String Theory Demystified

180 String Theory Demystifi ed
Space-Time Supersymmetry and Strings
We have introduced some of the basic ideas of space-time supersymmetry by
considering the point particle (known as the D0-brane). Now we move on to consider
the supersymmetric generalization of bosonic string theory. Recall that the action
for a bosonic string can be written as
1 2 αβ µ
SB =− ∫ d σ hh ∂αX ∂ X
2π
β µ (9.12)
Following the procedure used to write down the D0-brane action which was invariant
under SUSY transformations, we introduce a new fi eld:
µ µ µ
Π =∂ X −iΘ Γ ∂ Θ
α
α
A A (9.13)
α
µ
This approach is different than the RNS formalism discussed in Chap. 7. The Πα are actual fermion fi elds on space-time. In Chap. 7, we had spinors but the ψ µ were
space-time vectors and not genuine fermion fi elds.
It turns out that in string theory the number of supersymmetries is restricted to
N ≤ 2. If we consider the most general case allowed which has N = 2, then there
are two fermionic coordinates:
1a 2a
Θ and Θ
(9.14)
To get the full action we need to extend Eq. (9.12) in two steps. The fi rst step is to
simply add a corresponding piece containing the fermion fi elds defi ned in Eq. (9.13).
It has basically the same form:
1 2
S1=− d hh
2π
∫ σ
αβ
ΠΠ (9.15)
Now things get hairy for technical reasons. In supersymmetry, there is a local
fermionic symmetry called kappa symmetry. To avoid getting weighted down with
mathematical details in a “Demystifi ed series” book, we are going to leave it to you
to read about kappa symmetry in more advanced treatments. Here we simply take it
as a given that we need to preserve this kappa symmetry and that we can only do so
by adding the following unwieldy piece to the action:
1 2 αβ µ 1 1 2 2 αβ 1 µ 1 2 2
S = d ⎡−i
X
2 ∫ σ⎣ε ∂ ( Θ Γ ∂ Θ −Θ Γ ∂ Θ ) + ε Θ Γ ∂ Θ Θ Γ ∂ Θ ⎤ α
µ β µ β
α µ β
π
⎦
µ
α
βµ
(9.16)