Abstract
The fermionic string theory presented in this and the following chapter is the Neveu-Schwarz-Ramond spinning string. We present the world-sheet action and discuss its symmetries, most notably the local N = 1 world-sheet supersymmetry. The admissible periodicity and boundary conditions lead to the distinction between Neveu-Schwarz and Ramond sectors. The oscillator expansions of the world-sheet fermions differ in the two sectors. We close with an appendix on spinors in two dimensions. Quantisation of the fermionic string will be the subject of the following chapter.
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Notes
- 1.
We denote the amount of world-sheet supersymmetry by Nand reserve \(\mathcal{N}\)for space-time supersymmetry.
- 2.
Under general coordinate transformations, tensor indices are acted on with elements of \(GL(n, \mathbb{R})\).
- 3.
In any number of dimensions it is \({\overline{\chi }}_{\alpha }{\Gamma }^{\alpha \beta \gamma }{D}_{\beta }{\chi }_{\gamma }\)where \({\Gamma }^{\alpha \beta \gamma }\)is the anti-symmetrized product of three Dirac matrices which vanishes in two dimensions.
- 4.
The fields and parameters have mass dimensions \([\psi ]\,=\,[\chi ]\,=\,1/2\), \([\epsilon ]\,=\, - 1/2\), \([\xi ]\,=\,[X]\,=\, - 1\). All others are dimensionless. The fact that we choose Xto have dimension of length, rather than being dimensionless, as is more common for a scalar field in two dimensions, is the origin of the various powers of \(\sqrt{{ 2 \over \alpha \prime } }\).
- 5.
In later chapters we will also use the notations ψ L and \(\overline{\psi }\)for left-moving fermions and \({\psi }_{R}\)and ψ for right-moving fermions.
- 6.
For anticommuting variables they are defined as
$$\{F,{G\}}_{\mathrm{P.B.}} = \left ({ \partial F \over \partial {q}^{i}} { \partial G \over \partial {p}_{i}} -{ \partial F \over \partial {p}_{i}} { \partial G \over \partial {q}^{i}} \right ) + {(-1)}^{{\epsilon }_{F}}\left ({ \partial F \over \partial {\theta }^{\alpha }} { \partial G \over \partial {\pi }_{\alpha }} +{ \partial F \over \partial {\pi }_{\alpha }} { \partial G \over \partial {\theta }^{\alpha }} \right )$$(7.44)where (q, p) are the usual Grassmann even phase-space variables and \({\theta }^{\alpha }\)and \({\pi }_{\alpha } ={ \partial \mathcal{L} \over \partial {\theta }^{\alpha }}\)are Grassmann odd phase-space variables. Fand Gare functions on phase-space and \({\epsilon }_{F}\)is the Grassmann parity of F, i.e. \({\epsilon }_{F} = 0(1)\)for Feven (odd). All derivatives are left-derivatives. The canonical Hamiltonian is defined as \(H =\dot{ q}p +\dot{ \theta }\pi - L\)where the order in the second term matters.
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© 2012 Springer-Verlag Berlin Heidelberg
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). The Classical Fermionic String. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_7
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DOI: https://doi.org/10.1007/978-3-642-29497-6_7
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Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-642-29497-6
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