/sci/ - Science & Math

https://arxiv.org/abs/1607.08422
>It is natural to ask how to obtain global observables on a closed surface. The answer is provided by the theory of factorization homology. We compute the factorization homology of a closed surface [math]Σ[/math] with the coefficient given by a unitary modular tensor category, and show that the result is given by a pair [math](H,u_Σ)[/math], where [math]H[/math] is the category of finite-dimensional Hilbert spaces and [math]u_Σ∈H[/math] is a distinguished object that coincides precisely with the Hilbert space assigned to the surface [math]Σ[/math] in Reshetikhin-Turaev TQFT.

>>11341057
No. Classical fermions are sections of a spinor bundle whose spin structure is inherited from irreps of the central extensions of [math]SO(1,3)[/math], which contains two copies of the conventional "spin" group [math]SU(2)[/math]. One copy is the "usual spin"/helicity sector, while the other is the "fictitious spin"/charge sector. The -1 eigenspace of the latter of which is spanned by precisely the antiparticles. Fermion fields as a section into the associated vector bundle automatically takes into account the two sectors.
In fact, if the fermion is in addition massless then we have a chiral operator [math]\gamma_5[/math] decomposing the charge sector; this is where Weyl fermions come from. This decoupling between the charge and spin sectors allows us to move spins around independently of its charge copy, which is why Weyl fermions are the ones responsible for the chiral anomaly.

>>11281325
They transform differently, one as [math]\partial \mapsto g\partial[/math] and the other as [math]\partial \mapsto g^{-1}\partial[/math]; however,as bases of [math]T^{1,0}M[/math] and [math]T^{0,1}M[/math] which are naturally dual, that is the extent of their difference on flat spacetimes. However, on curved sapcetimes the Laplace-Beltrami operator [math]\Delta f= \operatorname{tr}H_f[/math] reads in coordinates [math]\partial_i (g^{ij}\partial_j f)[/math], where [math]H[/math] is the Hessian [math]H_f = (d^*\otimes d)f \in \operatorname{End}V[/math], and we use the metric tensor [math]g[/math] to identity [math]\operatorname{End}V \cong V^*\otimes V[/math]. Hence the order of the differentials matters unless [math]g[/math] is symmetric.