The spin-geometry tag has no usage guidance.

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### 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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340 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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321 views

### Spin and SO groups associated to a degenerate symmetric bilinear form

In "Spin geometry" by Lawson and Michelsohn it is defined the Clifford algebra $Cl(g)$ associated to a symmetric bilinear form $g$ in general, including the degenerate case. But the rest of the book ...

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999 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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### Relative index theorem for Clifford linear Dirac operators

Dear community,
there is relative index theorem due to Gromov and Lawson (Thm. 4.18 in POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS) which states that
...

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545 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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282 views

### Group action on spin^c 4-manifold.

[edit]
I'll try to be more precise.
In paper N.Nakamura, "Bauer–Furuta invariants under $Z_2$-actions" there is an assumption that $Z_2$ action "lifts to spin^c structure". What i think it means:
...

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122 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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335 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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646 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.
...

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676 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 $\bar{\epsilon}$ ...

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**1**answer

566 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

341 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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528 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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769 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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### 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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444 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 ...

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268 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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423 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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278 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 ...

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594 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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### 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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### 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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701 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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### 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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### 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 ...

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### 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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### 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 ...

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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.
...

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566 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 ...

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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 ...

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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 ...