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.

Filter by
Sorted by
Tagged with
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, ...
Zhiqiang's user avatar
  • 881
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 ...
Partha's user avatar
  • 759
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 ...
Partha's user avatar
  • 759
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 ...
Radeha Longa's user avatar
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 ...
domenico fiorenza's user avatar
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 $B Z_2$? ...
zeta's user avatar
  • 337
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^...
zeta's user avatar
  • 337
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{...
WillG's user avatar
  • 133
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 ...
Chris's user avatar
  • 255
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 ...
amilton moreira's user avatar
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)=...
B.Hueber's user avatar
  • 987
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 ...
Seewoo Lee's user avatar
  • 1,901
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 ...
Andrea Antinucci's user avatar
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,
maden's user avatar
  • 41
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 ...
zarathustra's user avatar
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, ...
Chris's user avatar
  • 255
2 votes
1 answer
212 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-...
Chris's user avatar
  • 255
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 ...
Partha's user avatar
  • 759
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 ...
Tanny Sieben's user avatar
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 ...
user302934's user avatar
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 $\...
Partha's user avatar
  • 759
-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?
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 ...
maden's user avatar
  • 41
6 votes
1 answer
296 views

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

$\newcommand{\GL}{\mathrm{GL}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\SU}{\mathrm{SU}}\newcommand{\Spin}{\mathrm{Spin}}\renewcommand{\O}{\mathrm O}\newcommand{\R}{\mathbb R}\newcommand\Z{\mathbb Z}$...
Arun Debray's user avatar
  • 6,756
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 ...
Eric's user avatar
  • 181
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) ...
Isaac's user avatar
  • 2,727
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 ...
Partha's user avatar
  • 759
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 ...
user128470's user avatar
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 $$ \omega_\mathbb{R}=c(e_1)\cdots c(e_m) $$ ...
Radeha Longa's user avatar
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 ...
Isaac's user avatar
  • 2,727
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\...
Partha's user avatar
  • 759
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 ...
Radeha Longa's user avatar
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,\...
Partha's user avatar
  • 759
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 ...
asv's user avatar
  • 21.1k
6 votes
1 answer
247 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 ...
B.Hueber's user avatar
  • 987
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 ...
Julian Seipel's user avatar
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 ...
Radeha Longa's user avatar
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 ...
Radeha Longa's user avatar
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 ...
wonderich's user avatar
  • 10.3k
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 ...
Radeha Longa's user avatar
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}\...
Partha's user avatar
  • 759
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)...
YJ Kim's user avatar
  • 313
5 votes
1 answer
322 views

Understanding the quadratic part in Seiberg Witten equation

Lets take a closed four manifold $M:=\Sigma_1\times \Sigma_2,$ where $\Sigma_i$s are compact Riemann surfaces. Now if $V$ and $W$ are Spin$^\mathbb{C}$ bundles on $\Sigma_1$ and $\Sigma_2$ ...
Partha's user avatar
  • 759
1 vote
0 answers
132 views

Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$

Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
wonderich's user avatar
  • 10.3k
8 votes
0 answers
215 views

Computation of the 3-dimensional $\mathbb{Z}/m$-equivariant spin cobordism group (with possibly non-empty fixed-point set)?

$\newcommand{\odd}{\mathrm{odd}}\newcommand{\ev}{\mathrm{ev}}$Consider tuples of the form $(Y,\mathfrak{s},\widehat{\sigma})$ where: $Y$ is a closed oriented 3-manifold, $\mathfrak{s}$ is a spin ...
Ian Montague's user avatar
6 votes
0 answers
189 views

A generalized Dirac operator

Let $(M^4,g)$ be a closed four-dimensional Riemannian manifold and $J$ be an almost complex structure on $M$. Then for normal coordinate $e_1,\dots e_4$ at a point $m,$ and for a section $\alpha$ of a ...
Partha's user avatar
  • 759
6 votes
3 answers
805 views

Is a spin structure on a knot complement the same thing as an orientation of the knot?

There are a variety of characterizations of spin structures on the tangent bundle of a manifold. Two facts about them: Spin structures on $TM$ are an affine space over $H^1(M; \mathbb{Z}/2\mathbb{Z})$...
Calvin McPhail-Snyder's user avatar
3 votes
1 answer
373 views

Pull back of Spin$^{\mathbb{C}}$ bundle

Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{\mathbb{C}}$ bundle, $V_+$ denote the ...
Partha's user avatar
  • 759
4 votes
1 answer
275 views

Identifying a $4$-form on a $6$-dimensional manifold

Let $M$ be a closed $6$-dimensional Riemannian manifold with a spin$^{\mathbb{C}}$ structure. It is known that real $4$-forms on $M$ act on the positive-spinors as trace-free hermitian endomorphisms ...
Partha's user avatar
  • 759
1 vote
1 answer
186 views

Injective group homomorphism on $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/2}$ or $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/4}\to U(2^{2k})$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$From Pierre Deligne's Notes on spinors, we can see that there ...
Марина Marina S's user avatar

1
2 3 4 5