# Zero modes in bosonic string theory

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 $$z^{kp+1}$$?

• You've cited some numbered equations. The answer to your question may hinge on the surrounding context. It would help if you edit into your question either links to these texts or, failing that, full names of them.
– J.G.
Commented Aug 6, 2021 at 17:46
• Thanks for the advice Commented Aug 6, 2021 at 18:09

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.