I've hit a stumbling block where I'm just not seeing how to get from line to line in the following calculation from David Tong's strings notes. Can someone spell out how line 1 becomes line 2 in the ∂X:eikX: calculation and how that result is actually being employed in the calculation for T:eikX:?
Answer
Contraction with exponential is actually more easy than polynomials. This is so because the exponential is the eigenfunctional of δδXμ. The cross contractions of (...)A and (...)B are given by
(...)A(...)B=exp(−α′2ημν∫d2z1∫d2z2ln|z12|2δδXμA(z1,ˉz1)δδXνB(z2,ˉz2))(...)AB
and we have the eigenfunctional equation
δδXμ(z1,ˉz1)exp(ik.X(z,ˉz))=ikμδ2(z−z1;ˉz−ˉz1)exp(ik.X(z,ˉz))
then if (...)B=exp(ik.X(z2,ˉz2)) we have
(...)A(...)B=exp(−iα′kμ2∫d2z1ln|z2−z1|2δδXμA(z1,ˉz1))(...)AB
just contractions on (...)A with:
−iα′kμ2ln|z2−z1|2
for each Xμ in (...)A. So, if (...)A=∂Xμ∂Xμ, we have one 0-contraction (0) plus two identical 1-contraction (1) plus one 2-contractions (2), and only this:
(0) :∂Xμ∂Xμ(z1)exp(ik.X(z2,ˉz2)):
(1) 2:∂1(−iα′kμ2ln|z2−z1|2)∂Xμ(z1)exp(ik.X(z2,ˉz2)):
(2) :∂1(−iα′kμ2ln|z2−z1|2)∂1(−iα′kμ2ln|z2−z1|2)exp(ik.X(z2,ˉz2)):
Adding all together you get the answer.
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