Abstract
In this chapter we will present a review of semi-classical string theory and classical integrability. In the first section we introduce the basis of classical bosonic string theory, where the present the Polyakov action and the coset construction. The last one is also used to introduce the concept of classical integrability and the Wess-Zumino-Witten model. We devote the second section to supersymmetric string theories, where we review the Neveu-Schwarz-Ramond superstring and the Green-Schwarz superstring. We also present the supersymmetrized version of the previous coset construction and apply it to the construction of the theory on \(AdS_5\times S^5\) and \(AdS_3 \times S^3\) backgrounds.
The motivation of formal string theory is to understand the truly fundamental ideas in string theory which is assumed by the practitioners to be the theory explaining or predicting everything in the Universe that may be explained or predicted. How may someone say that the motivation is similar to that of mathematicians?
Lubos Motls, Formal string theory is physics, not mathematics
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- 1.
Let us analyse the case of a massive relativistic particle to clarify this point. The action is given by
$$\begin{aligned} S=\frac{1}{2} \int _{s_0}^{s_1}{e \left( e^{-2} \dot{x}^\mu \dot{x}_\mu -m^2 \right) d\tau } \ ,\qquad {(2.1.2)} \end{aligned}$$where the dot represent the derivative with respect to \(\tau \). The reparametrization invariance can be used to fix \(e=1/m\). However the equation of motion we obtain, \(\ddot{x}^\mu =0\), wrongly includes time-like and light-like lines as solutions. This problem can be solved if we support it with the equation of motion for e, which reads \(\dot{x}^2+1=0\) after the “gauge-fixing” \(e=1/m\), so only the space-like lines are selected.
- 2.
The definition of left and right currents depends on the authors. This is because some authors define left-invariant and right-invariant currents (for example [7]) and other authors define Noether currents corresponding to multiplications of g by a constant element of the group from the left and from the right (for example [8]). This gives opposite definitions of these currents and create some confusions. Sometimes the definition of the left-invariant current has an extra global minus sign (for example [4]).
- 3.
The Lax connection of a system is not unique. Given an arbitrary matrix \(f(\tau , \sigma ,z)\), the flatness condition is invariant under the gauge transformation
$$\begin{aligned} L_\alpha \rightarrow L'_\alpha =f L_\alpha f^{-1} + (\partial _\alpha f )f^{-1} \ .\qquad {(2.1.10)} \end{aligned}$$If we choose the particular case \(f(\tau , \sigma ,z)\propto g(\tau , \sigma )\) we can relate the Lax connection written in the left and in the right currents.
- 4.
To prove this statement we should use the property \({{\mathrm{Tr}}}_{12} \{ A\otimes B\}={{\mathrm{Tr}}}(A) {{\mathrm{Tr}}}(B)\).
- 5.
Which one acts on the left and on the right depends on if we are working with the left or the right invariant current. In particular, for the left invariant current we have \(G_{\text {left}}(z) \times G_{\text {right}} (\bar{z})\).
- 6.
Actually things are more complex because the auxiliary field that is the world-sheet metric \(h_{\alpha \beta }\) also needs to be supersymetrized. Hence we don’t work with it, but with the zweibein \(e^\alpha _a\) defined as \(e^\alpha _a e^\beta _b \eta ^{ab}=h^{\alpha \beta }\) and their supersymmetric partner, a Majorana spinor-vector called gravitino. However, at the end we can gauge away these fields and fix a superconformal gauge where we put the gravitino to zero and the zweibein to identity.
- 7.
Sadly this quotient has no realization in terms of \(4n\times 4n\) matrices.
- 8.
We have left a general \(\gamma ^{\alpha \beta }=\sqrt{-h} h^{\alpha \beta }\) metric instead of directly choosing the conformal gauge \(\gamma ^{\alpha \beta }=\eta ^{\alpha \beta }\) because later we are going to consider a symmetry transformation with \(\delta \gamma \ne 0\).
- 9.
All the expressions of this subsection are written in the language of differential form to make the expressions more readable and compact.
- 10.
A different normalization for this Lax connection more fitted for the algebraic curve construction was presented in [15], which recovers the Noether currents for large values of the spectral parameter.
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Nieto, J.M. (2018). Strings in Coset Spaces. In: Spinning Strings and Correlation Functions in the AdS/CFT Correspondence. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-96020-3_2
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