# Questions tagged [spin-geometry]

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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### 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 ...
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### Manifold with totally geodesic boundary is spin if and only if its double is spin

Let $(M,g)$ be a Riemannian manifold with totally geodesic boundary $\partial M$. Let $(DM,Dg)$ be the double of $(M,g)$ obtained by reflection of across $\partial M$. I'm looking for a reference for ...
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### Question about Clifford volume element

I'm a little confused with the following: Let $M$ be a $m$ dimensional Riemannian manifold and $e_1,\cdots,e_m$ be a local orthonormal base of $TM$. Let $$\omega_\mathbb{R}=c(e_1)\cdots c(e_m)$$ ...
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### Dirac operator on 4-dimensional rectangle with the periodic boundary conditions is self-adjoint? What is its spectrum?

Let us think of the Euclidean Dirac operator $iD^k \gamma_k$ on the rectangle $[-1,1]^4$ with the periodic boundary conditions. The covariant derivative $iD^k$ carries a gauge potential term and we ...
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### Quadratic forms on $\mathbb{R}^{16}$ coming from octonions

$\DeclareMathOperator\RRe{Re}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sym{Sym}$Let $\mathcal{H}_2(\mathbb{O})$ denote the (10-dimensional) real vector space of octonionic Hermitian matrices ...
125 views

### Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard ...
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### Non-associative Clifford algebra

Let $V$ be a finite-dimensional $\mathbb{R}$ vector space equipped with a symmetric, bilinear form $b : V \times V \to \mathbb{R}$. My question is if there exists an analog of a Clifford algebra in ...
224 views

### Calculation of the top Chern class of spinor bundle over $S^{2n}$

It's well known that for a complex vector bundle $E$, we have $$c_n(E)=e_n(E_\mathbb{R})$$ But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class ...
1 vote
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### Relationship with between Clifford multiplication and pullback

Let $X$ be a smooth vector field on the even-dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
125 views

### Is spin cobordism an invariant for surgery of codimension $q\ge3$?

Recall that a surgery of codimension $q$ on an $n$-manifold $X$ is a modification of $X$ of the following type. Let $\Sigma^{n-q}\subset X$ be a smoothly embedded $(n-q)$-sphere with a trivialized ...
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Let $X$ be a smooth vector field on the even dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
154 views

### 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}\...
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$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $... 3 votes 0 answers 62 views ### Spin structures induced on embedded circles and choices of trivialisations I have a presumably basic question concerning spin structures that has me a bit confused. Let$C$be a circle embedded in a spun manifold$X^n$. Given a choice of trivialisation of the normal bundle ... 1 vote 1 answer 225 views ### The normalizer of$\operatorname{Spin}(2N)$in$\operatorname{U}(2^{N-1})$?$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$I can show that $$\U(2^{N-1})\supset \Spin(2N)$$ when$2N > 4$or a positive integer$N > 2$, so$\Spin(2N)$can be embedded in$\U(2^...
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$I am confused about the equivalence of some various definitions of spin structures and I was hoping for some help clearing out the fog. Let ...