The spin-geometry tag has no usage guidance.

**3**

votes

**0**answers

31 views

### $p$-forms acting on spinors

I'm trying to understand the paper by Atiyah, Hitchin and Singer called: ''Self-duality in four dimensional Riemannian geometry", available here.
I'm stuck at the point where it explains how the ...

**2**

votes

**2**answers

145 views

### Converse to Lichnerowicz Vanishing Theorem?

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies ...

**3**

votes

**1**answer

80 views

### Transitivity of $Spin(7)$ in triples of vectors

I have a simple question: transitivity of $Spin(7)$ in triples of orthogonal vectors. Let $Spin(7)\subset SO(8)$ act on $\mathbb{R}^8$, and $e_1,e_2,e_3$, $v_1,v_2,v_3$ be two triples of mutually ...

**6**

votes

**2**answers

234 views

### Index of Modified Dirac Operator

Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, ...

**5**

votes

**0**answers

100 views

### Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows:
Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...

**5**

votes

**0**answers

133 views

### The Operator $\overline{\partial} + \overline{\partial}^*$ on an Hermitian Manifold

Every compact Kähler manifold has a canonical $spin^c$ structure. Moreover, the associated Dirac operator is isomorphic to $\overline{\partial} + \overline{\partial}^*$, acting on ...

**3**

votes

**1**answer

91 views

### Parallel Transport on Hypersurface Spinor Bundle

So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link:
...

**7**

votes

**2**answers

225 views

### 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 ...

**1**

vote

**1**answer

57 views

### Spaces of Killing spinors for different orientation

Simply put, I want to understand how a change of orientation on a Riemannian spin manifold can change the space of Killing spinors.
To be more precise:
Let $M$ be a spin manifold (i.e. the first and ...

**22**

votes

**1**answer

790 views

### 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 ...

**5**

votes

**2**answers

167 views

### Spin Structures for Quaternionic-Kaehler and Hyper-Kaehler Manifolds

As is well-known (see Friedrich's book for example) every Kähler manifold is spin (or at least spin$^c$) and the Dirac is given (up to a twist) by $\partial + \partial^*$. What happens in the ...

**4**

votes

**1**answer

119 views

### Explanation that Twistor Space of $S^4$ is $\mathbb{C}P^3$?

I am attempting to read Atiyah's paper on self-duality in four-dimensional Riemannian geometry, and I came across the following basic example:
Let $S_-$ be the $SU(2)$-bundle of anti-self dual ...

**4**

votes

**0**answers

216 views

### 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 ...

**3**

votes

**0**answers

141 views

### The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$

It is commonly known that we have a chain of embeddings
$$SU(4)\subset Spin(7)\subset SO(8)$$
(there is more than one possible $Spin(7)$, just take one).
Which is the explicit analog for the Lie ...

**2**

votes

**1**answer

155 views

### Constant spinors from constant forms

Let $(X,g)$ be a $m$-dimensional complex, hermitian, spin manifold and let us denote by $S_{\mathbb{C}}$ its complex spinor bundle. Then:
$S_{\mathbb{C}}\simeq \Lambda_{\mathbb{C}}(X)$
Let $\nabla$ ...

**2**

votes

**0**answers

101 views

### The most general set-up for tensors and connections

This is maybe a too vague question, so I will try to be as specific as possible. My question is:
What is the most general set-up where one can define tensors and connections?
For example, we know ...

**1**

vote

**1**answer

83 views

### Cayley Subspaces in a Calibrated 8-Space

Suppose we are given $(\mathbb{R}^8,\Phi)$, where $\Phi$ is the self-dual 4-form that defines $Spin(7)\subset SO(8)$ (Cayley calibration, see Notes on the Octonians, page 23). Now some 4-subspaces $V$ ...

**4**

votes

**1**answer

195 views

### Frame-bundle reduction from spinor-bundle reduction

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times_{\rho}\Delta$ its spinor bundle, ...

**21**

votes

**1**answer

744 views

### 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 ...

**0**

votes

**0**answers

75 views

### Bispinors, polyforms, bilinears and supersymmetric manifolds

Let $(V,q)$ be a regular quadratic vector space, and let us denote by $Cl(V,q)$ the corresponding Clifford algebra. Then there exists an isomorphism of $\mathbb{Z}_{2}$-graded algebras:
...

**1**

vote

**0**answers

63 views

### Convention on Clifford Product

When studying the Clifford Algebra associated to some $(V,Q)$, one finds two basic identities differing by a sign:
$vv=Q(v)$ (see, for instance, Wikipedia)
$vv=-Q(v)$ (see, for instance, MathWorld ...

**1**

vote

**1**answer

150 views

### $Spin(7)$ as stabilizer of a $4$-form revisited

For a better understanding of this question, please see the question and answer here.
In $Spin(8)$ there are plenty of copies of $Spin(7)$; consider, for instance, the antiimage of $SO(7)<SO(8)$ ...

**2**

votes

**1**answer

234 views

### Octonions product: inversion in the right and identity in the left

Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena ...

**2**

votes

**0**answers

152 views

### Dirac operator in Generalized Geometry

I am wondering how the Dirac operator can be built in the context of Hichin's generalized geometry.
In particular, I have the following questions:
On a spin manifold, is the conventional spin ...

**1**

vote

**1**answer

176 views

### Isomorphisms of Positive and Negative Spinor Bundles

Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf, 1998):
2.6 Spin Bundles and the Dirac Operator
...

**5**

votes

**2**answers

361 views

### Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...

**1**

vote

**1**answer

124 views

### Twistors for spaces of $n-$dimensions

I was wondering if there is any general definition for twistors for spaces of any dimension with a definite (or indefinite) metric; for example, in $\mathbb{R}^3$.
Twistors are spinors of the ...

**14**

votes

**2**answers

1k views

### 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 ...

**5**

votes

**1**answer

328 views

### 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 ...

**1**

vote

**1**answer

170 views

### Equation for non-invertible elements in Clifford algebras

Suppose we have a Clifford algebra $Cl(V,q)$, $V\simeq \mathbb{R}^n$ and $q$ non-degenerate bilinear form. Then every non-zero element of $V\subset Cl(V,q)$ invertible, but they are not the only ones ...

**4**

votes

**2**answers

574 views

### Different definitions of spin structures

This is the definition of spin structure according to Wikipedia:
which is supposed to be the standard definition. But in the book The Geometry of Four-Manifolds (Donaldson-Kronheimer, page 76) one ...

**3**

votes

**1**answer

112 views

### Local index formula for >ungraded< elliptic operators

Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic ...

**1**

vote

**0**answers

113 views

### Norm of the operator acting on spinor bundle

Please forgive me if the question is too elementary, but however I was unable to manage by myself. The question comes from J.Varilly, H.Figueroa and J. Gracia-Bondia book "Elements of noncommutative ...

**4**

votes

**0**answers

140 views

### Element in spin group

I've got the following question: why is it true (if it really is?), that if I have a unitary element $u$ in the (real) Clifford algebra $Cl(V,g)$ which is even and the operator $\varphi(u)$ defined ...

**2**

votes

**1**answer

197 views

### Norms on Clifford algebra (C^* norm)

Basically I'm interested in operator algebras such as $C^*$ or von Neumann algebras. However I decided to learn a bit about noncommutative geometry (in particular spectral triples). Before doing this ...

**4**

votes

**0**answers

133 views

### spectral sequence differential for cobordism

From page 6 of these solutions:
the differential\begin{equation}d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})\end{equation}connecting the 1-st and the 2-nd row is the ...

**4**

votes

**1**answer

186 views

### Construction of a Bott manifold

I have been searching the literature for a construction of a simply connected spin manifold of dimension 8 with A-genus 1. I am not sure, but I think this is called a Bott manifold.
Can anybody help ...

**1**

vote

**1**answer

238 views

### A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$
is integral, for some almost complex structure ...

**3**

votes

**1**answer

179 views

### All symplectic manifolds have $Mp^c$-structures?

Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times_{\mathbb Z_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a ...

**1**

vote

**1**answer

165 views

### Connection on canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on symplectic manifold

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic 4-manifold $(M, \omega)$, with a compatible $J$ and $g$, so
\begin{equation}
{W_ + } = {T^{0,0}}{M^*} ...

**17**

votes

**2**answers

462 views

### 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 ...

**1**

vote

**0**answers

208 views

### Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...

**1**

vote

**1**answer

355 views

### How to understand two examples of spin bundle

I am confused by two examples of spinor bundles over 4-manifolds, which I saw in various places:
(1) The spinor bundle $S = S_+ \oplus S_-$ associated to a spin or spinc structure of Riemannian ...

**8**

votes

**2**answers

472 views

### 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 ...

**1**

vote

**1**answer

215 views

### Contact structures and adjunction inequality in 3-manifolds

It is a theorem of Eliashberg that in a tight contact 3-manifold $(M, \xi)$ we have the adjunction inequality $|\langle e(s),[\Sigma] \rangle| \leq -\chi(\Sigma) $ where $s=s(\xi)$ is the ...

**9**

votes

**4**answers

594 views

### Duality between K-theory and K-homology in the non-spin^c case.

I posted this question on Math.SE (http://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 ...

**5**

votes

**0**answers

167 views

### Duality between K-theory and K-homology in the non-compact, spin$^c$ case

Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong ...

**7**

votes

**2**answers

204 views

### is there an anyon structure analogous to spin structure for rank 2 bundle?

A spin structure on a Riemannian bundle of rank >2 is the lift of the structure group from $\text{SO}(n)$ to its universal cover $\text{Spin}(n).$ It may also be defined in the case $n=2$ as the lift ...

**1**

vote

**0**answers

140 views

### Orthogonal transformations with trivial spinor norm as product of reflections $r_w$ with $(w,w)=-2$

I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard I mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element of $O^+(\Lambda)$, ...

**14**

votes

**3**answers

1k views

### 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 $TM$
...