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4
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0answers
53 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
0answers
113 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
1answer
77 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: ...
6
votes
2answers
211 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
1answer
51 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
1answer
724 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
1answer
134 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
1answer
114 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
0answers
188 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
0answers
139 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
1answer
149 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
0answers
100 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
1answer
81 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
1answer
182 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, ...
20
votes
1answer
553 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
0answers
74 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
0answers
62 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
1answer
138 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
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0answers
145 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 ...
1
vote
0answers
148 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
1answer
160 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 ...
3
votes
1answer
263 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
1answer
119 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
2answers
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
1answer
262 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
1answer
163 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
2answers
524 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
1answer
104 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
0answers
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
0answers
133 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
1answer
185 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
0answers
130 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
1answer
168 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
1answer
230 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
1answer
166 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
1answer
163 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
2answers
444 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
0answers
205 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
1answer
328 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
2answers
452 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
1answer
213 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 ...
8
votes
4answers
555 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 ...
4
votes
0answers
157 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
2answers
200 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
0answers
137 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
3answers
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 $TM$ ...
0
votes
1answer
213 views

2Pi and 4Pi rotations in the Pin(1,3) group

Hi everyone, I'm currently studying the construction of the $Pin(1,3)$ group and given the definition I'm using to find its elements I'm having some problems with the signs associated with $2\pi$ and ...
15
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4answers
2k views

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. $Spin(8)$ ...
1
vote
1answer
297 views

Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal isotropics of $V\bigotimes \mathbb{C}$ ?

We say $L< (V\oplus V^{*})\bigotimes \mathbb{C}$ is isotropic when $< X,Y>=0$ for all $X,Y\in L$ Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal ...
9
votes
2answers
327 views

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 ...