STRING THEORY Problem Set # 1 January 23, 1997

Consider the quantum field theory of a single real scalar field x, on a 2-dimensional surface Σ with Riemannian metric g and action

S[x, g] = \frac{1}{4π} ∫_{Σ} dμ_{g}\{\frac{1}{2}g^{mn}∂_{m}x∂_{n}x + QR_{g}x\}.

Here, Q ∈ C and R_{g} is the Gaussian curvature of g. Introduce local complex coordinates z, z̄ in which g = 2g_{zz̄}|dz|^{2}.

1.) For Q = 0, define the renormalized stress tensor by

T (z) = −\frac{1}{2} lim_{w→z}(∂_{z}x∂_{w}x + \frac{1}{(z − w)^{2}} ).

a) Using the OPE of ∂_{z}x and ∂_{w}x, derive all singular terms in the OPE of T (z) and T (w), and confirm the form derived on general grounds

T (z)T (w) ∼ \frac{c/2}{(z − w)^{4}} + \frac{2}{(z − w)^{2}} T (w) + \frac{1}{z − w} ∂_{w}T (w).

b) Show, by explicit conformal reparametrization z → z′ = f(z), that T (z) transforms as a projective connection.

c) Use the result of b) to relate the stress tensor T (z) and the Virasoro generators L_{m} on the cylinder and on the annulus.

2.) For Q ≠ 0, consider the flat metric g = 2|dz|^{2}.

a) Derive the OPE of ∂_{z}x and ∂_{w}x.

b) Compute the stress tensor and show that T = T_{zz} is complex analytic at the classical level. Define a suitable renormalized stress tensor.

c) Use the result of b) to obtain the central charge of the theory.

d) Show that : exp{βx(z, z̄)} : is a primary field for any value of β ∈ C, and compute its conformal weight (h, h̄).

e) Is the field ∂_{z}x primary ? Derive its transformation law under conformal reparametrizations z → z′ = f(z).