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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Question about the covering space of a spin manifold

I'm a little confused about how to prove that a covering space $\widetilde{M}$ of a spin manifold $M$ is also a spin manifold. If not, is there some counterexample? Could you please give me some help ...
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Explicit equations describing complex spin groups as affine algebraic groups

Although there are tons of questions about spin groups here on math.SE, I could not find there what I want. What I want is this. Take the complex spin group $\operatorname{Spin}(2n,\mathbb C)$. It has ...
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Spinor bundle $\mathbb{S}(M)$ for connected sum $\mathbb{S}(M_1 \#M_2)$?

I was just reading This question Wherein the OP wants to know what the tangent bundle looks like in a general connected sum $$T\left(M_{1}\#M_{2}\right)$$ (where the connected sum is along some ...
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Schrodinger-Lichnerowicz-Weitzenböck formula for $Spin_G $ structures?

I understand that given a (pseudo-) Riemannian manifold admitting a spin structure, we may write the Schrodinger-Licnerowiz Formula in terms of the Dirac and Covariant derivative operators: $$D^{*}D\...
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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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Question about "unique" complex spin representations

Corollary 2.4.6 of Morgan's book on Seiberg-Witten invariants says: There is a $\textbf{unique complex representation}$ of $Spin(V)$ induced from any irreducible complex, finite-dimensional ...
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A simple example of a $G$-principal bundle that cannot be lifted to an $F$-principal bundle along an epimorphism $F\to G$.

I am trying to understand spin structures, in particular how they may fail to exist. To start with, I would like to see a most simple example of a $G$-principal bundle that cannot be lifted to an $F$-...
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Question about spin map

I'm confused with the following definition of a spin map. A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...
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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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Question about the implications of real vs quaternionic representations, for spin groups of indefinite signature.

Specifically, it's known that if there is a preserved anti-linear complex structure which squares to $-1$, we have a quaternionic representation. It is also commonly stated that if one can find a ...
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Lawson's Spin Geometry book - pieces that do not fit

This is from Lawson's book "Spin Geometry", there is a problem here, there are pieces that do not fit. In (4.3) and (4.4) shows how to construct the operator Q from P and its symbol. For ...
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First order geometric operator on symmetric spinors over 3-manifolds

Let $ \left( X^3,g \right)$ be a closed, oriented Riemannian $3$-manifold. Choose a spin structure, and denote by $\mathbb S $ the spinor bundle and $\mathbf c $ the Clifford multiplication. The Dirac ...
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Clifford multiplication with specific 2-forms in 6 dimensions

I am new to spin geometry and I am trying to understand spinor bundles in dimension 6 for something else I am reading. $\mathcal{Cl}(6,0)$ has a unique irreducible real 8-dimensional representation $S$...
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Notes on Spinors by Deligne

I was reading section 2 (Clifford Modules) of the Notes on Spinors by Deligne, and am a little puzzled by Proposition 2.2. In this proposition, it is assumed $V$ is a complex vector space and $Q$ a ...
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$Spin(6,2) = SU(2, 2, \mathbb{H})$

On Wikipedia https://en.wikipedia.org/wiki/Spin_group#Indefinite_signature, it says $Spin(6,2) = SU(2, 2, \mathbb{H})$. But I cannot find any reference. Does anyone know one, or any other references ...
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Hopf fibration from $SO(3)$ Lie algebra generators?

One can use the Pauli matrices $\sigma_i$ to generate $Cl_3(\mathbb{R})$ and taking commutators of these matrices gives the $SU(2)$ Lie algebra $\mathfrak{su}(2)=\biggl(\begin{matrix} ia&-z\\ z&...
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Clifford algebra and Spin

I am reading Riemannian Geometry and Geometric Analysis by Jurgen Jost. Let me mention the notations and what I have learned from the book. In the book, the Clifford algebra $Cl(V)$ of a real vector ...
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What are irreducible $n$-times antisymmetrized representations?

I came across the following statement while reading a string theory textbook: if $\boldsymbol{16}_s$ and $\boldsymbol{16}_c$ are the are respectively spinor and conjugate spinor representationsof $...
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Representation of $SU(4)$

I am trying to understand explicit description of spinors in dimension $6$. Now one can explain positive spinors in terms of the usual $4$-dimensional representation of $SU(4)\hookrightarrow GL(4).$ ...
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Theta characteristic and spin structures

In many places one can find the statement that the existence of of a Spin-structure on a (compact) Kähler manifold $M$ of (complex) dimension $n$ is equivalent to the existence of a $\theta$-...
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Quantum Mechanics- Spin

I start saying I'm Italian, so my English is not very well and I will probably make many grammar mistakes ( forgive me for that)... I have to find eigenvalues and eigenvectors of this Hamiltonian ...
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Calculation about Clifford multiplication

Let $X$ be a smooth vector field on the even dimensional sphere $S^n$ such that $|X|>0$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$. Let $v=c(X):S^+(TS^n)\to S^-(TS^n)$ ...
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cobordism for every spin structure on a boundary?

Let's consider two n-dimensional closed Riemannan manifolds N and M that are cobordant to an undetermined (n+1) dimensional manifold W (That is: N and M are the boundary of W). If all considered ...
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Relation between the two spin structures of $S^1$

I know that $S^1$ has two spin structures $s$ and $t$, corresponding to its two double covers. I am trying to understand which diffeomorphism $f \colon S^1 \rightarrow S^1$ sends $s$ to $t$, i.e. $f^*(...
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Definition of $Spin^c$ structure

The $Spin^{\mathbb{C}}$-structure is well-known to be defined as the principal $Spin^{\mathbb{C}}$-bundle that covers the oriented orthonormal frame bundle of an oreinted Riemannian manifold. I've ...
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Properties of $Spin^{\mathbb{C}}$ structures

For a given closed oriented smooth 4-manifold $M$, fix a metric on it, we can have the orthonormal frame bundle of its tangent bundle to be a principle $SO(4)$-bundle. This bundle can be lifted to a ...
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Pin(p,q) and Pin(q,p) are not isomorphic

I read from this answer saying that Pin(p,q) and Pin(q,p) are not isomorphic. https://physics.stackexchange.com/a/634272/42982 The $\text{Pin}(p,q)$ and $\text{Pin}(q,p)$ are said to be different ...
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$SO(2, 1)$ and $Spin(2, 1)$ in terms of $SL(2, R)$ and $GL(1, R)$

We know that $Spin(3)$ is a double cover of $SO(3)$, which means it is a double cover of $RP^3$. So both $Spin(3)=S^3$ and $SO(3)=RP^3$. They are not isomorphic. Now, how do we show that $SO(2, 1) = ...
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$SO(1, 1)$ and $Spin(1, 1)$ in terms of $GL(1, R)$

We know that $\mathsf{Spin}(2)$ is a double cover of $\mathsf{SO}(2)$, which means it is a double cover of $\mathsf{U}(1)$. So both $\mathsf{Spin}(2)$ and $\mathsf{SO}(2)$ are $\mathsf{U}(1)$. They ...
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Use the $S^3$ sphere geometry to visualize a $Spin(2k)$ group?

We know that the real 3-dimensional sphere is isomorphic and diffeomorphic to various Lie groups $$S^3 = Spin(3) = SU(2).$$ While the larger dimensional $Spin$ group has a close relation to $Spin(3)$, ...
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Restriction on $n$ for ${Spin(n)\times Spin(n+2)} \subset Spin(2n+2) $ to be true

We know that $$SO(n) \times SO(m)\subset SO(n+m) \tag{1}$$ $$\frac{Spin(n)\times Spin(m)}{{\mathbf{Z}/2}}\subset Spin(n+m) \tag{2}$$ are both true. Eq (1) is true for the obvious reason. Eq (2) can be ...
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visualize the embedding $\mathrm{GL}(2,\mathbb{C}) \supset \mathrm{SL}(2,\mathbb{C}) \supset \mathrm{SU}(2)$ in a larger $\mathbb{R}^{n}$?

How to visualize the following embedding: $$ \mathrm{GL}(2,\mathbb{C}) \supset \mathrm{SL}(2,\mathbb{C}) \supset \mathrm{SU}(2). \tag{1} $$ They are all Lie groups thus manifolds. By this "...
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$\frac{Spin(2n-1)\times Spin(2n+1)}{{\mathbf{Z}/2}}\subset SO(4n)?$

Is the following statement true? $$\frac{Spin(2n-1)\times Spin(2n+1)}{{\mathbf{Z}/2}}\subset SO(4n)? \tag{1}$$ For $n=1$, $SU(2)\subset SO(4)$ true. For $n=2$, $\frac{Spin(3)\times Spin(5)}{{\mathbf{...
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$Spin(2n+1) \subseteq SU(2^{n})$ for $n \geq 1$?

According to this post Embed a Spin group to a special unitary group It seems to me that there is an answer proposes that for odd integer $2n+1$: $$ Spin(2n+1) \subseteq SU(2^{n}). \tag{1} $$ Here $...
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$Spin(2n) \subseteq SU(2^{n-1})$ for $n \geq 5$?

According to this post Embed a Spin group to a special unitary group It seems to me that there is an answer proposes that for even integer $2n$: $$ Spin(2n) \subseteq SU(2^{n-1}). \tag{1} $$ Here $...
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algebraic structure of $ \otimes_{\Bbb R}$ in $\text{Spin}^c(V)\subset Cl(V)\otimes_{\Bbb R} \Bbb C$

I hope to know a clear explanation on the tensor product notation: $\otimes_{\Bbb R}$: Suppose we write $A \otimes_{\Bbb R} B$, what is the required algebraic structure of $A$ and $B$? What is the ...
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Relation between Homotopy $\pi_0(G)$ and Surfaces of transitivity --- of Lorentz group $G$

Homotopy $\pi_0(G)$ of Lorentz group $G$ We know the Lorentz group is an indefinite orthogonal group $O(1,3)$ with $\mathbf{R}$ coefficients, which Lie group topology has four connected pieces due to $...
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Explicit 45 Lie algebra generators as rank-16 matrix spinor representations of $𝑆𝑝𝑖𝑛(10)$

A simple Lie group $𝑆𝑝𝑖𝑛(10)$ has a spinor representations of 16 dimensions, which is distinct from the vector representation of 10 dimensions (coming from standard vector representation of SO(10))...
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Lie algebra matrix representations of $Spin(8)$: via traility - transform from vector representation to two half-spinor representations

A simple Lie group $𝑆𝑝𝑖𝑛(8)$ has 2 half-spinor representations and 1 vector representation (coming from standard vector representation of SO(8)), all of them have dimension 8. One can compose a ...
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Explicit 28 Lie algebra matrix representations of $Spin(8)$: Two half-spinor representations

A simple Lie group $𝑆𝑝𝑖𝑛(8)$ has 2 half-spinor representations and 1 vector representation (coming from standard vector representation of SO(8)), all of them have dimension 8. One can compose a ...
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Embedding $SU(2)\times SU(2)\to Cl(\Bbb R^4)$

Consider the spin group $\text{Spin}(4)$. By definition it is contained in the Clifford algebra $Cl(\Bbb R^4)$ (https://en.wikipedia.org/wiki/Spin_group#Construction). Since it is known that $\text{...
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Real / complex Lie algebra $\Rightarrow$ Real / pseudoreal/ complex representations (any correlation)?

I know that the distinction of real / complex Lie algebra. For example, The $su(2)=so(3)=sp(1)$ is a real Lie algebra. The $sl(2,\mathbf{C})=so(1,3)$ is a complex Lie algebra. Question 1: How are ...
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An embedding of $U(n)\to \text{Spin}^c(2n)$

Let $(V,\langle,\rangle)$ be an $2n$-dimensional real inner product space and consider its Spin$^c$ group $\text{Spin}^c(V)\subset Cl(V)\otimes_{\Bbb R} \Bbb C$. Suppose there is a compatible (...
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spin representations v.s. semi-spin representations from $\rho: \operatorname{Spin}(n, {\mathbb C})\to SL(N, {\mathbb C})$

I learned that the complex representations $$\rho: \operatorname{Spin}(n, {\mathbb C})\to SL(N, {\mathbb C})$$ of complex Spin groups $\operatorname{Spin}(n, {\mathbb C})$ helps to study the compact ...
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Number of inequivalent spin structures on $\#_{k} (S^2 \times S^1)$?

Every closed-orientable 3-manifold admits an $su(2)$ spin structure. I'm working with manifolds that are of the form: $$M=\#_{k}\left(S^{2}\times S^{1}\right)$$ For a Riemann surface I know the genus $...
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Relation between $SU(8)$, and $Spin(8)$ and $SO(8)/(\mathbf{Z}/2)$

It is east to use the special unitary group to contain the special orthogonal group so $$SU(n) \supset SO(n) .$$ For $n=8$, we have $$SU(8) \supset SO(8).$$ We know that the $SO(8)$ has a double cover ...
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2 votes
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Problem about the notation $\text{Spin}^c(V)\cong \text{Spin}(V)\times _{\{\pm1\}} S^1$

Lemma 2.6.1 of Morgan's book on Seiberg-Witten equations states that the group $\text{Spin}^c(V)$ is isomorphic to the group $\text{Spin}(V)\times _{\{\pm1\}} S^1$. The proof actually shows that $\...
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Spin groups and Clifford Algebra

Consider $\Bbb R^n$ with standard norm and its Clifford algebra $Cl(\Bbb R^n)$. It decomposes as $Cl(\Bbb R^n)=Cl_0(\Bbb R^n)\oplus Cl_1(\Bbb R^n)$ (even part and odd part). The group $\text{Spin}(n)...
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Lie algebra of the $\mathop{Spin}(4)$ group as 2-forms on tangent bundle

While trying to understand connections on spin manifolds, I stumbled over this (apparently obvious) statement. Consider a spin $4$-manifold $M$. I am struggling to understand the intuition of the ...
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Branched covering maps with branch set a knot or link, fermionlike spinor fields?

I've been learning and better articulating this question over time, so my apologies to those who've read earlier iterations. First a little background: I've been learning about mappings between spaces ...
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