Feynman Path Integral Formulation

42 1 Continuum Formulation
Fig. 1.3 A string vertex (left) and a closed string loop (right).
The Nambu-Goto string is not easy to work with due to the square root. One can
write down an action (Polyakov, 1981a,b) which is classically equivalent to it, but
does not involve the square root of the X variables, if one introduces the metric g ab
as a Lagrange multiplier. The Euclidean action
∫
I [g,X] = 1 2
d 2 σ √ gg ab ∂ a X μ ∂ b X μ . (1.195)
Variation of this action with respect to X μ gives Laplace’s equation for X μ , while
the variation with respect to g ab gives Eq. (1.192). One noteworthy feature of the
action in Eq. (1.195) is its large invariance group. It is invariant under world sheet
(a,b) diffeomorphisms, spacetime (μ,ν) Lorentz invariance, and invariance under
Weyl or conformal transformations
g ab (σ,τ) → e 2ω(σ,τ) g ab (σ,τ) , (1.196)
which implies that one is dealing with a two-dimensional conformal field theory.
Note also that the string so far is embedded in flat space, but one could consider
a more general embedding, with a suitable change in the spacetime metric
η μν → G μν (X), and possibly additional terms in the action such as curvature
[ 2 R(X)] contributions.
Note that in the gauge g ab = η ab the field X satisfies the wave equation
( ∂
✷X μ
2
≡
∂σ 2 − ∂ 2 )
∂τ 2 X μ = 0 , (1.197)
supplemented by the constraint equations T ab = 0 with
T 10 = T 01 = Ẋ · X ′ = 0 (1.198)
and
T 00 = T 11 = 1 2
(Ẋ 2 + X ′ 2 ) = 0 . (1.199)