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Divergence Theorem in Complex Coordinates

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The divergence theorem in complex coordinates,

\int_R d^2{z} (\partial_z v^z + \partial_{\bar{z}}v^{\bar{z}}) = i \oint_{\partial R}(v^z d\bar{z} - v^{\bar{z}}dz)

(where the contour integral circles the region R counterclockwise) appears in the context of two dimensional conformal field theory, to derive Noether’s Theorem and the Ward Identity for a conformally invariant scalar field theory (for example), and is useful in general in 2D CFT/string theory. This is equation (2.1.9) of Polchinski’s volume 1, but a proof is not given in the book.

This is straightforward to prove by converting both sides separately to Cartesian coordinates (\sigma^1, \sigma^2), through

z = \sigma^1 + i \sigma^2
\bar{z} = \sigma^1 - i \sigma^2

\partial_z = \frac{1}{2}(\partial_1 - i \partial_2)

\partial_{\bar{z}} = \frac{1}{2}(\partial_1 + i \partial_2)

d^2 z = 2 d\sigma^1 d\sigma^2 = 2 d^2 \sigma

and using the Green’s theorem in the plane

\oint_{\partial R}(L dx + M dy) = \int \int_{R} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dx dy

with the identifications

x \rightarrow \sigma^1, y \rightarrow \sigma^2
L \rightarrow -v^2, M \rightarrow v^1

There is perhaps a faster and more elegant way of doing this directly in the complex plane, but this particular line of reasoning makes contact with the underlying Green’s theorem in the plane, which is more familiar from real analysis.