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F + 1 2 S ... 2 i f + ··· , (72) where f is the path integral calculated using only the free action (70). 13b). It is the Weyl variation of the third term, gives rise to the term in βµν quadratic in H, that we are interested in, and in particular the part proportional to ¯ ν : . . f, d2z : ∂X µ ∂X (73) 31 3 CHAPTER 3 G . This third term is whose coefficient gives the H 2 term in βµν 1 2 S ... 2 i × f = 1 Hωµν Hω′ µ′ ν ′ 2(6πα′ )2 (74) ′ ′ ′ ¯ ν (¯ d2zd2z ′ : X ω (z, z¯)∂X µ (z)∂X z ) :: X ω (z ′ , z¯′ )∂ ′ X µ (z ′ )∂¯′ X ν (¯ z′ ) : .

38) that in this limit ϑ00 (ν, τ ) → 1, (34) ϑ10 (ν, τ ) → 1, (35) ϑ01 (ν, τ ) → 0, (36) ϑ11 (ν, τ ) → 0. (37) Note that all of these limits are independent of ν. 40a), we have ϑ00 (ν, τ ) = (−iτ )−1/2 e−πiν = (−iτ )−1/2 2 /τ ∞ ϑ00 (ν/τ, −1/τ ) 2 /τ e−πi(ν−n) . (38) n=−∞ When we take τ to 0 along the imaginary axis, each term in the series will go either to 0 (if Re(ν − n)2 > 0) or to infinity (if Re(ν − n)2 ≤ 0). Since different terms in the series cannot cancel for arbitrary τ , the theta function can go to 0 only if every term in the series does so: ∀n ∈ Z, Re(ν − n)2 > 0; (39) 57 7 CHAPTER 7 otherwise it will diverge.

The Laplacian in momentum space is − |k|2 = −|na ka + nb kb |2 = − |nb − na τ |2 ≡ −ωn2 a nb . 7) 2πα′ G′ (w, w′ ) = Xn n (w)∗ Xna nb (w′ ). 3) has the correct Fourier coefficients, that is, T2 T2 d2w X00 (w)G′ (w, w′ ) = 0, d2w Xna nb (w)G′ (w, w′ ) = (20) 2πα′ Xn n (w′ ), ωn2 a nb a b (na , nb ) = (0, 0). 3), which is adjusted (as a function of τ ) to satisfy equation (20). To prove (21), we first divide both sides by Xna nb (w′ ), and use the fact that G′ depends only on the difference w − w′ to shift the variable of integration: 2πα′ d2w ei(na ka +nb kb )·w G′ (w, 0) = 2 .

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