- #1

tom.stoer

Science Advisor

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## Main Question or Discussion Point

In another thread Ben indicated that string theory formulated as non-linear sigma model using world-sheet action is - in some sense - background independent. To discuss this I start with a generalization of the Polyakov action

##S_G[X] = \frac{1}{4\pi\alpha}\int d^2\sigma \, \sqrt{g} \, g^{ab} \, \partial_a X^\mu \, \partial_b X^\nu \, G_{\mu\nu}(X)##

Here g is the world sheet metric, X are scalar fields, and G is usually identified with the target-space metric.

So we do not have one single action S, but a class of actions S

This is what is usually called background-dependency.

What is wrong in my reasoning?

##S_G[X] = \frac{1}{4\pi\alpha}\int d^2\sigma \, \sqrt{g} \, g^{ab} \, \partial_a X^\mu \, \partial_b X^\nu \, G_{\mu\nu}(X)##

Here g is the world sheet metric, X are scalar fields, and G is usually identified with the target-space metric.

So we do not have one single action S, but a class of actions S

_{G}, labelled by G. My conclusion is that G is a non-dynamical background; the string X does not back-react on G; the dynamics of X does not connect different G-sectors; G is introduced by hand, it is not subject to the dynamics of the theory.This is what is usually called background-dependency.

What is wrong in my reasoning?

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