# Tagged Questions

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### 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^*} ...

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**1**answer

231 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 ...

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**3**answers

828 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 $TM$
...

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**2**answers

268 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 ...

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**1**answer

220 views

### Spin-c Structures with Near-Symplectic Forms

Consider a smooth compact oriented 4-manifold $X$. Although not all 4-manifolds admit a spin structure, they do admit spin-c structures. And if $X$ does admit a spin structure, then there is a ...

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**0**answers

315 views

### Low Dimensional Spin Manifolds

I am looking for examples of 2- and 3-dimensional flat spin manifolds with Euclidean and Lorentzian signatures, which admit parallel spinors and the dimension of the space of the parallel spinors is ...

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**1**answer

677 views

### Representation theory of (anti)self-dual tensors

I am using usual physics notations and I guess the physics motivations of this question are obvious.
Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the Lie algebra be, ...

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**1**answer

446 views

### 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 ...

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100 views

### Topological index and Dirac operator with a non compact group

A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...

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289 views

### Cotetrad, spin connection and Dirac operator

Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) ...

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395 views

### Global Lichnerowicz Formula Proof (in the Kahler case)?

For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...

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2k views

### 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 an ...

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**1**answer

319 views

### Spin-c Structures viewed w.r.t. Cell Decomposition

In my quest to understand spin representations, I am looking at the equivalent views of spin structures (on some oriented Riemannian $n$-manifold). Given such a manifold $M$, its tangent bundle $TM$ ...

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455 views

### spin structures on full flag manifolds

It is known that any full flag manifold $G/T$ is a spin manifold.
For example, we can prove it using that $G/T$ is a complex manifold,
by computing its 1st Chern class as follows:
For full flag ...

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610 views

### first chern class and spin structures

Let M be a compact complex manifold. Then is it true that if the first Chern class of M is even, then M admits a spin structure?

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2k views

### 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 ...

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249 views

### $Spin^c$ structure on the mapping torus of an automorphism of the torus

Let $\alpha$ be an orientation-preserving automorphism of the torus $T^2 = S^1 \times S^1.$ Since the mapping torus $M_{\alpha} = T^2 \times [0, 1] / (x, 0) \sim (\alpha(x), 1)$ is an orientable ...

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327 views

### 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 ...

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491 views

### equivariant index of Dirac Operator on $S^{2}$

First, I have to admit that I don't have much knowledge of Spin Geometry and Index Theory, the question could be too simple or naive and secondly there may be too many questions.
Let $D$ be the ...

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973 views

### 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},
...

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304 views

### Twisting Spinor Bundles with Line Bundles

In a paper I am reading, the following framework was given: Let $S$ be a spinor bundle, over a Riemannian manifold $M$, with Clifford action
$$
c:S \otimes \Omega^1(M) \to S.
$$
Moreover, let $E$ be ...

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**1**answer

582 views

### Clifford Action for Kahler Manifolds

I'm working with Kahler manifolds at the moment and looking at their spin$^c$ structure. I really don't know much about spin$^c$ structures in general and don't have enough time to learn it all at ...

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1k views

### 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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**1**answer

376 views

### Square of the Dirac and the Laplacian on a K\"{a}hler Manifold

In the Euclidean setting, the Dirac operator was constructed so as to give the square of the Laplacian. Now for a K\"{a}hler manifold with a spin$^c$ structure we have the a corresponding Dirac ...

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**2**answers

707 views

### Spin structures on the Grassmannians

I am trying to understand spin structures and am looking at the specific case of complex projective space (viewed as the quotient $SU(N)/U(N-1)$) and more generally the Grassmannians (viewed as the ...

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1k views

### 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 ...

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548 views

### 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 ...