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A Weyl transformation or Weyl rescaling is a conformal transformation of the worldsheet metric (see Chap. 5) of the form: h e ( , ) h (3.12)
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Since h h = it follows from Eq. (3.12) that h e ( , ) h . Now we recall two facts about determinants, where we let A, B be n n matrices: det( AB ) = det A det B det( A ) = det( I n A) = n det A
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CHAPTER 3 Symmetries and Worldsheet Currents
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In our case, we are working in two dimensions and so: det(e h ) = e2 det h This means that we have det h h e2 det h e h = det h h
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Therefore the Polyakov action is invariant under a Weyl transformation. Since Eq. (3.12) is dependent on the space-time coordinates ( , ) of the worldsheet, it is a local symmetry.
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Transforming to a Flat Worldsheet Metric
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Gauge freedom can be used to simplify the worldsheet metric. What this means is that we can use the symmetries of the action (i.e., utilize the transformations that leave the action and hence the physics unchanged) to write the worldsheet metric in a more convenient form, that is sometimes called the ducial metric h ( ). Here we consider a worldsheet with a vanishing Euler characteristic (a cylinder is relevant to our interest). The worldsheet has only two coordinates ( , ), and this means that h is a 2 2 matrix: h00 h = h10 h01 h11 (3.13)
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This is going to make things particularly easy for us. We immediately nd that there are only three independent components of the worldsheet metric. This is because, in general, the metric tensor g is symmetric, so that g = g The fact that the metric is just 2 2 means that the symmetry requirement xes the off-diagonal components: h01 = h10
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So, this means that we only have to specify three components of h . The choice can be simpli ed by using two of the local symmetries of the Polyakov action. From the previous section we recall that these are Reparameterization invariance Weyl transformations The rst case, reparameterization invariance, that is, a coordinate transformation, can be used to take the metric into a form that is proportional to the two-dimensional at Minkowski metric as follows: 1 0 h e ( , ) = e ( , ) 0 1 (3.14)
This form happens to be particularly useful, because now we can apply a Weyl transformation to get rid of the exponential factor. The end result is that it is possible to use the local symmetries of the Polyakov action to take the worldsheet metric into the at Minkowski metric: 1 0 h = 0 1 (3.15)
This is going to really simplify the situation at hand. First let s write down the Polyakov action [Eq. (2.27)] once again: SP = T d 2 h h X X 2
The rst thing to notice about Eq. (3.15) is that the determinant is just h = det h = det and so h = +1 Now since 1 0 h = 0 1 1 0 = 1 0 1
CHAPTER 3 Symmetries and Worldsheet Currents
we have h X X = h X X + h X X = X X + X X = X X + X X
Do you remember your quantum eld theory This equation should look familiar you might recognize the lagrangian (density) for a set of free massless scalar elds. Putting this into the Polyakov action, we see that using its local symmetries has taken it into the very simple form SP = T d 2 h h X X 2
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