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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What are "good" examples of spin manifolds?
I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:
What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin (...
43
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2
answers
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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 ...
29
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2
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How and why did mathematicians develop spin-manifolds in differential geometry?
First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that ...
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Triality of Spin(8)
Among simple Lie groups, $Spin(8)$ is the most symmetrical one in the sense that $Out(Spin(8))$ is the largest possible group. A description of this outer automorphism groups is as follows. $Spin(8)$ ...
26
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Harmonic spinors on closed hyperbolic manifolds
Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial?
I'm mainly interested in the 3-dimensional case ...
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answer
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Generalized geometry and spin structures
Let $(M,g)$ be a $d$-dimensional, oriented pseudo-Riemannian manifold, and $V$ the subbundle of $E=TM\oplus T^*M$ given by the graph of the musical linear isomorphism $g^\flat:TM\rightarrow T^*M$ ...
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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 ...
21
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Does Spin cobordism vanish in dimension $4k-1$?
For the purposes of a remark in a paper in preparation, I would like to know if anyone can confirm that $\Omega^{spin}_{4k-1} = 0$.
In the Atiyah-Patodi-Singer paper, Spectral asymmetry and ...
20
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3
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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.
...
19
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Exact Definition of Dirac Operator
Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ...
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what is a spinor structure?
There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$,
a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$
or ...
17
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An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space
Is there a closed, smooth, orientable manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$?
Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not ...
17
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Has Witten's perturbation on de Rham complex been studied on other elliptic complexes?
In his famous work, Supersymmetry and Morse theory, Witten perturbs de Rham complex by perturbing the exterior derivative
$$d_h=e^{-ht}de^{ht}.$$
And he proves Morse inequality using some spectral ...
16
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Open questions in "Spin geometry"
This is a very naive question. I have the impression that the area of "Spin geometry" is not an active research field. Sure Spin geometry is used in many different branches of mathematics and physics ...
15
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Spin structures on 7-dimensional spherical space forms
Background
Let $M$ be a spin manifold and let $\Gamma$ be a finite group acting freely and isometrically on $M$ in such a way that $M/\Gamma$ is a smooth riemannian manifold. The quotient will be ...
14
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1
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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{...
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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 ...
13
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Duality between K-theory and K-homology in the non-spin^c case.
I posted this question on Math.SE (https://math.stackexchange.com/questions/409444/), but got no answer. So I repost it here.
Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times ...
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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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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 ...
12
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1
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Spin structures on Sasakian manifolds and the Kähler analogy
A Sasakian manifold is often said to be the odd dimensional analogue of a Kähler manifold.
Now for a $2n$-dimensional Kähler manifold we know from Atiyah that it is spin exactly if the line bundle $\...
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6
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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 ...
11
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1
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Is a 4-dimensional submanifold of a spin manifold always spin?
Let $M^d$ be a $d$-dimensional orientable spin manifold, and $N^4$ is a closed $4$-dimensional orientable submanifold of $M^d$.
Is $N^4$ always spin?
If $d=5$, is $N^4$ always spin?
If $N^4$ is a ...
10
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5
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Dirac's Original Operator and the Hodge--Dirac Operator
For the usual $4$-dimensional Minkowski space $M$, the standard Dirac operator is given by
$$
D: C^{\infty}(M) \to C^{\infty}(M), ~~~~~ f \mapsto \sum_{i=1}^4 \gamma_i\frac{\partial f}{\partial x_i},
$...
10
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3
answers
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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 ...
10
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3
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Spin 4-manifold bounded by a mapping torus of tori
Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 &...
10
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2
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What is the "quaternionic" super Brauer group?
In addition to the two reasonably well-known categories $\mathrm{SuperVect}_{\mathbb R}$ and $\mathrm{SuperVect}_{\mathbb C}$ of real and complex super vector spaces, each of which is monoidally ...
10
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1
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356
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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 ...
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Reference request: Spin structures on surfaces and the spin mapping class group
I am looking for references on the following: Spin structures on surfaces, and particularly the spin mapping class group.
What is known about generating the spin mapping class group? Has anybody ...
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1
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Atiyah-Bott-Shapiro Orientation
Dear community,
there are so-called orientation maps $a:MSpin\to ko$ and $b:MSpin^c \to k$, "defined" in ABS's paper "Clifford modules". Unfortunately I am not familiar with representation theory.
...
9
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1
answer
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Commutative spectral triples
The corresponence between compact Hausdorff topological spaces and commutative unital $C^*$-algebras is rather well known: Gelfand Najmark theorem gives perfect correspondence between these categories....
9
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1
answer
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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 ...
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1
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Analog of "Spin" Chern-Simons Theory
3-dimensional Chern-Simons theories, with compact gauge group $G$, are determined by $H^4(BG)$. Looking at $U(1)$, with generator $c_1^2\in H^4(BU(1))=\mathbb{Z}$ for 1st Chern class $c_1$, there are ...
9
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Topology of the Universal Spinor Field Bundle
While reading article [1] below I came across the notion of a universal spinor bundle. This is defined at the beginning of section 6 (p.14) in [1] as follows: Let $M$ be a spin manifold and $\mathcal{...
9
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1
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$Spin^c$-Dirac-operator on the 3-torus
Consider the spinc structure on the flat standard 3-torus, which you get from the trivial (or any other) spin structure. Its associated vector bundle can be identified with a trivial bundle with fibre ...
9
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0
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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 ...
9
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0
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Sections of "forgetful" projections between flag manifolds
Given a subset $S\subseteq\{1,\cdots,n\}$ there is an associated flag manifold $F(S)$. Whenever $A\subseteq B$ there is a "forgetful" projection $F(A)\leftarrow F(B)$ (in fact I think its fibers are ...
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Second Stiefel-Whitney class as an obstruction to the existence of spin structure
Let $M$ be an oriented (closed) Riemannian manifold. Choose a good open cover and local trivialisations of the tangent bundle $U_i$. Then we get a system of transition functions $\varphi_{ij}: U_i \...
8
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2
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Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions
For an oriented $d$-manifold $M$, we can ask whether the manifold
admits a Spin structure, say, if the transition functions for the
tangent bundle, which take values in $SO(d)$, can be lifted to $\...
8
votes
2
answers
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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 ...
8
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2
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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 ...
8
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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{...
8
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1
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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 ...
8
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1
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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 ...
8
votes
1
answer
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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 ...
8
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1
answer
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Spin structure on mapping torus
I would like to know if, given a spin manifold $X$ and an orientation-preserving diffeomorphism $f : X \longrightarrow X,$ we can naturally endow the mapping torus $M_f = X \times [0, 1] / (x, 0) \sim ...
8
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1
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K-homology classes of Dirac operators on Hermitian manifolds
Given a compact Hermitian manifold $M$, we have three canonical pseudo-differential operators on the sections of complexified de Rham complex, namely
1) (d + d$^*,\Omega^{*})$
2) ($\partial$ + $\...
8
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0
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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 ...
8
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0
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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 ...
8
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0
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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 ...