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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Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for $n=2k+1$

Atiyah, Bott, and Shapiro paper on Clifford Modules around page 10 shows two facts. 1 - There is a lift $U(n) \to Spin^c(2n)$ from $U(n) \to SO(2n)\times U(1)$. Also an embedding (injective group ...
Марина Marina S's user avatar
8 votes
0 answers
277 views

Infinitely many nonempty Seiberg-Witten moduli spaces

The classic "finiteness" statement in Seiberg-Witten (SW) theory is that, for any smooth closed connected 4-manifold, there are only finitely many spin-c structures with nontrivial SW ...
Chris Gerig's user avatar
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8 votes
1 answer
349 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 ...
annie marie cœur's user avatar
1 vote
0 answers
38 views

Extending the Dirac operator on an open subset of a manifold and preserving positivity

Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
geometricK's user avatar
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1 vote
0 answers
75 views

Equivalence of $Spin^C$-Structures

I'm trying to understand the equivalence of $Spin^C(n)$-structures in the book "Dirac Operators in Riemannian Geometry" by Thomas Friedrich, p. 47 ff, but I got somehow stuck because I'm not ...
Peter Mischler's user avatar
5 votes
1 answer
530 views

A corollary of the non-existence of positive scalar curvature

I've been done some work with scalar curvature and managed to give a simple proof for the following result: Let $M$ be a closed manifold which do not admit a metric of positive scalar curvature. Then ...
L.F. Cavenaghi's user avatar
3 votes
0 answers
242 views

Research in spin geometry

I am currently learning differential geometry, but I have heard about the field of spin geometry and have skimmed through the book Dirac Operators in Riemannian Geometry by Thomas Friedrich. I have ...
Daniel Waters's user avatar
0 votes
0 answers
112 views

Dirac operator on a 5 dimensional tangent manifold with a $Spin(3)$-bundle

In p.3 of Witten paper from this Physics Letters B, Volume 117, Issue 5, 18 November 1982, Pages 324-328 Physics Letters B, 117(5), 324–328, he says that about the Dirac equation on a 5-dimensional ...
annie marie cœur's user avatar
7 votes
1 answer
263 views

Visualizing the wave operator in two dimensions

For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be ...
geometricK's user avatar
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0 answers
132 views

Clifford multiplication formula for $d\beta$ where $\beta$ is a $3$-form

Say $X$ be a $6$-dimensional compact Riemannian manifold which admits a $Spin^{\mathbb{C}}$ structure. Now I want to have a Clifford multiplication formula by $d\beta$ in terms of $\beta$ where $\beta$...
Partha's user avatar
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2 votes
0 answers
108 views

The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $(\Spin(...
wonderich's user avatar
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3 votes
1 answer
342 views

The normalizer of SU(n) in U(m)?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $...
wonderich's user avatar
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3 votes
0 answers
102 views

Spin structures induced on embedded circles and choices of trivialisations

I have a presumably basic question concerning spin structures that has me a bit confused. Let $C$ be a circle embedded in a spun manifold $X^n$. Given a choice of trivialisation of the normal bundle ...
user101010's user avatar
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1 vote
1 answer
264 views

The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?

$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that $$ \U(2^{N-1})\supset \Spin(2N) $$ when $2N > 4$ or a positive integer $N > 2$, so $\Spin(2N)$ can be embedded in $\U(2^...
wonderich's user avatar
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2 votes
1 answer
206 views

Clarification of different notions of spin structures

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$I am confused about the equivalence of some various definitions of spin structures and I was hoping for some help clearing out the fog. Let ...
user101010's user avatar
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2 votes
1 answer
342 views

Clifford multiplication formula on an almost complex manifold

$\DeclareMathOperator\End{End}$Following the deduction by John W. Morgan in his book The Seiberg–Witten equations and applications to the topology of smooth four manifolds, an almost complex manifold $...
Partha's user avatar
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1 vote
0 answers
169 views

Central extensions of orthogonal group by $C_2$

Suppose $(V,Q)$ is a quadratic space for definite quadratic form $Q$. It is stated in Pin groups that there are two central extensions of the orthogonal group $O(V)$ by the cyclic group $C_2$, ...
Ted Jh's user avatar
  • 191
2 votes
0 answers
222 views

Bryant-Salamon $G_2$ manifold on the spinor bundle over $S^3$

I am trying to understand the spaces constructed in R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy. My first problem is, essentially, about ...
F.T.'s user avatar
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2 votes
2 answers
229 views

Induced action by an involution on spinor bundle and Dirac operator

Let $M$ be a $4n$-dimensional spin manifold with a fixed Riemannian metric $g$. Let $S$ be a spinor bundle over $M$ and fix the Riemannian connection on it. There is a decomposition $S=S^+\oplus S^-$, ...
Kafka91's user avatar
  • 641
13 votes
2 answers
795 views

What is the relationship between spinors and supermanifolds and fermions?

I have the following two impressions about fermions in physics. I'm confused about their accuracy, and their compatibility: To consider the behavior of a fermion, whose intrinsic spin is described by ...
Tim Campion's user avatar
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22 votes
2 answers
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If the universal cover of a manifold is spin, must it admit a finite cover which is spin?

If $M$ is non-orientable, then it has a finite cover which is orientable (in particular, the orientable double cover). If $M$ is non-spin, then it does not necessarily have a finite cover which is ...
Michael Albanese's user avatar
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 ...
wonderich's user avatar
  • 10.3k
4 votes
0 answers
373 views

Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$ The $\Spin(1,3)$ is the ...
annie marie cœur's user avatar
7 votes
1 answer
194 views

Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$

For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e., $$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(...
user34104's user avatar
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3 votes
0 answers
457 views

Why in $S^2$ is there no spin structure? [closed]

For a Dirac fermion (spin half) on $S^2$, we have both the general covariant derivatives and the relativistic Hamitonian. What does the claim "in $S^2$ there is no spin structure" means? A reference ...
Quanhui Liu's user avatar
6 votes
0 answers
198 views

Spin structure using flag manifolds instead of a Riemannian metric

Let $(M,g)$ be an oriented Riemannian manifold of dimension $n$, and denote by $P_{\mathrm{SO}}\to M$ its oriented frame bundle. The usual definition of a spin structure is the data of a principal $\...
Pierre PC's user avatar
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9 votes
1 answer
293 views

Topological Spin manifolds in dimension 4

In his ICM Adress at Nice (Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, editeur, Paris 6 e ,1971, Volume 2, pp. 133-163.), Robion Kirby ...
Nicolas Boerger's user avatar
4 votes
0 answers
169 views

Understanding $w_2$ as an obstruction to trivializing the tangent bundle over 2-cells

I am reading through "A Geometric Proof of Rochlin's Theorem", and it is occurring to me, again, that I don't understand spin structures / $w_2$. My confusion arrises in, naturally, the proof of ...
user101010's user avatar
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7 votes
1 answer
353 views

Spectral gaps for spin manifold Laplace spectrum

For a (compact) spin manifold, we know that the eigenvalues $\lambda_n$ of the Dirac operator are countable, with finite multiplicity, and satisfy $$ |\lambda_n| \to \infty, ~~~ \text{ as } n \to \...
Fofi Konstantopoulou's user avatar
6 votes
0 answers
325 views

Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?

I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields. For example, the first Chern class of a complex line ...
Joe's user avatar
  • 535
3 votes
0 answers
45 views

Embedding of Riemannian symmetric spaces $E_I$ and $E_{IV}$ into Lie group $E_6$

In answer and comments to this mathoverflow question we have discussed possiblity of embedding Riemmanian symmetric spaces $E_I, E_{II}, E_{III},E_{IV}$ of dimension $42,40,32,26$ respectively into $...
user avatar
2 votes
0 answers
308 views

First Chern Class of Contact Structure which is not Torsion

Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...
Raffael's user avatar
  • 39
9 votes
0 answers
196 views

An equivalent definition for $\text{Spin}^c$-structures

I'm interested in proving the following proposition ([G], Remark page 48): Prop: A $\text{Spin}^c$-structure over an oriented vector bundle is equivalent (after stabilizing if the fiber dimension ...
Riccardo's user avatar
  • 1,998
2 votes
0 answers
194 views

Existence of a certain kind of compact spin manifold with boundary

For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...
M. L. Nguyen's user avatar
6 votes
0 answers
216 views

Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$

We know that the Arf-Brown-Kervaire (abk) invariant is a bordism invariant of $$ \Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z}), $$ where the $\mathbb{Z}/(8\mathbb{Z})$ is generated by a 2-manifold $M^...
annie marie cœur's user avatar
10 votes
1 answer
356 views

Discrete Pin structures

It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of ...
wonderich's user avatar
  • 10.3k
4 votes
1 answer
996 views

Spin groups in terms of matrices and/or linear operators

Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...
Libertron's user avatar
  • 329
5 votes
0 answers
156 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 $\...
Parisa Mahmoudi's user avatar
8 votes
2 answers
704 views

Lifting a diffeomorphism into a spinor bundle automorphism

I know several papers that treat this, but it seems that most of these papers do things very differently with quite different conclusions, so I am confused. Basically, when one tries to do classical ...
Bence Racskó's user avatar
2 votes
0 answers
92 views

Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry

We know well this short exact sequence $$ 1 \to \mathbb{Z}_2 \to Spin(N) \to SO(N) \to 1. $$ The $j$-th Stiefel-Whitney class of the associated vector bundle of $SO(N)$, as $w_j(V_{SO(N)})$, can be ...
wonderich's user avatar
  • 10.3k
2 votes
0 answers
122 views

The complex Clifford algebra

If $(E,g,w)$ is a vector space $E$ with a metric $g$ and a symplectic form $w$; then we can define the complex parts $(1,0)$ and $(0,1)$, so that the complex Clifford algebra is: $$e_1 . f_1 + f_1 . ...
AntBalan's user avatar
3 votes
0 answers
169 views

Pairing the Arf with Stiefel-Whitney class

The Arf invariant is a nonsingular quadratic form over a field of characteristic 2. The form that I looked at was: $$ S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;...
wonderich's user avatar
  • 10.3k
1 vote
0 answers
103 views

A generalization of the Clifford algebra

Let $(E,g)$ be a vector space with a symmetric bilinear form, and $a,b$ be two endomorphisms of $E$. The generalized Clifford algebra is defined by the free algebra of $E$ with quotient by the ...
A.Balan's user avatar
  • 187
3 votes
0 answers
151 views

The Dirac-Ricci operator

If we consider a spin manifold $M$, we can define the Ricci curvature $Ricc (X,Y)$ which is a symmetric tensor, moreover the spinors are defined, so that we can define a Dirac-Ricci operator: $$DR(\...
A.Balan's user avatar
  • 187
5 votes
0 answers
201 views

About mod 2 Index of Dirac Operators in 3D on Non-Orientable Manifold

I was reading Witten's paper "Fermion Path Integrals and Topological Phases" (https://arxiv.org/abs/1508.04715). He claimed that Indeed, on an orientable 3-manifold, the eigenvalues of the Dirac ...
Weicheng Ye's user avatar
2 votes
0 answers
73 views

Is there an analog of a Chern-Simons formula for the pfaffian $Pf(F)$ of a $SO(2n)$ curvature $F$?

..something similar to $tr(A \wedge dA + 2/3 * A \wedge A \wedge A)$ for $n = 2$ ?
user50311's user avatar
  • 305
14 votes
1 answer
565 views

Obstruction of spin-c structure and the generalized Wu manifods

Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the $$ H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
annie marie cœur's user avatar
10 votes
3 answers
708 views

Spin-H structures

Let us define a Spin-H structure as a reduction of a SO(n)-bundle by the group: $$Spin^H (n)=Spin(n) \times SU(2)/\{ 1,-1\}$$ The Spin-H structures are analogous to the well-known Spin-C structures ...
A.Balan's user avatar
  • 187
8 votes
0 answers
212 views

Are spin Hurwitz numbers $r$-spin Hurwitz numbers?

(I think the answer is no, but I'm not sure.) In Hurwitz theory, one counts $n$-fold branched covers $\Sigma'\to\Sigma$ of a Riemann surface $\Sigma$ with fixed ramification data around each branch ...
Arun Debray's user avatar
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1 vote
1 answer
68 views

Example of a certain partitioned manifold

I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$. (At first I ...
geometricK's user avatar
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