/sci/ - Science & Math

>>11552140
Do symplectic geometry.

>>11459528
>I recognise it
Lovely, hun. That's the paper I first learned about TQC from. I'm sure there are more pedagogical references out there but unfortunately I haven't looked too deeply into TQC.
>The IQC is in Waterloo right?
That's right! You'll be quite close to me too.

>>11429708
Those numerical factors come from the choice of the Verma module of the Virasoro labeled by the central charge [math]c[/math]. You can compute it by computing the first few terms in the OPE of the energy-stress tensor, which goes back to the fact I've iterated that [math]c[/math] counts chiral gapless states in the theory. Check out Streater and Velo's monographs on gauge theory, an article therein contains a pretty thorough derivation of the above; for elementary CFT, which it seems that you need, check out di Francesco.
Sorry I couldn't handhold you in a 4chan post.

>>11272300
Suppose your constraint gives rise to an embedding [math]g:M\hookrightarrow N[/math], then in general we can lift ([math]n[/math]-dimensional) Morse functions [math]f:N\rightarrow \mathbb{R}^n[/math] to a Morse function [math]h =g^* f:M\rightarrow \mathbb{R}^n[/math]. Since [math]g[/math] is a submersion, [math]dg[/math] is an injection of the tangent bundles [math]TM \hookrightarrow TN[/math] whence the critical loci satisfy [math]\operatorname{ker}dh \subset \operatorname{ker}df[/math], so we aren't "missing" any critical points on [math]M[/math] via the constraint [math]g[/math].
Now the stability at these critical loci are locally determined by the signs of the eigenvalues of the Hessian [math]H_h(x) = (d \otimes d) h (x)\in \mathbb{R}^{n\times n}[/math]; in particular, the index [math]\operatorname{ind}H_h(x) = \sum_{\lambda\in \operatorname{Spec}H_h(x)}\operatorname{sgn}\lambda[/math] is a topological invariant of Floer-type. Local minima are therefore characterized by the divisor [math]D_- \subset\operatorname{ker}dh[/math] containing points [math]x[/math] for which [math]\operatorname{ind}H_h(x) < 0[/math].
>>11272395
[math]\operatorname{grad}f[/math] is locally tangent to [math]M = \operatorname{im}g[/math], not normal.

https://arxiv.org/abs/1902.07437

>>11109222
Thank you.
Also to add to the thread a little, I'm wondering what HEP people think about https://arxiv.org/abs/1008.1917? It seems to be pretty interesting given the recent boom in popularity of holography in cond mat, which is my main field of physics.
By taking inspiration from cond mat there is also a result https://arxiv.org/abs/1806.11385 that generalizes [math]T[/math]-dualities in CFT to incorporate equivariant twists. I feel we can make some real progress in HMS with this.

>>11103491
Check out persistent homology and topological data analysis. ML is becoming increasingly useful in stat mech as well.
We do need people to crunch the numbers after all.

>>10924953
https://arxiv.org/pdf/1908.09820.pdf

>>10722976
Congrats anon.

>>10101832
Simplixes, since pretty much all category can be represented by its nerve.