By Headrick M

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**Example text**

The major difference comes from the fact that the Green’s function is doubled. 3): G′C2 (w, w′ ) = G′T 2 (w, w′ ) + G′T 2 (w, −w¯′ ). (49) However, since the boundary of C2 is given by those points that are invariant under the involution w → −w, ¯ the two terms on the RHS above are equal if either w or w′ is on the boundary. The renormalized self-contractions are also doubled, so we have n ⋆ ⋆ i=1 eiki ·X(wi ,w¯i ) ⋆⋆ = C2 iCCX2 (t)(2π)26 δ26 ( −3 ki ) i η(it) i

31) is d δ ( ki ) i

For D even but less than 26, the gamma function is infinite; even if one employs a “minimal subtraction” scheme to remove the infinity, the remainder has a logarithmic branch cut in k2 . For even D ≥ 26 there is indeed a pole, but this pole is simple (as one expects for a particle propagator) only for D = 26. 13 We follow the same steps in calculating the vacuum Klein bottle amplitude as Polchinski does in calculating the vacuum torus amplitude, starting with finding the scalar partition function: ZX (t) = 1 X,K2 = Tr(Ωe−2πtH ) ˜ = q −13/6 Tr(Ωq L0 +L0 ).

### A solution manual for Polchinski's String Theory by Headrick M

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