Search Results

In two dimensions, the Green's function of the Laplacian is the natural logarithm, $\nabla^2 \ln|z| = \delta(z)$, so we can take log of a polynomial the sum of delta-functions. \[ \nabla^2 \ln p(z) = …

The proof of the Wigner Semicircle Law comes from studying the GUE Kernel $$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} …

I am trying to understand better the quantization of the harmonic oscillator. Here are three ways of thinking about the harmonic oscillator. Eigenfunctions of the differential operator: $H = -\frac{ …

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix. \[ \int …

From what I had read, group characters can be "glued" together in a topological fashion and there is something to this effect in the paper by Dijkgraaf and Witten. TQFT seems to be a topological gener …

There are many examples of $q$-series identities being proven by interpreting them as generating series of geometric invariants like the Donaldson invariants. I would like to know if there are ways o …

The matrix-tree theorem states the number of spanning trees of a graph $G$ is equal to a modified determinant of the adjacency matrix or "graph Laplacian", $\Delta_G$: $$\#\{ \text{spanning trees of } …

In classical mechanics: If a Lagrangian $\mathcal{L}$ is preserved by an infinitesimal change in the state space variables $q_i \to q_i + \varepsilon K_i(q)$, this leads to only second order change in …

Is there a link between the theory of Domino Tilings and the Ising Model? In the global qualitative sense that physicists use, the answer is "yes". The connections could go like this: The dimer m …

What do these branching rules mean? \begin{eqnarray*} SO(6)_E &\to& SU(2)_\ell \times SU(2)_r \times U(1)_\Sigma \end{eqnarray*} I am taking these examples from a paper of Gukov (on p.51) but more …

This is the ABJM partition function on the 3-sphere, $$ Z(2) = \int \frac{d^2\mu}{(2\pi)^2} \frac{d^2\nu}{(2\pi)^2} \frac{\left[ 2 \sinh \frac{\mu_1 - \mu_2}{2}\right]^2\left[ 2 \sinh \frac{\nu_1 - …

The answer to my question is "yes". Technically, it's a spin-TQFT but now I am trying to make sense of that answer. Dimers on surface graphs and spin structures. I David Cimasoni, Nicolai Resheti …

Without any regards to the physics or the geometry used to generate this result, let's examine the formula of Gukov (see p. 32): $$ \mathcal{I}_{SU(2)}(q,t) = \frac{1}{2}\sum_{m \in \mathbb{Z}}\int \ …

Reading this paper of Masahito Yamazaki and Kazuya Yonekura, I am having trouble turning the physics jargon into mathematical statements. He is talking about Yang-Mills theory over $\mathbb{R}^4$. P …

Something I really enjoy about Tao's writing is that he proves the same theorem over and over. While I complain a bit sometimes about clarity, this is a heuristic that I very much believe in. This …