In section 21.6 "Moduli stabilization and the landscape" in Zwiebach's
string theory book, I read the sentence "Deriving the potential $V(R)$ associated with $R$ is a straightforward but technical calculation in general relativity". At this point I did not understand what the calculation was. I vaguely remembered a paper by Witten about instabilities in Kaluza-Klein spacetimes related to instantons. A calculation with instantons is indeed technical, but perhaps straightforward for experts. I found more information in a paper by Denef [1]. The calculation has nothing to do with instantons, but is indeed a straightforward calculation in differential geometry. In the rest of this blog post I set out the calculation in the form of a new exercise for Zwiebach's book.

# Pieter Jan De Smet Calculations

Various topics in mathematics and physics

## Wednesday, March 8, 2017

## Friday, February 3, 2017

### Comment about particle on a circle

The wave function of a particle on a circle is a solution of the Schrödinger equation
\begin{equation}\label{eq:20170129a}
i \frac{\partial \psi}{\partial t} = - \frac{1}{2 m} \frac{\partial^2 \psi}{\partial x^2}
\end{equation}
with $x \in [0 , 2 \pi]$ and $\hbar = 1$.
When \eqref{eq:20170129a} is solved in physics books, it is usually imposed that the wave function should be periodic [1]. I used to be puzzled why one has to impose the periodicity.
After all, I thought, only the probability density function $|\psi|^2$ has physical meaning, so one could as well impose that
\begin{equation}\label{eq:20170129b}
\psi(2 \pi) = e^{ i \alpha} \psi(0) \quad\text{with}\quad \alpha\in\mathbb{R}
\end{equation}

## Thursday, January 19, 2017

### A magnetostatic exercise in 10 dimensions

I calculate the electromagnetic field generated by electrical currents in 10 spacetime dimensions (9 space and 1 time). The set up is
as follows: the current flows down the positive $x_1$-axis, hits the origin and then spreads out
isotropically in the $x_2 x_3 x_4$ subspace, see figure 1 and 2. I wanted to calculate this because in string theory
a similar calculation is needed to obtain the Kalb-Ramond field generated by a string ending on a $D3$-brane [1]

## Wednesday, January 11, 2017

### A calculation in magnetostatics

I wanted to calculate the magnetic field generated by a current which flows down the positive $z$-axis,
hits the origin and then spreads out radially over the $xy$ plane, see figure 1.

## Friday, January 6, 2017

### Lorentz invariance of string theory in the light-cone gauge

On page 261 in his book [1] Zwiebach writes ''There is much at stake in this calculation. It is in fact, one of the most important calculations in string theory. [...]
The calculation is long and uses many of our previously derived results''. Then Zwiebach states the result
\begin{align}
\left[ M^{- I}, M^{- J}\right] = &- \frac{1}{\alpha'\ { p^+ }^2} \sum_{m=1}^{\infty}
\left(\alpha^I_{-m} \alpha^J_m - \alpha^J_{-m} \alpha^I_m \right)\nonumber\\
&\times \left\{ m \left[1 - \dfrac{1}{24} (D-2) \right] + \dfrac{1}{m} \left[ \dfrac{1}{24} (D-2) + a \right] \right\}\label{eq:20170103}
\end{align}
This is the commutator of two Lorentz transformations in the light-cone gauge.
The commutator should be zero for string theory to be Lorentz invariant.
The calculation of the commutator is indeed very tedious and after scribling too many pages I gave up.
I googled for a quick way to obtain \eqref{eq:20170103}. Here are the more interesting
results that I found.

## Monday, December 5, 2016

### On Killing spinors in general dimensions

The following property is true in four spacetime dimensions [1] [2]

If the electromagnetic field $F_{ab}$ satisfies Maxwell's equations
\begin{equation*}
\nabla_{[a}F_{bc]}= 0 \quad\text{and}\quad \nabla^a F_{ab} =0
\end{equation*}
and there is a spinor $\psi$ such that
\begin{equation}\label{eq:20161115b}
(\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi = 0
\end{equation}
and $i \bar \psi \gamma^{\mu} \psi$ is time-like

then the Einstein equations are satisfied as well:
\begin{equation*}
R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu}
\end{equation*}

*A*from now on) can for example be used to obtain the metric and electromagnetic field of the Israel-Wilson-Perjés (IWP) black holes. Because I wanted to generalize the IWP black holes to higher dimensions, I wanted to find the generalization of property*A*in higher dimensions.## Monday, November 21, 2016

### Extremal black holes via Killing spinors

The usual way to obtain the metric and electromagnetic field of a charged black hole in general relativity is to make a spherically symmetric ansatz,
insert this ansatz into the Einstein-Maxwell equations and then solve the resulting set of non-linear ordinary differential equations. In this post I explain
an alternative method that uses Killing spinors. This method can be used for extremal black holes. These are black holes with charge
equal to the mass.

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