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Results tagged with spin-geometry
Search options questions only
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user 24078
For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}^{\pm}(n)$ and $\operatorname{Spin}^c(n)$. This tag should also be used for any questions about the geometry of spin manifolds, including questions involving Dirac operators and the Lichnerowicz formula.
6
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1
answer
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Spinor bundle tensored with certain line bundle gives the dual spinor bundle
Let $E$ be a $spin^c$ bundle and $L_E$ be a (complex) line bundle defined using transition functions $\nu \circ g_{U,V}$ where $\nu:spin^c(n) \to \mathbb{T}$ is map such that $\ker \nu=spin(n)$ and $g …
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7
votes
1
answer
539
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First Chern class of a specific line bundle
Let $E$ be a spin$^c$ bundle and $spin^c(E)$ the corresponding $spin^c(n)$-principial bundle. Let $g_{U,V}: U \cap V \to spin^c(n)$ denote transition functions for this principial bundle and consider …
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8
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0
answers
306
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Dixmier-Douady class is the third integral Stiefel-Whitney class
Let $M$ be (say smooth) manifold. From the short exact sequence of groups $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}_2 \to 0$ (where the first map is multiplication by $2$) one obtain long exact …
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7
votes
1
answer
528
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Topological obstruction for the existence of spin$^c$ structure
Recently I asked on stack exchange the following question: https://math.stackexchange.com/questions/2088888/vanishing-of-certain-cohomology-class-and-existence-of-spin-structure
I would like to know …
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9
votes
0
answers
1k
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Second Stiefel-Whitney class as an obstruction to the existence of spin structure
Let $M$ be an oriented (closed) Riemannian manifold. Choose a good open cover and local trivialisations of the tangent bundle $U_i$. Then we get a system of transition functions $\varphi_{ij}: U_i \ca …
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9
votes
1
answer
1k
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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 categories …
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1
vote
0
answers
151
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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 g …
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6
votes
0
answers
256
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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 vi …
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4
votes
1
answer
813
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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 …
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