Questions tagged [spin-geometry]

For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}^{\pm}(n)$ 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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4 votes
1 answer
213 views

Spin$^c$ structures induced by an almost complex structure

Let $M$ be a closed spin$^c$ $4$-manifold with determinant line bundle $L$. If $c_1^2(L)=2\chi(M)+3\tau(M)$, where $\chi$ and $\tau$ denote the Euler characteristic and signature of $M$ respectively, ...
8 votes
2 answers
506 views

Is there a purely topological definition of $\text{Spin}(p,q)$?

I'm cross-posting this question from Math.SE, as it didn't get much attention there (even after a bounty). A common way to define the group $\text{Spin}(p,q)$ is via Clifford algebras. However, $\text{...
0 votes
0 answers
72 views

Does every Spin$(7)$-manifold has a unit-length spinor?

Say $M$ be a manifold with a Spin$(7)$-structure. $M$ is spin and hence spin$^c$. Say $S=S_+\oplus S_-$ be a spin$^c$-bundle on $M$. Does $S_+$ has a nowhere vanishing section? The result is true if ...
1 vote
0 answers
61 views

Spin(7)-instanton

Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the ...
5 votes
1 answer
386 views

A question about the existence of spin maps

Let $M, N$ be two smooth manifolds, not necessarily spin. My question is the following: How can we construct a non-constant spin map $f:M\to N$ of degree zero? Here spin map means that $f$ preserves ...
4 votes
1 answer
293 views

Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$

In Whitehead tower of $BO$, there is a induced fiber sequence: 1. $$ Z_2 \to B SO \to BO \overset{w_1}{\rightarrow} B Z_2 $$ How does this map $\overset{w_1}{\rightarrow}$ from $BO$ to $B Z_2$? ...
4 votes
0 answers
63 views

Pfaffian elements and anomalies

If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...
2 votes
2 answers
160 views

$String/CP^{\infty}=Spin$ or a correction to this quotient group relation

We know that there is a fiber sequence: $$ ... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ... $$ Is this fiber sequence induced from a short exact sequence? If so, is that $$ 1 \to B^2 Z = B S^...
3 votes
2 answers
475 views

Calculation of the top Chern class of spinor bundle over $S^{2n}$

It's well known that for a complex vector bundle $E$, we have $$c_n(E)=e_n(E_\mathbb{R}) $$ But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class ...
8 votes
1 answer
450 views

Definition of a spin group

$\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}$This follows on from Definition of Pin groups?, which notes there are three different definitions of the Pin group; thankfully, all of ...
1 vote
1 answer
103 views

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 ...
8 votes
1 answer
347 views

Euclidean and Minkowski Majorana spinors - inconsistency with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf Somehow I find some inconsistency with his Tables of Euclidean and ...
1 vote
1 answer
250 views

Uniqueness of spinor representation

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$I asked a similar question on math stack exchange here, but I wonder if it may be better received here. Let $n$ be ...
1 vote
1 answer
223 views

Spin connection vs. Cartan connection

I am studying the tetradic Palatini formalism of general relativity. In this formalism, one usually considers a manifold $M$, which is either non-compact or compact with Euler-characteristic $\chi(M)=...
2 votes
0 answers
64 views

Group of invertible elements in a degree 4 central simple algebra with symplectic involution with norm in a center

Let $A$ be a central simple algebra of degree 4 (i.e. dimension 16) over a field $F$ with $\mathrm{char}(F) \neq 2$. It is known that any such algebra is a tensor product $D_1 \otimes D_2$ of two ...
4 votes
1 answer
321 views

When the Pontryagin square is an even class?

Let $n$ be an even integer and $X$ a manifold. Given a cohomology class $B \in H^k(X,\mathbb{Z}_n)$, the Pontryagin square is a class $\mathfrak{P}(B)\in H^{2k}(X,\mathbb{Z}_{2n})$. Is it true that if ...
2 votes
0 answers
92 views

Chern number of positive spinor bundle

What is the second chern number $c_2(V_+)$ of the positive spinor bundle on a 4-manifold, in particular $S^4$? Why is it that $V_+$ is the same as the quaternion line-bundle? Thanks,
2 votes
0 answers
165 views

Proof of the Hirzebruch-Riemann-Roch theorem using the Atiyah-Singer index theorem

I am trying to read the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), but I do not understand the few last steps (theorem 4.11, page ...
3 votes
1 answer
317 views

Existence (or non existence) of principal bundle charts compatible with an $f$-reduction

I asked this question on math stack exchange here, but I wonder if it would be better received here. Let $\pi:P\rightarrow M$ and $\pi':P'\rightarrow M$ be principal $G$ and $H$ bundles respectively, ...
2 votes
1 answer
211 views

Necessary and sufficient conditions for pseudo Riemannian manifold to be time orientable

It is well known that a smooth manifold $M$ is orientable if the first Stiefel-Whitney class of the tangent bundle vanishes. In particular, this implies that if $M$ is equipped with a pseudo-...
1 vote
0 answers
62 views

Action of volume form on spinors in odd dimension

We know that for a smooth orientable manifold of dimension $2n, i^n$ times the volume form acts as identity on the positive spinors and acts as minus identity on the negative spinors via Clifford ...
13 votes
4 answers
2k views

Representations of Pin vs. Representations of Clifford

This may be total nonsense. But I need to know the answer quickly and I am too tired to think about it thoroughly. Let $k$ be a positive integer. Roe's "Elliptic Operators" claims that there ...
3 votes
0 answers
126 views

Spin structures on surfaces in terms of homology classes

It is well known that the spin structures on an oriented surface (with boundary) $M$ are in bijection with the set of cohomology classes $H^1(M,\mathbb{Z}/2)$. By Lefschetz duality, these correspond ...
8 votes
2 answers
360 views

Two different spin structures of the real projective space $\Bbb RP^3$

It is known that every orientable 3-manifold has a spin structure, because its tangent bundle is trivial. Also it is known that if a manifold $X$ has a spin structure, then the number of distinct spin ...
1 vote
0 answers
87 views

Norm of Killing spinor

A Killing spinor on a Riemannian spin manifold is a section of the spinor bundle satisfying the equation: \begin{align*} \nabla_X\phi=\lambda X\cdot\phi \end{align*} Here $X$ is a vector field and $\...
-4 votes
1 answer
202 views

What are the applications of spin geometry? [closed]

What are applications of spin geometry to physics? Does it have something to do with gravity?
6 votes
1 answer
246 views

Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard ...
2 votes
0 answers
78 views

Weitzenbock- Anti-selfdual

In "The Theory of Gauge Fields in Four Manifolds", B.Lawson proves the Bochner-Weitzenbock, for an anti-self-dual field $\Psi \in \Omega^2_-(\mathfrak{G}_E)$,where $\mathfrak{G}_E$ is the ...
4 votes
1 answer
1k views

Conformal Killing spinors

In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time). If $\epsilon$ and the $\bar{\epsilon}$ ...
43 votes
2 answers
5k views

Meaning/origin of Seiberg-Witten equations/invariants

Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from. We take ...
6 votes
1 answer
296 views

Is there a representation of $\mathrm{SU}_8/\{\pm 1\}$ that doesn't lift to a spin group?

$\newcommand{\GL}{\mathrm{GL}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\SU}{\mathrm{SU}}\newcommand{\Spin}{\mathrm{Spin}}\renewcommand{\O}{\mathrm O}\newcommand{\R}{\mathbb R}\newcommand\Z{\mathbb Z}$...
2 votes
0 answers
51 views

Orthonormal eigenspinors of the gauge-covariant dirac operator on $\mathbb{R}^4$, with extra conditions are possible?

Let $G$ be a simple Lie group, and $V$ a representation. Consider $\mathbb{R}^4$ with its flat Euclidean metric. Let $P$ be the trivial $G$-bundle on $\mathbb R^4$ equipped with some (non-trivial) ...
1 vote
0 answers
75 views

Spinors in dimension 6

From the representation of $\operatorname{Spin}(6)\cong \operatorname{SU}(4)$, one can deduce that on a $6$-dimensional manifold we get the postive spinor bundle from the usual $4$-dimensional ...
11 votes
6 answers
3k views

Explicit Spin Structures on the Torus

Basically, I am trying to build explicit examples of Dirac operators. To this end, I'm looking at the surface E = C/(Z + λZ) - for some λ in H \ SL(2,Z) - with the Euclidean metric and ...
2 votes
0 answers
61 views

Manifold with totally geodesic boundary is spin if and only if its double is spin

Let $(M,g)$ be a Riemannian manifold with totally geodesic boundary $\partial M$. Let $(DM,Dg)$ be the double of $(M,g)$ obtained by reflection of across $\partial M$. I'm looking for a reference for ...
6 votes
1 answer
271 views

Weitzenböck formula and comparison of norms

Let $M$ be a closed Riemannian manifold with a spin$^\mathbb{C}$ bundle $S$. Now for a spin connection $A,$ and a spinor $\phi,$ it can be shown that $C\lvert\nabla_A\phi\rvert^2\geq \lvert D_A\phi\...
2 votes
0 answers
106 views

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 $$ \omega_\mathbb{R}=c(e_1)\cdots c(e_m) $$ ...
0 votes
0 answers
124 views

Dirac operator on 4-dimensional rectangle with the periodic boundary conditions is self-adjoint? What is its spectrum?

Let us think of the Euclidean Dirac operator $iD^k \gamma_k$ on the rectangle $[-1,1]^4$ with the periodic boundary conditions. The covariant derivative $iD^k$ carries a gauge potential term and we ...
1 vote
0 answers
108 views

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 ...
1 vote
0 answers
79 views

Is Hodge decomposition detected in Clifford multiplication

This is a bit of a vague question, sorry for that. I am wondering if there's any detection of Hodge decomposition in terms of Clifford multiplication. For example if $\phi$ is a spinor and $\theta,\...
7 votes
2 answers
421 views

Quadratic forms on $\mathbb{R}^{16}$ coming from octonions

$\DeclareMathOperator\RRe{Re}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sym{Sym}$Let $\mathcal{H}_2(\mathbb{O})$ denote the (10-dimensional) real vector space of octonionic Hermitian matrices ...
20 votes
3 answers
3k views

Noncommutative smooth manifolds

Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples. Using his definition it is unclear how to separate the smooth structure from the metric. ...
2 votes
0 answers
129 views

Non-associative Clifford algebra

Let $V$ be a finite-dimensional $\mathbb{R}$ vector space equipped with a symmetric, bilinear form $b : V \times V \to \mathbb{R}$. My question is if there exists an analog of a Clifford algebra in ...
8 votes
1 answer
212 views

Isomorphisms of Pin groups

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 at ...
1 vote
0 answers
69 views

Relationship with between Clifford multiplication and pullback

Let $X$ be a smooth vector field on the even-dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
5 votes
1 answer
150 views

Is spin cobordism an invariant for surgery of codimension $q\ge3$?

Recall that a surgery of codimension $q$ on an $n$-manifold $X$ is a modification of $X$ of the following type. Let $\Sigma^{n-q}\subset X$ be a smoothly embedded $(n-q)$-sphere with a trivialized ...
0 votes
1 answer
181 views

Question about Clifford multiplication

Let $X$ be a smooth vector field on the even dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
3 votes
0 answers
167 views

Bound of the spinor element in Seiberg-Witten equation for a Kähler surface

Let's say we want to solve a perturbed version of SW equations on a closed Kähler manifold $(X,\omega):$ \begin{align*} &D_A\phi=0\\ &F_A+it\omega=q(\phi)=\phi\otimes\phi^*-\frac{|\phi|^2}{2}\...
4 votes
0 answers
82 views

Homomorphism from a product of spin groups to a bigger spin group

In the paper "Essential dimension of spinor and clifford groups" by Chernousov and Merkurjev, it says that there is a natural homomorphism $\operatorname{Spin}(n)\times \operatorname{Spin}(m)...
13 votes
2 answers
3k views

Spin^c structures on manifolds with almost complex structure

Let $M$ be a smooth even-dimensional manifold. Is it true that for each almost-complex structure $J$ on $M$ there exists a canonical spin$^c$ structure $S_J$ associated to $J$ ? (I've read this ...

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