Questions tagged [dirac-operator]
The dirac-operator tag has no usage guidance.
19
questions
0
votes
0answers
38 views
Invertible Dirac operator for generic metric
Let $(M,g)$ be an oriented closed spin Riemannian manifold.
We fix a spin structure of $M$. Suppose that the Dirac operator $D^g$ associated with $g$ is invertible, i.e.
$$(D^g)^2\geq\mu>0.$$
Q ...
6
votes
0answers
171 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
2answers
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
1answer
240 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 ...
17
votes
1answer
1k 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 ...
6
votes
0answers
81 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
0answers
25 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 } \...
9
votes
0answers
356 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
1answer
368 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
1answer
517 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
0answers
67 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\...
6
votes
1answer
353 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 := \...
4
votes
1answer
104 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
0answers
111 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
1answer
80 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 ...
4
votes
1answer
185 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
2answers
356 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
1answer
185 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}
...
4
votes
0answers
114 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 ...
