Abstract
This chapter summarizes the properties of some CFT systems. We focus on the free scalar field and on the first-order bc system (which generalizes the reparametrization ghosts). For the different systems, we first provide an analysis on a general curved background before focusing on the complex plane. This is sufficient to describe the local properties on all Riemann surfaces gāā„ā0.
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Notes
- 1.
To be identified with the string scale, such that Ī±ā²ā=āā 2.
- 2.
The group is \({\mathbb {R}}\) but the algebra is \(\mathfrak {u}(1)\) (since locally there is no difference between the real line and the circle).
- 3.
The š in the exponential is consistent with interpreting X and k as a contravariant vector.
- 4.
The Fourier expansion is taken to be identical for šā=āĀ±1 fields since āX is contravariant in target space. The difference between the two cases will appear in the commutators.
- 5.
In worldsheet Lorentzian signature, this becomes X(Ļ, Ļ)ā=āxā+āā 2 ptā+āā 2 wĻ as expected.
- 6.
- 7.
To simplify the discussion, we do not consider winding but only vertex operators of the form (7.33).
- 8.
- 9.
For the scalar field, the coupling of both sectors happened because of the periodicity condition (7.62).
- 10.
Other references, especially old ones, give it on the cylinder. This can be easily recognized if some ghost numbers in the holomorphic sector are half-integers: for the reparametrization ghosts, q Ī» is an integer such that the shift in (7.121) is a half-integer.
- 11.
Due to the specific structure of the inner product defined below, these subspaces are not orthonormal to each other.
- 12.
The reader should not get confused by the same symbol \(\mathcal H_{\text{gh},0}\) as in the case of the holomorphic sector.
- 13.
as in (
and 
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Erbin, H. (2021). CFT Systems. In: String Field Theory. Lecture Notes in Physics, vol 980. Springer, Cham. https://doi.org/10.1007/978-3-030-65321-7_7
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DOI: https://doi.org/10.1007/978-3-030-65321-7_7
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