A spin structure on a Riemannian bundle of rank >2 is the lift of the structure group from $\text{SO}(n)$ to its universal cover $\text{Spin}(n).$ It may also be defined in the cas …

There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$, a Riemannian metric is a section of positive definite symmetric bili …

I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard i mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element …

Among simple Lie groups, $Spin(8)$ is the most symmetrical one in the sense that $Out(Spin(8))$ is the largest possible group. A description of this outer automorphism groups is as …

I am trying to understand the Berline - Vergne Localization formula for the equivariant Index of the Dirac operator on a spin manifold M which states that the G - equivariant index …

We say $L< (V\oplus V^{*})\bigotimes \mathbb{C}$ is isotropic when $< X,Y>=0$ for all $X,Y\in L$ Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of …

I am puzzled by the following theorem that I found in the book "An introduction to Spinors and Geometry with applications to physics" by Benn & Tucker (Appendix A, pag 329): …

Hi everyone, I'm currently studying the construction of the $Pin(1,3)$ group and given the definition I'm using to find its elements I'm having some problems with the signs associ …

While reading article [1] below I came across the notion of a universal spinor bundle. This is defined at the beginning of section 6 (p.14) in [1] as follows: Let $M$ be a spin man …

Consider a smooth compact oriented 4-manifold $X$. Although not all 4-manifolds admit a spin structure, they do admit spin-c structures. And if $X$ does admit a spin structure, the …

I am looking for examples of 2- and 3-dimensional flat spin manifolds with Euclidean and Lorentzian signatures, which admit parallel spinors and the dimension of the space of the …

3-dimensional Chern-Simons theories, with compact gauge group $G$, are determined by $H^4(BG)$. Looking at $U(1)$, with generator $c_1^2\in H^4(BU(1))=\mathbb{Z}$ for 1st Chern cla …

Dear community, there are so-called orientation maps $a:MSpin\to ko$ and $b:MSpin^c \to k$, "defined" in ABS's paper "Clifford modules". Unfortunately I am not familiar with repre …

Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came fro …

In "Spin geometry" by Lawson and Michelsohn it is defined the Clifford algebra $Cl(g)$ associated to a symmetric bilinear form $g$ in general, including the degenerate case. But th …