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String Theory Demysti ed
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Volume elements in integrals can be written using coordinate transformations by including the determinant of the metric. Writing d 2 z = dzd z and using | det g| d 2 z = d d it follows that d 2 z = 2 d d Now consider the action S= 1 d 2 z X X 2 (5.14) (5.13)
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This is, in fact, the Polyakov action [Eq. (5.5)] in a much simpler mathematical form. To see this, we can use Eq. (5.7) together with Eq. (5.13). Notice that X X = 1 1 ( i ) X ( + i ) X 2 2 1 = ( X i X )( X + i X ) 4 1 = ( X X + i X X i X X + X X ) 4 1 = ( X X + X X ) 4
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To move from the third to the fourth line, we used the fact we can raise and lower indices with the Euclidean metric. That is, X = X = X , so i X X = i X X = i X X = i X X and the middle terms cancel. Therefore S= = = 1 1 2 d z X X = 2 2d d X X 2 1 1 2d d 4 ( X X + X X ) 2 1 d d ( X X + X X ) = SP 4
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CHAPTER 5 Conformal Field Theory Part I
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We ve shown that Eq. (5.14) is an equivalent way to write the Polyakov action. But it s much simpler, and it s much simpler to derive the equations of motion using this form. We can do this by varying the action [Eq. (5.14)] with respect to the coordinate X . This is done by letting X X + X . Then S = 1 1 2 2 d z X ( X + X ) = 2 d z X ( X + X ) 2
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1 1 2 2 d z X X + 2 d z X ( X ) = S + S 2
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We can obtain the equations of classical motion by requiring that S = 0. Integrating by parts and discarding the boundary term:
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1 d 2 z X ( X ) 2 1 = d 2 z X ( X ) 2
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We have used the fact that partial derivatives commute. This term must vanish for the action to be invariant. Therefore it must be the case that X ( z, z ) = 0 (5.15)
We ve written X = X ( z, z ) to emphasize that in general the coordinates can be a function of z and z . However, as you might guess from your studies of complex variables there is a special case of interest, that of analytic or holomorphic functions. A function f ( z, z ) is holomorphic if f =0 z That is, f = f ( z ) only. On the other hand, if f =0 z (5.17) (5.16)
and f = f ( z ), then we say that f is antiholomorphic. In string theory, if ( X ) = 0 then X is a holomorphic function which is called left moving. In the other case, where ( X ) = 0 , the function X is antiholomorphic and is called right moving.
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Generators of Conformal Transformations
To study the generators of a conformal transformation, we consider an in nitesimal transformation of the coordinates: x = x + Now consider an in nitesimal conformal transformation. That is if g ( x ) = ( x ) g ( x ) we take ( x ) = 1 f ( x ) where f ( x ) is some small departure from the identity. Then we have g ( x ) = (1 f ( x ))g ( x ) = g ( x ) f ( x )g ( x ). Using x = x + you can show that g = g ( + ) So, recalling that we are working with a conformal transformation about the at space metric, it must be the case that + = f ( x )g We can determine the form of f ( x ) by multiplying both sides of this equation by g . In d spacetime dimensions g g = d and so on the right we obtain g f ( x )g = d f ( x ). On the left side we have g ( + ) = g + g = + = + = 2 Hence f= And we have the relation 2 2 + = = ( ) d d (5.18) 2 d (raise indices with metric) i (relabel repeated indices which are dummy indices)
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