Local representations of a connection on the base manifold: Yang-Mills fields - Lec 22

[@Ghosted760]

So very glad that someone is recording these lectures.... so valuable, they should be archived. Thank you Dr. Schuller.

[@danielroddy2032]

Only wish the camera operator knew the importance and moved to capture the important parts…

[@philipschloesser]

I love it how in lecture 15 he spends all the time explaining that the Lie algebra of SL(2,C) does not consist of the traceless matrices but of much more complex objects, and now says that the Lie algebra of GL(n,R) is just all n x n matrices.

[@Spykoni1]

I think that in 21:23, the map of the pulled back h*ω should be T_m U x T_g G --> T_e G , since by its definition is also a Lie algebra valued 1-form

[@jackozeehakkjuz]

I bet that red chalk over the green board is a headache for colorblind. Even I struggle to see it. On the other hand, awesome lectures. These have changed completely my perspective on every area of mathematics and now I really feel I will be able to understand gauge theories.

[@samwinnick4048]

Joke's on you. The board is red and the chalk is green.

[@sainte5]

at 51:25 he IS happy!!

[@kapoioBCS]

Great lectures , but I think is this particular lecture he should have try to present more physics concrete examples for all the very interesting concepts here , like Yang Miles fields and Gauge theories etc!

[@prikarsartam]

I think it's more of an Geometric Anatomy here, those you mentioned are important limbs of it. It follows from the anatomy indeed, and he surely mentioned them in many cases

[@bh4542]

There is a mistake at 1:26 in the formula: the factor in partial derivatives mu above mu’ should only multiply the first term

[@bernardopicao267]

At 1:23:50, when he writes the first transformation, I believe he flipped the jacobian, since the index mu is covariant and thus the jacobian is “old over new”, ie del x/ del y. Thus the “correction” he then goes on to make is actually incorrect, in my opinion.

[@KevinFournier-xd3ub]

These are excellent lectures, I am glad I found them. I am curious, what is the text Professor Schuller is teaching from?

[@mashedpotatoez99]

These lectures seem to be very closely aligned with _Modern Differential Geometry for Physicists_ by Chris J Isham.

[@ivanphi]

The Lie algebra valued one form written in the first 3 minutes is not correct right? Shouldn't it map vector fields on the total space to Lie algebra valued functions on that total space.

[@letmeoffendyou]

I think it's correct, the connection-one w form takes a vector in TP and gives you its projection on g, thus w maps TP to T_e G=g. You can naturally extend w to Γ(TP) to g. In other words, w \in g tensor T*P. Once you know w, you automatically get the horizontal space as vectors X in TP s.t. w(X) = 0.

[@ivanphi]

@@letmeoffendyou How do you naturally extend w to Γ(TP) so that it has values in g? I don't think this is correct. For example, consider a smooth function A from M to g. Then we have an induced vertical vector field X^A which at p is given by X^{A(p)}. The latter is mapped by w at p to A(p). But then the one-form w maps the vector field X^A to A which is NOT an element of the Lie algebra g. It is a smooth function on M taking values in the Lie algebra. In fact, I've seen the problem extend throughout the rest of the lectures (compare the proof of the formula for the curvature in lecture 24 with Theorem 30.4 of Tu's Differential Geometry: Connections, Curvature, Characteristic Classes).

[@vivalibertasergovivitelibe4111]

@@ivanphi So I took a look at the Wikipedia article on vector valued differential forms and it says that the vector valued differential forms are indeed isomorphic to the tensor product of the differential forms over the base manifold tensor the vector space. Now there is no reason given but after a bit of thinking I came up with a simple argument why this is true. It is true by the simple fact that we can write the vector valued forms as the tensor product of the forms on the base manifold M and sections of the bundle MxV where V is the respective vector space. This tensor product however is C^infity(M,IR) bilinear. Choosing a basis on V we can now express any section of the bundle MxV as a map p->(p,f1(p),...,fn(p)) where fi are smooth functions. Now this is just a linear combination of sections (p,0,...,fi(p),0,...). Since the tensor product as mentioned before is C^infinity(M,IR) bilinear that means that the fi can simply be absorbed into the form over M yielding the same vector valued form.

[@niamcd6604]

​@@vivalibertasergovivitelibe4111Are you kidding!??! Wikis@tsia!??

[@marijaturk5994]

4 years later and now I'm stuck on this. I think you're right, I think it is wrong.

[@Alpasonic]

It would be great to have a link to notes for these lectures (if any exist)

[@joelcurtis7447]

This guy did us all a favor and put together these fantastic notes. Extremely detailed, and faithful to the video lectures. mathswithphysics.blogspot.com/2016/07/lectures-on-geometric-anatomy-of.html

[@joelcurtis7447]

My only complaint is that he didn't include enough of Dr Schuller's very nice pictures. But hey, what do you want for nothing, right? And easy enough to supplement with your own.

[@LillianRyanUhl]

I have to disagree with the esteemed professor on his remark concerning the the connection 1-form ω having sort Γ(TP) → Tₑ G =: 𝔤 at around 8:20 and before Indeed, TP ≃ VP ⊕ ᵢₙₜ HP by the connection induced from ω, and this connection is equivalently described as a fiber-wise projection operation Π : TP → VP which is technically a retract of the short exact sequence of vector bundles over P VP ↪ TP dπ:-> TB (The final map is of course the pushforward, what the professor has brrn calling π_*) Indeed, the map ι : 𝔤 ≅ VP ↪ TP that the professor discussed in the previous lecture has the property that ω ∘ ι = Id_𝔤 and that ker ω = HP Hence, any global vector field X ∈ Γ(TP) directly decomposes into ver(X) + hor(X), and we see ω(X) = ι⁻¹ ∘ Π (X) = ι⁻¹(ver(X)) ∈ Γ(VP) ≅ Γ(𝔤 × P), the final isomorphism is precomposition with ι⁻¹ (pointwise) Now certainly 𝔤 has the same dimension as G while Γ(VP) is an infinite dimensional vector space, so immediately we see there is an issue and the professor's sort cannot be correct What the professor would want to write on the left hand side is that ω restricted to the space of fundamental vector fields on P, those vector fields Xᴬ for A ∈ 𝔤 which are exactly the G-translation invariant vector fields on P, takes values in exactly 𝔤 The discrepancy is that a vector field can everywhere take values only in VP yet not bet invariant under translation by G, in just the same way that only very few vector fields on G are translation invariant under the regular left and right actions of G on itself. Indeed, *defining* the Lie algebra 𝔤 of G to be this very space of fundamental vector fields on G is a very common approach to the fundamentals of Lie theory, as this makes the adjoint actions (whence the all important Lie bracket) much more transparenrlty defined Thus, summarily, the sort of ω as considered herein is nothing more or less than Γ(TP) → Γ(𝔤 × P) ≅ Γ(VP)

[@LillianRyanUhl]

Sincerely, a master's student whose defense is two days whom argued with her advisor extensively and repeatedly on this very point

[@LillianRyanUhl]

Of course the most elementary correct version is to not describe ω as acting on sections at all and instead mere tangent vectors, i.e. ω : TP → 𝔤 Indeed this more faithfully mirrors the interpretation of a (typical) one-form as a (TP → P) -fiberwise linear map ω : TP → ℝ and justifies on its face by its form how our connection one-forms ω can be regarded as "𝔤-valued one-forms"

[@bernardopicao267]

Hi! Since the domain of iota is the Lie algebra, why do you say ω(X) is a section of the vertical bundle?

[@nmdwolf]

If the Yang-Mills fields are defined as the pull-back of a one-form and hence are also one-forms, then why do they not transform as tensors?

[@nmdwolf]

Yes, this sounds correct. I forgot about the fact that they are indeed defined using (completely) different sections. Thanks for you answer

[@shubhamkadian9345]

Could someone elaborate on the step at 42:51?

[@tanchienhao]

the (abstract) exp map is defined as a map from Lie algebra to Lie group. for matrix Lie groups the exp map is just the matrix exponential. So in that step, he is applying the matrix exp and using matrix multiplication (denoted using einstein summation over indices)

[@armantavakoli7926]

10:32 Globalization is another issue!

[@ericfay5531]

What are some good physics books to read, where one can apply these structures?

[@letmeoffendyou]

depends what you're looking for, i liked *John Baez - Gauge fields, Knots & Gravity*

[@Spykoni1]

Probably you will find useful: Nakahara - Geometry, Topology and Physics

[@VPN14494]

@@letmeoffendyou thumbs up for 'Joan' Baez

[@niamcd6604]

​@@VPN14494For a moment I thought he could be her son❤

[@yuridas1]

Where are the proofs to the two theorems mentioned at 26:20 and 1:00:00 ?

[@Tomaplen]

They're true, I swear to god. QED

[@rahularvind5078]

for future viewers, you can find the exact proofs in isham’s book on diff geom, chapter 6.

[@vivgm5776]

@@rahularvind5078 glad I find out about this book, thanks!

[@millerfour2071]

7:40, 16:40, 34:11, 1:10:50, 1:27:53

[@DerekJason3000]

So all this math is summing up the gap between small neutron stars and a black hole? Could someone give me a heads up an where to look so I can understand this?

[@mrnarason]

?