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Appendix H Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. General Formula for Central Charge

Using the general formula for the central charge of the \(bc\) ghost system

\[ c_\lambda^{bc} = -2(6\lambda^2 - 6\lambda + 1) \]

calculate the central charge for each of the following cases.

(a) Reparametrization ghosts of the bosonic string: \(\lambda = 2\) (b) The \(\beta\gamma\) system for superconformal symmetry: \(\lambda = 3/2\) (c) Free fermion: \(\lambda = 1/2\) (d) \(\lambda = 0\) (trivial example)

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B-2. Ghost Energy-Momentum Tensor and Verification of OPE

The energy-momentum tensor of the \(bc\) ghost system is given by

\[ T_{\text{ghost}}(z) = -2\,b(z)\,\partial c(z) - \partial b(z)\,c(z)\]

The fundamental OPE of \(b\) and \(c\) is $$

b(z)\,c(w) \sim \frac{1}{z - w}$$

(a) Compute the OPE of \(T_{\text{ghost}}(z)\,b(w)\) and show that \(b(w)\) is a primary field with conformal weight \(h_b = 2\), namely

\[ T_{\text{ghost}}(z)\,b(w) \sim \frac{2\,b(w)}{(z-w)^2} + \frac{\partial b(w)}{z-w}\]

(b) Similarly, compute the OPE of \(T_{\text{ghost}}(z)\,c(w)\) and verify that the conformal weight of \(c(w)\) is \(h_c = -1\).

(c) Verify that \(h_b + h_c = 1\), and explain how this relation is consistent with the conformal weight of the ghost number current \(j(z) = -b(z)c(z)\).

Hint

Use Wick's theorem. In \(T_{\text{ghost}}(z)\,b(w)\), use the contraction \(\langle c(z)b(w)\rangle = -1/(z-w)\). The contraction with \(\partial c(z)\) gives \(1/(z-w)^2\).

Medium

M-1. \(T_{\text{ghost}} b\) OPE

\[ T_{\text{ghost}}(z)\, b(w) \sim \frac{\lambda\, b(w)}{(z-w)^2} + \frac{\partial b(w)}{z-w} \]

Determine the coefficients of \(T_{\text{ghost}}\) such that the above OPE holds (with \(\lambda\) kept as a general value). Specifically, assume the form \(T_{\text{ghost}} = \alpha\, :bc': + \beta\, :b'c:\), and express \(\alpha, \beta\) in terms of \(\lambda\) from the condition that this OPE takes the above form.

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Advanced

A-1. Derivation of the Superstring Critical Dimension \(D=10\)

Derive the critical dimension \(D = 10\) of the superstring from the condition that the total central charge of the matter fields and ghosts vanishes. The matter fields consist of \(D\) free bosons (\(c = D\)) and \(D\) worldsheet fermions (each \(c = 1/2\), totaling \(c = D/2\)), and the ghost fields consist of the \(bc\) system (\(c = -26\)) and the \(\beta\gamma\) system (\(c = +11\)).

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A-2. Reduction of Matter Field Central Charge

The result of Problem H.3 can be interpreted as follows: "Since the superconformal ghosts \(\beta\gamma\) have central charge \(+11\), fewer matter fields are needed by that amount." Verify that the total central charge of the matter fields is reduced from the bosonic string (\(c_{\text{matter}} = 26\)) to the superstring (\(c_{\text{matter}} = 15\)).

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