The Quantized Fermionic String
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 proj
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The Quantized Fermionic String
Abstract The fermionic string is quantized analogously to the bosonic string, though this time leading to a critical dimension d D 10. We first quantize in lightcone 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 d dimensions.
8.1 Canonical Quantization We proceed in the same way as in the bosonic theory by making the replacement (3.1) and, in addition, by replacing the Dirac bracket for the anticommuting world-sheet fermions by an anti-commutator: f ; gD:B: !
1 f ; g: i
(8.1)
We find f
C .; /;
0 C . ; /g
D 2 ı. 0 /;
f
.; /; C .; /;
0 . ; /g
D 2 ı. 0 /;
0 . ; /g
D0
f
(8.2)
or, in terms of oscillators1
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.
R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 8, © Springer-Verlag Berlin Heidelberg 2013
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8 The Quantized Fermionic String
fbr ; bs g D ırCs :
(8.3)
We define oscillators with positive mode numbers as annihilation operators and oscillators with negative mode numbers are creation operators. We have seen in Chap. 3 that ˛0 and ˛ 0 correspond to the center of mass momentum of the string. We will see below how the fermionic zero mode operators b0 and b 0 in the R-sector are to be interpreted. But we observe already here that they satisfy a Clifford algebra: fb0 ; b0 g D : (8.4) The level number operator is N D N .˛/ C N .b/ ;
(8.5)
where N .˛/ D
1 X
˛m ˛m ;
mD1
X
N .b/ D
r br br :
(8.6)
r2ZC>0
The oscillator expressions of the super-Virasoro generators are again undefined without giving an operator ordering prescription. As in Chap. 3 we define them by their normal ordered expressions, i.e. .b/ Ln D L.˛/ n C Ln
(8.7)
with L.˛/ n D
1X W ˛m ˛mCn W; 2 m2Z
L.b/ n
n 1 X rC W br bnCr W D 2 2
(8.8)
r2ZC
and Gr D
X
˛m brCm :
(8.9)
m2Z
Obviously, normal ordering is only required for L0 and we include again an as yet undetermined normal ordering constant a in all formulas containing L0 .
Unless stated otherwise, the expressions for the open string coincide with the ones for the rightmoving sector of the closed string.
8.1 Canonical Quantization
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The algebra satisfied by the Ln and Gr can now be determined. Great care is again required due to normal ordering. One finds d ŒLm ; Ln D .m n/ LmCn C m .m2 2/ ımCn; 8 m r GmCr ; ŒLm ; Gr D 2 d 2 r ırCs : fGr ; Gs g D 2 LrCs C 2 2
(8.10)
This is the super-Virasor
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