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Extra info for A solution manual for Polchinski's String Theory

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We will denote the product of these two traces T (w1 , . . , wn ), and of course it will also depend implicitly on the Chan-Paton factors themselves λai . 1), we can write the n-tachyon amplitude as SC2 (k1 , a1 ; . . ; kn , an ) = 26 26 ki )gon i(2π) δ ( i ∞ 0 dt (8π 2 α′ t)−13 η(it)−24 2t × T (w1 , . . 5 n dwi ∂C2 i=1 −3 η(it) ϑ1 i

As we will see, these conditions will be necessary to allow us to regularize the infinite products that will arise in carrying out the functional integrals in (1), and then to renormalize them by introducing a counter-term action, in a way that respects the symmetries of the action (2). For f and g scalars, the inner product satisfying these two conditions is 1 (f, g)e ≡ dτ ef g. 0 (7) 36 5 CHAPTER 5 We can express the matter action (2) using this inner product: lm2 1 . Sm [X, e] = (e−1 ∂X µ , e−1 ∂Xµ )e + 2 2 (8) We now wish to express the path integral (1) in a slightly less formal way by choosing a fiducial einbein el for each point l in the moduli space, and replacing the integral over einbeins by an integral over the moduli space times a Faddeev-Popov determinant ∆FP [el ].

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.

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


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