Zero modes in bosonic string theory

Question

I recently began studying string interactions, my main references have been David Tong and Kevin Wray lecture notes. While going trough chapter 6 of the former one, I came across with the zero modes concept which would be responsible of the introduction of the Dirac Delta function ensuring momentum conservation (eq 6.9). In order to understand more, what's behind, I found an alternate formulation for vertex operators in GSW, that factors the zero modes from the others, in eq 7.1.6-7.1.7 from GSW.

First of all, I am not seeing how is RHS of 7.1.6 obtained, it would seem to me that (which is possibly wrong)

$$ Z_0=e^{ik\cdot x+k\cdot p \ln z}=e^{ik\cdot x}e^{\ln z^{k\cdot p }} $$

Could somebody help me understanding the origin of that expression and also how is it possible to switch $z^{kp-1}$ by $z^{kp+1}$?

Answer 1

We can prove$$e^{ikx+kp\ln z}=e^{ikx}e^{(kp+1)\ln z}=e^{(kp-1)\ln z}e^{ikx}$$with the BCH formula$$e^Ae^B=e^{A+B+[A,\,B]/2+[A,\,[A,\,B]]/12+\cdots}.$$For the first equation, take$$A=ikx=ik_ax^a,\,B=(kp+1)\ln z=(k^bp_b+1)\ln z$$ so$$k_ak^a=2\implies[A,\,B]=ik_ak^b(i\delta^a_b)\ln z=-2\ln z\implies[A,\,[A,\,B]]=0.$$You should see that finishes proving the first result. I'll leave the other one as an exercise.