All Questions
Tagged with riemannian-geometry spin-geometry
45 questions
1
vote
0
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48
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Question on gamma matrices
Let $(M,g)$ be a pseudo-Riemannian spin manifold and let us denote by $S$ the spinor bundle, i.e. the associated vector bundle with respect to the spin representation. Usually, the "gamma ...
- 1,171
4
votes
0
answers
223
views
+50
A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ ...
- 954
2
votes
0
answers
30
views
Trace-free Hermitian endomorphisms in dimension $7$
Let $M$ be a spin-manifold of dimension $7$. Let $S\rightarrow M$ be a spin-bundle on $M$. Then Clifford multiplication ($c$) gives us the following isomorphism:
\begin{align*}
c:i\Lambda^2\oplus\...
- 954
0
votes
0
answers
134
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Positive mass theorem and Seiberg-Witten equations
Apologies for not a very rigorous question. I came across this PhD thesis by XIAO ZHANG, a student of Yau. From the thesis:
"We also investigate some
basic facts on Spin$^c$ structure on $4$-...
- 954
3
votes
0
answers
118
views
Decomposition of forms in $\operatorname{SU}(4)$-manifold
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let $(X,\Omega,\omega,J)$ be a manifold with an $\SU(4)$ structure. Since $\SU(4)\subset\Spin(7)$, $X$ also has a $\Spin(7)$-structure. I ...
- 954
1
vote
0
answers
100
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Dirac operator on $\operatorname{Spin}(7)$, $G_2$ and $\operatorname{SU}(3)$ manifolds
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let's take a $\Spin(7)$ manifold $M$ (the $\Spin(7)$ structure can have torsion), then the standard Dirac operator from negavtive spinors to ...
- 954
3
votes
0
answers
281
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A question in $\operatorname{Spin}(7)$ geometry
$\DeclareMathOperator\Spin{Spin}$I am looking for a proof of a fact (I think it's true intuitively due to representation theory) in $\Spin(7)$ geometry. Let's take a closed $\Spin(7)$-manifold $(M^8,g)...
- 954
1
vote
0
answers
72
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Spin(7)-instanton
Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the ...
- 954
2
votes
0
answers
86
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Weitzenbock- Anti-selfdual
In "The Theory of Gauge Fields in Four Manifolds", B.Lawson proves the Bochner-Weitzenbock, for an anti-self-dual field $\Psi \in \Omega^2_-(\mathfrak{G}_E)$,where $\mathfrak{G}_E$ is the ...
- 41
2
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0
answers
89
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Manifold with totally geodesic boundary is spin if and only if its double is spin
Let $(M,g)$ be a Riemannian manifold with totally geodesic boundary $\partial M$. Let $(DM,Dg)$ be the double of $(M,g)$ obtained by reflection of across $\partial M$. I'm looking for a reference for ...
- 308
2
votes
0
answers
126
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Question about Clifford volume element
I'm a little confused with the following:
Let $M$ be a $m$ dimensional Riemannian manifold and $e_1,\cdots,e_m$ be a local orthonormal base of $TM$. Let
$$
\omega_\mathbb{R}=c(e_1)\cdots c(e_m)
$$
...
- 563
6
votes
1
answer
299
views
Weitzenböck formula and comparison of norms
Let $M$ be a closed Riemannian manifold with a spin$^\mathbb{C}$ bundle $S$. Now for a spin connection $A,$ and a spinor $\phi,$ it can be shown that $C\lvert\nabla_A\phi\rvert^2\geq \lvert D_A\phi\...
- 954
1
vote
0
answers
116
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Existence of a local spinor bundle
I am confused about the existence of a local spinor bundle.
My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...
- 563
3
votes
2
answers
604
views
Calculation of the top Chern class of spinor bundle over $S^{2n}$
It's well known that for a complex vector bundle $E$, we have
$$c_n(E)=e_n(E_\mathbb{R}) $$
But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class ...
- 563
1
vote
0
answers
73
views
Relationship with between Clifford multiplication and pullback
Let $X$ be a smooth vector field on the even-dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
- 563
0
votes
1
answer
209
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Question about Clifford multiplication
Let $X$ be a smooth vector field on the even dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
- 563
3
votes
0
answers
171
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Bound of the spinor element in Seiberg-Witten equation for a Kähler surface
Let's say we want to solve a perturbed version of SW equations on a closed Kähler manifold $(X,\omega):$
\begin{align*}
&D_A\phi=0\\
&F_A+it\omega=q(\phi)=\phi\otimes\phi^*-\frac{|\phi|^2}{2}\...
- 954
5
votes
1
answer
355
views
Understanding the quadratic part in Seiberg Witten equation
Lets take a closed four manifold $M:=\Sigma_1\times \Sigma_2,$ where $\Sigma_i$s are compact Riemann surfaces. Now if $V$ and $W$ are Spin$^\mathbb{C}$ bundles on $\Sigma_1$ and $\Sigma_2$ ...
- 954
6
votes
0
answers
201
views
A generalized Dirac operator
Let $(M^4,g)$ be a closed four-dimensional Riemannian manifold and $J$ be an almost complex structure on $M$. Then for normal coordinate $e_1,\dots e_4$ at a point $m,$ and for a section $\alpha$ of a ...
- 954
3
votes
1
answer
428
views
Pull back of Spin$^{\mathbb{C}}$ bundle
Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{\mathbb{C}}$ bundle, $V_+$ denote the ...
- 954
4
votes
1
answer
282
views
Identifying a $4$-form on a $6$-dimensional manifold
Let $M$ be a closed $6$-dimensional Riemannian manifold with a spin$^{\mathbb{C}}$ structure. It is known that real $4$-forms on $M$ act on the positive-spinors as trace-free hermitian endomorphisms ...
- 954
1
vote
0
answers
38
views
Extending the Dirac operator on an open subset of a manifold and preserving positivity
Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
- 1,903
5
votes
1
answer
566
views
A corollary of the non-existence of positive scalar curvature
I've been done some work with scalar curvature and managed to give a simple proof for the following result:
Let $M$ be a closed manifold which do not admit a metric of positive scalar curvature. Then ...
- 1,774
7
votes
1
answer
266
views
Visualizing the wave operator in two dimensions
For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be ...
- 1,903
2
votes
0
answers
230
views
Bryant-Salamon $G_2$ manifold on the spinor bundle over $S^3$
I am trying to understand the spaces constructed in R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy.
My first problem is, essentially, about ...
- 213
2
votes
2
answers
243
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Induced action by an involution on spinor bundle and Dirac operator
Let $M$ be a $4n$-dimensional spin manifold with a fixed Riemannian metric $g$. Let $S$ be a spinor bundle over $M$ and fix the Riemannian connection on it. There is a decomposition $S=S^+\oplus S^-$, ...
- 641
2
votes
0
answers
197
views
Existence of a certain kind of compact spin manifold with boundary
For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...
- 121
10
votes
3
answers
757
views
Spin-H structures
Let us define a Spin-H structure as a reduction of a SO(n)-bundle by the group: $$Spin^H (n)=Spin(n) \times SU(2)/\{ 1,-1\}$$ The Spin-H structures are analogous to the well-known Spin-C structures ...
- 187
7
votes
0
answers
171
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Vanishing of K-theoretic index and positive scalar curvature
I'm confused about a seemingly basic point about a classical result on positive scalar curvature and would appreciate it if an expert could help me out.
Let $M^n$ be a closed spin manifold with ...
- 1,903
8
votes
0
answers
220
views
Regularilty of Commutative Spectral Triples
In Connes' approach to non-commutative geometry, the notion of a spectral triple is said to generalize compact Riemannian manifolds to the non-commutative setting. Motivating classical examples ...
- 173
9
votes
0
answers
1k
views
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 \...
- 9,330
3
votes
2
answers
339
views
Converse to Lichnerowicz Vanishing Theorem?
The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \...
- 787
6
votes
2
answers
476
views
Index of Modified Dirac Operator
Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, ...
- 787
5
votes
0
answers
179
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$ ...
- 1,407
4
votes
1
answer
382
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:
https://projecteuclid.org/...
- 787
1
vote
1
answer
107
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 ...
- 71
4
votes
2
answers
396
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 ...
- 41
19
votes
2
answers
4k
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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 ...
- 2,091
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....
- 9,330
1
vote
1
answer
418
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^*} \...
- 857
2
votes
1
answer
1k
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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 four-...
- 857
4
votes
1
answer
1k
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}$ ...
- 3,541
46
votes
4
answers
10k
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 (...
5
votes
3
answers
1k
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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 ...
- 3,218
2
votes
2
answers
763
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 ...
- 1,453