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Is the Polyakov action invariant under general coordinate transformation on the target space?

  1. Oct 1, 2014 #1
    The Polyakov action,

    [tex]
    S=\frac{1}{4\pi\alpha^\prime}\int d^2\sigma\sqrt{-h}h^{\alpha\beta}G_{ij}(X)\partial_\alpha X^i\partial_\beta X^j
    [/tex]

    has the local symmetries, diffeomorphism on world sheet and the Weyl invariance.
    But is diffeomorphism on the target space also a symmetry?
    The target space metric transforms

    [tex]
    G_{ij}(X)\to G_{ij}^\prime(X^\prime)=\frac{\partial X^k}{\partial X^i}\frac{\partial X^l}{\partial X^j}G_{kl}(X).
    [/tex]

    But I don't know how to deal with the part, [itex]\partial_\alpha X^i\partial_\beta X^j[/itex].
     
  2. jcsd
  3. Oct 2, 2014 #2

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    Apply the chain rule as usual, it should work.
     
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