Questions tagged [spin-geometry]
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.
224
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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 $\Sigma_2$ ...
8
votes
0
answers
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Computation of the 3-dimensional $\mathbb{Z}/m$-equivariant spin cobordism group (with possibly non-empty fixed-point set)?
$\newcommand{\odd}{\mathrm{odd}}\newcommand{\ev}{\mathrm{ev}}$Consider tuples of the form $(Y,\mathfrak{s},\widehat{\sigma})$ where: $Y$ is a closed oriented 3-manifold, $\mathfrak{s}$ is a spin ...
1
vote
0
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132
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Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$
Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
6
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189
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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 ...
6
votes
3
answers
805
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Is a spin structure on a knot complement the same thing as an orientation of the knot?
There are a variety of characterizations of spin structures on the tangent bundle of a manifold. Two facts about them:
Spin structures on $TM$ are an affine space over $H^1(M; \mathbb{Z}/2\mathbb{Z})$...
3
votes
1
answer
373
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 ...
4
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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 ...
4
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1
answer
287
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Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for $n=2k+1$
Atiyah, Bott, and Shapiro paper on Clifford Modules around page 10 shows two facts.
1 - There is a lift $U(n) \to Spin^c(2n)$ from $U(n) \to SO(2n)\times U(1)$. Also an embedding (injective group ...
1
vote
1
answer
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Injective group homomorphism on $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/2}$ or $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/4}\to U(2^{2k})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$From Pierre Deligne's Notes on spinors, we can see that there ...
8
votes
0
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276
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Infinitely many nonempty Seiberg-Witten moduli spaces
The classic "finiteness" statement in Seiberg-Witten (SW) theory is that, for any smooth closed connected 4-manifold, there are only finitely many spin-c structures with nontrivial SW ...
7
votes
2
answers
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What is a formal definition of a Fermionic quantum field?
I could not locate a definition of Fermionic quantum field (like for an electron!) in even Kevin Costello's book, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.4961&rep=rep1&...
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vote
0
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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
vote
0
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Equivalence of $Spin^C$-Structures
I'm trying to understand the equivalence of $Spin^C(n)$-structures in the book "Dirac Operators in Riemannian Geometry" by Thomas Friedrich, p. 47 ff, but I got somehow stuck because I'm not ...
5
votes
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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 ...
3
votes
0
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Research in spin geometry
I am currently learning differential geometry, but I have heard about the field of spin geometry and have skimmed through the book Dirac Operators in Riemannian Geometry by Thomas Friedrich. I have ...
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0
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Dirac operator on a 5 dimensional tangent manifold with a $Spin(3)$-bundle
In p.3 of Witten paper from this Physics Letters B, Volume 117, Issue 5, 18 November 1982, Pages 324-328 Physics Letters B, 117(5), 324–328, he says that about the Dirac equation on a 5-dimensional
...
7
votes
1
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263
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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 ...
0
votes
0
answers
131
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Clifford multiplication formula for $d\beta$ where $\beta$ is a $3$-form
Say $X$ be a $6$-dimensional compact Riemannian manifold which admits a $Spin^{\mathbb{C}}$ structure. Now I want to have a Clifford multiplication formula by $d\beta$ in terms of $\beta$ where $\beta$...
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Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$We know that
$$
\Spin(1,3)=\SL(2,\mathbb C)
$$
and
$$
\Spin(4)=\SU(2) \times \SU(2).
$$
The $\Spin(1,3)$ is the ...
3
votes
1
answer
342
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The normalizer of SU(n) in U(m)?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $...
2
votes
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answers
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The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $(\Spin(...
3
votes
0
answers
102
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Spin structures induced on embedded circles and choices of trivialisations
I have a presumably basic question concerning spin structures that has me a bit confused.
Let $C$ be a circle embedded in a spun manifold $X^n$. Given a choice of trivialisation of the normal bundle ...
1
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1
answer
263
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The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?
$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that
$$
\U(2^{N-1})\supset \Spin(2N)
$$
when $2N > 4$ or a positive integer $N > 2$, so $\Spin(2N)$ can be embedded in $\U(2^...
2
votes
1
answer
202
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Clarification of different notions of spin structures
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$I am confused about the equivalence of some various definitions of spin structures and I was hoping for some help clearing out the fog. Let ...
2
votes
1
answer
340
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Clifford multiplication formula on an almost complex manifold
$\DeclareMathOperator\End{End}$Following the deduction by John W. Morgan in his book The Seiberg–Witten equations and applications to the topology of smooth four manifolds, an almost complex manifold $...
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0
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169
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Central extensions of orthogonal group by $C_2$
Suppose $(V,Q)$ is a quadratic space for definite quadratic form $Q$. It is stated in Pin groups that there are two central extensions of the orthogonal group $O(V)$ by the cyclic group $C_2$, ...
2
votes
2
answers
229
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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 $S=S^+\oplus S^-$, ...
2
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0
answers
221
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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 ...
17
votes
2
answers
603
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An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space
Is there a closed, smooth, orientable manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$?
Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not ...
22
votes
2
answers
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If the universal cover of a manifold is spin, must it admit a finite cover which is spin?
If $M$ is non-orientable, then it has a finite cover which is orientable (in particular, the orientable double cover).
If $M$ is non-spin, then it does not necessarily have a finite cover which is ...
13
votes
2
answers
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What is the relationship between spinors and supermanifolds and fermions?
I have the following two impressions about fermions in physics. I'm confused about their accuracy, and their compatibility:
To consider the behavior of a fermion, whose intrinsic spin is described by ...
10
votes
3
answers
708
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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 ...
6
votes
2
answers
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Action of the spin covariant derivative on gamma matrices?
How does the spin covariant derivative $\nabla^S_{\mu}$ act on gamma matrices satisfying: $\{\gamma^{\mu},\gamma^{\nu}\} = g^{\mu\nu}$, i.e.
$$\nabla^S_{\mu}\gamma^{\nu} = ?$$
where $\nabla^S := \...
7
votes
1
answer
194
views
Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$
For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e.,
$$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(...
3
votes
0
answers
453
views
Why in $S^2$ is there no spin structure? [closed]
For a Dirac fermion (spin half) on $S^2$, we have both the general covariant derivatives and the relativistic Hamitonian. What does the claim "in $S^2$ there is no spin structure" means? A reference ...
7
votes
1
answer
350
views
Spectral gaps for spin manifold Laplace spectrum
For a (compact) spin manifold, we know that the eigenvalues $\lambda_n$ of the Dirac operator are countable, with finite multiplicity, and satisfy
$$
|\lambda_n| \to \infty, ~~~ \text{ as } n \to \...
6
votes
0
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198
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Spin structure using flag manifolds instead of a Riemannian metric
Let $(M,g)$ be an oriented Riemannian manifold of dimension $n$, and denote by $P_{\mathrm{SO}}\to M$ its oriented frame bundle. The usual definition of a spin structure is the data of a principal $\...
9
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Topological Spin manifolds in dimension 4
In his ICM Adress at Nice (Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, editeur, Paris 6 e ,1971, Volume 2, pp. 133-163.),
Robion Kirby ...
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Understanding $w_2$ as an obstruction to trivializing the tangent bundle over 2-cells
I am reading through "A Geometric Proof of Rochlin's Theorem", and it is occurring to me, again, that I don't understand spin structures / $w_2$. My confusion arrises in, naturally, the proof of ...
6
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0
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Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?
I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields.
For example, the first Chern class of a complex line ...
21
votes
2
answers
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Does Spin cobordism vanish in dimension $4k-1$?
For the purposes of a remark in a paper in preparation, I would like to know if anyone can confirm that $\Omega^{spin}_{4k-1} = 0$.
In the Atiyah-Patodi-Singer paper, Spectral asymmetry and ...
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 ...
3
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Embedding of Riemannian symmetric spaces $E_I$ and $E_{IV}$ into Lie group $E_6$
In answer and comments to this mathoverflow question we have discussed possiblity of embedding Riemmanian symmetric spaces $E_I, E_{II}, E_{III},E_{IV}$ of dimension $42,40,32,26$ respectively into $...
2
votes
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308
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First Chern Class of Contact Structure which is not Torsion
Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...
4
votes
2
answers
831
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$spin_{\mathbb{C}}$ Connection and Charge Parity
From the paper "Gapped Boundary Phases of Topological Insulators via Weak Coupling" on page 11,
https://arxiv.org/abs/1602.04251
the authors states that on a curved manifold with a $spin_{\mathbb{C}}...
2
votes
0
answers
194
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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, ...
9
votes
0
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196
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An equivalent definition for $\text{Spin}^c$-structures
I'm interested in proving the following proposition ([G], Remark page 48):
Prop: A $\text{Spin}^c$-structure over an oriented vector bundle is equivalent (after stabilizing if the fiber dimension ...
10
votes
1
answer
355
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Discrete Pin structures
It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of ...
6
votes
0
answers
215
views
Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$
We know that the Arf-Brown-Kervaire (abk) invariant is a bordism invariant of
$$
\Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z}),
$$
where the $\mathbb{Z}/(8\mathbb{Z})$ is generated by a 2-manifold $M^...
4
votes
1
answer
990
views
Spin groups in terms of matrices and/or linear operators
Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...