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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55 views

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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0 votes
0 answers
114 views

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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6 votes
1 answer
210 views

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\...
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  • 455
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Existence of a local spinor bundle

I am confused about the existence of a local spinor bundle. My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...
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1 vote
0 answers
63 views

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,\...
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7 votes
2 answers
384 views

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 ...
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4 votes
1 answer
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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  • 351
2 votes
0 answers
77 views

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 ...
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3 votes
1 answer
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 ...
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58 views

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 ...
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5 votes
1 answer
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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1 answer
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Question about Clifford multiplication

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 ...
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3 votes
0 answers
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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4 votes
0 answers
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Homomorphism from a product of spin groups to a bigger spin group

In the paper "Essential dimension of spinor and clifford groups" by Chernousov and Merkurjev, it says that there is a natural homomorphism $\operatorname{Spin}(n)\times \operatorname{Spin}(m)...
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1 answer
255 views

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 $\Sigma_2$ ...
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1 vote
0 answers
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Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$

Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
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8 votes
0 answers
183 views

Computation of the 3-dimensional $\mathbb{Z}/m$-equivariant spin cobordism group (with possibly non-empty fixed-point set)?

$\newcommand{\odd}{\mathrm{odd}}\newcommand{\ev}{\mathrm{ev}}$Consider tuples of the form $(Y,\mathfrak{s},\widehat{\sigma})$ where: $Y$ is a closed oriented 3-manifold, $\mathfrak{s}$ is a spin ...
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6 votes
0 answers
166 views

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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5 votes
3 answers
712 views

Is a spin structure on a knot complement the same thing as an orientation of the knot?

There are a variety of characterizations of spin structures on the tangent bundle of a manifold. Two facts about them: Spin structures on $TM$ are an affine space over $H^1(M; \mathbb{Z}/2\mathbb{Z})$...
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3 votes
1 answer
238 views

Pull back of Spin$^{\mathbb{C}}$ bundle

Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{\mathbb{C}}$ bundle, $V_+$ denote the ...
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4 votes
1 answer
260 views

Identifying a $4$-form on a $6$-dimensional manifold

Let $M$ be a closed $6$-dimensional Riemannian manifold with a spin$^{\mathbb{C}}$ structure. It is known that real $4$-forms on $M$ act on the positive-spinors as trace-free hermitian endomorphisms ...
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1 vote
1 answer
154 views

Injective group homomorphism on $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/2}$ or $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/4}\to U(2^{2k})$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$From Pierre Deligne's Notes on spinors, we can see that there ...
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4 votes
1 answer
262 views

Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for $n=2k+1$

Atiyah, Bott, and Shapiro paper on Clifford Modules around page 10 shows two facts. 1 - There is a lift $U(n) \to Spin^c(2n)$ from $U(n) \to SO(2n)\times U(1)$. Also an embedding (injective group ...
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8 votes
0 answers
245 views

Infinitely many nonempty Seiberg-Witten moduli spaces

The classic "finiteness" statement in Seiberg-Witten (SW) theory is that, for any smooth closed connected 4-manifold, there are only finitely many spin-c structures with nontrivial SW ...
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6 votes
1 answer
232 views

Euclidean and Minkowski Majorana spinors - inconsistency with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf Somehow I find some inconsistency with his Tables of Euclidean and ...
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35 views

Extending the Dirac operator on an open subset of a manifold and preserving positivity

Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
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1 vote
0 answers
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Equivalence of $Spin^C$-Structures

I'm trying to understand the equivalence of $Spin^C(n)$-structures in the book "Dirac Operators in Riemannian Geometry" by Thomas Friedrich, p. 47 ff, but I got somehow stuck because I'm not ...
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5 votes
1 answer
428 views

A corollary of the non-existence of positive scalar curvature

I've been done some work with scalar curvature and managed to give a simple proof for the following result: Let $M$ be a closed manifold which do not admit a metric of positive scalar curvature. Then ...
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3 votes
0 answers
194 views

Research in spin geometry

I am currently learning differential geometry, but I have heard about the field of spin geometry and have skimmed through the book Dirac Operators in Riemannian Geometry by Thomas Friedrich. I have ...
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0 votes
0 answers
82 views

Dirac operator on a 5 dimensional tangent manifold with a $Spin(3)$-bundle

In p.3 of Witten paper from this Physics Letters B, Volume 117, Issue 5, 18 November 1982, Pages 324-328 Physics Letters B, 117(5), 324–328, he says that about the Dirac equation on a 5-dimensional ...
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7 votes
1 answer
252 views

Visualizing the wave operator in two dimensions

For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be ...
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0 votes
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115 views

Clifford multiplication formula for $d\beta$ where $\beta$ is a $3$-form

Say $X$ be a $6$-dimensional compact Riemannian manifold which admits a $Spin^{\mathbb{C}}$ structure. Now I want to have a Clifford multiplication formula by $d\beta$ in terms of $\beta$ where $\beta$...
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2 votes
0 answers
80 views

The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $(\Spin(...
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2 votes
1 answer
293 views

The normalizer of SU(n) in U(m)?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $...
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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 ...
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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^...
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2 votes
1 answer
130 views

Clarification of different notions of spin structures

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$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 ...
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2 votes
1 answer
254 views

Clifford multiplication formula on an almost complex manifold

$\DeclareMathOperator\End{End}$Following the deduction by John W. Morgan in his book The Seiberg–Witten equations and applications to the topology of smooth four manifolds, an almost complex manifold $...
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104 views

Central extensions of orthogonal group by $C_2$

Suppose $(V,Q)$ is a quadratic space for definite quadratic form $Q$. It is stated in Pin groups that there are two central extensions of the orthogonal group $O(V)$ by the cyclic group $C_2$, ...
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2 votes
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Bryant-Salamon $G_2$ manifold on the spinor bundle over $S^3$

I am trying to understand the spaces constructed in R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy. My first problem is, essentially, about ...
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2 votes
2 answers
199 views

Induced action by an involution on spinor bundle and Dirac operator

Let $M$ be a $4n$-dimensional spin manifold with a fixed Riemannian metric $g$. Let $S$ be a spinor bundle over $M$ and fix the Riemannian connection on it. There is a decomposition $S=S^+\oplus S^-$, ...
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13 votes
2 answers
539 views

What is the relationship between spinors and supermanifolds and fermions?

I have the following two impressions about fermions in physics. I'm confused about their accuracy, and their compatibility: To consider the behavior of a fermion, whose intrinsic spin is described by ...
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19 votes
2 answers
1k views

If the universal cover of a manifold is spin, must it admit a finite cover which is spin?

If $M$ is non-orientable, then it has a finite cover which is orientable (in particular, the orientable double cover). If $M$ is non-spin, then it does not necessarily have a finite cover which is ...
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7 votes
1 answer
177 views

Isomorphisms of Pin groups

My goal is to identify the $Pin$ group $$ 1 \to Spin(n) \to Pin^{\pm}(n) \to \mathbb{Z}_2 \to 1 $$ such that $Pin^{\pm}(n)$ are isomorphisms to other more familiar groups. My trick is that to look at ...
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4 votes
0 answers
288 views

Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$ The $\Spin(1,3)$ is the ...
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5 votes
1 answer
148 views

Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$

For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e., $$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(...
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3 votes
0 answers
338 views

Why in $S^2$ is there no spin structure? [closed]

For a Dirac fermion (spin half) on $S^2$, we have both the general covariant derivatives and the relativistic Hamitonian. What does the claim "in $S^2$ there is no spin structure" means? A reference ...
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6 votes
0 answers
175 views

Spin structure using flag manifolds instead of a Riemannian metric

Let $(M,g)$ be an oriented Riemannian manifold of dimension $n$, and denote by $P_{\mathrm{SO}}\to M$ its oriented frame bundle. The usual definition of a spin structure is the data of a principal $\...
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