# Questions tagged [dirac-operator]

The dirac-operator tag has no usage guidance.

19
questions

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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**

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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

**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

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

**1**answer

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 ...

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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

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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**

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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

**1**answer

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

**1**answer

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**

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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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

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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 ...