Covariant Vertex Operators, BRST and Covariant Lattices
We reexamine the 10-dimensional type II and heterotic superstring theories using the bosonic language. The aim of this bosonic formulation is the construction of the covariant fermion vertex operators, which involves a proper treatment of the \((\beta ,\g
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Covariant Vertex Operators, BRST and Covariant Lattices
Abstract We reexamine the 10-dimensional type II and heterotic superstring theories using the bosonic language. The aim of this bosonic formulation is the construction of the covariant fermion vertex operators, which involves a proper treatment of the .ˇ; / ghost system, This will in turn lead to the introduction of the so-called covariant lattices.
13.1 Bosonization and First Order Systems Recall1 the action of the fermionic string in superconformal gauge: Z 1 d 2 z @X @X C @ C @ ; SD 4
(13.1)
where we have only written the matter (X , ) part. We will turn to the ghost part below. These fields generate a superconformal field theory with cO D 10 (c D 15) where from now on we discuss only the right-moving, holomorphic part of the theory. The two-dimensional supercharge, also called supercurrent, is TF .z/ D
i @X .z/ 2
.z/:
(13.2)
Applying the techniques of bosonization as introduced in Chap. 11, we replace the ten real fermions .z/ ( D 1; : : : ; 10) by five chiral bosons i .z/ (i D 1; : : : ; 5) with momentum eigenvalues being lattice vectors of the D5 weight lattice. The bosonization is performed by converting the ten real fermions .z/ to the complex Cartan-Weyl basis: 1 ˙ i .z/ D p . 2
2i 1
˙i
2i
/.z/ ;
i D 1; : : : ; 5:
(13.3)
1 The first part of this section is a review of material from Sect. 11.4 with a slight change of notation: X i ! i .
R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 13, © Springer-Verlag Berlin Heidelberg 2013
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13 Covariant Vertex Operators, BRST and Covariant Lattices
The action for the complex fermions is SD
1 4
Z
d 2 z . C i @ i C i @ C i /:
(13.4)
The part of the generators of the Wick rotated Lorentz group SO.10/ which are constructed from the world-sheet fermions are bosonized according to J Ci;i .z/ D W Ci i .z/ W D i @ i .z/ J ˙i;˙j .z/ D W ˙ i ˙ j W D W e ˙i
i ˙i j
.z/ W
.i < j /;
(13.5)
where the complex fermions themselves are expressed as ˙i .z/ D W e ˙i
i .z/
W:
(13.6)
Here and in the following we drop cocycle factors. They are necessary to get anticommuting fermions ˙i for i ¤ j and to produce manifestly covariant results, as e.g. in Eq. (13.8) below. The states of the spinning string theory are created from the SL.2; C/ invariant vacuum state by vertex operators which contain the five bosons i .z/. Let us concentrate on expressions which are exponentials of these bosons; possible derivative terms @ i .z/ play only a trivial role in the following. We will not discuss the X dependent part of the vertex operators either. In the NS sector the states are space-time bosons. The ground state is the NS vacuum j0i which is a tachyon as discussed in Chap. 8. The first excited state is the massless ten-dimensional vector j i D b1=2 j0i with the corresponding vertex operator .z/. Thus, the vector vertex operator in the bosonized version
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