The spin-geometry tag has no usage guidance.

**6**

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**1**answer

574 views

### 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.
...

**2**

votes

**1**answer

633 views

### Conformal Killing spinors

In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time).
If $\epsilon$ and the $\bar{\epsilon}$ ...

**2**

votes

**1**answer

521 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 ...

**32**

votes

**2**answers

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 ...

**2**

votes

**1**answer

339 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 $TM$ ...

**3**

votes

**2**answers

512 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 ...

**3**

votes

**1**answer

734 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?

**27**

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**4**answers

3k 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 ...

**4**

votes

**1**answer

431 views

### Spin structures and quadratic forms on surfaces

In his paper "Spin structures and quadratic forms on surfaces", Johnson constructs a bijection between the set of spin strucutres on a smooth closed orientable surface $S$ and the set of quadratic ...

**2**

votes

**1**answer

262 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 ...

**6**

votes

**1**answer

403 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 ...

**7**

votes

**0**answers

277 views

### Directed arcs on a surface

This question is a little odd. I have specific class of structures on a surface, which satisfy several nice properties, and I want to know if they are more natural geometric structures in disguise ...

**5**

votes

**3**answers

564 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 ...

**4**

votes

**5**answers

1k 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},
...

**2**

votes

**1**answer

345 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 ...

**6**

votes

**1**answer

675 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 ...

**23**

votes

**1**answer

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 $g^\flat:TM\rightarrow T^*M$ ...

**0**

votes

**1**answer

394 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 ...

**4**

votes

**2**answers

819 views

### Complex Projective Space Spin and Dirac: Part II

(1) Let $M$ be a complex manifold of real dimension $2n$, and denote the line bundle of complex $(N,0)$-forms by $\Omega^{(N,0)}(M)$. When $M = CP^N$, the line bundles are indexed by the integers, and ...

**4**

votes

**2**answers

764 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 ...

**7**

votes

**5**answers

1k views

### 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 is a ...

**16**

votes

**3**answers

1k views

### 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.
...

**8**

votes

**1**answer

553 views

### $Spin^c$-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 ...

**8**

votes

**6**answers

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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votes

**2**answers

577 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 ...