Lectures on String Theory

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representations of the transversal (and simultaneously little) Lorentz group SO(24)
as follows
(
α−1ᾱ i −1|0〉 j = α −1ᾱ [i −1|0〉
j] + α
} {{ }
−1ᾱ (i j)
−1 − 1 )
24 δij α−1ᾱ k −1
k |0〉 + 1
} {{ } 24 δij α−1ᾱ k −1|0〉
k .
} {{ }
276
299
singlet
The massless excitation of spin two transforming in representation 299 of SO(24)
was proposed to be identified with a graviton, the quantum of the gravitational
interaction. To make this identification one has to relate the string scale α ′ with the
Planck scale G = M −2
P
, where G is the Newton constant and M P is the Planck mass:
α ′ = M −2
P .
Since the masses of the massive string modes are proportional to 1/α ′ = MP 2, these
string excitations are extremely heavy due to the large value of MP 2 and, by this
reason, they do not show up at the energy scales of the Standard Model.
As in the opens string case the higher massive states of closed string are combined
at a given mass level into representations of the little Lorentz group SO(25). The
relation between maximal spin and the mass is now
4.3 BRST quantization
J max = α′
2 M 2 + 2 .
The path integral approach proved to be a very useful tool for quantizing the theories
with local (gauge) symmetries. The starting point is the Polyakov action and a new
BRST (Becchi-Rouet-Stora-Tyutin) symmetry. We know that the induced metric
Γ αβ = ∂ α X µ ∂ β X µ and the intrinsic metric h αβ are related classically through the
condition T αβ = 0. However, quantum-mechanically this is not the case.
The basic idea is to define the path integral using the Polykov action and integrate
over
h αβ , X µ
being considered as independent variables:
∫
Z = Dh αβ (σ, τ)DX µ (σ, τ)e iSp[X,h]
Due to the gauge invariance the last integral is ill-defined. This occurs because we
integrate infinitely many times over physically equivalent, i.e. related to each other
by gauge transformations, configurations. This can be understood looking at a much
simpler example of the two-dimensional integral
Z =
∫ +∞ ∫ +∞
−∞
−∞
dxdy e −(x−y)2 .