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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29 views

Spinors and derivatives of spinors

Spinors $\psi$ rotate vectors. For instance $a'=\psi a\psi^{-1}$. In physics, one often encounters Lagrangians with spinors. For instance, the Weyl Lagrangian is $$ L=i\psi^t _R \sigma^u \partial_u ...
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Can $\operatorname{Spin}(n)$ be defined through the short exact sequence?

Define spin group $\operatorname{Spin}(n)$ to be the double cover group of $SO(n)$. Then it fits into the short exact sequence $0 \to \mathbb{Z}_2 \to \operatorname{Spin}(n) \to SO(n) \to 0$. My ...
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if a spin group is not simply connected, can the manifold be? [on hold]

Suppose we have a spin group with indefinite signature so that it is not simply connected. Does that mean the manifold is also not simply connected?
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Is $SU(n) \subset \text{Spin}(2n)$?

I believe this should be true for the following reason: The injection $SU(n) \hookrightarrow SO(2n)$ induces a map of Lie algebras $\mathfrak{su}(n) \hookrightarrow \mathfrak{so}(2n) \cong \mathfrak{...
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43 views

Existence of a spin structure in Kaehlerian manifolds

I have a few questions regarding the existence of a spin structure on Kaehlerian and hyperKaehlerian manifolds. I cannot seem to provide a reference for proofs or counterexamples, so references are ...
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Embed a Spin group into an Orthogonal group

How do we embed a Spin group to an Orthogonal group? Say a Spin($n$) group is a Lie group with $\frac{n \cdot (n-1)}{2}=$ Lie algebra generators. Say a Spin(10) group is a Lie group with $\frac{10 \...
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Embed a Spin group to a special unitary group

How do we embed a Spin group to a Unitary group? Say a Spin(10) group is a Lie group with $\frac{10 \cdot 9}{2}=45$ Lie algebra generators. Say a special unitary group SU($n$) has a $n^2-1$ Lie ...
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Is a “spinor” an element of the Spin group, or an object that transforms under the Spin group—or both?

I've been researching spinors, and I'm a bit confused by some of the terminology. In some cases, spinors seem to be presented as elements of the Spin group, whereas in others they seem to be presented ...
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Why consider this homology relation on vector fields?

I am studying Turaev's work on torsions of manifolds, specifically the paper Euler Structures, Nonsingular Vector Fields, and Torsions of Reidemeister Type. (I cannot find an open-access version of ...
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The definition of the Dirac operator

I apologize for asking what may be a very basic question. But I have recently begun a more mathematical introduction to spinors and I am stuck at the formal definition of the Dirac operator. ...
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What is the relation between structure groups or transition functions of two isomorphic vector bundles?

Vector bundle $S=\Lambda^{\bullet}T^*M \otimes |\det TM|^{\frac{1}{2}}$ is called an spinor bundle for the bundle $TM\oplus T^*M$. it is an associated bundle to a $Spin(n,n)$-principal bundle where ...
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Explicit homeomorphism $SL(2,\mathbb{C})\cong SU(2)\times\mathbb{R}^3$.

Take a look at Gallier, pg. 207. There is a homeomorphism $$ SO^+(p,q)\cong SO(p)\times SO(q)\times \mathbb{R}^{pq}. $$ Hence the universal cover of $SO^+(p,q)$ is $$ \text{Spin}(p)\times\text{Spin}(q)...
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When is the spin group a universal covering?

The double covering $$ \text{Spin}(n)\to SO(n) $$ is the universal covering for $n\geq 3$ because $\pi_1(SO(n)) = \mathbb{Z}_2$ in this case. Analogously, $$ \text{Spin}(1,n)\to SO^+(1,n) $$ is the ...
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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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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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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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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, $\psi C\...
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$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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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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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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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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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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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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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 ...