String Theory and M-Theory

102 Conformal field theory and string interactions
gauge theory (without matter fields) is a free theory, but the string extension
has nontrivial interactions.
In the case of Yang–Mills theory, two fields can be multiplied by the rule
Aik ∧ Bkj = Cij. (3.140)
k
This is a combination of matrix multiplication and antisymmetrization of
the tensor indices (the wedge product of differential geometry). This multiplication
is associative but noncommutative. A corresponding rule for string
fields is given by a ∗ product,
A ∗ B = C. (3.141)
This infinite-dimensional matrix multiplication is depicted in part (b) of
Fig. 3.7. One identifies the coordinates of the right half of string A with those
of the left half of string B and functionally integrates over the coordinates
of these identified half strings. This leaves string C consisting of the left
half of string A and the right half of string B. It is also necessary to include
a suitable factor involving the ghost coordinates at the midpoint σ = π/2.
A fundamental operation in gauge theory is exterior differentiation A →
dA. In terms of components
dA = 1
2 (∂µAν − ∂νAµ)dx µ ∧ dx ν , (3.142)
which contains the abelian field strengths as coefficients. Exterior differentiation
is a nilpotent operation, d 2 = 0, since partial derivatives commute and
vanish under antisymmetrization. The nonabelian Yang–Mills field strength
is given by the matrix-valued two-form
or in terms of tensor indices,
F = dA + A ∧ A, (3.143)
Fµν = ∂µAν − ∂νAµ + [Aµ, Aν]. (3.144)
Let us now construct analogs of d and F for the open-string field. The
operator that plays the roles of d is the nilpotent BRST operator QB, which
can be written explicitly as a differential operator involving the coordinates
X(σ), c(σ). Given the operator QB, there is an obvious formula for the
string-theory field strength, analogous to the Yang–Mills formula, namely
F = QBA + A ∗ A. (3.145)
The string field A describes physical string states, and therefore it has ghost
number −1/2. Since QB has ghost number +1, it follows that F has ghost