In order to find the Liouville equations, we have to make a transformation on the action (7), and such a transformation is considered as a perturbation with a weak coupling $\lambda =-\frac{k^2}{4\pi {\alpha }^{\prime }}$ that we propose as follows

$$S_0\to S_0+\lambda \int d^2\sigma \left(e^{2\phi }+i\overline{\psi }\psi e^{\phi }\right).$$E8

Using standard variational calculus, we can show that the corresponding equations of motion have the following form

$${\partial }_a{\partial }^a\phi =e^{2\phi }+\frac{i}{2}\overline{\psi }\psi e^{\phi },{\rho }^a{\partial }_a\psi =\psi e^{\phi },{\partial }_a\overline{\psi }{\rho }^a=-\overline{\psi }{e}^{\phi }.$$E9

These equations define the Liouville equations of motion. However, we need to find the supersymmetric version of such equations, that is why we will introduce in the following the superfield formulation of the theory.

2.3 Superfield formulation

By construction, super-Liouville field theory is the supersymmetric generalization of bosonic Liouville theory, which is considered a two-dimensional matter-induced theory of gravity. However, the super-Liouville theory describes 2D supergravity induced by supersymmetric matter [5]. Such a construction is based on the introduction of superfields associated with the supersymmetry $N=1$ which can be taken as follows [7].

$$\mathrm{\Phi }=\phi +\overline{\theta }\psi +\theta \overline{\psi }+\theta \overline{\theta }F$$E10

where $\phi$, $\psi$, and $F$ are the component fields of $\mathrm{\Phi },$ and in this case, we can prove that the super-Liouville field equations can be expressed in terms of the supersymmetric derivative $D$ as follows

$$\overline{D}D\mathrm{\Phi }=exp\left(\mathrm{\Phi }\right)$$E11

where $D={\partial }_{\theta }+\theta \partial$ and $\overline{D}={\partial }_{\overline{\theta }}+\overline{\theta }\partial$. Indeed, straightforward computations lead to

$$\overline{D}D\mathrm{\Phi }=-F-\overline{\theta }\overline{\partial }\overline{\psi }+\theta \partial \psi +\theta \overline{\theta }\partial \overline{\partial }\phi$$E12

Performing the exponential expansion of the superfield $\mathrm{\Phi }$, we find

$$exp\left(\mathrm{\Phi }\right)=\left[1+\overline{\theta }\psi +\theta \overline{\psi }+\theta \overline{\theta }\left(-\overline{\psi }\psi -exp\left(\phi \right)\right)\right]exp\left(\phi \right)$$E13

By inserting (12) and (13) in the Eq. (11) and using the complex transformations $z=\sigma +\mathit{i\tau }$ and $\overline{z}=\sigma -\mathit{i\tau }$, we can rewrite the super-Liouville eq. (11) as follows

$$\partial \overline{\partial }\phi =-\overline{\psi }\psi exp\left(\phi \right)-exp\left(2\phi \right);\partial \psi =\overline{\psi }exp\left(\phi \right);\overline{\partial }\overline{\psi }=-\psi exp\left(\phi \right)$$E14

where ${\partial }_z=\partial$ and ${\partial }_{\overline{z}}=\overline{\partial }$.

Note that the only constant of motion of this theory is the momentum energy tensor $T_2$ of conformal weight 2 verifying [19, 20].

$$T_2\left(z\right)={\partial }^2\phi -{\left(\partial \phi \right)}^2,\overline{\partial }{T}_2\left(z\right)=0$$E15

In what follows, we will study the Lie algebra structures and the integrability criterion of our theory.

2.4 Integratability criterion

As mentioned in [21], the Lie superalgebra associated with the super-Liouville equations is given by the superalgebra $\mathit{osp}\left(1|2\right)$ which is formed by three bosonic generators $e^{\pm }$ and $h$ of the Lie algebra $\mathit{sl}\left(2\right)$ and of two fermionic generators $f_{\pm }$. The structure of the superalgebra $\mathit{osp}\left(1|2\right)$ is given by the following (anti)commutation relations:

$$\begin{array}{c}\left[h,e_{\pm }\right]=\pm 2e_{\pm };\left[h,f_{\pm }\right]=\pm f_{\pm };\left[e_{+},e_{-}\right]=h;\left[f_{\pm },e_{\pm }\right]=0\\ \\ \left[f_{\pm },e_{\mp }\right]=-f_{\mp },\left\{f_{\pm },f_{\pm }\right\}=\pm {\mathit{ie}}_{\pm },\left\{f_{+},f_{-}\right\}=\frac{-i}{2}h\end{array}$$E16

in terms of ${\mathrm{\mathbb{Z}}}_2$-graded vector space the superalgebra $\mathit{osp}\left(1|2\right)$ is written in the form

$$\mathit{osp}\left(1|2\right)={\left(h,e_{+},e_{-}\right)}_{\overline{0}}\oplus {\left(f_{+},f_{-}\right)}_{\overline{1}}$$E17

here $\overline{0}$ and $\overline{1}$ represent the bosonic and fermionic generators, respectively.

One of the integrability test methods is given by the existence of the Lax pair [18]. In this system, the determination of the Lax pair helps us to study the integrability criterion of the obtained super-Liouville theory. Indeed, the equation defining the null curvature of our system is given by

$${\mathit{DA}}_{\overline{\theta }}+\overline{D}{A}_{\theta }+\left\{A_{\theta },A_{\overline{\theta }}\right\}=0$$E18

here the generators of the Lax pair $\left(A_{\theta },A_{\overline{\theta }}\right)$ are given as functions of the generators of the super Lie algebra $\mathit{osp}\left(1|2\right)$. A possibility to do this realization is given by

$$A_{\theta }=f_{+}+D\mathrm{\Phi }h;A_{\overline{\theta }}=-2iexp\left(\mathrm{\Phi }\right)f_{-}$$E19

after a simple calculation, we get

$${\mathit{DA}}_{\overline{\theta }}=-2\mathit{iD}\mathrm{\Phi }exp\left(\mathrm{\Phi }\right)f_{-};\overline{D}{A}_{\theta }=\overline{D}D\mathrm{\Phi }h$$E20

$$\left\{A_{\theta },A_{\overline{\theta }}\right\}=-exp\left(\mathrm{\Phi }\right)h+2\mathit{iD}\mathrm{\Phi }exp\left(\mathrm{\Phi }\right)f_{-}$$E21

Now, by using the null curvature condition (18) and the commutation relations (16) of the super Lie algebra $\mathit{osp}\left(1|2\right)$, we obtain the following super-Liouville equation of motion

$$\overline{D}D\mathrm{\Phi }=\mathit{exp}\mathrm{\Phi }$$E22

In the next paragraph, we treat the case of superstring field theory in the existence of gauge superfields on the worldsheet superspace.

2.5 Gauged superstring action

The gauged superstring theory has been the subject of several studies [22]. It is obtained by the introduction of abelian gauge fields on the worldsheet. Such fields appear as spacetime coordinates. Let us start with the following bosonic action

$$S_1=-\int d^2\sigma \left(\frac{1}{4\pi {\alpha }^{\prime }}{\partial }_a{X}^{\mu }{\partial }^a{X}_{\mu }+\frac{1}{4g^2}{F}_{\mathit{ab}}{F}^{\mathit{ab}}\right),$$E23

where $g$ is the gauge coupling constant, $F_{\mathit{ab}}={\partial }_a{A}_b-{\partial }_b{A}_a$ is the abelian field strength associated with gauge field $A$, $a;b=0,1$ in the two-dimensional worldsheet surface, and $\mu =0,..,9$ in the ten-dimensional spacetime. Therefore, we have the following gauge condition

$${\partial }_a{A}^a=0.$$E24

the supersymmetric action is obtained by inserting the following superfields

$$Y^{\mu }\left(\sigma ,\tau ;{\theta }^1,{\theta }^2\right)=X^{\mu }\left(\sigma ,\tau \right)+\overline{\theta }{\psi }^{\mu }\left(\sigma ,\tau \right)+\frac{1}{2}\overline{\theta }\theta B^{\mu }\left(\sigma ,\tau \right),$$E25

$${\mathcal{A}}^a\left(\sigma ,\tau ;{\theta }^1,{\theta }^2\right)=A^a\left(\sigma ,\tau \right)+\overline{\theta }{\rho }^a\chi \left(\sigma ,\tau \right)+\frac{1}{2}\overline{\theta }\theta W^a\left(\sigma ,\tau \right),$$E26

Here, the fields $B^{\mu }$ and $W^a$ represent auxiliary fields, and the field $\chi$ is a Majorana-like spinor that represents the supersymmetric partner of $A_a$. With the two Grassmann variables ${\theta }^1$ and ${\theta }^2$, we can form a resultant Majorana spinor variable as follows:

$$\theta =\left(\begin{array}{c}{\theta }^1\\ {\theta }^2\end{array}\right),\overline{\theta }=\left(\begin{array}{c}{\overline{\theta }}^1\\ {\overline{\theta }}^2\end{array}\right)$$E27

We must also modify our covariant derivatives by introducing the following super covariant derivatives

$${\mathcal{D}}^a={\mathit{\kappa \epsilon }}^{\mathit{ab}}{\rho }_b\mathfrak{D},\mathfrak{D}=\frac{\partial }{\partial \overline{\theta }}-i{\rho }^a\theta {\partial }_a,$$E28

where ${\epsilon }^{01}=-{\epsilon }^{10}=1,$${\epsilon }^{00}={\epsilon }^{11}=0$, and $\kappa \in \left\{\pm \frac{1}{\sqrt{2}},\pm \frac{i}{\sqrt{2}}\right\}$ is a constant which verifies ${\overline{\mathcal{D}}}^a{Y}_{\mu }{\mathcal{D}}_a{Y}^{\mu }=\overline{\mathfrak{D}}{Y}_{\mu }\mathfrak{D}{Y}^{\mu }$.

However, such existence of gauge fields implies that the action is transformed into

$$S=-\frac{1}{4{\mathit{\pi \alpha }}^{\text{'}}}\int d^2\sigma \left({\partial }_a{X}^M{\partial }^a{X}_M-i{\overline{\psi }}^M{\rho }^a{\partial }_a{\psi }_M-B^M{B}_M\right)$$E29

where $X^M$, $B^M$ are the bosonic fields of conformal weight, respectively, 1 and 2, ${\psi }^M$ is the fermionic field of conformal weight $\frac{3}{2}$ and $M\in \left\{\mu ,a\right\}$ such as

$$\left\{X^M\right\}=\left\{X^{\mu }\right\}\cup \left\{X^a\right\},$$E30

Our hypothesis is to assume that the fields can be written in the following way

$$X^M={\xi }^M.\phi ,{\psi }^M={\xi }^M.\psi ,B^M={\xi }^M.B$$E31

where ${\xi }^M$ is the Lorentz constant field of conformal weight 1, and verifying ${\xi }^M{\xi }_M=1$, $\phi$ and $B$ are two bosonic fields of conformal weight 0 and 1, respectively, and $\psi$ represents a fermionic field of conformal weight $1/2$. Based on the assumption (31), the action (29) takes the following form

$$S_1=-\frac{k^2}{4\pi {\alpha }^{\prime }}\int d^2\sigma \left({\partial }_a\phi {\partial }^a\phi -i\overline{\psi }{\rho }^a{\partial }_a\psi -B.B\right),$$E32

the equations of motion of the different fields are given by

$${\partial }_a{\partial }^a\phi =0;{\rho }^a{\partial }_a\psi =0;{\partial }_a\overline{\psi }{\rho }^a=0;B=0.$$E33

In order to find the Liouville field equations, we will have to consider a specific value for the B field, namely [22].

$$B=e^{\phi }+\frac{i}{2}\overline{\psi }\psi .$$E34

Finally, we find the same equations of motion of Liouville’s theory

$${\partial }_a{\partial }^a\phi =e^{2\phi }+\frac{i}{2}\overline{\psi }\psi e^{\phi };{\rho }^a{\partial }_a\psi =\psi e^{\phi };{\partial }_a\overline{\psi }{\rho }^a=-\overline{\psi }{e}^{\phi }$$E35

The study of the superfield formalism and the integrability criterion is treated in the same way as the previous section.

3. Toward super Toda field theory

Today’s theories represent a general framework of Liouville’s theory, which are completely integrable classical theories associated with finite-dimensional semi-simple Lie algebras [23, 24]. Of course, Toda’s theory appears in the effective action of two-dimensional string theory with a conforming gauge [25, 26]. However, in the following, we will extend the (super) Toda theory based on a special conformal transformation.

3.1 Bosonic Toda field theory

Let us consider the superstring action with worldsheet super-symmetry which is given by (1). The main goal of our work is to obtain the super-conformal Toda field theory derived from superstring theory related to the action (1). However, as for the ${\psi }^{\mu }$ fermionic fields, we need the $\phi$ scalar fields satisfying the following assumption

$$X^{\mu }=\frac{k^{\mu }}{\sqrt{2}}\sum _{i=1}^n{\mathcal{K}}_{\mathit{ii}}{\phi }_i;{\psi }^{\mu }=k^{\mu }\psi$$E36

the action (1) will now become

$$S_0=-\frac{k^2}{4\pi {\alpha }^{\prime }}\int d^2\sigma \left(\frac{1}{2}\sum _{i,j=1}^n{\mathcal{R}}_{\mathit{ij}}{\partial }^a{\phi }_i{\partial }_a{\phi }_j-i\overline{\psi }{\rho }^a{\partial }_a\psi \right)$$E37

$$\text{with}{\mathcal{R}}_{\mathit{ij}}={\mathcal{K}}_{\mathit{ii}}{\mathcal{K}}_{\mathit{jj}},i,j=1,..,n$$E38

where $k^{\mu }$ is a constant vector that satisfies $k^{\mu }{k}_{\mu }=k^2$, $\mu =0,1,\dots ,9$, ${\mathcal{K}}_{\mathit{ij}}$ is the Cartan matrix of the associated Lie algebra, $\beta$ represents a coupling constant, and ${\phi }_i$ are scalar fields.

In order to find the super Toda field theory, we take the action (38) and make the following transformation

$$S_0\to S_0+\lambda \int d^2\sigma \left[\frac{1}{\beta }\sum _{i=1}^{n}exp\left(\beta \sum _{j=1}^n{\mathcal{R}}_{\mathit{ij}}{\phi }_j\right)\right]$$E39

with $\lambda =-\frac{k^2}{4\pi {\alpha }^{\prime }},$ and the resulting action is

$$S=-\frac{k^2}{4\pi {\alpha }^{\prime }}\int d^2\sigma \left(\frac{1}{2}\sum _{i,j=1}^n{\mathcal{R}}_{\mathit{ij}}{\partial }^a{\phi }_i{\partial }_a{\phi }_j+\frac{1}{\beta }\sum _{i=1}^{n}exp\left(\beta \sum _{j=1}^n{\mathcal{R}}_{\mathit{ij}}{\phi }_j\right)-i\overline{\psi }{\rho }^a{\partial }_a\psi \right)$$E40

By making the action vary with regard to the field $\phi$ its derivatives, we find the following equations of motion

$${\partial }_a{\partial }^a{\phi }_i=exp\left(\beta \sum _{j=1}^n{\mathcal{R}}_{\mathit{ij}}{\phi }_j\right)$$E41

However, we have found the classical Toda field equation [18, 20, 27] with the matrix elements ${\mathcal{R}}_{\mathit{ij}}={\mathcal{K}}_{\mathit{ii}}{\mathcal{K}}_{\mathit{jj}}.$

3.2 Super field analysis

Concerning the superfield formulation associated with the $N=1$ supersymmetric theory, we can establish the following components

$${\mathrm{\Phi }}_i={\phi }_i+\overline{\theta }{\overline{\psi }}_i+{\mathit{\theta \psi }}_i+\theta \overline{\theta }{F}_i;i=1,\dots$$E42

where $D=\frac{\partial }{\partial \theta }+\theta \partial$ and $\overline{D}=\frac{\partial }{\partial \overline{\theta }}+\overline{\theta }\partial .$ Indeed, straightforward computations lead to

$$D\overline{D}{\mathrm{\Phi }}_i=-F_i-\overline{\theta }\partial {\psi }_i+\theta \partial {\overline{\psi }}_i+\theta \overline{\theta }{\partial }^2{\phi }_i$$E43

expanding the exponential of the superfield $\mathrm{\Phi }$, we find

$$D\overline{D}{\mathrm{\Phi }}_i=exp\left(\beta \sum _{j=1}^n{\mathcal{R}}_{\mathit{ij}}{\mathrm{\Phi }}_j\right)$$E44

in the case of $\mathit{osp}\left(3|2\right)$ Lie algebra, we have this expression for the famous Cartan matrix $\mathcal{K}=\left(\begin{array}{cc}1& -1\\ -1& 0\end{array}\right)$ which implies $\mathcal{R}\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$, and in this case, the super Toda equation becomes

$$D\overline{D}{\mathrm{\Phi }}_1=exp\left(\beta {\mathrm{\Phi }}_1\right)$$E45

our theory admits two conserved currents, $T^{\left(2\right)}$ of bosonic character with conformal weight 2 and $T^{\left(3/2\right)}$ of fermionic character which plays the role of the supersymmetric partner of $T^{\left(2\right)}$ and which has a conformal weight of 3/2. Explicitly we have

$$T^{\left(3/2\right)}={\overline{D}}^2{\mathrm{\Phi }}_1\overline{D}{\mathrm{\Phi }}_1-\frac{1}{\beta }{\overline{D}}^3\left({\mathrm{\Phi }}_1+{\mathrm{\Phi }}_2\right)$$E46

$$T^{\left(2\right)}={\overline{D}}^2{\mathrm{\Phi }}_1\overline{D}{\mathrm{\Phi }}_1\overline{D}{\mathrm{\Phi }}_2-\frac{1}{\beta }{\overline{D}}^3{\mathrm{\Phi }}_1\overline{D}{\mathrm{\Phi }}_2-\frac{1}{2\beta }{\left({\overline{D}}^2{\mathrm{\Phi }}_2\right)}^2-\frac{1}{{\beta }^2}{\overline{D}}^4{\mathrm{\Phi }}_2$$E47

Now, the question that arises is this an integrable theory?

3.3 Criterion of integrability

A key step in proving integrability of Toda superstring theory is to establish explicitly the generators of Lax pairs. The null curvature condition is defined as follows

$$\mathit{DA}\overline{{}_{\theta }}+\overline{D}{A}_{\theta }+\left\{A_{\theta },A\overline{{}_{\theta }}\right\}=0$$E48

with the super Toda Lax pair is given by:

$$A_{\overline{\theta }}=\sum _{i=1}^n\overline{D}{\mathrm{\Phi }}_i{h}_i+\sum _{i=1}^n{e}_i;A_{\theta }=\sum _{i=1}^n{f}_{i}exp\left(\beta \sum _{j=1}^n{\mathcal{R}}_{\mathit{ij}}{\mathrm{\Phi }}_j\right)$$E49

where

$$\left[e_i,f_j\right\}={\delta }_{\mathit{ij}}{h}_j,\left[h_i,e_j\right\}={\mathcal{R}}_{\mathit{ij}}{h}_j,\left[h_i,f_j\right\}=-{\mathcal{R}}_{\mathit{ij}}{f}_j,$$E50

By using the commutation relations of the Lie superalgebra $\mathit{osp}\left(3|2\right)$ and also the zero curvature condition, we directly obtain

$$D\overline{D}{\mathrm{\Phi }}_i=\sum _{i,\mathit{\text{fermionic}}}exp\left[\beta \sum _j{\mathcal{R}}_{\mathit{ij}}{\mathrm{\Phi }}_j\right]+\overline{\theta }\theta \sum _{i,\mathit{\text{bosonic}}}exp\left[\beta \sum _j{\mathcal{R}}_{\mathit{ij}}{\mathrm{\Phi }}_j\right]$$E51

which are exactly the equations of motion of the super Toda field theory (44). Therefore, we can have the same results of the papers [25, 26, 28] by replacing ${\mathcal{K}}_{\mathit{ij}}$ by ${\mathcal{R}}_{\mathit{ij}}.$

4. Conclusion

In this paper, we have discussed in detail the notion of 2D integrable models that live in the worldsheet of the Ramond–Neveu–Schwarz superstring type theory. The resulting nonlinear model can describe the motion or oscillation of this worldsheet in a ten-dimensional spacetime.

First, we recall the action of the superstring theory, its equations of motion as well as its supersymmetric transformations.

However, we studied how we can extract the Liouville equations from the action of 2D worldsheet superstring theory, we expressed the supersymmetric model by introducing the superfield formalism, and we examined the integrability via the Lax equation with the existence of the associated Lax pair. In the case where our theory presents gauge fields on the worldsheet manifold, we have treated this case by taking into account this coupling.

Afterward, we have the opportunity to remake our study for a general case, namely the Toda field theory, and we use the same approach to link this model to the worldsheet of the RNS superstring theory and study the integrability criterion.