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14

You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3). 6. Conclusion From this very brief summary of Witten's achievements it should be clear that he has made a profound impact on contemporary mathematics. In his hands physics is once ...


7

The question is not clearly stated, but if appropriately interpreted it does make sense. Let $S$ be a spinor bundle over a pseudo-Riemannian manifold $(M,g)$, whose bundle of Clifford algebras is denoted by $Cl(M,g)$. Since $S$ is a spinor bundle, there exists a morphism of bundles of associative, unital algebras: $\gamma \colon Cl(M,g)\to End(S)$ to the ...


7

Unlike boson fields, the fermion fields do not posses a classical limit. This complicates their construction in a spacetime manifold. The approach developed by DeWitt is to introduce a "superclassical" limit, in which the fermion fields become anticommuting Grassman variables (by taking a Majorana representation the fields can be assumed to be real). This is ...


6

No, we cannot (completely) hear the shape of a drum, even if it is spinorial. Two metric fields with the same collection of eigenvalues are called isospectral. There exist Dirac isospectral deformations; continuous 1-parameter families of mutually non- isometric metrics with the same Dirac spectrum have been constructed. They are of the form $M_s =...


6

I think this is how a physicst would treat spin_c structures: Suppose $(M,g)\cong(\mathbb{R}^n,g_{ij}\mathrm dx^i\mathrm dx^j)$ is a coordinate chart. We can then define $n$ complex $2^{\lceil\frac{n}{2}\rceil}$-"gamma matrices" $(\gamma_{i\alpha}^\beta)_{1\le i\le n,1\le \alpha,\beta\le 2^{\lceil\frac{n}{2}\rceil}}$ satisfying the anticommutation relation $\...


4

Let $c : \mathbb{C}\mathrm{l}_n \to \operatorname{End}(S_n)$ be your irrep, so that the “dual” irrep $c^\ast : \mathbb{C}\mathrm{l}_n \to \operatorname{End}(S_n^\ast)$ is defined by $$ \forall x \in \mathbb{C}\mathrm{l}_n, \quad c^\ast(x) := c(x^!)^t. $$ Thus, if $\pi := c\vert_{\operatorname{Spin}^\mathbb{C}(n)} : \operatorname{Spin}^\mathbb{C}(n) \to U(...


2

First some historical comments: The first sufficiently clear reference that I know for the spinorial Weierstrass is a preprint by Rob Kusner and Nick Schmitt link. From the point of view of the results, Friedrich does not add much, the value of Friedrich's article mostly lies in its very explicit presentation in invariant language. Friedrich's article was ...


2

Most of the questions raised above are answered in the article "The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles" by Christian Bär and myself published in Annals of Global Analysis and Geometry June 1998, Volume 16, Issue 3, pp 221–253. http://link.springer.com/article/10.1023/A:1006553302362. A preprint version is available on the arxive as ...


2

Posted from the comments (1 2 3), by request. Of course, the spinor formula itself arguably is an explicit formula, for some values of explicit, so I'll take explicit to mean polynomial in the entries, as you suggest—in which case I imagine one can prove rigorously that the answer is ‘no’. If some special formulæ are instead of interest, Jessica Fintzen, ...


1

One very hand-waving way to think about it is as follows: the obstruction to a spin structure is the second SW class, which if non-trivial means that there are some problematic $-1$ signs that occur when moving spinors around the manifold. A spinc structure is a spin structure but with $\pi$ fluxes threaded through the manifold in a way such that a charge $1$...


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