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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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Spinor representation for indefinite signature

Recall that for a $n$-dimensional vector space with positive definite quadratic form, we have the spin representation thanks to the following inclusions $$\Delta_n : \mathrm{Spin}(n) \rightarrow{} \...
Integral fan's user avatar
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Spinors on (euclidean signature) spacetime

Let's consider a spacetime $M$ which is a also spin manifold. In Euclidean signature We have that the frame bundle is a principal $GL(4,\mathbb{R})$ bundle over $M$. Even dimensional spin manifolds ...
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Can two different spin structures on a manifold induce the same spin$^c$ structure?

Let $(M,g)$ be an oriented Riemannian $n$-manifold with transition functions $g_{\alpha\beta}:U_{\alpha \beta}\to SO(n)$. A spin structure on $M$ is a lift of $g_{\alpha\beta}$'s to functions $\tilde{...
user302934's user avatar
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Parametrizing Rotations of Minkowski Space $\mathbb{R}^{1,1}$ Using $\text{Spin}(1,1)$

I was able to derive that $$\text{Spin}(1,1)=\{a+be_{12}|a^2-b^2=\pm1\},$$ which has 4 connected components. Individually, they parametrize rotation in each quadrant. Moreover, the map $a+be_{12}\...
Miles Gould's user avatar
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Exterior algebra and space of spinors

The approach I am used to define spinors in arbitrary dimensions is based on finding a matrix representation of the Clifford algebra $$\Gamma_a\Gamma_b+\Gamma_b\Gamma_a = 2\delta_{ab}\mathbf{1}\tag{1}....
Gold's user avatar
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Apparent or real contradiction is in Eberlein's paper?

The questions I pose here concerns the paper "The Spin Model of Euclidean 3-Space" by W. F. Eberlein (The American Mathematical Monthly, Vol. 69, No. 7 (Aug. - Sep., 1962), pp. 587-598) (...
mma's user avatar
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Self-adjoint unbounded Toeplitz operator

Let $M$ be a complete manifold and $E\to M$ a Dirac bundle (Hermitian vector bundle equipped with a Clifford action $c\colon T^*M\to \rm{End}(E)$ and a Clifford connection $\nabla$). The Dirac ...
amnesiac's user avatar
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Equivariant Multiplicative Formula for Sphere Bundles in the Index Theorem

I am reading through the proof of the Atiyah-Singer index theorem, out of Lawson-Michelson, and I'm a bit confused about the proof of multiplicative property for indices of sphere bundles, where I ...
Elie Belkin's user avatar
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Stiefel-Whitney Classes for Physicists

I am a theoretical physicist and i am trying to understand better how spin structure works. I understand fairly decently Riemannian geometry but have little to zero knowledge of algebraic/differential ...
LolloBoldo's user avatar
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How to express spinors with an arbitrary change of coordinates?

A spinor field in classical differential geometry is defined as a section of a spinor bundle, which is by definition the associated bundle constructed from the spinor representation and the ...
A. J. Pan-Collantes's user avatar
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Question about Cartan's Theory of Spinors, Section 53 a spinor is a Euclidean tensor

Context I'm studying spinors in detail as part of research project. I'm working through Cartan's Theory of Spinors [1]. In section 53, A spinor is a Euclidean tensor, Cartan asks us to, "Consider ...
Michael Levy's user avatar
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Why is Spin$(n)/\pm1 \cong$ SO$(n)$?

I know that there is a short exact sequence (where all maps are continuous): \begin{align} \mathbb{Z}/2\mathbb{Z} \rightarrow \text{Spin}(n) \rightarrow \text{SO}(n) \end{align} Showing that $...
Henry T.'s user avatar
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How to show that Spin(n) is connected.

I want to show that Spin$(n)$ is connected (for $n\geq2$) by the same argument as explained in the quote from the book in this question: link. I tried to justify why it suffices to find a path between ...
Henry T.'s user avatar
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Derivation of a spin connection in general relativity

On my journey to understand the mathematical structure behind general relativity, I came across the concept of a spin connection. which (I understand) is a connection defined in a spinor bundle. The ...
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Motivating spinors via the Dirac equation

I'm trying to motivate spin through Dirac's equation. So far, here is what I understand: Upon trying to take the "square root" of the space-time Laplacian (i.e. find an operator such that $D ...
Integral fan's user avatar
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Spinor bundles and spin structures

I want to check my understanding of the relationship between spinor bundles and spin structures. So, as far as I understand, given a principal $\operatorname{SO}(n)$-bundle $\pi : P_{SO} \rightarrow M$...
Tomás's user avatar
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Is a spin manifold automatically orientable and time-orientable?

I'm reading Hamilton's Mathematical Gauge Theory. He defines a manifold to be spin if the second Stiefel–Whitney class of the manifold vanishes (p. 379). Does this condition already implies that the ...
Níckolas Alves's user avatar
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Special Vector Fields Preserving Harmonic Spinors

Let $(M,g)$ be a Riemannian manifold and $(\Sigma,\mathrm{cl}^g,h,\nabla^h)$ be a Hermitian Dirac bundle and $D$ its Dirac operator. Let $\psi$ be a harmonic spinor, i.e. $D\psi=\mathrm{cl}^g\circ\...
Viktor Majewski's user avatar
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What is the topology of a Clifford algebra?

I'm reading Hamilton's Mathematical Gauge Theory and I'm currently studying the Pin and Spin groups. Hamilton defines them as specific subsets of a Clifford algebra, and it is understood that the ...
Níckolas Alves's user avatar
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I am confused about a certain property of spinor transformations that seems to be inconsistent based on my current understanding.

I am considering the transformation of a two dimensional Weyl spinor $\lambda^\alpha$ given by a matrix transformation of the form $p(\lambda)^\beta = \lambda^\alpha m_\alpha^{\ \ \beta}$. Let's say ...
Teddy Baker's user avatar
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Cohomology ring of $BPin(n)$ with Z coefficients

I cannot find any reference for the characteristic classes of $Pin(n)$ with coefficients in $\mathbb Z$. What is the cohomology ring $H^*(BPin(n); \mathbb Z)$ ? Is it true that $H^*(BPin(2); \mathbb ...
Overflowian's user avatar
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Intuition about r-spin structures

I should preface this by saying that I am a physicist. My question pertains to the paper https://arxiv.org/abs/1802.09978, where $r$-spin structures are defined (Def. 2.1). To orient the discussion, ...
johnny's user avatar
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Connection on spinor bundle induced by Levi-Civita Connection

Given a Levi-Civita connection $\nabla$ on the tangent bundle with bundle metric $g$, a spinor representation $(\rho, S)$ and a spin structure $P$, what is the induced connection on a spinor bundle $...
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Irreducible representations of the identity component of the Spin group

Let $Spin(p,q)$ be the real spin group to the quadratic space $\mathbb{R}^{p,q}$ defined via Clifford algebras as $Pin(\mathbb{R}^{p,q}) \cap Cl^0(\mathbb{R}^{p,q})$, meaning as the subset of the ...
anonymous250's user avatar
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Proof double covering of $SO(V,Q)$ by $Spin(V,Q)$ is local homeomorphism

Let $(V,Q)$ be a finite-dimensional quadratic space over either the real or complex numbers with a non-degenerate, but not necessarily positive-definite quadratic form $Q$. We define the Spin group $...
anonymous250's user avatar
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Is the Eigen spectrum of a matrix completely defined by the algebra of its parts?

Consider two vector spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, where $0<n<m<\infty$. Now, I'd like to define matrices $A\in\mathbb{C}^{n\times n}$ and $B\in\mathbb{C}^{m\times m}$ in the ...
Jun_Gitef17's user avatar
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1 answer
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how to show all spin groups are double covers?

If that's the definition, then how do we know double covers of $SO(n)$ exist?
Alex's user avatar
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Introducing internal spin into the mechanical equations of an orbiting particle

In the preface of Lawson & Michelsohn's Spin Geometry, they mentioned some physics relevant to their book which I find interesting: The theory of Dirac had another interesting feature. In the ...
Dasheng Wang's user avatar
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First few cases of the Clifford algebra $\text{Cl}_{r,s}$

I have been reading Lawson & Michelsohn's Spin Geometry, and in the beginning of Chapter $1$ $\S 4$ on classification of the Clifford algebra $\text{Cl}_{r,s}$, they say that With little ...
Dasheng Wang's user avatar
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Graded representation of the Clifford algebra $\text{Cl}_{r,s}$

I have been reading Lawson & Michelsohn's Spin Geometry, and there is a remark about the decomposition of $\text{Cl}_{r,s}$ in terms of $\mathbb Z_2$-graded tensor product below the propostion $3....
Dasheng Wang's user avatar
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Associated graded algebra of Clifford algebra $\text{Cl}(V,q)$

I have been reading Lawson & Michelsohn's Spin Geometry, there is a sentence in the proof of Proposition $1.2$ that I don't understand. The proposition is the following: For any quadratic form $q$...
Dasheng Wang's user avatar
2 votes
1 answer
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Deducing inequality from exact triangle in Heegaard Floer homology

In Hom's lectures on Heegaard Floer homology, pages 8 and 9 contain a proof that $rk \widehat{HF}(Y) \geq |H_1(Y, \mathbb{Z})|$ for rational homology spheres. The proof involves using an exact ...
horned-sphere's user avatar
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Question on double covers of Lie groups and Spin groups isomorphisms

In physics, both $SO(3)$ and $SO^{+}(3,1)$ Lie groups are of paramount importance. Things start to become very curious when we realize that in Quantum Physics (Non-Relativistic Quantum mechanics and ...
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What justifies the statement that a Dirac spinor can be written as two Weyl spinors?

I am cross listing this from physics SE in case it is more appropiate here. That post can be found here: https://physics.stackexchange.com/questions/794843/what-justifies-the-statement-that-a-dirac-...
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Why are Dirac spinor representations defined as a projection onto the first factor?

Let $\mathbb{C}l(n)$ denote the Clifford algebra over $\mathbb{C}^n$ with the standard bilinear form. Then $$\mathbb{C}l(n) \cong \begin{cases} \text{End}(\mathbb{C}^N) \quad n \text{ is even}\\ \text{...
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What is the relationship between the kernel of a homomorphism and its degree as a covering map?

Let $\phi: A \rightarrow B$ be a continuous homomorphism between Lie groups (or any appropriate generalization). Is there any relationship between the degree of $\phi$ as a covering map of and the ...
CBBAM's user avatar
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Explicit triality representation in Spin(8)

First of all, I'm from a physics background, so pardon my probable lack of mathematical rigor in my question. Let's say I have an 8-dimensional Clifford algebra $C\ell_8$ with generators $e_i$, for ...
user38680's user avatar
1 vote
1 answer
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Adjoint representation of $V\subset \text{Cl}(V,q)$

I have been trying to learn Clifford algebra from Lawson & Michelsohn's Spin Geometry. There is a geometric explanation for the adjoint representation formula ($V$ is a vector space, $q$ is a ...
Dasheng Wang's user avatar
1 vote
0 answers
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Torsion spin-c structures on $S^1\times S^2$

I have been reading the paper https://arxiv.org/pdf/1902.04050.pdf by Zemke and at some point we have the following: My question is, how can one make sense of torsion $\mathrm{Spin}^c$ structures on $...
horned-sphere's user avatar
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Seeking Correct 2x2 Matrix Representation for CL(1,1) in Clifford Algebra

I'm working with the Clifford algebra $ \text{CL}(1,1) $ and attempting to find an appropriate $ 2 \times 2 $ matrix representation. This algebra corresponds to a space with the metric signature $ (1, ...
Anon21's user avatar
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Formulating $Spin^c(3,1)$ Connection and Curvature on a $GL+(4,R)/Spin^c(3,1)$ Structured Manifold

I am exploring a geometric framework where the usual metric tensor role (as in $GL^+(4,\mathbb{R})/\text{SO}(3,1) $) is replaced by a structure defined by the quotient $ GL^+(4,\mathbb{R})/\text{Spin}^...
Anon21's user avatar
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Doubts on definition of spin structure

In Friedrich's book "Dirac operators in Riemannian geometry" there is this definition for spin structure. Let $X$ be a CW-complex and $(Q, \pi, X, SO(n))$ a $SO(n)$-principal bundle on $X$. ...
Wukong's user avatar
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2 answers
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Understanding's Wikipedia's definition of a spinor

I am trying to understand spinors from a mathematical view. I've seen similar questions on this website but I'm still unclear on what they are exactly. On Wikipedia they state: Although spinors can ...
CBBAM's user avatar
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4 votes
1 answer
284 views

Topological definition of Spin$(p,q)$?

In short, How can we define Spin(p, q) without referencing Clifford algebras? The answer should be something like "Spin$(p, q)$ is the unique double cover of SO$^+(p, q)$ such that ...". ...
WillG's user avatar
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4 votes
1 answer
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Are spinors in 3 dimension quaternions, or are they elements of $\mathbb C^2$?

In Wikipedia both statement is present, which one is true? In one hand, here stands: def 1. Thus the (real) spinors in three-dimensions are quaternions, and the action of an even-graded element on a ...
mma's user avatar
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How does the following representation of $\mathbb{C}l(3)$ decompose into irreducibles?

Consider the three dimensional complex Clifford algebra $\mathbb{C}l(3)$ and the following representation $$S=\text{Span}\{|0\rangle,c^\dagger|0\rangle,\gamma^3|0\rangle,\gamma^3c^\dagger|0\rangle\},$$...
Ivan Burbano's user avatar
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Is there any spin group $Spin(n)$ for any value of $n$ which is isomorphic to $SU(3)$?

I was wondering about the spin group and its relations with the unitary groups. Like we have $Spin(2)\simeq U(1)$, $Spin(3) \simeq SU(2)$. I was wondering whether this relation can be taken further to ...
Eden Zane's user avatar
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Union of spin-c structures

Let's say we have two 4-manifolds with boundary $M_1$ and $M_2$, with spin-$\mathbb{C}$ structures $s_1$ and $s_2$. Let's say also there is an orientation reversing diffeomorphism $f:\partial M_1 \to \...
horned-sphere's user avatar
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Does $Spin(n,m)$ double cover only $SO(n,m)^+$

Subtleties like this are often glossed over in physics texts so I thought I'd ask here. Again, Does $Spin(n,m)$ double cover only the component of $SO(n,m)$ connected to the identity (often denoted ...
R. Rankin's user avatar
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How to solve spinorial differential equations.

I've asked this question in the Physics Exchange Forums, but I think this is a more appropaite site to ask this question. How would you solve the following differential equation, and find its ...
Álvaro Rodrigo's user avatar

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