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Apologies for not a very rigorous question. I came across this PhD thesis by XIAO ZHANG, a student of Yau. From the thesis: "We also investigate some basic facts on Spin$^c$ structure on $4$-dimension …

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let $(X,\Omega,\omega,J)$ be a manifold with an $\SU(4)$ structure. Since $\SU(4)\subset\Spin(7)$, $X$ also has a $\Spin(7)$-structure. I wa …

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let's take a $\Spin(7)$ manifold $M$ (the $\Spin(7)$ structure can have torsion), then the standard Dirac operator from negavtive spinors to …

I am trying to understand Taubes' paper on SW$\Rightarrow$ Gr. I don't understand how either of the equations 2.16 or 2.17 appears, I would be happy to understand how the curvature term $F_a$ appears …

I am reading Taubes' paper on SW$\Rightarrow$ Gr and lost in some analysis, can anyone help me to see how to get equation 2.19 from equation 2.18? Is this some version of Kato for the Laplacian?

It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see som …

Let's take a closed $G_2$ manifold $(M,\Phi)$. $\Phi$ denoting the three-form which defines the $G_2$ structure on $M$. Let's take a closed two form $\theta\in\Omega^2(M).$ Is \begin{align} d(\theta_{ …

I am trying to understand the dimensional reduction of Seiberg-Witten equations from dimension $4$ to $3$, more specifically my concern is about ellipticity of the new equations in dimension $3$ under …

Let's take a $G_2$ manifold $(M,\Phi)$, then we get a Spin$(7)$ manifold by taking $(M\times\mathbb{R},\Psi:=\Phi\wedge dt+*_M\Phi),$ where $t$ is the coordinate in the $\mathbb{R}$-direction. $\Phi\i …

Let $M$ be an oriented closed $6$-manifold. $V$ be an hermitian complex vector bundle of dimension $4$ on $M.$ Hence $c^3(V)\in H^6(M,\mathbb{Z})\cong \mathbb{Z}$ can be thought of as an integer and o …

A Killing spinor on a Riemannian spin manifold is a section of the spinor bundle satisfying the equation: \begin{align*} \nabla_X\phi=\lambda X\cdot\phi \end{align*} Here $X$ is a vector field and $\p …

It's a well known result by Kazdan and Warner that on a closed Riemannian manifold the pde: \begin{align*} \Delta f+ge^f=c \end{align*} has a unique solution for $g\geq 0,$ and $c$ a positive constant …

It is known that $S^6$ has (real) killing spinor. I am looking for an explicit description of such a spinor. More generally, we can choose an almost complex structure say, $J$ on $S^6$. Now, we can de …

Let $M$ be a closed manifold of dimension $6$, and we look at the collection of smooth functions on $M$ which satisfy: $$ \|f\|_{C^0}\leq C\big(\|f\|_{L^{14}}^2+1\big) $$ for some fixed $C>0$. Can we …

This is a bit of a vague question, sorry for that. I am wondering if there's any detection of Hodge decomposition in terms of Clifford multiplication. For example if $\phi$ is a spinor and $\theta,\al …