Questions tagged [spin-geometry]

For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}(n)$ 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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30 views

Connection between double cover and spinor fields when there are ramification points?

Apologies, I'm very new and naive to topology. I have been learning about different covering maps between manifolds and recently was learning about the map $T^2 \rightarrow S^2$ which is a two-to-one ...
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50 views

$\operatorname{Spin}^+(s,t)$ is a group

I'm reading materials on spin groups. Let $Cl(s,t)$ be the Clifford algebra of $\mathbb{R}^{s+t}$ with standard bilinear form $\eta$ with signature $(s,t)$. The book then defines, $$\operatorname{Spin}...
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34 views

Index of Dirac Operator

Let $X$ be a compact Kähler manifold, then there exists a canonical Spin$^\mathbb{C}$ bundle on $X$ for any given complex line bundle $L$: \begin{align*} S^+=\bigoplus_i\Lambda^{2i}(X;L)\text{ and } S^...
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Spin structures on a Riemann surface with boundary

A spin structure on a Riemann surface $X$ is a line bundle $L$ on $X$ such that $L^2$ is isomorphic to the cotangent bundle $T^*X$. A genus $g$ Riemann surface is known to have $2^{2g}$ distinct spin ...
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Spin derivative on a submanifold

I am having some trouble proving the following expression. For a 3-manifold $Y$, spinor bundle $S$, codimension 1 submanifold $M\subset Y$, subbundle $S_M$, and normal vector $N$, the covariant ...
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Is it possible to derive the spinor for $Cl_{1,3}(\mathbb{R})$?

Formally we have $$Cl_{1,3}(\mathbb{R})\otimes_\mathbb{R} \mathbb{C} \cong Cl_4(\mathbb{C})\cong M_4(\mathbb{C})\cong \text{End}(S)$$ and on a more explicit level (see wiki) we can exhibit the spinor ...
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67 views

$SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3,\mathbb{R})?$ or $SL(2,\mathbb{C})\cong \operatorname{Spin}(3,1,\mathbb{R})?$ [closed]

Which one of the isomorphism is correct? $$SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3,\mathbb{R})?$$ $$SL(2,\mathbb{C})\cong \operatorname{Spin}(3,1,\mathbb{R})?$$ If $SL(2,\mathbb{C})\cong \...
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$\operatorname{spin}^c$ structure through the Pauli matrices

Let $M$ be a n-dimensional compact oriented smooth manifold. As far as I know, a $\operatorname{spin}^c$ structure on $M$ (or $TM$) is either a principal $\operatorname{spin}^c(n)$-bundle $P$ such ...
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Reference request: characteristic numbers of manifolds

I am wondering if there is a good resource out there that computes characteristic numbers of common manifolds. For example, the $\hat{A}$-genus of the tori $T^n$ or spheres $S^n$ for various $n$. It ...
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Changing the Riemannian metric does not affect the $Spin^c$-structure

It is well known that changing the Riemannian metric on a manifold does not change the $Spin$-structure. I suppose the same should be true for $Spin^c$-structures, but I am unable to prove it or to ...
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Conflicting definitions of a spinor

I've come across two definitions of "spinors" that I'm having a hard time reconciling: Spinors are the "square root" of a null vector (see here, and also Cartan's book "The ...
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106 views

Prove or disprove $SO(4n) \supseteq \frac{(Sp(1)\times Sp(n))}{\mathbb{Z}_2}$?

I suspect that this is true $$ \boxed{SO(4n) \supseteq \frac{(Sp(1)\times Sp(n))}{\mathbb{Z}_2}.} $$ How do we prove it? When $n=1$, we have $$ SO(4) \supseteq \frac{(SU(2)\times Sp(1))}{\mathbb{Z}_2}...
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How to visualize the double cover of $SO(4)$ as two copies of $S^3$?

How to visualize the double cover of the rotational symmetry group of $S^3$ (which is $Spin(4)$, namely the double cover of $SO(4)$) as two copies of $S^3$? This is due to $Spin(4) = SU(2) \times SU(...
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Connection $A$ on Seiberg Witten equation

Serberg Witten equation requires a given orthogonal frame bundle $P_{SO} \rightarrow X$, we give a $Spin^{c}$ lifting $P_{Spin^{c}}$ and the assoiated spinor bundle $S=P_{Spin^c}\times _{\rho}\Delta_{...
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Definition of the twisted spin covariant derivative for twisted spinor bundles

Let $S\to M$ be the spinor bundle and \begin{equation} \nabla^S\colon\Omega^0(M,S)=\Gamma(M,S)\to\Omega^1(M,S)=\Gamma(M,T^*M\otimes S) \end{equation} the spin covariant derivative. In addition, let $E\...
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153 views

Could the discovery of a counterexample to the Unit Conjecture change mathematicians’ understanding of spinors in general?

Recently it has been reported that Giles Gardham has found a counterexample to the Unit Conjecture for group rings, as given in https://arxiv.org/abs/2102.11818 (and also in a popular article https://...
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Classification of $\operatorname{spin}^c$ structures

I'm reading the notes about Seiberg-Witten invariants by Salomon (see https://people.math.ethz.ch/~salamon/PREPRINTS/witsei.pdf). In Theorem 5.5 he gives an interesting classification of $\...
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Why these are representations of ${\rm Spin}(D)$ and not ${\rm SO}(D)$?

In a course on supersymmetry I have learned the following way to construct spinor representations in $D$ dimensions: we first find matrices $\Gamma^A$ which obey the Clifford algebra $$\{\Gamma^A,\...
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Construction of $C\ell(V,q)$ for a quadratic space $(V,q)$ and proving an injection $V\hookrightarrow C\ell(V,q)$ exists.

I've been reading Lawson & Michelsohn's Spin Geometry, and struggled particularly with their (apparently wrong) proof that a quadratic space embeds into its Clifford algebra. I did a bit of ...
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1answer
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Isomorphism of representation spaces

The 4-dimensional spin group $Spin(4)=SU_{+}(2) \times SU_{-}(2)$, denote a typical element as $(A_+,A_-)$. We have for the 4-dimensional Euclidean space $V=\mathbb{R}^4 \simeq \mathbb{H} $ we can ...
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The maximal (special) unitary subgroup contained in the Spin group: $Spin(2N)\supset U(N) \supset SU(N) ?$

Is it true that given a Spin group $Spin(2N)$, the maximal special unitary subgroup that it can contain is $SU(N)$? So $$ Spin(2N) \supset SU(N)? $$ Is it true that given a Spin group $Spin(2N)$, the ...
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43 views

Inner products and Spinors

Let $X$ be the three sphere $S^3$. Then it has a spin structure and its spinor bundle is an $SU(2) = Spin(3)$ vector bundle. Let $(e^1, e^2, e^3)$ be an oriented, local orthonormal basis for $T^* X$. ...
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If the bundle of orthonormal frames has a continuos/smooth global section will the bundle of spin frames also have one?

Let $(M,g)$ be a semi-Riemannian manifold with metric of signature $(p,q)$. I believe the signature of the metric is not relevant for this discussion so I leave it arbitrary (corrections to this are ...
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Unimportance of the metric for spin structures

I have some pedantic confusions when it comes to spin structures. Let $B$ be a nice space (like a CW complex) and let $\xi$ be an oriented vector bundle on $\xi$. If I choose a metric on $\xi$, then ...
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1answer
67 views

Demonstrate $\mathrm{Spin}(3, 3) = \mathrm{SL}(4, R)$

It seems that there is an accidental isomorphism $$ \mathrm{Spin}(3, 3) = \mathrm{SL}(4, R). $$ Some facts I am aware is that $\mathrm{SL}(4, R)$ has the Lie algebra of 15 generators. the homotopy ...
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1answer
60 views

Lie algebra of SO(V) is isomorphic to spin(V)

where $a\in Cl^2(V)$, and $\tau(a)(v)=[a,v]$ for $v\in V$. I was wondering is the first equality correct as written? Jost earlier defined the clifford algebra as the quotient of the tensor algebra by ...
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456 views

What makes spinors mysterious?

Everyone familiar with spinors presumably knows the quote by Sir Michael Atiyah, that spinors are mysterious in spite of their algebra being formally understood. I have heard this sentiment echoed in ...
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Can spinors be generalized from $\mathbb{Z}_2$ to arbitrary polyhedral symmetry groups via topological gluing?

The classical example of a spinor is to consider a path around the center of a mobius strip, and attach to each point in the path a perpendicular vector that locally points "up". This ...
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1answer
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Why Spin(V) is a connected group?

I want to know why $Spin(V)$ is connected. I am watching $Spin(V)$, where $V$ is a real vector space with a positive defined metric, as the products of elements of the Clifford algebra $Cl(V)$ of the ...
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How to take the pullback of a Witt basis onto a subspace

I'm interested in how to take the pullback of a Clifford algebra onto an induced metric. Suppose we take the topological three-sphere embedded in $\mathbb{R}^{4}$ and we allow for a complexified ...
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50 views

Triangulable Spin 5-manifolds but do not admit any smooth structure

I noticed the discussion in this post, which states that there are examples of D=5 spin manifolds which are triangulable, but do not admit any smooth structure. What are some of these examples of D=...
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55 views

How to understand the spin structures on $T^d$ and $\mathbb{RP}^d$

How to understand the spin structures on $T^d$ and $\mathbf{RP}^d$? We know that $H_1(T^d,\mathbb{Z}_2)=(\mathbb{Z}_2)^d$. It looks that there are $2^d$ choices on 1-cycle of $T^d$. But it seems ...
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Quadratic enhancements of the intersection form and spin structures

If we search spin structures and quadratic enhancements of the intersection form, there are many papers discussing the issue. What are the relations and basic intuitions between the quadratic ...
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What do characteristic classes of spinor bundle depend on?

Let $M^n$ be a smooth manifold. Equip $M$ with a Riemannian metric and let $S$ be a spinor bundle. We can consider characteristic classes of $S$ (or $S_+,S_-$ for when $n$ is even), for example the ...
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1answer
86 views

Obstruction to a Spin structure on a bundle ξ, and ξ ⊕ $n$ det ξ

In Ref, it says that: The obstruction to putting a Spin structure on a bundle $ξ (= Rn → E → B)$ is $w_2(ξ) \in H^2(B;Z/2Z)$. Pin± structures is that Pin− structures on ξ correspond to Spin ...
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Does only the group $SO(n)$ admit a double cover which is $\operatorname{Spin}(n)$?

The double cover of the group $SO(n)$ is the spin group $\operatorname{Spin}(n)$. Do any of the other groups $SU(n)$, $\operatorname{Sp}(n)$, $G_2$, $F_4$, $E_6$, $E_7$, $E_8$ have double covers? If ...
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References for learning real analysis background for understanding the Atiyah--Singer index theorem

I am interested in learning the Atiyah--Singer theorem, and its version for families of operators. For this purpose, I have tried to read the recent book by D.Bleecker et.al.. However I have ...
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39 views

Supergroup for SO groups vs Spin group

We know that the $\mathbb{Z}/2$ central extension of $ SO(d) $ can give a nontrivial double/universal cover of $SO(d)$ known as the $Spin(d)$ group. They have this relation $$ 1 \to \mathbb{Z}/2 \to ...
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Examples of non-spin 2-manifolds

Do we have some examples of 2d manifolds, which are non-spin (thus $w_2(M) \neq 0$)? thus they must be non-orientable $w_1(M)^2 \neq 0$ thus $w_1(M) \neq 0$. (p.s. every 2d differentiable/triangulable ...
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Why does the determinant of the Dirac operator on $S^n$ approach $1$ as $n\to\infty$?

Bär and Schopka (reference below) present an interesting conjecture regarding the determinant of the Dirac operator on spheres $S^n$. The conjecture is simple: in the limit $n\to\infty$, the ...
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Are these two Seiberg-Witten equations the same?

In Jurgen Jost's book on Riemannian Geometry, he describes the following equation as one of the Seiberg-Witten equations: $$F^{+}_A = \frac{1}{4}\langle e_j \cdot e_k \cdot \varphi , \varphi \rangle e^...
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A spinorial generalization of the Hopf map

If $V = \mathbb{R}^3$ with the Euclidean inner product $g$, and $S = \mathbb{C}^2$ is the corresponding space of spinors, then there is a quadratic map $h: S \to V^*$, which maps $\psi \in S$ to $h(\...
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Harmonic Spinors with Vanishing Christoffel Symbols

The Dirac equation can be defined as $$D\psi = \Sigma_{i=1}^3 c(e_i) \nabla_{e_i},$$ where $$\nabla_{e_i} = e_i + \frac{1}{4} \Sigma_{j,l=1}^3 \Gamma^l_{ij} c(e_j) c(e_l).$$ $c$ is the standard ...
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Spinors and Klein-Gordon Equation

I'm currently working through Chapter 13 of Wald's General Relativity and spinors are being a little illusive to me. The question is pretty much: Using the Klein-Gordon equation in the form: $$\...
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Compare two definitions of equivariant index of Dirac operator

Let $(M^{2n},g)$ be a Riemannian manifold with spin structure. Let $G$ is a compact Lie group. Suppose that $G$ acts on $M$ smoothly and the metric is $G$-invariant. Also, assume that the $G$ action ...
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1answer
62 views

“enlarged” form and Aut(Spin(8))

In https://en.wikipedia.org/wiki/SO(8)#Spin(8), it says that "Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a ...
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39 views

Why no Majorana spinors in Spin(4)?

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$ We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$ It is also said $\Spin(1,...
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42 views

$16 \otimes_s 16 = [1] \oplus [5]^+$ from spinor representation

A teacher says that in the spinor representation of $Spin(10)$, there is $$ 16 \otimes_s 16 = [1] \oplus [5]^+ $$ here $16$ is irreducible spinor representation of $Spin(10)$. And $[n]$ denotes the $n$...
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61 views

Invariant bilinear forms on Dirac spinors

Let $V$ be an even dimensional complex vector space $\dim V =2m$. Equip $V$ with non-degenerate symmetric bilinear form $g$, then Clifford algebra $\operatorname{Cl}(V,g)$ is simple and thus has ...
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Formulas for the Spinor Representation Product Decompositions $2^{[\frac{N-1}{2}]} \otimes 2^{[\frac{N-1}{2}]}=?$ and …

We know that given the dimension $N$, we can construct the corresponding spinors for the $Spin(N)$ group (which has $Spin(N)/\mathbb{Z}_2=SO(N)$ so $Spin(N)$ is a double cocver of the spatial ...

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