## Harmonic oscillator partition function using path integrals

Two pages of notes explaining how to derive the partition function of the harmonic oscillator using the path integral. Care is needed in normalizing the measure, and relatedly, to deal with the Matsubara zero mode of the free particle.

Harmonic oscillator path integral

## Schwarzschild free energy

Two pages of notes on computing the Schwarzschild free energy in 4 dimensions, following the 1977 paper of Gibbons and Hawking.

Schwarzschild Free Energy

## Videos of some talks I’ve given

2015 Kavli Institute, Santa Barbara, California

http://online.kitp.ucsb.edu/online/entangled15/mahajan/

http://perimeterinstitute.ca/videos/transport-chern-simons-matter-theories

2018 Kavli Institute, Santa Barbara, California

http://online.kitp.ucsb.edu/online/chord18/mahajan/

## Degree of the identity map on S^2

The winding number of the identity map from $S^2$ to $S^2$ should be one, and we can check this using the following Mathematica code.

Denoting the triple product of 3-vectors by square brackets, the general expression is

$\int d\theta d\phi\, \left[\, \vec{f}, \dfrac{\partial \vec{f}}{\partial \theta}, \dfrac{\partial \vec{f}}{\partial \phi} \,\right] \in 4\pi \mathbb{Z}$.

## Comparing Sch. black holes with different CC

We will restrict to 3+1 dimensions, and compare the emblackening factors.

## Internal spin of a dyon

The normalization of $U(1)$ gauge fields means that the magnetic charge of the fundamental magnetic monopole is $2\pi$. (It’s magnetic field is $\widehat{r}/(2r^2)$, so the flux through the sphere is $2\pi$.)

Dirac quantization then tells us that the electric charge is an integer.

Now let us look at the time derivative of the angular momentum of an electric charge $q$ moving in the field of a magnetic monopole with field $B \widehat{r}/r^2$.

$\dfrac{d}{dt} m \vec{r} \times \vec{v} = \vec{r} \times \vec{F} = qB \vec{r} \times \left(\vec{v}\times \dfrac{\widehat{r}}{r^2}\right) = qB \left( \dfrac{\vec{v}}{r} - \dfrac{\widehat{r}}{r^2} \vec{r}\cdot\vec{v} \right) = qB \dfrac{d}{dt} \widehat{r}$.

Thus, even though angular momentum is not conserved, we can define a new quantity $m \vec{r}\times \vec{v} - qB \widehat{r}$, which is conserved.

Now if the electric charge and the monopole were to form a bound state, the above calculation strongly suggests that we assign an internal spin to the dyon of magnitude $qB$. For the fundamental values $q=1$ and $B=1/2$, we get spin $1/2$.

## Fibonacci and Ising anyons

### Fibonacci

There are two types of anyons: 1 and $\tau$. The nontrivial fusion rule is

$\tau \star \tau = 1 \oplus \tau$.

This is as if we projected down to the $\mathbb{Z}_2$-even sector of the 3D Ising model, and $\tau$ is thought of as the $\epsilon$ operator.

These anyons are related to the $p=3$ Read-Rezayi state and are thought to have something to do with the $\nu=12/5$ quantum hall state. Fibonacci anyons are non-abelian and provide a gate set which is universal for quantum computation.

### Ising

There are three types of anyons: 1, $\sigma$ and $\epsilon$. The fusion rules are the same as the OPE fusion rules of the 2D Ising model. (In the 2D Ising model, we have $\epsilon \star \epsilon = 1$ instead of the more generic $\epsilon \star \epsilon = 1 \oplus \epsilon$ that we might expect from the $\mathbb{Z}_2$ symmetry. In the 3D Ising model, it is the second relation which is true.)

The correct Chern-Simons description of Ising anyons is the WZW model $SU(2)_{k=2}$. This is also relevant for the $\nu=5/2$ Moore-Read state.

This system does have “non-abelian” anyons, but the gate set that one gets is not universal for quantum computation.