String Theory and M-Theory

3.5 The structure of string perturbation theory 89
3.5 The structure of string perturbation theory
The starting point for studying string perturbation theory is the world-sheet
action with Euclidean signature. Before gauge fixing, it has the general form
SWS = L(hαβ; X µ ; background fields) d 2 z . (3.107)
M
As usual, hαβ is the two-dimensional world-sheet metric, and X µ (z, ¯z) describes
the embedding of the world sheet M into the space-time manifold
M. Thus z is a local coordinate on the world sheet and X µ are local coordinates
of space-time. Working with a Euclidean signature world-sheet metric
ensures that the functional integrals (to be defined) are converted to convergent
Gaussian integrals. The background fields should satisfy the field
equations to be consistent. When this is the case, the world-sheet theory
has conformal invariance.
Partition functions and scattering amplitudes
Partition functions and on-shell scattering amplitudes can be formulated as
path integrals of the form proposed by Polyakov
Z ∼
DX µ · · · e −S[h,X,...] . (3.108)
Dhαβ
Here Dh means the sum over all Riemann surfaces (M, h). However, this
is a gauge theory, since S is invariant under diffeomorphisms and Weyl
transformations. So one should really sum over Riemann surfaces modulo
diffeomorphisms and Weyl transformations. 11
World-sheet diffeomorphism symmetry allows one to choose a conformally
flat world-sheet metric
hαβ = e ψ δαβ. (3.109)
When this is done, one must add the Faddeev–Popov ghost fields b(z) and
c(z) to the world-sheet theory to represent the relevant Jacobian factors in
the path integral. Then the local Weyl symmetry (hαβ → Λhαβ) allows one
to fix ψ (locally) – say to zero. However, this is not possible globally, due
to a topological obstruction:
ψ = 0 ⇒ R(h) = 0 ⇒ χ(M) = 0. (3.110)
So, such a choice is only possible for world sheets that admit a flat metric.
11 In the case of superstrings in the RNS formalism, discussed in the next chapter, the action also
has local world-sheet supersymmetry and super-Weyl symmetry, so these equivalences also need
to be taken into account.