/sci/ - Science & Math

>>11553752
I had written up a couple paragraphs on this trying to simplify things (via Gysin/complexification/principal fibrations) further but I couldn't find anything satisfactory, so I opted to not say anything. Unless one has a specific [math]f[/math] in mind I don't think it's worth the effort, hence the lack of literature. At this point you may as well just brute force the integral via an embedding [math]SO(n) \hookrightarrow S^{n-1}[/math]

>>11466780
>published in math as an undergrad
>"guys should I give up?"
What was the paper about though

>>11409304
I'd say Pontjagyn

>>11267992
>philosophy
Sorry that's not my area of expertise so I can't comment on that.

>>11010362
Nope, it'll only get slower as you age.

>>10899745
I don't understand what you're asking exactly because you're using terms that don't mean what you think they mean. But I think it's rude to not at least answer you under a generous interpretation of what you might be asking.
At least in free theories, particles are sections of the associated vector bundle of a [math]\operatorname{Pin}^c_{1,n}[/math]-bundle. For [math]n\geq 3[/math], spin-statistics applies and bosons correspond to integer irreps and fermions are half-integer irreps. In other words, under the central extension [math]1\rightarrow U(1) \rightarrow \operatorname{Pin}^c_{1,n} \rightarrow O(n) \rightarrow 1[/math] bosons are irreps that pull back to [math]1 \in U(1)[/math] and fermions are those that pull back to [math]-1 \in U(1)[/math].
Physical theories require unitarity and the so-called "topological spin-statistics". The former states that the fields have non-negative norm (the associated vector bundle is actually a Hermitian bundle), while the latter states that Dehn twists of the underlying manifold by [math]\pi/2[/math] gets mapped to a "fermionic supercharge operator" [math]\operatorname{tr}(-1)^F[/math] that counts the number of fermions on the [math]\operatorname{Vect}[/math] side.
Aside from these algebraic/topological requirements, we may wish to equip the bundle with a [math]G[/math]-structure in order to gauge certain symmetries. This requires the underlying [math]G[/math]-manifold to be Hamiltonian and the existence of a moment map such that the [math]G[/math]-structure is compatible with the [math]\operatorname{Pin}^c[/math]-structure. This allows Atiyah-Bott localization.
In addition, you require either the Wightman axioms to be satisfied, or the Osterwalder-Schrarder (slightly weaker) axioms in addition to reflection positivity. Regularizability is given by the Strocchi-Swieca regularity conditions on the space of sections.

So please, study these (or at least read some Landau-Lifshitz) and ask coherent questions next time.

>>10220862
Geometric topology

>>10198159
He's kind of cute tho

>>10156270
Depends on what you're doing, but there are results in PDE/functional analysis where if a term like [math]\epsilon S_1[/math] explicitly breaks the symmetry of your action functional [math]S_0[/math] on some Banach space, then for sufficiently small [math]\epsilon[/math] and sufficiently regular [math]S_1[/math] the kernel of the first-variation map for [math]S_0 + \epsilon S_1[/math] is [math]o(\epsilon)[/math]-away from that of [math]S_0[/math]. This means that critical points (if they exist) of [math]S_0 + \epsilon S_1[/math] is close to those of [math]S_0[/math].
Of course you have the standard results like Noether and Goldstone that tell you what happens when the compact Lie group [math]G[/math] of symmetries is broken into a subgroup [math]H[/math], but they do not guarantee the existence and "closeness" of the critical points.