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.
317
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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, ...
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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}^...
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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 $X$. ...
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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 ...
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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 ...". ...
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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 ...
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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
$$S=\text{Span}\{|0\rangle,c^\dagger|0\rangle,\gamma^3|0\rangle,\gamma^3c^\dagger|0\rangle\},$$...
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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 ...
- 305
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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 \...
- 805
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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 ...
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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 ...
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Spinors in Spin Geometry
First of all I am a physicist with a decent knowledge of graduate-level geometry.
I'm studying Spin Geometry from Bär Lecture Notes and I have some trouble understanding what spinors are from his ...
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Coefficient conditions for square root of the Laplacian
I am confused about what's going on in the attached picture (from the introduction of Friedrich's Dirac Operators in Geometry). The author claims that for an operator $P$ to satisfy $P = \sqrt{\...
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Is the total space of the frame bundle of a smooth manifold $M$ always spin?
I seem to remember reading in Kobayashi that the total space of a frame bundle is always parallelizable. If I'm remembering this correctly then the total space of the frame bundle $FM$ is always spin, ...
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A way to realize the $H^1(B;\mathbb{Z}/2)$ action on Pin- and Spin- structure
A Pin-structure on a real vector bundle $F^n\to B$ is a $Pin_n$ principal bundle $P^\#\to B$ with an isomorphism $P^\#\times_{Pin_n}\mathbb{R}^n\cong F$. A familiar property shows that $H^1(B;\mathbb{...
- 127
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Uniqueness of Spinor representation
Consider the group $\text{Spin}(n)$ with $n$ even, i.e. the double cover of $SO(n)$, with Lie group homomorphism $\lambda:\text{Spin}(n)\rightarrow SO(n)$. We know that the spin group admits a ...
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Why Representation of Clifford algebra are constant for an orthonormal frame?
Let $e_\alpha$ be a basis of the tangent bundle $TM$ and $ \rho: T_x M \rightarrow \operatorname{End}\left( W\right)$ a representation of a Clifford algebra.
In this text Field theory from a bundle ...
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Definition of Spin structure?
I've encountered the following definition of spin structure:
"Let $p:P\rightarrow B$ a principal $SO(p,q)$-bundle, a spin structure" (over the associated vector bundle with metric and ...
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Deriving an expression for the wave operator acting on the gravitational spinor: $\square \Psi_{ABCD} = 6\Psi_{EF(AB} \Psi_{CD)}{}^{EF}$
I am working through the exercises in the ‘Spinors’ section of Stewart’s book ‘Advanced General Relativity’.
Problem 2.5.7 asks to derive the spinor formula
$$
\square \Psi_{ABCD} = 6\Psi_{EF(AB} \...
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What is the representation that makes invariant Dirac equation?
In physics, the Dirac equation has the invariance under the representation of $SL(2,\mathbb{C})$ (see, wiki). So I expect the following to happen, in general mathematics:
Let $M$ be an oriented ...
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What's the explicit map building the double cover of SO(p,q)?
I've recently encountered the following statement:- "The map $r:Spin(p,q)\rightarrow SO(p,q)$ is a double cover". However in the proof there's no explicit construction of that map. Does ...
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What's a double cover?
I encountered the following definition:- "The $Spin^{\mathbb{C}}(p,q)$ group is
$$Spin^{\mathbb{C}}(p,q):=Spin(p,q)\times U(1)\bigg/(a,e^{i\theta})\sim(-a,-e^{i\theta})$$
and I also found the ...
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Stiefel-Whitney classes in Čech cohomology
Let M be a smooth manifold and Let G be a Lie group. We have the sheaf $\mathcal{F}_G=(F,\rho)$ of groups defined by
$$F(U)=C^{\infty}(U,G)$$
for open sets $U\subset M$ (in particular set $F(\emptyset)...
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Clifford algebra $Cl_1 \cong \mathbb{C}$
I'm reading Spin Geometry by Lawson and Michelsohn. In general Clifford algebras are defined by $Cl(V,q)=\mathcal{T}(V)/\mathfrak{I}(V)$, where $\mathcal{T}(V)$ is the tensor algebra of $V$ and $\...
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Definition of $\operatorname{Spin}^c(n)$ group
I have trouble with the definition of the $\operatorname{Spin}^c(n)$-group.
First off (for example in this paper), the group $\operatorname{Spin}^c(n)$ is defined as the double cover of $\operatorname{...
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Kernel for the Hodge chain homotopy
Let $M$ be a Riemannian manifold of dimension $n$. For $p\in \mathbb Z$, let
$C^p=A^p(M;\mathbb R)$,
$d:C^p\to C^{p+1}$ be the differential,
$*:C^p\to C^{n-p}$ be the Hodge star, satisfying $**=(-1)^{...
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Exponential of a spin connection belongs to what group?
This is probably a silly question but please humor me. In an orthonormal tetrad formulation of GR we define (co)frame fields $e^{a},e_{a}$ that everywhere obeys $e^{a}e_{b}=\delta_{b}^{a}$.
Choosing a ...
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Spinor representation of $SL_2(\mathbb{C})$ on Weyl Spinors.
Let $Cl(1,3)$ be the Clifford algebra for $\mathbb{R}^{1,3}$, i.e. $\mathbb{R}^4$ equipped with a Minkowski scalar product of signature $(-,+,+,+)$. The complexification of the Clifford algebra is $\...
- 2,164
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What is the definition of $\mathrm{Hom}_{\mathrm{Cl}}(S,E)$?
Let $V$ be an euclidean vector space, $\mathrm{Cl}$ its Clifford algebra, $S$ the spinor module and $E$ some Clifford module. The space $\mathrm{Hom}_{\mathrm{Cl}}(S,E)$ appears both in proposition $3....
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Lie algebra isomorphism $\mathfrak{spin}^+(t,s)\rightarrow so^+(t,s)$
Let $\eta$ be a pseudo Euclidian inner product of signature $(t,s)$ (i.e. in matrix notation $t$ $-1$'s down the diagonal, followed by $s$ $+1$'s down the diagonal, or rather in any orthonormal basis ...
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For every local section of a $SO(M)$ there exists precisely two local sections of $\text{Spin}(M)$
Let $M$ be an $n$ dimensional oriented riemannian spin manifold, then the bundle of oriented orthonormal frames admits a spin structures. That is a principal $\text{Spin}(n)$ bundle denoted $\text{...
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Representation of spin group carrying a symplectic form
I am currently studying the paper
Parker, Thomas; Taubes, Clifford Henry, On Witten’s proof of the positive energy theorem, Commun. Math. Phys. 84, 223-238 (1982). ZBL0528.58040. and have a question ...
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Nice way to show that a covering space of a Lie group is also a Lie group?
I apologize if this question is naive.
Suppose you know nothing about covering spaces, and they are introduced to you in the context of smooth manifolds, and Lie groups, of which you have a working ...
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Can we get a Spin-like connection from a group reduction ${\rm GL}^+(4,\mathbb{R})/{\rm Spin}^c(3,1)$ or $\exp {\rm CL}(3,1)/{\rm Spin}^c(3,1)$?
I understand from 1 that reducing the structure group ${\rm GL}^+(4,\mathbb{R})$ of the frame bundle $FX$ of a world manifold $X^4$ to the ${\rm SO}(3,1)$ group entails the Levi-Civita connection as ...
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The Spin version of ${\rm LX}/{\rm SO}(3,1)$ - is it $\widetilde{LX}/ {\rm Spin}(3,1)$?
Let X be an orientable smooth manifold and let LX be its frame bundle. It is well-known that the global section of the quotient bundle between the frame bundle LX (with structure group GL$^+$(4)) and ...
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Lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$: Stiefel-Whitney class and non/spin manifold
Define the lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$.
What is the property of Stiefel-Whitney class $w_1(TM)$ and $w_2(TM)$ for $M= L^k(n)$?
What is the spin or nonspin manifold property? Is ...
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Family of spin Dirac operators coming from path of metrics
Let $\mathbb{R}\ni t\mapsto g_t$ be a continuous path (in the $C^{\infty}$-topology of $\Gamma(\text{Sym}^2(T^*M))$) of smooth metrics on a compact spin manifold $M$. For each $t\in \mathbb{R}$, we ...
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An exponential map from CL(4,R) to Spin(3,1) which is surjective?
I am trying to find an algebra whose exponential maps surjectively to Spin(3,1).
I understand that its lie algebra is $\mathfrak{so}(3,1)$, but I think it leaves elements out.
I am working with the ...
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An attempt to derive the spin connection from a structure reduction.
In differential geometry, a structure reduction of the frame bundle $FX$ to ${\rm SO}(3,1)$ associates the pseudo-Riemannian metric to the global section of $FX/{\rm SO}(3,1)$.
I like the efficiency ...
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Equivalence of homotopical definition of spin structures with the "classical" one
Let $M$ be an oriented smooth surface, and let \begin{equation}\mathbb{Z_2} \to \operatorname{Spin}(2) \overset{\theta}{\to} \operatorname{SO}(2) \end{equation} be the usual central extension defining ...
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Difficult question on the Reduction of Dirac operators thru quantum symplectic reduction
Ok, so... we know from the literature how the $n$-particle version of the center of mass Calogero with coupling $k(k+1)$ arises from the reduction of the Laplace operator on $su(n)$. Namely, we have ...
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An explicit isomorphism between $\mathbb{R}^+ \times {\rm Spin}^c(3,1)$ and ${\rm GL}^+(2,\mathbb{R})\times {\rm GL}^+(2,\mathbb{R})$? [closed]
I am interested in the following isomorphism
$$
\begin{align}
\mathbb{R}^+\times {\rm Spin}^c(3,1)& \cong \mathbb{R}^+\times {\rm Spin}(3,1) \times {\rm U}(1) \tag{1}\\
&\cong \mathbb{C}^+\...
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Representing the spin group with rotors
In these notes 1, the author claims on page 7 that
One can define the Lie algebra of Spin(n) in terms of quadratic
elements of the Clifford algebra. This is what we will do here.
The author then ...
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Can the spin connection be defined entirely from a structure group reduction?
I understand from 1 that reducing the structure group $GL^+(4,\mathbb{R})$ of the frame field $FX$ of a world manifold $X^4$ to the $SO(4,\mathbb{R})$ group entails the Levi-Civita connection as the (...
- 2,551
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In what sense are Pauli matrices "intertwiners"?
I'm trying to understand the following quote from the Wikipedia article on the Pauli matrices.
More formally, this defines a map from $\mathbb {R} ^{3}$ to the vector space of traceless Hermitian $2\...
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Isotropic Subspace Implying the Existence of a Linear Equation Issue
Given a quadratic form in $n = 2 \nu + 1$ complex dimensions written as
$$F = (x^0)^2 + x^1 x^{1'} + ... + x^{\nu} x^{\nu'}$$
a vector $x = (x^0,x^1,...,x^{\nu},x^{1'},...,x^{\nu'})$, with $x^0,x^i,x^{...
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Recommended books for Spin Geometry [closed]
I would like you to tell me about the best introduction book for spin geometry, in particular, the following topics:
Clifford Algebras from the foundation to the classification and their ...
- 454
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Does SO$(V, Q)$ have a unique connected double cover?
Let $V$ be a real or complex finite dimensional vector space with nondegenerate quadratic form $Q$. According to the spin representation Wikipedia article,
Up to group isomorphism, SO$(V, Q)$ has a ...
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Are $\mathrm{Pin}(3, 1)$ and $\mathrm{GL}(2, \mathbb{C})$ isomorphic?
In physics, it’s very common to utilize the group (exceptional) isomorphism
$$
\mathrm{Spin}^{+}(1, 3) \approx \mathrm{SL}(2, \mathbb{C})
$$
in problem solving. I’m working with $\mathrm{Pin}$ ...
- 338
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Wedge product of representations of the Lorentz group
The (projective) irreducible representations of the Lorentz group which I denote by $\rho_{m,n}$ are classified by two nonnegative half-integers. I would like to decompose the grades $$
\wedge^k\rho_{...
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