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Bosonic Bc Ghost Central Charge

Physical scope

One chiral bc system with the exact weights, vacuum, stress tensor, and normal ordering in def-st-bosonic-bc-ghost-system.

Statement

For the fermionic bc system with weights (2,-1), OPE b(z)c(w)~1/(z-w), SL(2)-invariant vacuum, and stress tensor Tgh=−2:b∂c:−:∂b c:, the TT OPE has central charge cgh=−26 on the algebraic finite-particle domain.

Statement references

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§5.2–5.3.2 (PDF file pp.121–126; printed pp.114–119); printed pp.114–119); printed pp.114–119); printed pp.114–119); printed pp.114–119); printed pp.114–119); printed pp.113–120); bc stress tensor and displayed TT double-contraction calculation.

Facts & Assumptions

[F1]

Bosonic Bc Ghost System fixes odd modes with {bm,cn}=δm+n,0, {bm,bn}={cm,cn}=0, normal ordering relative to the SL(2)-invariant vacuum, and the adopted stress tensor Tgh=−2:b∂c:−:∂b c:.

[A1]

The OPE calculation is formal radial ordering on the algebraic finite-particle domain. Fermionic Wick signs are retained; no analytic completion is used.

Given: Use Tgh(z)=−2:b(z)∂c(z):−:∂b(z)c(z): and b(z)c(w)∼(z−w)−1.

Proof

1.1givenF1A1algebra

Put u=z−w. Differentiating the basic contraction gives ∂b(z)c(w)∼−u−2, b(z)∂c(w)∼u−2, and ∂b(z)∂c(w)∼−2u−3. The reversed-order contractions have the signs required by the odd CAR.

2.1step 1.1F1A1algebra

Write A=:∂b c: and B=:b ∂c:, so T=−A−2B. Wick's rule gives the double-contraction coefficients of AA, AB, BA, and BB respectively as −1,−2,−2,−1 times u−4; the two cross terms each carry coefficient (−1)(−2)=2 from T. Thus the total fourth-order coefficient is (−1)+2(−2)+2(−2)+4(−1)=−13. The single contractions, using the same differentiated kernels and Taylor expansion about w, give 2T(w)u−2+∂T(w)u−1; all remaining terms are regular.

3.1step 2.1algebra∎

Therefore T(z)T(w)∼−13(z−w)−4+2T(w)(z−w)−2+∂T(w)(z−w)−1. By the convention T(z)T(w)∼cgh/2(z−w)4+⋯, cgh/2=−13 and cgh=−26.

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