I am writing my dissertation on heterotic string theory. Do you think this Ed Witten guy can help me?
SUCKMYMUSCLE
I don't know that I believe you, but if you can answer this basic question from an introduction to string theory course (first graduate course in string theory, so it should be pretty easy), I'll answer your question.
Consider the Polyakov action of a string moving in D-dimensional Minkowski space background (with metric η
μν taking the typical 4x4 matrix form):
S= -T/2 ∫dσ
2√h(σ)h
αβ(σ)∂
αX
μ∂
βX
μσ
α = {στ}; X
μ = X
μ(σ
α); σ∈[o,π]; τ ∈(-∞,∞)
∑ = the world sheet; h
αβ(σ
γ) = metric on the world sheet
h(σ
γ) = -det(h
αβ); α = 1,2; μ = 0,...,D-1
(I used a short hand notation for partial derivatives, but didn't use the pre-subscript comma that is often used in tensor analysis in case your profs don't regularly use it)
Obviously, I used summation suppression as usual.
Now:
1) Show the LOCAL symmetry transformations:
(HINT: one will be reparametrization, the other will be weyl transformations)
2) Compute the energy momentum tensor T
αβ defined by:
h
αβ ⟶ h
αβ + δh
αβ ⇒ S ⟶ S + δS
δS = -T/2 ∫dσ
2√h(σ)h
αβ(σ)T
αβWhich condition on T
αβ does invariance under Weyl transformation imply?
(see...we're not even considering a space-time where the curvature tensor doesn't vanish, so you only have to deal with a flat geometry for space-time)
I don't have Latex on this computer, so I had to do it with the damned "special characters" on the mac fonts. You can reply with latex or post a scan of your scratch work if it's easier.
Good luck on your dissertation! I don't know why you chose Heterotic string theory, but I guess you have to go with what you're interested in.