Questions tagged [dirac-operator]

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-2 votes
1 answer
65 views

Partial derivative in terms of Kronecker delta and the Laplacian operator [closed]

How can the following term: $$ T_{ij} = \partial_i \partial_j \phi$$ be written in terms of Kronecker delta and the Laplacian operator $\mathbin\bigtriangleup = \nabla^2$? I mean is there a relation: $...
4 votes
1 answer
379 views

"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 ...
4 votes
0 answers
63 views

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 ...
2 votes
0 answers
97 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 ...
5 votes
1 answer
222 views

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^\...
0 votes
1 answer
116 views

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}...
2 votes
0 answers
127 views

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|\...
-1 votes
1 answer
139 views

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, ...
2 votes
1 answer
214 views

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\}=...
-4 votes
1 answer
103 views

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 ...
1 vote
0 answers
27 views

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: ...
1 vote
0 answers
64 views

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 ...
4 votes
1 answer
238 views

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 ...
2 votes
0 answers
152 views

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,\...
22 votes
2 answers
3k views

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 ...
31 votes
6 answers
4k views

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 "...
2 votes
1 answer
267 views

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}^{\...
6 votes
0 answers
189 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 ...
0 votes
0 answers
112 views

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 ...
3 votes
0 answers
311 views

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 ...
4 votes
1 answer
486 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 ...
2 votes
2 answers
229 views

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 $S=S^+\oplus S^-$, ...
6 votes
2 answers
897 views

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 := \...
8 votes
0 answers
400 views

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 ...
6 votes
1 answer
648 views

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 :...
6 votes
0 answers
306 views

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 ...
2 votes
0 answers
68 views

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 ...
6 votes
0 answers
325 views

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 ...
17 votes
2 answers
2k views

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 ...
5 votes
1 answer
272 views

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 ...
7 votes
0 answers
119 views

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 ...
1 vote
0 answers
33 views

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 } \...
13 votes
0 answers
1k views

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-...
13 votes
1 answer
457 views

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 ...
10 votes
1 answer
1k views

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:...
2 votes
0 answers
73 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\...
4 votes
1 answer
114 views

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 ...
4 votes
0 answers
131 views

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 ...
1 vote
1 answer
115 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 ...
5 votes
1 answer
239 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 ...
7 votes
2 answers
590 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 ...
1 vote
1 answer
349 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} ...
5 votes
0 answers
156 views

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 ...