Lectures on String Theory

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As we will see the states in the R sector give the space-time fermions, while the
states in the NS sector are the space-time bosons. This further gives that (R,R) and
(NS,NS) sectors are space-time bosons while (R,NS) and (NS,R) are fermions.
For the open string case we have that
ψ + δψ + − ψ − δψ −
must vanish at σ = 0 and σ = π. If we assume that ψ + = αψ − at the end point of
string, then
(α 2 − 1)ψ − δψ − = 0
which allows for α = ±1. Thus, at each end of the string we should have ψ + = ±ψ − .
We can always agree to choose ψ + (0, τ) = ψ − (0, τ) as it is a matter of convention,
then on the other hand of the string we have two possibilities
6.6 Superconformal algebra
ψ + (π, τ) = ψ − (π, τ) (Ramond) ,
ψ + (π, τ) = −ψ − (π, τ) (Neveu − Schwarz) .
In order to compute the Poisson bracket between the components of the stress tensor
and the supercurrent we need the fundamental Poisson bracket for the fermions. The
Dirac action leads to the following bracket
{ψ µ +(σ), ψ ν +(σ ′ )} = −2πiδ(σ − σ ′ )η µν
{ψ µ −(σ), ψ ν −(σ ′ )} = −2πiδ(σ − σ ′ )η µν .
Using these brackets together with brackets between the bosonic fields one find the
following Poisson algebra of the constraints
(
)
{T ++ (σ), T ++ (σ ′ )} = −2π 2T ++ (σ ′ )∂ ′ + ∂ ′ T ++ (σ ′ ) δ(σ − σ ′ ) ,
3
)
{T ++ (σ), G + (σ ′ )} = −2π(
2 G +(σ ′ )∂ ′ + ∂ ′ G + (σ ′ ) δ(σ − σ ′ ) ,
{G + (σ), G + (σ ′ )} = −iπT ++ (σ)δ(σ − σ ′ ) .
This is the so-called N = 1 superconfomal algebra in 2dim. Here N = 1 refers to
the fact that supersymmetry transformations are performed with the help of one
Majorana spinor.
The action of the supercurrent on the bosonic and fermionic fields generate supersymmetry
transformations (bosonic field transforms into fermionic one and vice-versa):
{G + (σ), X µ (σ ′ )} = −π ψ µ +(σ)δ(σ − σ ′ ) ,
{G + (σ), ψ µ (σ ′ )} = −iπ ∂ + X µ +(σ)δ(σ − σ ′ )