Questions tagged [dirac-operator]

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Pfaffian elements and anomalies

If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...
domenico fiorenza's user avatar
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78 views

Definition of Clifford super-connections

I have some questions concerning the definition of Clifford super-connections in Heat Kernels and Dirac Operators: Definition 3.39. If $A$ is a super-connection on a Clifford module $E\to M$, we say ...
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Proof that $[[D^2,f],f]=2[D,f]^2$

Let $E$ be a Clifford module with Clifford multiplication $c$. On page 117 of Heat Kernels and Dirac Operators it is claimed that "any operator satisfying \begin{equation}\tag{1} \forall f \in C^\...
Filippo's user avatar
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Can any Clifford module bundle be extended to a Dirac bundle?

I assume that the question in the title is clear, so let me talk about its relevance: According to theorem 4.3 in Heat Kernels and Dirac Operators the index theorem \begin{equation}\tag{1} \mathrm{ind}...
Filippo's user avatar
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"The index is independent of the Dirac operator"

Fix a Clifford module bundle $E$ on a compact Riemannian manifold $M$ and let $D_0$ and $D_1$ be two Dirac operators (compatible with the Clifford action). The proof of the Atiyah-Singer index theorem ...
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Heat kernels and Dirac operators - Why are half densities invoked in the definition of heat kernels?

The authors of Heat kernels and Dirac operators chose to consider a generalized Laplacian $H$ on a bundle $E\otimes|\Lambda|^{1/2}$ and heat kernels of $H$ are defined to be sections of $(E\otimes|\...
Filippo's user avatar
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Classification of real Clifford algebras

$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, ...
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On Dirac/ Clifford matrices

Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$. The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition $$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$ where $\{A,B\}=...
asv's user avatar
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An integral similar to the Delta function [closed]

I have an integral on the form $\int_{-\infty}^{\infty} e^{-k \omega' |\tau|} e^{i \tau(\omega'-\omega)} d\tau$ that I would like to simplify (or basically solve). This indeed comes from a problem ...
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Need to prove $uD\psi=\frac{n}{2} \text{grad}(u)\cdot\psi$

I'm reading Thomas Friedrich's article "On the conformal relation between twistors and Killing spinors" and I'm stuck in a part of the proof of proposition 3 (page 64), which states: ...
Ricardo M. S.'s user avatar
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A proof that the analytic index for families is multiplicative

I am looking for a detailed proof of the analytic index for families (of geometrically defined Dirac operators is good enough and assuming the existence of the kernel bundle) is multiplicative. Any ...
Ho Man-Ho's user avatar
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Yamabe operator, conformal transformations and square of the Dirac operator

On an $(M,g)$ compact n-dimensional Riemannian manifold the Yamabe operator $Y = 4(n-1)/(n-2)L + s$, where $L$ is the Laplacian and $s$ is the scalar curvature is conformally covariant, which for me ...
Fetchinson0234's user avatar
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An equality regarding Dirac operator

Let S be a spinor bundle on a closed Riemannian manifold M, with a spin connection A. Then for a spinor field $\phi$, we know \begin{align*} \frac{1}{2}\Delta|\phi|^2=\langle \nabla_A^*\nabla_A \phi,\...
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Dirac operator on Kähler manifold

Reference: John Morgan's book on Seiberg-Witten theory. (pg 110) I was working out the computational details of formulation of Dirac operator on Kähler manifold. If we choose the $\mathrm{Spin}^{\...
Lamda8's user avatar
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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 ...
Partha's user avatar
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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 ...
annie marie cœur's user avatar
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Discrete spectrum of Dirac operator

It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete. For example at least for $d=4$, this ...
annie marie cœur's user avatar
4 votes
1 answer
438 views

Path integral presentation of solutions of Dirac equation

It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold). Is there a way to present solutions of the Dirac equation using path ...
asv's user avatar
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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 $S=S^+\oplus S^-$, ...
Kafka91's user avatar
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Atiyah-Singer theorem in heat kernels and Dirac operators

I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne. I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the ...
user267839's user avatar
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6 votes
1 answer
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McKean-Singer formula in Heat Kernels and Dirac Operators book

I'm reading "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. The setting is: $E \to M$ is a $\mathbb{Z}_2$-graded vector bundle on a compact Riemannian manifold $M$ and $D :...
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Spectral flow of Dirac operator twisted by instanton

Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...
Gorapada Bera's user avatar
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Realization of limit of discrete series using Dirac operators

I wonder if there is a geometric realization of limit of discrete series in the flavor of Atiyah-Schmid or Parthasarathy realizing discrete series using Dirac operators on G/K. I know you can see ...
GTA's user avatar
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6 answers
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What's "geometric algebra"?

Sometimes one bumps into the name "geometric algebra" (henceforth "GA"), in the sense of this Wikipedia article. Other names appear in that context such as "vector manifold", "pseudoscalar", and "...
Qfwfq's user avatar
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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 ...
Joe's user avatar
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17 votes
2 answers
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Quantum corrections to geometry

In this video Alain Connes made a comment about the ,,quantum corrections'' of the geometry. I would like to understand this notion in some details since I haven't found anything about this in the ...
truebaran's user avatar
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5 votes
1 answer
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Dirac operator on a Morita equivalent algebra

Let $(A,H,D)$ be a spectral triple and let $B$ be an algebra which is Morita equivalent to $A$. Then there exists a finitely generated, projective $A$ module $E$ such that $B=End_A(E)$. Endow $E$ with ...
truebaran's user avatar
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22 votes
2 answers
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Is Witten's proof of the positive mass theorem rigorous?

I noticed that the only official reason given for awarding Edward Witten the Fields medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully ...
Hollis Williams's user avatar
7 votes
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Relate two different mod 2 indices: $\eta$ invariant and the number of zero modes of Dirac operator, associated to SU(2)

My major question in this post here is that: How can we relate the following two mod 2 indices: $\eta$ invariant, the number of the zero modes of the Dirac operator $N_0'$ mod 2, associated to ...
wonderich's user avatar
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Optimal control problem with spike source and "split" state

For $p \in \mathbb{R}$, consider the following problem: \begin{equation} \label{1} \begin{cases} \operatorname{div}(a \nabla u ) = p\delta_{x_0} \quad \text{in } \Omega \\ u=0 \quad \text{on } \...
GaC's user avatar
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Eta-Invariant and Atiyah-Patodi-Singer Index Theorem

In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-...
Valac's user avatar
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1 answer
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One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8. Suppose ...
DLIN's user avatar
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10 votes
1 answer
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Bochner formula in different forms

I am looking for a reference (better a book) that contain integral Bochner formulas for domains with boundary (I need it for 1-forms and functions only). For example I will need the following formula:...
Anton Petrunin's user avatar
2 votes
0 answers
72 views

Solution of lift spinor

Let $\pi:M\to X$ be a fibration map between two spin manifolds, i.e. the fiber $\pi^{-1}(x)$ is a manifold, suppose $s:X\to M$ is an embedding. Let $\Phi$ be a solution of Dirac equation, i.e. $D^X\...
DLIN's user avatar
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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 := \...
phydev's user avatar
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1 answer
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Question on a paper by U. Krähmer ("Dirac operators on quantum flag manifolds")

I don't know if this is an adequate question for MO. But I cannot understand many aspects of the said paper https://link.springer.com/content/pdf/10.1023%2FB%3AMATH.0000027748.64886.23.pdf by ...
hänsel's user avatar
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4 votes
0 answers
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Idea of Dirac operator on quantum groups

This is a somewhat unexact question. I would like to now more on the principle of the Dirac operator, especially for quantum groups. I have learned in some articles about the Dirac operator on the ...
hänsel's user avatar
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1 vote
1 answer
107 views

Let $\mathcal{M}(\Xi)$ set of all probability destributions on $\Xi$. Supremum over $\mathcal{M}(\Xi)$ is equal to sup over Dirac distributions

This doubt is born because I am reading an article in this link in pag 12 in order to use these ideas to prove another problem that raised me. My doubt is following: Let $(\Xi,\mathcal{E})$ be a ...
matematicaActiva's user avatar
5 votes
1 answer
238 views

The first eigenfunction of Dirac operator for surface

Let $M$ be a spherical oriented surface with Riemannian metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar ...
Z. Ye's user avatar
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7 votes
2 answers
568 views

General questions on the eigenfunctions of Laplacian and Dirac operators

We know that the eigenvalues of the Laplacian contains a lot of information of a Riemannian manifold, but they do not determine the full information ( Hearing the shape of a drum). And the ...
Z. Ye's user avatar
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1 vote
1 answer
341 views

Finite Element Method and Dirac eigenvalue problem

Consider Dirac equation in 2D with Hamiltonian given by (arb. units) \begin{equation} H=-i \begin{pmatrix} 0&\partial_x-i\partial_y\\ \partial_x+i\partial_y & 0\\ \end{pmatrix}. \end{equation} ...
drszdrsz's user avatar
5 votes
0 answers
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Eta invariants of fiber bundles

The general question is: What is known about the eta invariants of fiber bundles? The particular case I am interested in is the following. The fiber bundle is a bundle $S$ of even-dimensional round ...
Samuel Monnier's user avatar