Gravity and Strings

412 String theory
This term does not change the classical equations of motion because the two-dimensional
Einstein–Hilbert Lagrangian density is just the curvature 2-form, which is locally a total
derivative: the spin-connection 1-form is ω ab = ɛ ab ω and R ab = ɛ ab dω.Actually, this term
is the constant φ0 times a topological invariant: the Euler characteristic χ =2−2g−b−c,
where g is the genus (number of handles) of the two-dimensional worldsheet, b the number
of boundaries, and c the number of crosscaps (which are present only when the worldsheet
is non-orientable). As we are going to see, φ0 is the vacuum expectation value of the dilaton,
a massless scalar field present in all string theories, and g = e φ0 can be interpreted as the
string coupling constant that counts loops in string amplitudes.
14.1.1 Superstrings
The string theories we have studied so far are called bosonic because they include only
bosonic worldsheet fields and, furthermore, their spectra only contain spacetime bosons, as
we will see. To construct a string theory whose spectrum includes fermions, we can generalize
the action of a spinning point-particle (although the historical order did not follow
this logic). The action for a spinning particle contains, in addition to the commuting variables
X µ (ξ) that describe the position of the particle, anticommuting variables ψ µ (ξ) that
describe the spin degrees of freedom and were first proposed in [105, 219] and studied and
generalized in [70, 86–8, 106, 188, 190, 251]. The simplest action one can write has global
worldline supersymmetry transformations that relate X µ and ψ µ , which form a scalar supermultiplet
[188, 190], and the natural generalization, which is invariant under worldline
reparametrizations, is also naturally invariant under local worldline supersymmetry transformations.
This generalization requires the introduction of auxiliary fields: an Einbein
e(ξ), asinthe bosonic case (e2 = γ ), and a gravitino χ, which form a one-dimensional
supergravity multiplet that has no dynamics. The action for the massless case is
S =− 1
2
dξe e−2 ˙X µ ˙Xµ + e−1ψ µ ˙ψµ − e−2χψ µ
˙Xµ . (14.15)
and the supersymmetry transformations that leave it invariant are
δɛ X µ = ɛψ µ , δɛe = ɛχ,
δɛψ µ =−ɛ( ˙X µ − 1
2χψµ )e−1 , δɛχ = 2˙ɛ. (14.16)
Invariance under worldline reparametrizations and supersymmetry transformations leads
to constraints (gauge identities) that are necessary for consistency. Furthermore, it can be
shown that the quantization of this model leads to the Dirac equation for spin- 1
2 particles
[86, 106, 190] (for more recent references, see [424, 529]) and this action can be used to
obtain path-integral representations of propagators and Feynman diagrams (see e.g. [537]
and references therein) for spin- 1
particles. It is also remarkable that, when it is coupled to
2
gravity [87], the action leads to the Dirac equation in curved spacetime as given in [187]