66
2 Worldsheet Path Integral: Vacuum Amplitudes
target spacetime is completely equivalent to matter minimally coupled to Einstein–
Hilbert gravity with a cosmological constant (tuned to impose Weyl invariance at
the quantum level). The advantage of describing the coupling power in this fashion
is that it directly generalizes to scattering amplitudes and to open strings. The
parameter 0 is interpreted as the expectation value of the dilaton. Replacing it
by a general field μ ) is a generalization of the matter non-linear sigma model,
but this topic is beyond the scope of this book.
2.6
Summary
In this chapter, we started with a fairly general matter CFT—containing at least
D scalar fields X μ —and explained under which condition it describes a string
theory. The most important consequence is that the matter 2d QFT must in fact
be a 2d CFT. We then continued by describing how to gauge fix the integration
over the surfaces and we identified the remaining degrees of freedom—the moduli
space M g —up to some residual redundancy—the conformal Killing vector (CKV).
Then, we showed how to rewrite the result in terms of ghosts and proved that they
are also a CFT. This means that a string theory can be completely described by
two decoupled CFTs: a universal ghost CFT and a theory-dependent matter CFT
describing the string spacetime embedding and the internal structure. The advantage
is that one can forget the path integral formalism altogether and employ only CFT
techniques to perform the computations. This point of view will be developed for
off-shell amplitudes (Chap. 11) in order to provide an alternative description of how
to build amplitudes. It is particularly fruitful because one can also consider matter
CFTs which do not have a Lagrangian description. In the next chapter, we describe
scattering amplitudes.
2.7
Suggested Readings
Numerous books have been published on the worldsheet string theory. Useful (but
not required) complements to this chapter and subsequent ones are [23, 33] for
introductory texts and [3, 6, 7, 22, 29] for more advanced aspects.
• The definition of a field measure from a Gaussian integral and manipulations
thereof can be found in [18, sec. 15.1, 22.1, 26, chap. 14, 11, 28].
• The most complete explanations of the gauge fixing procedure are [18, sec. 15.1,
22.1, 3, sec. 3.4, 6.2, 29, chap. 5, 7, 21, chap. 5]. The original derivation can be
found in [10, 25].
• For the geometry of the moduli space, see [26, 27].
• Ultralocality and its consequences are described in [11, 28] (see also [17,
sec. 2.4]).
• The use of a Weyl ghost is shown in [31, sec. 8, 32, sec. 9.2].
2 Worldsheet Path Integral: Vacuum Amplitudes
target spacetime is completely equivalent to matter minimally coupled to Einstein–
Hilbert gravity with a cosmological constant (tuned to impose Weyl invariance at
the quantum level). The advantage of describing the coupling power in this fashion
is that it directly generalizes to scattering amplitudes and to open strings. The
parameter 0 is interpreted as the expectation value of the dilaton. Replacing it
by a general field μ ) is a generalization of the matter non-linear sigma model,
but this topic is beyond the scope of this book.
2.6
Summary
In this chapter, we started with a fairly general matter CFT—containing at least
D scalar fields X μ —and explained under which condition it describes a string
theory. The most important consequence is that the matter 2d QFT must in fact
be a 2d CFT. We then continued by describing how to gauge fix the integration
over the surfaces and we identified the remaining degrees of freedom—the moduli
space M g —up to some residual redundancy—the conformal Killing vector (CKV).
Then, we showed how to rewrite the result in terms of ghosts and proved that they
are also a CFT. This means that a string theory can be completely described by
two decoupled CFTs: a universal ghost CFT and a theory-dependent matter CFT
describing the string spacetime embedding and the internal structure. The advantage
is that one can forget the path integral formalism altogether and employ only CFT
techniques to perform the computations. This point of view will be developed for
off-shell amplitudes (Chap. 11) in order to provide an alternative description of how
to build amplitudes. It is particularly fruitful because one can also consider matter
CFTs which do not have a Lagrangian description. In the next chapter, we describe
scattering amplitudes.
2.7
Suggested Readings
Numerous books have been published on the worldsheet string theory. Useful (but
not required) complements to this chapter and subsequent ones are [23, 33] for
introductory texts and [3, 6, 7, 22, 29] for more advanced aspects.
• The definition of a field measure from a Gaussian integral and manipulations
thereof can be found in [18, sec. 15.1, 22.1, 26, chap. 14, 11, 28].
• The most complete explanations of the gauge fixing procedure are [18, sec. 15.1,
22.1, 3, sec. 3.4, 6.2, 29, chap. 5, 7, 21, chap. 5]. The original derivation can be
found in [10, 25].
• For the geometry of the moduli space, see [26, 27].
• Ultralocality and its consequences are described in [11, 28] (see also [17,
sec. 2.4]).
• The use of a Weyl ghost is shown in [31, sec. 8, 32, sec. 9.2].