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In string theory , What exactly are right movers and left movers ?Are they waves propagating along the string to the right and to the left respectively ? Can some-one please show me how to derive in details the general solution of the equation of motion for the string?

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On what background? I presume you mean the flat background, otherwise the question has no answer. The right movers are the fields that move to the right and the left movers are the fields that move to the left, just as you would think. – Ron Maimon Jun 18 '12 at 5:57
Thanks , Ron I mean The flat background . – Nabil Jun 18 '12 at 10:54
In any background they are called right and left movers, Ron. – Ernesto Ulloa Jun 19 '12 at 18:18

I hope the instructors don't mind me posting their materials here but :

The problem sets (which have solutions) walk you through the derivation of the equation of motion for the string (i'm assuming you mean the derivation of the wave equation and then the derivation of the form X must take to solve that wave equation). The left moving wave is just X(t+s) part and the right moving wave is X(t-s) part. (Where of course by t i mean tau and s i mean sigma).

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This is a very good answer. – Ernesto Ulloa Jun 19 '12 at 18:19

The solutions to the classical string equations of motion are given by the embedding functions as functions of the string world sheet parameters (σ,τ). The solutions are given in terms of arbitrary vector functions of σ±τ. The positive sign are called right movers and the negative sign left movers. Being functions of this particular combination of parameters they represent wavelike oscillations propagating in opposite direction in the world sheet parameter space of the string. Left or right is just referring to the relative sign of the space-like and time-like word sheet parameters. You can think about them like the opposite traveling waves in an ordinary standing wave.

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