String Theory Demystified

176 String Theory Demystifi ed
which are the bosonic fi elds X µ ( σ, τ).
This can be done by adding new fi elds,
typically denoted by Θ a ( σ, τ ) , which map the worldsheet to fermionic coordinates.
Taking the X µ ( σ, τ)
together with the Θ a ( σ, τ ) will enable us to map the worldsheet
to superspace. This approach to superstring theory is known as the Green-Schwarz
(GS) formalism.
To summarize, when applying worldsheet supersymmetry
• We extend the coordinates ( τ, σ ) by introducing fermionic coordinates
1 2
θ and θ . This gives us super-worldsheet coordinates.
In this case:
• We are developing an extension of space-time itself, creating a superspace
described by the pair X µ ( σ, τ)
and Θ a ( σ, τ ) .
• An N = m supersymmetric theory will have a= 1, ..., m,
or m fermionic
coordinates.
A SUPERSYMMETRIC POINT PARTICLE
We introduce the formalism by going back to the simplest case we can describe—a
point particle. This will allow us to go over the main ideas without getting bogged
down by the formalism. It turns out this approach actually has some direct relevance
to string theory anyway. In modern parlance, a point particle is called a D0-brane.
So the physics we will lay out here is known as the D0-brane action (this is a
Dp-brane with p = 0 ). This type of object can be found in the type IIA superstring
theory.
The action for a relativistic point particle of mass m can be written as
1 ⎛ 1 2 2⎞
S = ∫ dτx
−em
⎝
⎜
e ⎠
⎟
(9.4)
2
As noted in Chap. 2, e is called the auxiliary fi eld. The action written in this form is
well suited to the study of massless particles. Letting m → 0 gives
1 1 2
S = ∫ dτx (9.5)
2 e
To make the jump to superspace, we consider the space defi ned by the pair of
coordinates:
µ Aa
x , θ
(9.6)