Questions tagged [dirac-operator]
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42
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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 ...
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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^\...
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1
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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}...
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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|\...
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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\}=...
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-4
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1
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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:
...
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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 ...
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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 ...
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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}^{\...
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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 ...
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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
...
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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 ...
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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 ...
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2
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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 $S=S^+\oplus S^-$, ...
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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 ...
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1
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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 ...
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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 ...
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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 "...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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0
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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 } \...
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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-...
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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 ...
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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:...
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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\...
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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 := \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}
...
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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 ...
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