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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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Adjoint of the Coupled Covariant Derivative on Spinors

I want to understand the proof for a $C^0$ bound of solutions to the Seiberg-Witten equations. Among other places, it can be found in Kronheimer, Mrowka: "The genus of embedded surfaces in the ...
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Spin structure and characteristic classes

I do not know if anyone can help me with these doubts of spin structures and characteristic classes. 1) Is there an orientable manifold that is not spin? 2) Is there a finite group $ G $ such that ...
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Spin structure and bordism

I have some questions about bordism and spin structures on manifolds. If you have any examples or references I would appreciate it. Is there a 3-manifold $ M $, orientable, which does not support 3 ...
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1answer
64 views

spinor representation

I am trying to study Special unitary group of order 2 and some textbooks mention objects transform under special unitary group are called Spinors then How can we represent a spinor using matrix ?
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Spinors - Groups and Double Cover of Lorentz Group

As part of a project, I keep coming across a small nit-picking area regarding the spinor group $SU(2)$. The Lorentz group can be thought of as the group of rotations in $SO(1,3)$. I am under the ...
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Spinor bundle of a fibration over a circle

I have a question: I am reading an article and the author is looking at a submersion $Z \rightarrow M \xrightarrow{\pi} S^1$ with $Z$ is of even dimension. We have $TM = TZ \oplus T S^1$. Given ...
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53 views

Does the triviality of the orthonormal frame bundle imply the triviality of the spin bundle?

Let $M$ be a space and time orientable spin semi-Riemannian manifold of signature $(p,q)$, ${\rm Fr}(M)$ be its bundle of space and time oriented pseudo-orthonormal frames, $\Lambda : P\rightarrow {\...
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Can we 'build' spinor structure not only from a Riemann Manifold but 'extract it' also from another algebraic structures?

I want to understand what type of structures are Spin Structure: are a monoids, ringoids, groups? Can we build spinor structure find also from another structures not 'extract it' only from a Riemann ...
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82 views

Computing $\Omega_1^{\text{Spin}}\cong \Bbb Z_2$

I'm trying to understand why $\Omega_1^{\text{Spin}}\cong \Bbb Z_2$. I know it's a pretty standard computations but I'd like to have an explicit description (and explanation) of what's going on. As ...
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Why do the $\Gamma$ matrices behave like vectors and tensors in the spinor representation of SO groups?

One of the things that confuse me most when I study group theory and quantum field theory is that I constantly run into the situations where $\psi C \Gamma_M\chi$ are treated like vectors, $\psi C\...
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1answer
74 views

$W$ spin implies $\partial W$ spin

Let $M$ be a compact orientable manifold with the first two Stieffel-Whitney numbers equal to zero (this is my definition of SPIN manifold). Let $B$ be the boundary of $M$; I want to show that $B$ is ...
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Any spin $M^3$, exists a natural induced $\text{Pin}^-$ structure on Poincare dual PD

It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural ...
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Double cover of $\operatorname {SO}(V\oplus V^*) $

We know that $\operatorname{Spin}(V \oplus V^*)$ is the double cover of $\operatorname {SO}(V\oplus V^*)$ via the map $$\rho: \operatorname {Spin}(V \oplus V^*)\rightarrow \operatorname {SO}(V\oplus V^...
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Elliptic boundary condition eigenvalue problem

In Chapter 1, section 1.5 of "The Dirac Spectrum" by Nicolas Ginoux, different elliptic boundary conditions for Dirac operators are introduced. On page 24, there is the following theorem Theorem 1....
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1answer
78 views

Spinor representation for $\operatorname{Spin}(V \oplus V^*)$

I'm studding Hitchin's Generalized Calabi-Yau Manifolds https://arxiv.org/abs/math/0209099 and I've stuck here: Suppose that $V$ is a vector space and denote its dual by $V^*$. Now we know that the $\...
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49 views

references for physical gauge theory and spinors.

Does anyone know of any good references for physical examples of gauge theory (as a mathematically precise theory of connection on principal bundles). Simple examples will do (e.g the $U(1)$ ...
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$H^1(M,\mathbb{Z}_2)$: 1st Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure

$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient, it is often to see that we say the 1st Stiefel Whitney class ...
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Pin group isomorphisms

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 ...
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Clifford algebra is isomorphic to exterior algebra, proof in Lawson's

This Proposition 1.2, pg 10 of Lawson's Spin Geometry. Context: We have a homomorphism of graded algebra $$ \wedge^*(V) \rightarrow G^*$$ induced on each component from the map $$\wedge^r(V) \...
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Clifford Multiplication and Spinor Bundles

I am trying to follow the discussion of Clifford multiplication on page 384 of The Wild World of 4-Manifolds, by Alexandru Scorpan (link, although I hope this will be totally self-contained), and I'm ...
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67 views

spin mapping class group of circles

The MCG of the circle is $\mathbb{Z}/2$, generated by an orientation-reversing diffeomorphism. The circle has two spin structures, a periodic one and an anti-periodic one. Each has a nontrivial ...
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Pinors vs Spinors

I was reading the paper "The Pin Groups in Physics: C, P, and T" by M. Berg, C. Morette-DeWitt et al. in which they analyze the (double) covering groups of (Lorentzian) orthogonal groups $\...
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Spin groups and the Quaternionic Representation

Quaternionic representation is also known as the pseudo real representation. For the $Spin(n)$ group, it looks that only $Spin(3+8k), Spin(4+8k), Spin(5+8k)$ representation are pseudo real. ...
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Transitive action of $H^2(M;\Bbb Z)$ on $Spin^c$ structures over $M$

I’ve a problem understanding why the action of the second cohomology group (integer coefficients) of an oriented smooth manifold $M$ is free and transitive on the set of $Spin^c$. I’m following these ...
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Characteristic classes of spinor bundle

Given a spin structure on a oriented Riemannian manifold $(M,g)$, a spinor is a section of the spinor bundle $\pi:\mathbf{S}\to M$. I am trying to calculate the characteristic classes of the spinor ...
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Dependence of spinor bundle on choice of metric

For an oriented Riemannian manifold $(M^n,g)$ with spin structure, one can define the spinor bundle $\pi_g:\mathbf{S}_g\to M$. The space of metrics is convex. So if $g_t=(1-t)g_0+tg_1$ is a family of ...
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Is this operator pseudodifferential or trace-class?

Let $M$ be a closed manifold and $D$ a Dirac operator on $M$. Then (the closure of) $D^2+1$ has a bounded inverse $$(D^2+1)^{-1}:L^2\rightarrow H^2,$$ where $H^2$ is the second Sobolev space. In ...
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some questions on spin group

The spin group of an inner product space $V$ is defined in terms of the Clifford algebra of $V$, which is spanned by products of vectors in $V$. Does any vector in $V$ correspond to a ...
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A calculation from Berline-Getzler-Vergne

Let $M$ be a manifold. Suppose that $D$ is a Dirac-type operator on a $\mathbb{Z}_2$-graded Clifford module $E\rightarrow M$, in the sense that $D^2$ is a generalised Laplacian. Define the the action ...
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348 views

Do oriented null cobordant manifolds admit spin structures?

Let $M$ be an oriented null cobordant manifold. Since $M$ is oriented its first Stiefel-Whitney class vanishes. Since $M$ is null cobordant all of its Stiefel-Whitney numbers vanish. Is it known if ...
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Reference request for the irreps of the Spin group

I'm searching for a reference request where all irreducible representations of the Spin group or of $\mathfrak{so}(n)$ are classified. It seems to be 'well-known' that the Lie algebras correspond to ...
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“Square root” of a decomposition of a homogeneous polynomial to harmonic and $x^2 q$.

It is well known that any homogeneous polynomial $f \in \mathbb R[x_1, \ldots, x_n]$ can be uniquely split as $f = f_0 + x^2 f_1$, where $x^2 \equiv (x_1)^2 + \ldots + (x_n)^2$ and $f_0$ harmonic: $\...
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1answer
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On Proposition 2.6 Gualtieri Thesis Generalized complex geometry

I'm working with Gualtieri's thesis about Generalized complex Geometry and I don't understand the proof of the Proposition 2.6 (p. 7). It says Every maximal isotropic subspace (maximal totally null ...
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1answer
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Spinor chiral transformation by $\psi \to \gamma^5 \psi$

Let $\psi$ be a spinor. Let $\gamma^0,\gamma^1, \gamma^2, \gamma^3$ be the usual gamma matrices and the fifth $\gamma^5 : = i\gamma^0\gamma^1\gamma^2\gamma^3.$ Then if we define $\psi \to \psi' := \...
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Understanding of Spin(n) and SO(n)

I want to make sure I understand the relation between spin and rotation (mainly between SU(2) and SO(3), but also in general). (I am a physics major, so I apologize if my statements are not very ...
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Inclusion between spin groups?

I think this should have an answer, but I can't see what it is. It's inspired by the section labelled "Spinors" in Parker's and Taubes's paper, "On Witten's Proof of the Positive Energy Theorem." Here'...
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Handle attachment and spin$^c$ structures

My apology for the uninformative title; I don't think my question can be compressed into one line. I'm trying to understand the relation between handle attaching and spin$^c$ structures. A particular ...
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143 views

Spinor bundle of a spin Manifold is a Clifford bundle

I'm following Gompf and Stipsicz book about $4$-Manifold and Kirby Calculus. Here (page 34) they claim that the spinor bundle $S\to X$ over a spin compact manifold $X$ of dimension $n$ (even) is a ...
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Meaning of notation in John Roe's Elliptic Operators book

In the section about the asymptotic expansion for the heat kernel in the "Heat and wave equations" chapter of John Roe's book, he includes the following in one of this proposition: Suppose that $M$ ...
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Spin Bundle and Connection on $R^3$?

What is the spin connection on the spin bundle $S$ over $R^3$? Let metric be $dx_1^2+dx_2^2+dx_3^2$ with orientation $dx_1 \wedge dx_2 \wedge dx_3$. From my understanding, the spin bundle over $R^3$ ...
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Majorana flip relation

In supergravity book by Freedman and Proyen, enter image description here but I couldn't derive the first equation by using last two.
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$spin(n)$ equivariance of Dirac operator

Let $D$ be the Dirac operator on $\mathbb{R}^n$ i.e. $D=\sum_{j=1}^nE_j\frac{\partial}{\partial x_j}$ where $E_j$ are $2^r \times 2^r$ matrices (where $n=2r$ or $n=2r+1$) satisfying $E_i^2=I, \ \ ...
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Obstruction theoretic approach to complex spin structures

It seems this question is extremely well-known in literature, but I have poor understanding on these stuffs, and it seems nobody asks this question before in math.stackexchange, so I decided to ask ...
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Can somebody show an explicit example of a spinor?

I've seen a lot of articles of spinors defined using representation theory, Clifford Algebras, etc. But i want to see (if its possible) how a spinor "looks like" and how it changes sign with a 2pi ...
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Clifford action of exterior derivative of self-dual 2-form

I am confused about a claim on page 30 of the lectures notes of Hutchings and Taubes on Seiberg-Witten theory (https://math.berkeley.edu/~hutching/pub/tn.pdf). Background: Let $M$ is a Riemannian 4-...
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The correspondence between complex spin structures and associated clifford module structure

The usual definition of a $spin^c$ structure on a riemannian manifold $M$ is often given by (the equivalence class of) the lifting of the frame bundle $P\to M$ of the tangent bundle $TM$ via the ...
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1answer
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Spin structure and restriction to fibers

Let us assume that $E$ is an oriented Riemannian vector bundle equipped with spin structure. Therefore there is a $spin(n)$ principial bundle $spin(E)$ and an equivariant map $\eta:spin(E) \to SO(E)$ ...
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187 views

Representations of $S O( n )$ coming from $GL( N ,\mathbb{ R})$

I would like to show that "the finite-dimensional spinor representation of $SO(N)$ does not arise from a finite-dimensional representation of $GL(N)$" , as stated here and here. Apparently we should ...
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Swing-Twist decomposition for quaternion rotations (verification)

PRZEMYSLAW DOBROWOLSKI has written a paper that (I think) can be applied to swing-twist decompositions for quaternion rotations called "SWING-TWIST DECOMPOSITION IN CLIFFORD ALGEBRA" . I tried to ...
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Confusion with two different notations for heighest weights of spin representations

I don't understand how the two notations for $SO(n)$ heighest weights in fact 1 and 2 are related; moreover, what are the weigths of all other irreducible spinorial representations in notation 2? ...