String Theory and M-Theory

104 Conformal field theory and string interactions
of X. A ghost factor has been inserted at the midpoint. φ(σ) is the bosonized
form of the ghosts described earlier. This ensures that contributes −3/2
to the ghost number, as required. This definition of integration satisfies the
important requirements
QBY = 0 and [Y1, Y2] = 0. (3.152)
(a)
(b)
σ=0
σ=π/2 σ=π
L R
R
R L
Fig. 3.8. Integration of a string functional requires identifying the left and right
halves as depicted in (a). The three-string vertex, shown in (b), is based on two
multiplications (star products) and one integration and treats the three strings
symmetrically.
We now have the necessary ingredients to write a string action. Trying
to emulate the Yang–Mills action runs into a problem, because no analog of
the metric g µρ has been defined. Rather than trying to find one, it proves
more fruitful to look for a gauge-invariant action that does not require one.
The simplest possibility is given by the Chern–Simons form
S ∼ A ∗ QBA + 2
A ∗ A ∗ A . (3.153)
3
In the context of ordinary Yang–Mills theory the integrand is a three-form,
whose variation under a gauge transformation is closed, and therefore such
a term can only be introduced in three dimensions, where it is interpreted as
giving mass to the gauge field. In string theory the interpretation is different,
though the mathematics is quite analogous, and the formula makes perfectly
good sense. In fact, in both cases extremizing the action gives rise to the
deceptively simple classical field equation F = 0.
The fact that the string equation of motion is F = 0 does not mean the
theory is trivial. Dropping the interaction term, the equation of motion
L