STRING THEORY Problem Set # 6 March 6, 1997

In this problem set, we investigate a number of properties of the combined matter (for flat space-time of dimension D) and ghost system. Recall that the worldsheet action in locally flat complex coordinates z and z̄ is given by

S = \frac{1}{2π} ∫_{Σ} d^{2}z(\frac{1}{2}∂x · ¯∂x + b¯∂c + b̄∂c̄).

The matter stress tensor is denoted by T^{(x)}(z) and the quantum ghost stress tensor is given by

T^{(bc)}(z) = lim_{w→z}(−∂b(z)c(w) − 2b(z)∂c(w) + \frac{1}{(z − w)^{2}} ).

The classical BRST current is given by

j^{B}(z) = c(z)(T^{(x)}(z) + \frac{1}{2}T^{(gh)}(z)) + \frac{3}{2}∂^{2}c(z).

Problem 1

a) Give a suitable definition of the quantum BRST current, by subtracting OPE singularities.

b) Compute the OPE of j^{B} with b, c and x, and deduce the (anti-) commutators of the BRST operator

Q^{B} = ∮ \frac{dz}{2πi}j_{z}^{B}

with these fields.

c) Show that the resulting transformation laws on the fields b, c and x give rise to a classical symmetry of S for any dimension D of space-time, and that j^{B} is related to the Noether current of this symmetry.

d) Show that Q^{B} is nilpotent as a quantum mechanical operator if and only if D = 26, using your favorite method. (Hint : It is easy to show that when D ≠ 26, Q^{B} cannot be nilpotent, by using the relation between the full stress tensor T = T^{(x)} + T^{(bc)} and the anti-commutator of Q^{B} with b.)

Problem 2

Henceforth, set D = 26, and consider correlation functions of vertex operators V_{i}, including ghosts :

⟨V_{1} ⋯ V_{N}⟩ = ∫ Dx ∫ D(bb̄)D(cc̄)V_{1} ⋯ V_{N} ∏_{j=1}^{ν} |(μ_{j}, b)|^{2}e^{−S}

For simplicity, we restrict to surfaces Σ of genus h > 1; μ_{j} is a basis of Beltrami differentials, associated with moduli coordinates m_{j}, and ν = 3h − 3.

Assume that V_{i} are physical vertex operators, so that V_{i} commutes with the BRST operator Q^{B}. Now assume that one vertex operator, say V_{N}, is BRST exact, i.e.

V_{N} = ∫_{Σ} d^{2}w ∮_{C} \frac{dz}{2πi}j^{B}(z)W_{N}(w)

for some operator W_{N}, which is not necessarily a physical vertex operator. The curve C is a simple contour around the point w. By deforming the contour C, show that ⟨V_{1} ⋯ V_{N}⟩ is a total derivative in the moduli m_{j}.