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Results tagged with spin-geometry
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user 131004
For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}^{\pm}(n)$ and $\operatorname{Spin}^c(n)$. This tag should also be used for any questions about the geometry of spin manifolds, including questions involving Dirac operators and the Lichnerowicz formula.
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A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ man …
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2
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28
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Trace-free Hermitian endomorphisms in dimension $7$
Let $M$ be a spin-manifold of dimension $7$. Let $S\rightarrow M$ be a spin-bundle on $M$. Then Clifford multiplication ($c$) gives us the following isomorphism:
\begin{align*}
c:i\Lambda^2\oplus\Lamb …
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0
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123
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Positive mass theorem and Seiberg-Witten equations
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 …
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3
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116
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Decomposition of forms in $\operatorname{SU}(4)$-manifold
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$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 …
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97
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Dirac operator on $\operatorname{Spin}(7)$, $G_2$ and $\operatorname{SU}(3)$ manifolds
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let's take a $\Spin(7)$ manifold $M$ (the $\Spin(7)$ structure can have torsion), then the standard Dirac operator from negavtive spinors to …
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3
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0
answers
279
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A question in $\operatorname{Spin}(7)$ geometry
$\DeclareMathOperator\Spin{Spin}$I am looking for a proof of a fact (I think it's true intuitively due to representation theory) in $\Spin(7)$ geometry. Let's take a closed $\Spin(7)$-manifold $(M^8,g …
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1
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71
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Spin(7)-instanton
Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the deter …
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1
vote
0
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67
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Action of volume form on spinors in odd dimension
We know that for a smooth orientable manifold of dimension $2n, i^n$ times the volume form acts as identity on the positive spinors and acts as minus identity on the negative spinors via Clifford mult …
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1
vote
0
answers
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Norm of Killing spinor
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 …
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1
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answers
77
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Spinors in dimension 6
From the representation of $\operatorname{Spin}(6)\cong \operatorname{SU}(4)$, one can deduce that on a $6$-dimensional manifold we get the postive spinor bundle from the usual $4$-dimensional represe …
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6
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1
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292
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Weitzenböck formula and comparison of norms
Let $M$ be a closed Riemannian manifold with a spin$^\mathbb{C}$ bundle $S$. Now for a spin connection $A,$ and a spinor $\phi,$ it can be shown that $C\lvert\nabla_A\phi\rvert^2\geq \lvert D_A\phi\rv …
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1
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Is Hodge decomposition detected in Clifford multiplication
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 …
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3
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answers
170
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Bound of the spinor element in Seiberg-Witten equation for a Kähler surface
Let's say we want to solve a perturbed version of SW equations on a closed Kähler manifold $(X,\omega):$
\begin{align*}
&D_A\phi=0\\
&F_A+it\omega=q(\phi)=\phi\otimes\phi^*-\frac{|\phi|^2}{2}\text{Id} …
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5
votes
1
answer
345
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Understanding the quadratic part in Seiberg Witten equation
Lets take a closed four manifold $M:=\Sigma_1\times \Sigma_2,$ where $\Sigma_i$s are compact Riemann surfaces. Now if $V$ and $W$ are Spin$^\mathbb{C}$ bundles on $\Sigma_1$ and $\Sigma_2$ respectivel …
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6
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A generalized Dirac operator
Let $(M^4,g)$ be a closed four-dimensional Riemannian manifold and $J$ be an almost complex structure on $M$. Then for normal coordinate $e_1,\dots e_4$ at a point $m,$ and for a section $\alpha$ of a …
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