Kiefer C. Quantum gravity

THE FREE BOSONIC STRING 85
In the standard treatment of the bosonic string, the ‘gauge freedom’ (with
respect to two diffeomorphisms and one Weyl transformation) is fixed by the
choice h αβ = η αβ = diag(−1, 1). Instead of (3.51) one has then
S P = − 1 ∫
4πα ′ d 2 ση αβ ∂ α X µ ∂ β X µ , (3.58)
that is, the action for D free scalar fields in two dimensional Minkowski space.
In two dimensions there is a remaining symmetry which leaves the gauge-fixed
action invariant—the conformal transformations. These are angle-preserving coordinate
transformations 4 which change the metric by a factor e 2ω(σ,τ ) ;theycan
therefore be compensated by a Weyl transformation, and the action (3.58) is invariant
under this combined transformations. A field theory with this invariance
is called a ‘conformal field theory’ (CFT). A particular feature of two dimensions
is that the conformal group is infinite-dimensional, giving rise to infinitely many
conserved charges (see below).
Consider in the following the case of open strings where σ ∈ (0,π). (Closed
strings ensue a doubling of degrees of freedom corresponding to left- and rightmovers.)
The Hamiltonian of the gauge-fixed theory reads
H = 1 ∫ π (Ẋ2 4πα ′ dσ +(X ′ ) 2) . (3.59)
0
Introducing the components of the energy–momentum tensor with respect to the
lightcone coordinates σ − = τ − σ and σ + = τ + σ, it is convenient to define the
quantities (m ∈ Z)
L m = 1 ∫ π
2πα ′ dσ ( e imσ T ++ +e −imσ )
T −− . (3.60)
0
One recognizes that L 0 = H. Because the energy–momentum tensor vanishes
as a constraint, this holds also for the L m ,thatis,L m ≈ 0. The L m obey the
classical Virasoro algebra
{L m ,L n } = −i(m − n)L m+n , (3.61)
exhibiting that they generate the group of conformal transformations (the residual
symmetry of the gauge-fixed action). The {L n } are the infinitely many conserved
charges mentioned above. It turns out that quantization does not preserve
this algebra but yields an additional term called ‘anomaly’, ‘central term’, or
‘Schwinger term’,
[ˆL m , ˆL n ]=(m − n)ˆL m+n + c2
12 (m3 − m)δ m+n,0 , (3.62)
where c is the central charge. For the case of the free fields X µ ,itisequalto
the number of space–time dimensions, c = D. Due to the presence of this extra
4 In GR, the term ‘conformal transformation’ is usually employed for what is here called a
Weyl transformation.