The Classical Fermionic String
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 perio
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The Classical Fermionic String
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 D 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.
7.1 Motivation for the Fermionic String So far we have only discussed bosonic strings. That means that all physical world-sheet degrees of freedom had been described by bosonic variables. We have treated the classical and the quantum theory, the algebra of the constraints (the Virasoro algebra) and we have found that at the quantum level the theory makes sense only in the critical dimension which was found to be d D 26. The spectra of the open and closed, oriented and unoriented theories were found to contain a tachyon, a fact which is at least alarming. Let us recall that its negative .mass/2 arose from the (regularized) zero point energy of an infinite set of bosonic harmonic oscillators. The problem with the tachyon may be cured if we introduce, on the world sheet, fermionic degrees of freedom which are quantized with anti-commutators. Then there is a chance that the zero point energies cancel and the tachyon is absent. One basic symmetry principle that guarantees the absence of a tachyon in the string spectrum is space-time supersymmetry. It is important to keep in mind the distinction between world-sheet and space-time supersymmetry. The fermionic string theories that we will discuss all possess world-sheet supersymmetry but not necessarily space-time supersymmetry and are not all tachyon-free. Whether a particular string theory is space-time supersymmetric or not will manifest itself, for instance, in the spectrum. Especially the existence of one or more massless gravitinos will signal space-time supersymmetry. R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 7, © Springer-Verlag Berlin Heidelberg 2013
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7 The Classical Fermionic String
Here we will present the Neveu-Schwarz-Ramond (RNS) superstring, which features manifest world-sheet supersymmetry but lacks manifest space-time supersymmetry. We should mention that there exists also the so-called Green-Schwarz (GS) formalism in which space-time supersymmetry is manifest at the cost of manifest world-sheet supersymmetry. It uses in a crucial way the triality property of SO.8/, the transverse Lorentz group in ten dimensions, which is, as we will see later in this chapter, the critical dimension for the fermionic string. The fermionic degrees of freedom are world-sheet scalars which carry an SO.8/ spinor index. We will not discuss the GS formulation of the superstring but it
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