@jeffgian.bsky.social Do you ever need to use spinors when doing Lattice QCD? To consider quarks, and not just gluons?

Xavier Crean, @jeffgian.bsky.social and Biagio Lucini worked on analysing the topological data arising from lattice gauge theory, like Lattice QCD. See : arxiv.org/abs/2501.19320

I think this is one of the super meaningful applications of TDA, because the data is intrinsically topological (SU(3)).

If you apply the path integral approach to a discretised computer simulation of QCD, you get Lattice QCD. When doing Lattice QCD, I think you assign SU(3) elements to each edge of a 4D cubical grid. That now gives you ... some topological data. Tada..!

The QCD Lagrangian has a Dirac operator which makes quarks and gluons interact. With this Lagrangian, you can either write down scattering amplitudes or Feynman path integrals. My understanding is that the path integral approach still keeps the differential geometry intact.

I forgot what a spinor is.. a Spin group is a subset of a Clifford algebra. You need this stuff to define fermions, which are electrons and quarks (and other stuff like neutrino). To define quantum chromodynamics (QCD), you get the spin bundle and also a SU(3)-bundle and define a Lagrangian.

A connection 1-form is also called a gauge field. This formalises, for example, gluon fields. For a full formal account of the standard model, you still need a lot more abstract machinery. Next up is spinors.

A connection 1-form omega is a g-valued 1-form on P, satisfying some well-behavedness wrt G. Ehremsann connection and connection 1-form are equivalent notions, and the horizontal subspace specified in Ehresmann connection is exactly the kernel of the projection map specified by a connection 1-form.

An Ehresmann connection H is a sub-bundle of the tangent bundle of a principal G-bundle P->M (double bundle??) The rank of H is equal to the dimension of the base manifold M. H is chosen so that it's "horizontal" and well-behaved wrt G-action.

helo

would u accept "shittification"
also theres a math word called "sheafification"

me soon

더불어 저 본인도 upper middle class으로 세상을 다분히 비뚤어지게 봤는데, 여러 이유로 이제는 좌파로 정체화하게 됐어요. 더 정직하고 바르게 살 수 있게 됐어요.

카투사로 일할때 미국 유학간 애들이 다 그런 느낌이었어서 여러모로 인상이 안좋게 박힌 기억이 나네요.

he seemed viral in south korea, what's big about him?

reminds me of the "The Boring Side" that the ingredient lists get written as for the "hip" companies like Oatley and Aussie