The h Bar - 2024-04-24

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1:04 AM

It depends how projects are selected by the funding organizations. When writing grant applications, there are question boxes that now ask for technological applications and time frame etc which essentially puts a complete end to research in more fundamental areas.

I get this for hep-th (which is why most of funding comes from the department for those projects), but it's gotten to a point where even standard areas of condensed matter theory are being heavily targeted.

Honestly, I kind've feel bad for a lot of the condensed matter theory profs. There used to be several in my old departments…

not me misreading a table of definitions for coefficients and performing 20+ integrals using my incorrect misreadings...

it happens to the best of us

@DIRAC1930 one program i visited i was excited about, but then when i talked to the CMT people in the department, they either had $0$ funding or were planning to take a single student. sadge.

2 hours later…

2:55 AM

@ACuriousMind this is the kind things I've been asking about on this chat for a while

3 hours later…

5:37 AM

who started this bundle theoretic approach to field theory? or what are some of its origins

6:06 AM

why are papers so expensive to read

like most journals charge around $40 per paper

naturallyInconsistent

6:19 AM

@SirCumference isnt this highly criticised?

@SirCumference there is a hub of science similar to the genesis of libraries

naturallyInconsistent

lol

6:40 AM

@SillyGoose Cartan I think?

dang that's a long time ago

well okay i guess not that long ago

I mean bundles themselves are at most from the 30's, if you go really far back

I think it got into physics in like the 50's, but as a pretty marginal use

You don't really see bundles that much until the 60's I think

During the whole GR renaissance

i see

Probably inspired by the use of bundles in famous differential geometry books like Kobayashi

oh mayn that's the K in CKM

6:45 AM

CKM?

oh maybe not the same kobayashi

ckm matrix related to standard model

Oh yeah no, Kobayashi is a super common name in Japan

ohh i see

It's like Smith

You may also remember it from Star Trek

The Kobayashi Maru test

i never watched star trek but many physics students and professors seem to have

6:49 AM

I have been looking into like the trends in styles of differential geometry through history

The sleek modern method is mostly due to Kobayashi et al

@naturallyInconsistent I mean paying to read research as a whole is highly criticized

but the price alone just boggles my mind

if arxiv (or "other means") didn't exist you could spend hundreds of dollars a month just to keep up with the latest research

I think people in the olden days just borrowed them from the library :p

naturallyInconsistent

7:07 AM

@SirCumference monopolies do the shit that they do

that is true sadly

paying $40 so i can read a 4 page paper sounds like a ripoff, except that there isn't an alternative

naturallyInconsistent

capitalism gotta ruin everything

@SirCumference technically, this is totally false. We can easily implement a different system and not be beholden to the status quo

@naturallyInconsistent so what's the hold up

i mean academia has a weird system where we value prestige over money

i feel like switching that trend would probably be difficult at this point

Hello Everyone...

I mean by now I think most people in physics will just read the preprint :p

7:15 AM

@Slereah well thank god that's an option lol

i don't really understand something. we set up classical chern-simons theory and then find "gauge" transformations that do not leave the classical action invariant, but will turn into gauge transformations post-quantization of the theory?

i'm surprised journals haven't tried to lobby for that being a copyright violation or something

so is classical chern-simons theory not really anything but a set up for a quantum theory?

@SirCumference How can it be, this is published before the journal

@Slereah well that's true

i guess they can't do much about it even though it takes away almost all their income

7:18 AM

They don't really make their money on individual sales of articles

They mostly sell subscriptions to universities

naturallyInconsistent

@SirCumference what part of "nobility culture" is weird to the hooman condition? Hoomans have had monarchy for millennia.

i guess per this post it seems the point of singling out large gauge transformations is to have in mind the quantized state space

7:36 AM

Here's something I was wondering about... Does evolution prefer almost parity symmetric organisms? How would one go about showing this is the case?

@SirCumference usually libraries would buy them, and you'd just ask for your local university library membership

Like if I asked what is the probability of an assymetric organism surely there would be a non zero number?

@SirCumference they dont need to, academics will publish in journals regardless of preprints

blebs but ACM's answer says that classically large transformations are still genuine gauge transformations. but in the classical chern-simons theory it seems like they are not because they genuinely change the action

8:00 AM

Well then it's not a symmetry of the action :p

8:12 AM

hm is being a symmetry of the action not a necessary condition for being a gauge transformation?

8:59 AM

"The combination of push forward and coordinate transformation is an example of a diffeomorphism. A diffeomorphism is a one-to-one mapping between the manifold and itself. ... In a diffeomorphism, we shift the point at which a tensor is evaluated by pushing it forward using a vector field and then we transform (pull back) the coordinates so that the shifted point has the same coordinate labels as the old point."

Is there another name for the specific operation of "shifting the point on the manifold combined with changing coordinates"?

12

A: In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are isometric, or just diffeomorphic?

Indeed, "diffeomorphism invariance" of GR in physics in this context means in proper mathematical parlance that isometric (pseudo-)Riemannian manifolds are physically equivalent. In my view, this confusion between "diffeomorphism" and "isometry" is probably due to physicists usually looking at a ...

"define the diffeomorphism to be an isometry so that the fields on the target with the new coordinates are equivalent to the fields on the source."

is it an "isometry"?

An isometry is just a diffeomorphism that preserves the metric

because the coordinates of the new points are the same as the coordinates of the old points, then the distance between two new points must be the same as the distance between the two old points under this operation, right?

and the pushforward by itself is also an example of a diffeomorphism, but not an isometry?

9:17 AM

Isomorphisms fundamentally preserve norms, but it is also true that it preserves distances, yes

Flow of a curve by an isometry leads to a curve of the same length

hi

9:37 AM

would u say u r more mathematical than most physicists or less mathematical

wait. pushfowards act on vectors, not points according to baez? bertschinger's notes are confusing.

Yes

Diffeomorphisms act on points

Their pushforward on vectors

this quote "The combination of push forward and coordinate transformation is an example of a diffeomorphism. A diffeomorphism is a one-to-one mapping between the manifold and itself. ... In a diffeomorphism, we shift the point at which a tensor is evaluated by pushing it forward using a vector field and then we transform (pull back) the coordinates so that the shifted point has the same coordinate labels as the old point."

no longer makes sense to me

given this

10:16 AM

in physics, pushing forward everything on the original manifold is part of diffeomorphism

in mathematics, diffeomorphism is just a smooth map between manifolds

let's say the math diffeomorphism is a map between manifolds, and the physics diffeomorphism is a map from a manifold with tensor fields to a manifold with tensor fields

@lucabtz yes, I noticed the D-module/monodromy connection

in math, a diffeomorphism is an equivalence relation on smooth structures. in physics, a diffeomorphism is an equivalence relation on smooth structures with tensor fields

10:33 AM

@qwerty I think this is a confusing way to talk about things, and it is caused by physicists being unable to talk about anything without using coordinates :P

A diffeomorphism is just a smooth invertible map $f : M \to M$

a coordinate chart is a smooth invertible map $\phi : U_i\to M$ for some $U_i\subset \mathbb{R}^n$

what the snippet you quote there is trying to say is that they're applying the diffeomorphism $f$ to $M$, but then using $f\circ \phi : U_i \to M$ as a coordinate chart - the image $\phi(U_i)$ gets "shifted" by $f$, so the "same" coordiantes now denote different points

naturallyInconsistent

@ACuriousMind it should be the other way around, $M\to U_i$

@naturallyInconsistent different conventions are possible; since the map is invertible (on its image) it doesn't matter which direction you call the chart

@naturallyInconsistent this is incorrect. the chart is not defined on all manifold points

in $\phi _i : U_i \rightarrow M$, the map need not be onto

it is onto for a subset of $M$

the first sentence about "a combination of coordinate change and pushforward is a diffeomorphism" is also a mathematical nightmare, but what they mean is that if you look at $f$ in terms of your new coordinate chart $f\circ \phi$, what's happening is that you just "changed the coordinates" by shifting $\phi(U_i)$ around but all your tensors additionally changed through $f$s pushforward

naturallyInconsistent

@ACuriousMind I know it is invertible so that it is somewhat doesnt matter. My point, however, is that there is a unique $c\in U_i$ that is selected by each single chart for every $p\in M$ (where the chart is defined over) whereas for the inverse function, we often extend its domain of applicability by identifying, say, $\forall n\in\mathbb Z\qquad2n\pi=0$

@RyderRude stop replying to my stuff.

10:43 AM

@naturallyInconsistent I'm not sure what you mean. Perhaps the issue is that what we often casually call "coordinates" are not actually coordinate charts in the proper mathematical sense, e.g. $\mathbb{R}\to S^1, x\mapsto \mathrm{e}^{\mathrm{i}x}$ is not a coordinate chart - the circle needs at least two charts to be covered

naturallyInconsistent

@ACuriousMind yes, I am also aware and was trying to allude to this

maybe the issue is that I should have also written the target as $V_i\subset M$ when I said the map is invertible?

naturallyInconsistent

@ACuriousMind that is to reply to RR. I was not onto that

@ACuriousMind What you have deduced in this comment, is what I was trying to mean.

Anyway, since we are both on the same page here, it is time for meow to exit the convo. It is time for debauchery miehehehe

ok, time for me to read through this carefully. thanks for your input everyone.

$R$ to $S^1$ is a covering in technical terms

this doesnt have to do with charts. it is defined in the absence of co ordinates

11:00 AM

@ACuriousMind yeah. They also mention higgs bundles which I don't know much about but people have been mentioning in the papers I'm reading lately

11:19 AM

would u say u r more mathematical than most physicists or less?

more mathematical means more rigorous

1 hour later…

Relativisticcucumber

12:21 PM

@SirCumference if u think 40$ is expensive try submitting to nature XD

i never understood why this is really a problem tho. sure papers and journals are expensive, but i feel like anyone who would understand them/be able to read them and do anything with them is in academia or uni and to my understanding every uni has full access to all legit journals, making reading free to everyone with an edu email. so i feel the amount of people this system harms is actually quite small, no? im definitely open to learning new info if im wrong

@Relativisticcucumber it's not true only those in academia or uni are interested in scientific papers. there's ex-academics/ people with doctorates working in industry, teaching, tech etc.

it's also not true every uni has full access to all legit journals

i have been at a uni in an european country which was on the poorer side that didnt have access to all the journals i needed

Relativisticcucumber

@qwerty i know they arent the only but im more speaking about the number of people this issue hurts being quite small in my rough estimation

bc as far as problems go, a problem that hurts a handful of people in terms of their leisure reading is a p small problem in my mind

but ppl get v mad ab this issue so i am curious what other points of view are

@qwerty oh i did not know this bleh

i am sure it "disproportionately affects" people including researchers who are already impacted for other reasons. socio-economic i would guess being the most obvious

@Relativisticcucumber If I saw a reason for them to be so expensive I probably wouldn't care at all, but as it stands I do not see that any part of the labor chain here that actually contributes value sees any of the money: As someone outside academia, I'm supposed to pay for a paper, but none of that money goes to the person who wrote it and none of that money goes to the people who reviewed it - what am I paying for here, exactly?

2

Double that question if the author(s) received governmental funding to begin with, since I think if the state funds research, the public should have unconditional access to the results of that research

Relativisticcucumber

@ACuriousMind yeah i have heard this argument raised also when people talk about submission fees. i just think its never the case in the world that money/fees work like this so why should academia

12:31 PM

@ACuriousMind agreed

Relativisticcucumber

i think the cost of things are what people will pay and since the majority of ppl paying for journals is unis with billions of dollars why not jack up the price

@Relativisticcucumber when I buy a "normal" book, then the author gets a share

fiction authors don't pay publishers to sell their books and then actually live off government writing grants :P

@Relativisticcucumber monopolies are usually considered bad. and most people would rather the unis spent that money on research and teaching

journalists are (usually) paid by the news outlets that publish their stories, too

so no, I don't agree that "it's never the case in the world that money/fees work like this", academic publishing is predatory and parasitic in a way and to a degree no comparable industry is

Relativisticcucumber

well i mean more broadly speaking prices are hiked up as much as the market will pay. this just seems to be how things work, no?

of course there are counter examples but i believe this statement to be generally accurate

12:34 PM

youtube.com/watch?v=PZ_qnBwSejE There is a new interview of Witten

i suppose you're now asking more fundamentally "monopolies should have the right to exist"

Relativisticcucumber

i also look at it like the world works based off of which systems will thrive. clearly this is a thriving system. rarely are the worlds most thriving systems morally correct. thats kind of why they thrive -- you dont become successful to that degree by sharing and being nice

@ACuriousMind i can name quite a few XD

entertainment in every degree for starters

@Relativisticcucumber the core of my objection isn't that papers cost money (except for the government funding thing) - it's that literally zero percent of that money goes to the people who actually produce the product I'm buying. Yes, many other industries underpay their workers from my point of view, too, but paying the crucial workers literally nothing is quite unique

Relativisticcucumber

idk i just dont find this unique. my mom worked off of a minimum wage job at a taco place for my young life making not even enough for us to buy food. the CEO of del taco is probably quite rich XD sure she got paid smth but i mean really its the same system. i guess to me this difference of "i give you pennies versus i give you nothing at all" isnt really meaningful

your take seems to be "modern slavery exists makes academic publishing just another predatory system that shouldnt bother us"

Relativisticcucumber

12:40 PM

@qwerty no my stance is that when people dont have food or parents or health care or rights to education we shouldnt sit here crying about if we can read nature papers or not we should care about the more extremely meaningful consequences of these systems and how our society is influenced by them

@Relativisticcucumber people can care about more than one issue

anyway, i should go back to reading about diffeos

@Relativisticcucumber I don't think one injustice existing makes another disappear; and I do think the systems that make academic publishing bad are very different from the systems that make minimum wage labor bad. If you gave me the choice whether to eliminate problems with academic publishing or with low-wage labor, sure, I would probably choose low-wage labor, doesn't mean I suddenly feel only "meh" about academic publishing

Relativisticcucumber

@ACuriousMind hm interesting

i feel meh about all of it but still the perspective is interesting

well another thing is im not sure why people feel entitled to scientific information. i mean education is not free, books are not free, and publishing is no different. so idk this issues feels to me a lot like people complaining that a valuable commodity is expensive. i do get the objection that you stated about the cost not being justified in terms of who you think the money should [...]

[...] go to and if that is the main objection of people then i can see that but also i think its the right of a company who has a successful business model to determine that sort of thing and most authors agree to this system and it benefits them as well as far as i understand

Is ACM a morally grey character in this story

@Relativisticcucumber "education is not free" is a very US-centric statement

6

12:55 PM

Who's the villain though?

Relativisticcucumber

@ACuriousMind yes i am american so that is where my mentality is centered but also living in china for some years, that seems to be true here as well

but i do acknowledge that that is not true worldwide

in the basic individual sense, education in my country is free - I have never paid to attend any school or university, and I received government funds to ensure I could attend university without having to hold a job

What are you American now?!

Relativisticcucumber

@Mr.Feynman is this a joke

sorry my sense is off lol

No, I'm that dumb

Relativisticcucumber

12:57 PM

@ACuriousMind ok coming from a place where health care and education is free i do see ur stance more

lucky XD

Genuinely surprised because at the end of our last conversation I'd acknowledge you were not or maybe I'm just confused about the past

Don't hate me @Relativisticcucumber plz

You're my favorite non-ACM user

@Relativisticcucumber because science belongs to all of humanity and should be made as accessible as possible

and so the idea that this entire education system is funded by the public precisely to be free to individuals like me to me is in insoluble contradiction to the idea that corporate publishers sit on top of this humanistic publicly-funded system and just skim billions of profit off the top of it without contributing anything just...doesn't fit. It's literally a parasite attached to an otherwise well-meaning (though often in details quite flawed) machine

Relativisticcucumber

@Mr.Feynman <3

@ACuriousMind ok this makes more sense

i can see it definitely from the pov of the european system

@Mr.Feynman im afraid ppl will hate me after my comments here XD i dont consider myself to be a bad person but now im having doubts XD

i guess i didnt realize how much of this "nothing is free" mentality was engrained in me. like i feel growing up in the US its more like "ur entitled to nothing" idk maybe its just me but i never had this idea like "knowledge or science belongs to everyone"

or its always been like "fight tooth and nail to get smth" so i am used to these high prices of things

man wish i was german

@ACuriousMind would we ever have a manifold+tensors, apply a diffeomorphism but not have a pushforward from the diffeomorphism ?

Relativisticcucumber

1:04 PM

this is also funny bc i have heard a stereotype that americans are entitled so now im just confused

@Relativisticcucumber perhaps because you could view it as entitled/elitist to think that knowledge should only belong to a wealthy/privileged few

kmowledge would b free in an ideal world

unfortunately, everything about the modern world is rigged against the poor. it's because the rich r friends with lawmakers

1:19 PM

@Relativisticcucumber I mean, it's not as if there aren't people with different opinions over here, too.

But there's a long tradition of viewing education in terms of what's called the Humboldtian model, see also Weimar Classicism and renaissance humanism - that education and knowledge are ends in themselves and a precondition for being a fully realized and moral person.

the high school I went to explicitly called itself humanist in reference to these ideas and taught Latin and Ancient Greek and Philosophy - not because these were economically valuable skills but because these are, in the humanist view, valuable sources of knowledge for any human, to know and understand the past and to be taught how to reason about what one can know and what one should do

@Relativisticcucumber I thought you were a Ditto

1:53 PM

British Topography and Saxon and Northern Literature vs Weinberg's QFT, which is the real keeper?

@DIRAC1930 Weinstein mentioned around 5 times in the comments to this video, absolutely crazy

"The integral curves of a vector field provide a continuous one-to-one mapping of the manifold back to itself, called a pushforward. The pushforward generalizes the simple translations of flat spacetime. A finite translation is built up by asuccession of infinitesimal shifts."

surely this is not a pushforward because it acts on curves, not vectors?

it sounds like just a diffeomorphism?

what is that from

same document, from bertschinger

the integral curves give you the flow, i.e. a family of diffeomorphisms

you can of course then consider the pushforward along those diffeomorphisms

@ACuriousMind web.mit.edu/edbert/GR/gr5.pdf

this is such a shame because this document covers exactly what im interested in but the language is so garbled

@ACuriousMind that makes much more sense

2:02 PM

just sounds like physicists abusing mathematical terminology, which is par for the course (cf. e.g. the horrible confusion in "physics group theory" around $\otimes$ vs. $\times$)

i have no idea what a student by themselves is supposed to do if all the resources are so frustratingly inconsistent and garbled

@ACuriousMind big eeeeew

@qwerty reject them and use math books/ping ACM

@Mr.Feynman maths books are written in language that is too technical for me and dont cover physics topics... although this baez book has been very helpful

@ACuriousMind i am only slightly embarrassed about how many months of life i once lost to being confused about the most basic of mathematical abuse: the conflation of a function with the value of a function at a point. or a function with a related functional in action principles.

no reason to be embarrassed, many sources really aren't very clear about that

@ACuriousMind why is it a family of diffeos and not just one diffeo applied everywhere on the manifold?

2:13 PM

@qwerty see en.wikipedia.org/wiki/… - the diffeomorphism is "moving each point along its integral curve a set amount" and for each "set amount" you get a different diffeomorphism

@qwerty the author is trying to pretend a pushforward is some magical new thing that is different to a tensor transformation law arising from a coordinate transformation, he makes this confusion clear below equation (10), you likely know what a tensor transformation is but now you're confused about something you already know because big words

@bolbteppa i think it is somewhat different becuz it need not be induced by a diffeomorphism.

one can map to a manifold of a different dimension

this idea is useful in defining an embedding @bolbteppa

That's not relevant here, but what you're talking about is a simple change of variables e.g. in a vector field which is the exact same idea

yeah.. the math still looks identical to a change of co ordinates. it's just that the manifolds are not diffeomorphic

it was strange for me to know that pushforward was just a big word for the good old tensor transformtation

but they define it in the abstract

@bolbteppa could you please spell out what your objections are below eqn 10, just to make sure im following?

2:34 PM

@ACuriousMind oh so the set up is sort of different to the case where you have a single diffeomorphism which maps points from a manifold M to a manifold N. here we start by thinking about tangent vectors along curves, and each tangent vector generates a different diffeomorphism from a point in M to a point later in time in M?

Not really: You have the whole vector field, and it has its integral curves - the curves to which the vector field is tangent at every point. These curves don't intersect, so it is well-defined to take any point and say "I move it by $t$ along its integral curve". When you do this to every point at the same time, you have a diffeomorphism where every point moves by $t$ along the integral curve its on,

and the "family" is because you can freely choose $t$, getting a different diffeomorphism for each $t$

a vectior field defines a time dependent diffemorphism usin the curves @qwerty

at each $t$, u hav a different diffeomorphism from M to M

@ACuriousMind hm, i'm not sure how this is different to what I wrote?

In (9) he's studying the scalar $S(x^{\mu})$ at a new point $x'^{\mu} = \frac{\partial x'^{\mu}}{\partial x^{\nu}} x^{\nu} = x^{\mu} + \xi^{\mu} \lambda$ infinitesimally separated from the original point $x^{\mu}$, i.e. is studying $S$ at a new point $x'^{\mu}$ obtained from his original $x^{\mu}$ via a(n infinitesimal) coordinate transformation, $S(x'^{\mu})$, and he calls this a pushforward, then he infinitesimally expands (9) to get (10)

[i.e. he evaluates the scalar at the new point $x'$, infinitesimally expanding in terms of the original point $x$, which he frames as studying $S$ at the pushforward of $x$...],

then below (10) he pretends his coordinate transformation aka pushforward can be cancelled by a coordinate transformation revealing a symmetry - it's technically incoherent, but remember big words = rigor. What a normal person would say is, if the action is invariant under a coordinate transformation, we have a symmetry and the extra term would vanish, but big words...

@qwerty I don't know what you mean by "each tangent vector generates a different diffeomorphism" - in my description, the tangent vector field is fixed, it doesn't change, what changes is how much you move along its integral curves

2:42 PM

also note that all the diffeomorphisms are completely determined by the infinitesimal diffeomorphism

Unfortunately that is a very problematic statement

oh

why is this

@bolbteppa well, i agree with the incoherency part. its very disappointing, i thought bertschinger was very reliable given his reviews in the 90s on cosmological dynamics

Is the diffeomorphism group connected

it is not typically

For a start you will usually have the orientation preserving and orientation reversing component

2:44 PM

@qwerty en.wikipedia.org/wiki/Course_of_Theoretical_Physics

oh i didnt mean that every diffeomorphism can be generated like this

All the parts of the connected component of the diffeomophism group can be generated by (maybe more than one) vector field flow, though

@ACuriousMind i suppose i was thinking each tangent vector in the tangent vector field would tell each point how it moves in a time dt, and so each tangent vector generates a map in that way?

@bolbteppa LL covers diffeos? / gauge symmetries?

the full diffeomorphism group is extremely hard to talk about imo becuz it would be an infinite dimensional lie group

Here we are explicitly studying continuous symmetries and studying the consequences in the action, so all this connectedness to the identity stuff is irrelevant

2:46 PM

my statement is about a sub-group of the group which is generated by the field

@qwerty ah, yes, sure - but that's just how you get to the integral curves themselves - you're solving the differential equation where $\phi(0,x) = x_0$ and $\dot{\phi}(t,x) = X$, where $X$ is your vector field

@qwerty Yes

i think lie algebra of the full diffeomorphism group would be the commutator algebra of all vector fields on the manifold

and the latter is an infinite dimensional vector space

@bolbteppa which volume?

2

2:51 PM

@bolbteppa mutters and i deluded myself thinking i had read it cover to cover already

pretty sure LL is the analogue of mary poppins' carpet bag

@bolbteppa ive looked through the index and the ToC and i dont see anything on flows/ gauge symmetries in general / diffeos?

They don't use those words but they do the same thing, e.g. in section 94 they use this to set up the energy-momentum tensor and derive its conservation law in curved space, a diffeomorphism is a (curvilinear) coordinate transformation etc

I suppose im interested in getting to the bottom of this confusion of diff geom language not just the coordinate version

and i want to understand symmetry and gauge transforms etc which is not covered much

3:06 PM

@bolbteppa I think those comments are because Weinstein has said on Joe Rogan's podcast that he wanted to have a debate with the leading string theorists and then said that he would be willing to debate Witten even though he is afraid of him because of how smart Witten is

3:18 PM

@qwerty you might be looking for something like the EGH review? (pdf link)

@ACuriousMind thank you!!!

1 hour later…

4:33 PM

Let P be a principal bundle. Do both its structure group and base manifold constrain whether P can be trivial or not?

i am confused because in Nakahara there are two theorems. One is that a P is trivial iff it admits a global section. Two is that a fiber bundle is trivial if its base manifold is simply connected.

But then there is three from some chern simons notes that if G is simply connected and M is dim $\leq 3$ that P must be trivial (or trivializable)

@SillyGoose the second theorem is untrue, i.e. you must have misread it: The sphere $S^2$ is simply connected but its tangent bundle is non-trivial (this is the hairy ball theorem)

Hm okay then i think i misunderstand a word

it says more verbatim that the manifold is contractible

contractible means all homotopy groups are trivial

simply connected means the first homotopy group is trivial

I see

okay, then what is the question? None of your three statements contradict each other

4:44 PM

I am wondering what causes there to not exist global sections in the A-B effect then

@ACuriousMind tbh I wouldn't even mind paying like $10 per paper. But $40 is just nuts

It's almost like it's designed to prevent anyone unaffiliated with a uni from reading research papers

Granted I'm still affiliated with one, but who knows if I will be in the long term

@SillyGoose the location of the solenoid is removed from consideration; your base is not simply connected

but i thought we just had that the base being not simply connected does not necessarily mean anything for the bundle

maybe I am conflating things. i (perhaps mis)understand that if there exists a global section on P, then we can define a single gauge potential all over P (no need to take "local sections")

well, none of the other two theorems applies, either: U(1) is not simply connected, and you don't have a global section because for the $F$ of a solenoid you can't find a global $A$ such that $F=\mathrm{d}A$

we've been over the latter point already in terms of not being able to find a global vector field whose curl the magnetic field of the solenoid is; I'm not really sure what the question is

hm then im confused because i would have thought a bit of the point of chern-simons theory would be to capture global data of the bundle. but we're assuming from the get go (in this case) that the bundle is trivial. what global data is there, then?

there is data of the base manifold maybe

4:52 PM

I don't understand where you think we're assuming that the bundle is trivial "in this case"

what case?

where are we assuming its trivial?

well in witten's paper and also in the first part of freed's notes to my understanding

and what does C-S theory have to do with the A-B effect?

we have $M$ is dim $3$ and $G$ is simply-connected in freed notes, which to my understanding he states implies that $P$ must be trivial

that may be; what's the issue?

@ACuriousMind well i thought it might have been a simpler situation and it is a concrete example of when the bundle surely cannot be trivial

@ACuriousMind then isnt there No global data if P is trivial

4:57 PM

you still have the choice of connection on it

But then in the abelian theory even though there are distinxt classes of connections all are physically the same “field strength”

but the abelian case is U(1), which is not simply-connected

you're contradicting yourself in the setup here

Ohh

man

wait why must an abelian group be taken to be U(1)

what other Abelian Lie groups do you think there are?

$(\mathbb{R},+)$

5:52 PM

you know, now that i'm finishing up my last courses, i'm gonna miss doing proofs for homework problems

felt like doing a puzzle

i guess i could do it on my own, but it's kinda lonely to know no one else is gonna read them

@ACuriousMind it looks like we could have some that are not $U(1)$ as feynman mentions en.m.wikipedia.org/wiki/Abelian_Lie_group

@SirCumference i don’t think i’ll ever miss writing proofs for a math course XD

@SillyGoose huh, i guess to each their own

those are the one kind of homework i actually enjoyed

there was a nice sense of satisfaction when things suddenly line up

@SirCumference there will be a nice sense of satisfaction for me once i no longer have to prove things about the product topology xD

Bleb then i don’t understand what is the point of this immense generality of freed’s part 1 notes on C-S theory when it doesn’t even cover $U(1)$

Why not just present the entirely general chern-simons

All at once

@SillyGoose well, you still gotta do the problem sets if you're learning something on your own :P

it irons out any misunderstandings, after all

6:07 PM

True

I would like problem sets from a mathematical physics course. Like prove that projective representations of SO(3) are equivalent to representations of its lie algebra

@SillyGoose yes, uniqueness up to isomorphisms holds for abelian Lie algebras though

hm but why does a unique lie algebra matter for us in this case

i thought the primary object of interest in this case is the genuine lie group

hm well $\mathbb{R}$ is not compact, which might be a different problem lol

golly gee

6:27 PM

The real problem is you haven't studied Atiyah-Pododi-Singer, spend 5 years on that then C-S will simply be the $k=2$ case

18

A: How to understand Chern-Simons action

Here is a mid 1970s point of view, courtesy of Atiyah-Patodi-Singer. Suppose you have a complex vector bundle $E$ of rank $r$ over a smooth manifold $M$. A polynomial function $P$ on the space of $r\times r$ matrices is called invariant if $P(T AT^{-1})=P(A)$ for any $r\times r$ co...

i wish XD

okay but ill take a look at the MO thanks

> Secretly this higher principal connection structure also governs the first, seemingly simpler case. The action functional of Chern-Simons theory is always the volume holonomy of a 3-connection, the Chern-Simons circle 3-connection.

but i guess per feynman and acms comments above, given the restriction that $G$ is compact and simply-connected, there does not exist an abelian $G$ satisfying those restrictions. so i guess freed in his part 1 notes is considering non-abelian chern-simons with this restricted class of non-abelian lie groups.

In fact, I take my joke back, the real problem is you haven't studied n-lab madness first, spend a lifetime on that then it will become obvious as the circle 3-connection secret

well i at least agree with you that n-lab is madness ;) (at least to present day me)

hm well to my understanding these papers or notes are just trying to introduce or work with the umbrella notion of chern-simons theory in a simplified setting. so is it correct to understand that (1) let's assume the principal bundle is trivial. actually, this can be encoded in restrictions on the structure group: $G$ is simply-connected and compact. (2) then, we have global sections and can use the "naive" Chern-Simons action as writ in Witten's Jones polynomial paper.

(3) all global features of the theory are then trivial/uninteresting except for the choice of connection over the bundle

7:06 PM

I think the (1) and (2) are not so problematic, but I am trying to better understand the true statement corresponding to the statement I am trying to make in (3)

in the abelian Chern-Simons theory, if we naively take $F = dA = 0$ as the equation of motion and identify the gauge equivalence class of on-shell potentials with the first de Rham cohomology group $H^1(M)$, then we have some identifications leading to $H^1(M) \cong \text{Hom}(H_1(M, \mathbb{Z}), \mathbb{R})$ cf. Then, we can identify $H_1(M, \mathbb{Z})$ with the abelianization of $\pi_1(M)$.

This, at least to me, is more directly topological information about $M$. We have that the space of states literally is classified by (indirectly) by the fundamental group of the base manifold

hm amybe this is more appropriate for a q on the main site :P

I’m confusing myself, but. The original “splitting the atom” experiment amounted bombarding lithium-7 with protons to get two alpha particles. But the proton has +1 charge and the alpha particles each have +2 charge. Where did the +3 charge come from?

I’d assume there’s electrons being left behind but all of the sources I’ve consulted ignore this

naturallyInconsistent

7:30 PM

@Semiclassical you are bombarding 7Li with p. The +3 comes from Li

naturallyInconsistent

7:46 PM

@Relativisticcucumber The issue I'm bringing up is alleviated by technology of VPNs these days, but it is not that long ago that, say, a botanist / zoologist / archaelogist goes to Africa, and immediately loses access to even their own taxpayer funded work. Shipping, paper and editing costs money, but it is never prohibitively expensive

@naturallyInconsistent so only the lithium nucleus is being bombarded?

i think a chat gpter just put an answer on one of my questions XD

8:10 PM

@naturallyInconsistent Hi, I have a question. If you recall at some point we talk about bound and stationary states. If I remember correctly you said, that a system can be characterized by stationary states, which are not bound right?

2 hours later…

10:00 PM

@SillyGoose my point is there are only U(1), R and products of them, and R is non-compact and boring - when in physics we talk about "Abelian gauge theory" we always mean U(1)

10:15 PM

okay i see

well i think i have narrowed down my confusion/question

(1) is the point of the bundle formalism of field theory to keep track of global data: e.g. not to look at what we would now call pullbacks of a connection $1$-form via local sections, but to look at the connection $1$-form itself.

(2) Take the perspective of Witten in his Jones paper. Is the point of Chern-Simons theory to (i) quantize it and then (ii) use Wilson loop observables in the theory to compute link invariants? Namely, the point is not to actually compute any data about the bundle, the manifold, or the group. It is to compute invariants of links drawn in the space?

@SillyGoose the data contained in the local connection forms and in "the bundle" is equivalent; it's just two ways of looking at the same information

@SillyGoose the Wilson loop observables are the holonomy of the connection along the loop; I do not understand the distinction that would make this not "data about the bundle"

well i guess what does "perturbative field theory" look at? I was imagining it looks at a single local section on the bundle or something, not all local sections

bit of a strange question - usually we do perturbative QFT in contractible flat space where you don't have any non-trivial bundles anyway

watt then why even do perturbative qft in that case?

maybe i don't know what it means for the qft to be perturbative

who does perturbative QFT where?

you seem to be under the impression I have the entire detail of the Jones polynomial paper memorized - I have not :P

10:26 PM

perturbative QFT on a trivial bundle

well this is just in reference to what you were saying

so what's the question?

why do perturbative qft on a triviable bundle

where else would you do it???

why would the bundle have to be non-trivial for perturbative QFT

what do you think "perturbative QFT" means?

it's the standard run-of-the-mill kind of QFT you learn in every intro to QFT book

ever seen a bundle there?

i thought perturbative QFT would mean you are working in some neighborhood of the actual underlying bundle structure

which would mean you are forgetting everything but that neighborhood and treating it as what's important

To me, "perturbative QFT" without any further qualifiers means the default way of doing QFT with Feynman diagrams and a perturbation expansion around the free Lagrangian in Minkowski space

I do not understand why it enters into this discussion about bundles :P

10:30 PM

well yes this is a bit tangential

The first point at which, in standard QFT, you might encounter hints of the bundle formalism are instantons, which are famously non-perturbative effects

i think then my question is how (or in what resource would it be described) does the holonomy of a connection of a loop depend on the base manifold, bundle, and structure group

because i see that it depends on the connection

and maybe what connections are available depend on the bundle and manifold, but i am not really sure the precise relation and how it is a "topological dependence"

I don't see what this has to do with "perturbative QFT" :P

but anyway, for a flat connection, the holonomy around two homotopic loops is the same, i.e. nullhomotopic loops always have trivial holonomy

the proof of this is either Ambrose-Singer or the non-Abelian Stokes theorem

so your base needs to allow for homotopically non-trivial loops for the holonomy to be interesting in the case of flat connections

which is a topological property

hmm i see

so this is about like the fundamental groupoid of the base manifold

that's how you get to the statement that the flat connections are characterized fully by homomorphisms from the fundamental group to the gauge group

10:39 PM

also so to see if i understood...this is about the topology of the base manifold not about the bundle, or no?

sure

it's interesting to note that the Chern classes of bundles that are not trivial but admit flat connections must all be torsion and hence all flat bundles over bases where the respective cohomology does not have torsion are trivial, see e.g. math.stackexchange.com/q/156493/143136

blebs okay thank you. i have been trying to sort out the dependencies between the triviality of the bundle, the connection being flat, and the topological properties of $M$, $P$, and $G$

so i guess maybe the "simple" cases of Chern-Simons is abelian, i.e. $U(1)$ Chern-Simons. Then there is trivial bundle non-abelian Chern-Simons. And then, finally, there is non-trivial bundle non-abelian Chern-Simons.

where $U(1)$ Chern-Simons can take place, I assume, on both trivial and non-trivial bundles

and then i guess there are generalizations depending on how many restrictions you put on the base manifold too...

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