Least Action

Nontrivializing triviality..and vice versa.

Archive for the ‘String Theory’ Category

Fun with Homology

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There is a theorem (currently being attributed to Wikipedia, but I’m sure I can do better given more time) which states that

All closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2n-gons) can be glued to make different manifolds.

Conversely, a closed surface with n non-zero classes can be cut into a 2n-gon.

Two interesting cases of this are:

  1. Gluing opposite sides of a hexagon produces a torus T^2.
  2. Gluing opposite sides of an octagon produces a surface with two holes, topologically equivalent to a torus with two holes.

I had trouble visualizing this on a piece of paper, so I found two videos which are fascinating and instructive, respectively.

The two-torus from a hexagon

The genus-2 Riemann surface from an octagon

I would like to figure out how one can make such animations, and generalizations of these, using Mathematica or Sagemath.

There are a bunch of other very cool examples on the Youtube channels of these users. Kudos to them for making such instructive videos!

PS – I see that $\LaTeX$ on WordPress has become (or is still?) very sloppy! 😦

Written by Vivek

October 20, 2016 at 21:57

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

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This is an unofficial errata for the book Basic Concepts of String Theory by Ralph Blumenhagen, Dieter LĂźst 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

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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

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.