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Appendix H Solutions

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Basic

Medium

Advanced

Basic

B-1. General Formula for Central Charge

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Problem:

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)

Solution:

Substitute into the general formula \(c_\lambda^{bc} = -2(6\lambda^2 - 6\lambda + 1)\).

(a) \(\lambda = 2\): $$ c = -2(6 \cdot 4 - 6 \cdot 2 + 1) = -2(24 - 12 + 1) = -2 \cdot 13 = \boxed{-26} $$

(b) \(\lambda = 3/2\): $$ c = -2\left(6 \cdot \frac{9}{4} - 6 \cdot \frac{3}{2} + 1\right) = -2\left(\frac{27}{2} - 9 + 1\right) = -2 \cdot \frac{11}{2} = \boxed{-11} $$

Note: Since the \(\beta\gamma\) system consists of bosonic anticommuting fields (reversed statistics), the overall sign of the formula is flipped, and the effective central charge becomes \(\boxed{+11}\) (see H.7 "The General \(\lambda\) Case — The Formula \(c = -2(6\lambda^2 - 6\lambda + 1)\)" in the text).

(c) \(\lambda = 1/2\): $$ c = -2\left(6 \cdot \frac{1}{4} - 6 \cdot \frac{1}{2} + 1\right) = -2\left(\frac{3}{2} - 3 + 1\right) = -2 \cdot \left(-\frac{1}{2}\right) = \boxed{+1} $$

To be consistent with the central charge \(c = 1/2\) for a single free fermion, this formula applies to a complex fermion (2 real components).

(d) \(\lambda = 0\): $$ c = -2(0 - 0 + 1) = \boxed{-2} $$

Medium

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

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Problem:

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

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.

Solution:

Assume \(T_{\text{ghost}}(z) = \alpha\, :b(z)\partial c(z): + \beta\, :\partial b(z)\, c(z):\) and compute the \(T_{\text{ghost}}(z)\, b(w)\) OPE using Wick's theorem.

From the fundamental OPE \(b(z)c(w) \sim 1/(z-w)\):

\[ \langle \partial c(z)\, b(w)\rangle = \partial_z\frac{1}{z-w} = -\frac{1}{(z-w)^2} \]
\[ \langle c(z)\, b(w)\rangle = \frac{1}{z-w} \]

First term \(\alpha :b\partial c:(z)\, b(w)\):

Contract \(\partial c(z)\) with \(b(w)\), leaving \(b(z)\) uncontracted. Taking into account the sign from anticommuting fields:

\[ \alpha \cdot (-1) \cdot \left(-\frac{1}{(z-w)^2}\right) \cdot b(z) = \frac{\alpha\, b(z)}{(z-w)^2} \]

Expanding \(b(z) = b(w) + (z-w)\partial b(w) + \cdots\):

\[ = \frac{\alpha\, b(w)}{(z-w)^2} + \frac{\alpha\, \partial b(w)}{z-w} + \cdots \]

Second term \(\beta :\partial b\, c:(z)\, b(w)\):

Contract \(c(z)\) with \(b(w)\), leaving \(\partial b(z)\) uncontracted:

\[ \beta \cdot (-1) \cdot \frac{1}{z-w} \cdot \partial b(z) = -\frac{\beta\, \partial b(z)}{z-w} \]

Expanding \(\partial b(z) = \partial b(w) + (z-w)\partial^2 b(w) + \cdots\):

\[ = -\frac{\beta\, \partial b(w)}{z-w} + \cdots \]

Total:

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

Comparing with the primary field form \(\frac{\lambda\, b(w)}{(z-w)^2} + \frac{\partial b(w)}{z-w}\):

  • \(\alpha = \lambda\)
  • \(\alpha - \beta = 1 \Rightarrow \beta = \lambda - 1 = -(1-\lambda)\)

Therefore:

\[ \boxed{T_{\text{ghost}} = \lambda\, :b\partial c: - (1-\lambda)\, :\partial b\, c:} \]

Note: This differs in overall sign from the notation in the main text H.5 "The Energy-Momentum Tensor of the \(bc\) System", \(T_{\text{ghost}} = -\lambda :bc': + (1-\lambda):b'c:\), but this is due to differences in the definition of conjugate fields (\(b \leftrightarrow \bar{b}\) or the ordering convention of \(:bc:\)). The physical conclusion (central charge \(-26\)) remains the same.

Advanced

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

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Problem:

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 (\(c = +11\)).

Solution:

Total central charge of the superstring:

\[ c_{\text{total}} = c_{\text{matter}} + c_{bc} + c_{\beta\gamma} \]

Writing out each term:

  • Matter fields: \(D\) bosons (each \(c = 1\)) and \(D\) fermions (each \(c = 1/2\))
\[ c_{\text{matter}} = D \cdot 1 + D \cdot \frac{1}{2} = \frac{3D}{2} \]
  • Reparametrization ghosts: \(c_{bc} = -26\)
  • Supersymmetry ghosts: \(c_{\beta\gamma} = +11\)

Condition \(c_{\text{total}} = 0\):

\[ \frac{3D}{2} - 26 + 11 = \frac{3D}{2} - 15 = 0 \]
\[ \frac{3D}{2} = 15 \;\Longrightarrow\; \boxed{D = 10} \]

A-2. Reduction of Matter Field Central Charge

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Problem:

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 (\(c_{\text{matter}} = 15\)).

Solution:

  • Bosonic string: The matter fields consist only of \(D = 26\) bosons $$ c_{\text{matter}}^{\text{bosonic}} = 26 \cdot 1 = 26 $$ The ghosts contribute only \(c_{bc} = -26\). \(c_{\text{total}} = 26 - 26 = 0\) ✓

  • Superstring: From Problem H.3, \(D = 10\) $$ c_{\text{matter}}^{\text{super}} = \frac{3 \cdot 10}{2} = 15 $$ The ghosts contribute \(c_{bc} + c_{\beta\gamma} = -26 + 11 = -15\). \(c_{\text{total}} = 15 - 15 = 0\) ✓

Interpretation: The central charge \(+11\) of the \(\beta\gamma\) system "reduces" the total ghost contribution to \(-15\). Correspondingly, the matter fields can also be reduced from \(26 \to 15\), so the required spacetime dimension decreases from \(D = 26 \to 10\). This reveals that the introduction of supersymmetry and the reduction of spacetime dimensions are inseparably linked.