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55) ϑ1 (x|iu) = 2q 1/8 sin(πx) (1 − (1 + 2 cos(2πx))q + . . ) ; ∞ ϑ4 (x|iu) = (−1)k q k 2 /2 k=−∞ (56) e2πikx = 1 − 2 cos(2πx)q 1/2 + . . (57) For η and ϑ1 this power series involves only integer powers of q, whereas for ϑ4 it mixes integer and half-integer powers. Substituting (55)-(57) into (54) yields ∞ 2α′ ki ·kj i

During these transformations (which always begin with (44)), the region (39), in which ϑ00 goes to 0, will repeatedly be shifted by 1/2 and rescaled by τ0 (under (46)). ) It is easy to see that this cumulative sequence of rescalings will telescope into a single rescaling by a factor q, where p/q is the final value of τ0 in reduced form. As for the case when τ0 is irrational, I can only conjecture that the theta functions diverge (almost) everywhere on the ν plane in that limit. 4) for closed string tachyon vertex operators on the torus.

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′ ) : . . f , where we have normal-ordered the interaction vertices. The Weyl variation of this integral will come from the singular part of the OPE when z and z ′ approach each other.