Least Action

Nontrivializing triviality..and vice versa.

Posts Tagged ‘String Theory

Errata for Basic Concepts of String Theory by Blumenhagen, Lüst and Theisen

leave a comment »

This is an unofficial errata for the book Basic Concepts of String Theory by Ralph Blumenhagen, Dieter Lüst and Stefan Theisen. I couldn’t find an official errata, but I’ll probably discontinue this at some point when I do run into one.

Chapter 2: The Classical Bosonic String

  • Page 8. The line below equation 2.3. There should be two dots, one each on x^\mu and x^\nu in the definition of \dot{x}^2.

Chapter 8: The Quantized Fermionic String

  • Page 213: Equation 8.56. There is only one charge conjugation matrix in odd dimension d = 2n+1, either C_+ or C_-. To find out which one it is, for odd d, determine d(d-1)/2: if this is even, C_+ exists; if it is odd, C_- exists. So, equation 8.56 is wrong: one should use C_- for odd n, and C_+ for even n.To derive this criterion, compute C \gamma_c C^{-1} and observe that it equals (-1)^{d(d-1)/2} \gamma_c in general, which determines whether C = C_+ or C = C_-. For a quick list of charge conjugation matrices in various dimensions and their symmetry properties, see page 11 of http://www.nikhef.nl/~t45/ftip/AppendixE.pdf.

Chapter 14: String Compactifications

  • Page 510: Equation 14.262. The e^{e} on the right-hand side in the local Lorentz transformation of the vielbein should be e^{b}.

Chapter 18: String Dualities and M-Theory

  • Page 690: The expression for \tilde{F}^{(p+2)} in the paragraph below equation (18.26) has extra indices. It is the contraction of \tilde{F}_{M_{0}\ldots M_{p+1}} with \tilde{F}^{M_{0}\ldots M_{p+1}}.

Written by Vivek

December 27, 2014 at 15:05

Posted in Errata, String Theory

Tagged with

Divergence Theorem in Complex Coordinates

leave a comment »

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.