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 \((\beta ,\gamma )\)ghost system, This will in turn lead to the introduction of the so-called covariant lattices.
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Notes
- 1.
The first part of this section is a review of material from Sect. 11.4 with a slight change of notation: \({X}^{i} \rightarrow {\phi }^{i}\).
- 2.
We can also have \(a = \frac{1} {2}\)for integer \(\lambda \)corresponding to a twisted sector.
- 3.
The generalization of (5.13) is \({L}_{0} = {\sum olimits }_{n=1}^{\infty }n({b}_{ -n}{c}_{n} + {c}_{-n}{b}_{n}) -{ 1 \over 2} \lambda (\lambda - 1)\).
- 4.
Alternatively, we require that \({\delta }_{\alpha }\int DbDc\,{b}^{{N}_{b}}\,{c}^{{N}_{c}}{e}^{-S[b,c]} = 0\)where the band cinsertions are necessary to absorb the zero modes. This emphasizes the meaning of Qas a background charge, cf. below.
- 5.
A completely equivalent way is to consider \(\langle {\oint olimits }_{C}{ dz \over 2\pi i} j(z)...\rangle\)where. . . stands for some operator insertions on the sphere which the contour Cencloses. In the absence of a background charge this would be zero as we can contract the contour on the back of the sphere. This requires a change of variables z = 1 ∕ uand the contour now encircles u = 0. However, due to the inhomogeneous term in (13.30) we get a non-zero contribution. This means that only correlation functions of operators with a total charge Qcan be non-zero.
- 6.
- 7.
Using the integral representation of the δ-function and formal manipulations, one can show, by verifying the OPEs, that \({e}^{\phi (z)} = \delta (\beta (z)),{e}^{-\phi (z)} = \delta (\gamma (z)),\eta (z) = \partial \gamma (z)\delta (\gamma (z))\)and \(\partial \xi (z) = \partial H(\beta (z))\)(His the Heaviside step function). Another way to verify this is to show that the states \(\delta (\beta (0))\vert 0\rangle\)and \(\delta (\gamma (0))\vert 0\rangle\)have the properties (13.52) for ε = − 1 and \(\lambda = 3/2\)of the | + 1⟩ and | − 1⟩ vacua, respectively. Note that the zero-mode of ξ cannot be expressed in terms of \(\beta ,\gamma \).
- 8.
The BRST cohomology in the large algebra is trivial because Qis invertible there. Indeed, \(\{Q,c\xi \partial \xi {e}^{-2\phi }\} = \text{ constant}\).
- 9.
In the derivation of Eq. (13.82) we need the subleading term of the first of the operator products in Eq. (13.8). One can show that in ddimensions
$${k}_{\mu }{u}^{\alpha }{\psi }^{\mu }(z){S}_{\alpha }(w) \sim \frac{1} {\sqrt{2}}{(z - w)}^{1/2} \frac{1} {\frac{d} {2} - 1}{u}^{\alpha }{k}_{\mu }{\psi }^{\mu }{\psi }^{u }{({\Gamma }_{u }){}_{\alpha }}^{\dot{\beta }}{S}_{\dot{ \beta }}$$(13.81)for on-shell \({u}^{\alpha }\).
- 10.
In Chap. 5we wrote \(\phi \)for integrated and ψ = cϕ for unintegrated vertex operators.
- 11.
If we include the zero mode of the \(\eta \), \(\xi \)system, we also have to neutralize its background charge and get \(\langle 0\vert {e}^{3\sigma -2\phi +\chi }\vert 0\rangle = 1\). \({e}^{3\sigma -2\phi +\chi }\vert 0\rangle\)is however not BRST invariant.
- 12.
We use C = C + in which case Majorana spinors and the Dirac matrices are real in the Majorana representation.
- 13.
Using the boundary condition (13.100) on the real axis, we can analytically continue the spin field to the lower half plane and use the doubling trick to represent the supercharge by a closed-contour integral. The superconformal ghost is continuous across the real axis.
- 14.
The condition is \(\det ({\Gamma }^{0}\cdots {\Gamma }^{p} - {R}^{-1}{\Gamma }^{0}\cdots {\Gamma }^{p}R) = 0\).
- 15.
To see this, refer to the decomposition (14.40).
- 16.
This is for the relevant part of P + 1. The other pieces also correspond to vectors in the (0) conjugacy class of D 5, 1.
- 17.
These lattice maps will be discussed in more detail below.
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). Covariant Vertex Operators, BRST and Covariant Lattices. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_13
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DOI: https://doi.org/10.1007/978-3-642-29497-6_13
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