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I am trying to understand what is a Higgs bundle as defined in this paper by Gukov and Pei. They say it is a pair $(E, \Phi)$ $E$ is a holomorphic principal $G^\mathbb{C}$ bundle $\Phi \in H^0(\S …

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 …

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) = …

If $h(z)$ is analytic on the disk centered at 0 of radius r, by the Cauchy Residue formula \[ \int \int_D h(z)\, dx dy = \pi r^2 h(0) \] The disk is the simplest example of a quadrature domain since …

In a paper by Yuji Tachikawa, I found a q-deformed "2d Yang-Mills paritition function for a cylinder". Here it is (adapted): $$ Z(q, x_L, x_R) = \mu(q, x_L)^{-1/2} \langle x_L | \bigg\[ \sum_{R \in …

Is there a way to classify (and build) the principal SU(2) bundles over a given topological surface up to homeomorphism? In the end, I would like to examine the associated bundle whose fiber is a giv …

To phrase the question in a concrete way, I read in a paper: The super Poincare subalgebra of osp(6,2|4) has bosonic part $so(5,1) \oplus usp(4) \simeq so(5,1) \oplus so(5)$. It's hard to unpack thi …

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 …

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{ …

Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$: $$ \int_M e^{n \mathbf{e}} e …

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 …

Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of a random Unitary matrix. I am interested in the expected value: $$\mathbb{E}_{X \in U(N)}\left[ \prod_{i=1}^n \frac{1}{(1 - \lambda_i)^2}\righ …

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 \ …

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 - …

For any Riemann surface with punctures $C$, and Lie group $G$, the character variety is the space of maps $\mathrm{Hom}(\pi_1(C), G)$. I know that if $G= S_n$ (not a lie group), then $\mathrm{Hom}( …