arXiv:hep-th/9304011 v1 Apr 5 1993

for arbitrary α 0 . For particular values of α 0 , it turns out to contain a consistent unitary
subspace.
Taking the operator product with the modified T of (1.15), we find that the conformal
weight of e βφ is shifted to ∆ = 1 2 β(β−2α 0). The same conformal weights would be inferred
from two-point functions calculated in the presence of a ‘background charge’ − √ 2α 0 at
infinity, so the modification of T (z) in (1.15) is interpreted as the presence of such a
background charge. This formalism was originally used by Chodos and Thorn in [24], and
was more recently revived by Feigen and Fuks in a form used in [25] to calculate correlation
functions of the c < 1 conformal field theories.
Exercise. Momentum Shift from Background Charge
Consider a Gaussian field with background charge Q = 2iα 0 . Derive the relation
ip φ = α − Q 2 , (1.16)
for the momentum p φ of the state created by the vertex operator exp(αφ) by considering
the state created by the path integral on the disk, analogous to the example of the
Gaussian model in (1.9).
2. 2D Euclidean Quantum Gravity I: Path Integral Approach
In this chapter, we shall consider the implementation of quantum gravity as a theory
which makes precise and well-defined sense out of a path integral over metrics on some
spacetime.
2.1. 2D Gravity and Liouville Theory
Using the principle of general covariance, any quantum field theory S matter [X i ] in any
number of dimensions may be coupled to gravity, resulting in an action S[g, X i ], where X i
refer to “matter” fields in the theory and g is the metric.
Classically, the theory S[g, X i ] with gravity coupling in two dimensions is always a
conformal field theory. To see this, recall that the stress energy tensor of the theory is
δ
δg αβ S = T αβ . Defining the Liouville mode as the overall scale of the metric, g = e γφ ĝ,
we see that the Liouville equation of motion is T α α = 0. This defines a classical conformal
field theory. In two dimensions, we may pass to local complex coordinates and write this
equation as T z¯z = 0. Conservation of energy-momentum then shows that the theory has a
holomorphic energy momentum tensor T zz = T (z).
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