Abstract
The fermionic string is quantized analogously to the bosonic string, though this time leading to a critical dimension d = 10. We first quantize in light-cone gauge and construct the spectrum. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the type IIA and the type IIB superstring. We also present the covariant path integral quantization. The chapter closes with an appendix on spinors in ddimensions.
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- 1.
Again, we will only write down the expressions for the right-moving sector of the closed string. The left-moving expressions are easily obtained by simply putting bars over all mode operators. Unless stated otherwise, the expressions for the open string coincide with the ones for the right-moving sector of the closed string.
- 2.
In fact, the zero modes generate a finite group of order 2d + 1under multiplication. Using results from the representation theory of finite groups, one can show that for deven there is only one inequivalent irreducible representation of this group which is not one-dimensional. This representation must therefore be the one in terms of Dirac matrices. The one-dimensional representations of the group clearly violate the Dirac algebra. For dodd, there are two representations but they only differ by a sign of one of the Dirac matrices.
- 3.
We anticipate that the critical dimension will be even.
- 4.
In this way the states are presented by their weight vector; cf. Chap. 13.
- 5.
The general statement is that we can impose Majorana and Weyl conditions simultaneously on spinors of SO(p, q) if and only if \(p - q = 0 {\rm mod}\,\,8\). For Minkowski space-times (q = 1) this is the case for \(d = 2 + 8n\)and for Euclidean spaces (q = 0) for d = 8n. More details can be found in the appendix of this chapter.
- 6.
In Euclidean signature the labelling is usually \({\gamma }^{1},\ldots ,{\gamma }^{d}\), but for unity of notation we use \(\mu = 0,\ldots ,d - 1\)in both cases.
- 7.
In Chap. 7we used a different representation for \(d = 2\).
- 8.
Another common notation is \({b}^{i} = {\gamma }^{i}\)and \({b}_{i} = {\gamma }^{\bar{\imath }}\).
- 9.
Another common choice is \(D = i{\gamma }^{0}\)to make \({\overline{\psi }}_{D}\psi \)Hermitian.
- 10.
Proven e.g. in B. Zumino, Normal forms of complex matrices. J. Math. Phys. 3, 1055 (1962).
- 11.
In later chapters we will also use \((\alpha ,\dot{\alpha })\)for chiral spinor indices.
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© 2012 Springer-Verlag Berlin Heidelberg
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). The Quantized 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_8
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DOI: https://doi.org/10.1007/978-3-642-29497-6_8
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