Least Action

Nontrivializing triviality..and vice versa.

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

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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 453: below equation 14.70. There’s an extra q after \Lambda^q.
  • Page 454: Equation 14.76. There’s an extra $\wedge$ after \bar{\partial}_j \omega_{i_1,\ldots i_p \bar{j}_1\ldots \bar{j}_p}.
  • 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}}.
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Written by Vivek

December 27, 2014 at 15:05

Posted in Errata, String Theory

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Los Alamos Science

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Continuing the expository theme of my last post, I want to bring to your attention a collection of beautiful, crisp and entertaining articles by Richard Slansky in a 1984 publication of Los Alamos Science. They are available at

http://www.fas.org/sgp/othergov/doe/lanl/pubs/number11.htm

and are (in my opinion) must readings for students of theoretical physics, particularly those specializing in high energy theory, string theory, etc. In case you didn’t know, Slansky is also the author of a definitive review on group theory, which is a standard resource for particle physicists. It is worth having a printout of the review at your desk (and also an online copy in your tablet, smartphone, etc.) if you are interested in doing anything serious with group theory.

I should also take this opportunity to bring a list of Physics articles that have appeared over the years in Los Alamos Science. The comprehensive list of these articles with hyperlinks to online versions is at

http://la-science.lanl.gov/cat_physics.shtml

Written by Vivek

December 9, 2014 at 18:49

Review Articles – Particle Physics, String Theory, Supersymmetry and Supergravity

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[to be updated]

A list of useful reviews on various aspects of string theory, branes, etc. is at http://www.nuclecu.unam.mx/~alberto/physics/stringrev.html. There are links to TASI lectures as well as review articles by prominent string theorists.

Another useful list of string theory papers and reviews is http://web.mit.edu/redingtn/www/netadv/Xstring.html.

Additionally, a list of books and useful review articles for supersymmetry and supergravity is at http://www.stringwiki.org/wiki/Supersymmetry_and_Supergravity.

A useful list of references for Collider Physics is at http://tigger.uic.edu/~keung/me/class/collider/web-docs.html.

 

Written by Vivek

December 3, 2014 at 15:07

An inexpensive way to connect an HDMI screen to a Macbook

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I am not a huge Mac fan, but I do happen to have access to a Macbook Air, and since I find it hard to carry two laptops, a tablet and at least 5-6 books daily, I wanted to figure out a cheap way to get my Macbook Air to work with my HP Pavilion 27xi HDMI monitor. After some research, I went ahead and purchased an AmazonBasics Mini DisplayPort (Thunderbolt) to HDMI Adapter cable for about $10. There are many more cables available on Amazon, and most of them are much more expensive. But they’re all manufactured in roughly the same areas of our planet, so I wasn’t convinced about buying a costlier adapter especially for something I wouldn’t be heavily relying on. But this adapter is very nifty and works well.

The only other problem was to get the resolutions to match. For this, click on the leftmost button on the touch-activated panel of the HP Pavilion 27xi monitor. A menu opens up. Use the third soft key from left (with a “-” symbol) to navigate down to Image Control. When Image Control is highlighted, tap on the leftmost soft key again. Now, select Custom Scaling and press the leftmost soft key. Then, select Overscan and select “Off”. This should eliminate the problem of screen contents on the HDMI monitor being chopped. In my case at least, the default setting (Auto) does not work well with the Macbook Air I am working on.

Written by Vivek

November 6, 2014 at 21:53

Posted in Electronics

Tagged with , , ,

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.

If I were a Springer-Verlag Graduate Text in Mathematics….

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If I were a Springer-Verlag Graduate Text in Mathematics, I would be William S. Massey’s A Basic Course in Algebraic Topology.

I am intended to serve as a textbook for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind the various concepts is emphasized.

Which Springer GTM would you be? The Springer GTM Test

Written by Vivek

October 10, 2014 at 17:52

Posted in Uncategorized

Calligraphic symbols in LaTeX

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If you’re unhappy with the way the standard amsmath/amssymb packages display a caligraphic L, and you want a more curly/loopy L for your Lagrangian density, then here’s a way out.

1. Install the package texlive-calrsfs. I had to run “yum install texlive-calrsfs” as superuser to accomplish this on Fedora 20.

2. Declare the following headers above your document

\documentclass{article}
\usepackage{calrsfs}
\DeclareMathAlphabet{\pazocal}{OMS}{zplm}{m}{n}
\newcommand{\La}{\mathcal{L}}
\newcommand{\Lb}{\pazocal{L}}

 

3. The more loopy version of L is now mapped to \La, whereas the conventional version is \Lb.

Hope this helps!

Adapted from: http://tex.stackexchange.com/questions/69085/two-different-calligraphic-font-styles-in-math-mode

Written by Vivek

July 18, 2014 at 10:35

Posted in LaTeX, Linux