1:51 AM

2:36 AM
skullpatrol says, hi pal @RyanUnger

3:19 AM
Hello World.

4:01 AM
@DanielSank Huh? They publish astro papers
though tbh most of the astro papers i've read from nature have been very old

din din
boooooow

🙏🏼🙏🏼🙏🏼

Namaste

2 hours later…
5:47 AM
@bolbteppa do you know any good resources on 2D CFTs?

So what I am confused about here is that we need to add the proper spin projections and get the unknown right?
Now Addition of spin implies these spin projections can take a range of values...Are we supposed to try all...or is there some other trick?
Because if we tried all there would be so many possibilities and so many answers( some like 7/2 which is not possible...but some even like 1/2 which is not mentioned in the correct answer)..
For example $2-1/2=x+1$ gives x=1/2 so it can be a spin 1/2 particle right?
But it cannot be according to the answer!!!
So what's wrong with the above equation?

I think I am missing some concept.(I read addition of spin angular momentum from Zettili's book of QM and Kamal and Griffith's book on particle physics...)

6:06 AM
3 messages moved from Problem Solving Strategies
Can anyone help with Manas' question. It was asked in the JEE room but none of us there can figure out the answer.

1 hour later…
7:18 AM
@RyanUnger Thx

7:34 AM
@SirCumference I meant that I heard that astrophysicists largely avoid Nature, not that Nature disallows astrophysics.

8:11 AM
@NiharKarve why not Di Francisco

@Slereah last I checked there didn't seem to be an online version

There are always online versions
if you know where to look
wink wink
nudge nudge
say no more

ok I'll see
but I'll probably have to settle for some lecture notes
thanks

@NiharKarve This book ?

8:49 AM
@JohnRennie that's the one, yes

@NiharKarve cough ... Z-Library ... cough

@JohnRennie thanks, never heard of that one

I created a room by mistake and want it to be removed please :(

these online PDF troves are a great use case for 10minutemail

@Jasmine just don't use it and it'll get deleted for inactivity automatically in a while

9:05 AM
@ACuriousMind Is there exist a procedure too delete old rooms?

Rooms that have more than 15 messages by at least 2 different users will be frozen if they are inactive, but not deleted. SE doesn't seem worried about the space the rooms take up on their servers, so there's no real deletion - if needed, all rooms can be unfrozen or undeleted after an indefinite amount of time.

9:31 AM
It's not like SE is overrun with chat
There aren't a lot of active chats here

9:42 AM
@ManasDogra I don't really see a reason why either 1/2 or 7/2 should be impossible

9:58 AM
I was calculating the multiplicity of N oscillators, get the same result as Kettle, $g(N,n)=\frac{(N+n-1)!}{n!(N-1)!}$; however, when I plug in N=2 (2 oscillators), n=2 (total energy is 2hw), I get a $g=3$!!

@JohnRennie Since yu said that you will look into the standard state problem when you have time,when is the earliest that I ping you?

shouldn't $g=1$?

@user586228 I probably won't have time today.

No probelm tomorrow?
problem*

10:52 AM
@NiharKarve DI Francesco is considered the standard reference so it's probably worth getting a copy. I think most string theory books/notes have at least a chapter on CFT, eg Weigand's and Tong's notes
these notes are also often cited, but I confess I never read them arxiv.org/abs/hep-th/9108028
for a slightly different perspective (mostly different applications) there's the statphys literature, Cardy's book has some of the basics, but I think he also has some notes specifically on CFT
Mussardo's book also has CFT chapters. Then there are more maths heavy/rigorous expositions.

Yeah, the yellow book is worth getting if you are going to devote any serious time to cft, but there's four sets of notes cited here that are often recommended, and try some of BPZ to get a sense of what cft is really about which is comparatively scattered through Di F, a random thesis on ads-cft will also do the bare minimum cft without going into the weeds on minimal models

@fqq haha, my main motivation was to find something more comprehensive than Ch4 Tong String Theory. But yeah, that's where I'm coming from
Thanks so much for the refs guys, really appreciate it

The perimeter videos follow Di F pretty closely and it's useful to get some extra input on this stuff

@NiharKarveup for a game of chess again sometime?

1 hour later…
12:10 PM
@bolbteppa I'm afraid I can never look at your name and think it's "Bob"
Bob Teppa

To make matters worse that b actually stands for a V (and the second one is silent, and the p's are actually r's)

Bob will do

@bolbteppa Cyrillic?

Yeah
Professor Vito Volterra (, Italian: [ˈviːto volˈtɛrra]; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. == Biography == Born in Ancona, then part of the Papal States, into a very poor Jewish family: his father was Abramo Volterra and mother, Angelica Almagia.Volterra showed early promise in mathematics before attending the University of Pisa, where he fell under the influence of Enrico Betti, and where he became professor of rational mechanics...

12:49 PM
@MoreAnonymous sure :)

@bolbteppa how did л -> L

1:18 PM
El (Л л; italics: Л л) is a letter of the Cyrillic script. El commonly represents the alveolar lateral approximant /l/. In Slavic languages it may be either palatalized or slightly velarized; see below. == Allography == In some typefaces the Cyrillic letter El has a grapheme which may be confused with the Cyrillic letter Pe (Пп). Note that Pe has a straight left leg, without the hook. An alternative form of El (Ʌ ʌ) is more common in Russian, Ukrainian, Belarusian, Bulgarian, Macedonian, and Serbian. == History == The Cyrillic letter El was derived from the Greek letter lambda (Λ λ). I...
eL google translate agrees with me :p

1:31 PM
@bolbteppa but if you go by letter-relations, then you have to transcribe the cyrillic r also with r, i.e. bolbterra
although the modern P looks like the ancient $\rho$, it is not the case that P developed from $\rho$ - $\rho$ developed into R, the similarity between Latin P and Greek $\rho$ is accidental

Not according to google translate :p

1:47 PM
I'm sorry to hear Google translate doesn't understand the history of letters :P
but I'm not sure how you're using it, when I enter Вольтерра into Translate, it outputs Volterra
in contrast, bolbteppa becomes something different:

What does (English) Volterra output as in Russian

keep iterating
see if you find a fixed point
@bolbteppa Volterra -> Вольтерра, like in your profile

There we go

2:03 PM
Aha! We finally know who you are!
are were :P

2:18 PM
I have to use the mathjax method to do diagrams on my site
Using the way that nlab does it
It works but boy it's not pretty

\array{
&T_xX&\\
\overset{f_1}{\swarrow}&\downarrow_{f_1 \oplus f_2}&\overset{f_2}{\searrow}\\
T_{q_1} M_1 & T_{q_1} M_1 \oplus T_{q_2} M_2 & T_{q_2} M_2 \\
\underset{g_1}{\searrow} & \downarrow_{ g_1 \oplus g_2} & \underset{g_2}{\swarrow} \\
&T_yY&}
Maybe I shall put them as png instead

2:47 PM
Also I'm trying to define the exterior tensor product properly but that involves too many projectors
I'm running out of letters
they are getting unwieldy

Start a new trend
Use Klingon letters

@Slereah what's the "exterior tensor product"?

@ACuriousMind the structure defined for bitensors

@ACuriousMind you know K theory don't you?

@Slereah I'm not sure what that is either :P
@RyanUnger not really

2:58 PM
Given X1,X2∈H the external tensor product over these is the functor

⊠:Mod(X1)×Mod(X2)⟶Mod(X1×X2)
given on A1∈Mod(X1) with A2∈Mod(X2) by

A1⊠A2≔(p∗1A1)⊗X1×X2(p∗2A2)∈Mod(X1×X2),
where p1,p2 denote the projection maps out of the Cartesian product X1×X2∈H.

@ACuriousMind A tensor but bi
Roughly speaking a tensor defined at two points of the same manifold
Your bundle is something like $$\pi : V_1 \boxtimes V_2 \to M \times M$$

I feel like this is one of those things where if you just write it in coordinates there's no issue

@RyanUnger pretty much

it's an object $v_i(x)w_\mu(y)$ etc

That's what most papers do certainly

3:00 PM
doesn't need to be $M\times M$ right
just some product

Well it could be any two manifolds, yeah

yeah

Although in the case I need it for it's the same one
the most famous bitensor being the worldfunction
Also the parallel propagator, I guess

3:23 PM
@ACuriousMind Yes you are right...By the way this question is from an entrance exam to TIFR,India.

3:46 PM
For an equation like $(\mathbf{\nabla u })\cdot \mathbf{S}$
eek
what's dot product in mathjax
Is there an ambiguity in which index is dotted
oh actually never mind

4:33 PM
3I was calculating the multiplicity of N oscillators, get the same result as the Thermal Physics wirtting by K&K, $g(N,n)=\frac{(N+n-1)!}{n!(N-1)!}$; however, when I plug in N=2 (2 oscillators), n=2 (total energy is 2hw), I get a $g=3$!! I think $g$ should be $1$?
There should be only one state for min energy for a system of two oscillators. No idea what I have gone wrong.

Hi All..
Pls see the link the what is sentence of underline.
In mechanics what is the meaning of "To avoid reaction at point we take moments about it."

@Shing why are you using $n=2$ and think that's the minimal energy?
If I understand correctly, in your formula $n$ represents the sum of the excitation numbers of all the harmonic oscillators involved. The lowest possible excitation number of a QHO is 0, so $n=0$ is the non-degenerate ground state of a system of oscillators, too
for $n=2$, you have three possible distributions of the excitations among 2 oscillators, namely $(2,0), (1,1), (0,2)$, so $g=3$ is the expected result

5:03 PM
Hey I was a bit curious - I have some questions that are too ill formed for main sites.
In particular I am kind of interested what "algebraic quantum field theory" refers to today and what its "status" is. I've been told that the program has kind of "failed" (and I think the person talking to me was referring to the stuff in Haag's book "Local quantum fields") is that accurate?
I also found a book by Baez Segal and another guy that is literally called "Introduction Algebraic and Constructive Quantum Field Theory", but its content / perspective is so different from Haag's book.

5:34 PM
Depends what you mean by "failed"

I don't know - maybe something like "people no longer think its a good approach"?

It's used in some fields, most QFT people do not use it because the "classic" methods work well enough, but also it's not enough to solve standard model calculations by itself

@ACuriousMind Yes, you are correct, thank you so much! I was having each's oscillator's energy in mind, and I just "shifted" the 1/2 to 1. this is something very wrong, I think. the total energy of two oscillators (if I use n+1/2) in the states (0,0), (1,1) are $1\hbar\omega$ and $3\hbar\omega$ respectively. However, if I use n+1, then (0,0), (1,1) are $2\hbar\omega$ and $4\hbar\omega$.

But modern methods for it will also use methods from AQFT
It's not lost to the wind

do you understand AQFT to be the net of local operator algebras approach?

5:36 PM
okay, now I remember why the "1/2", I think it is for symmetry of the ladder?

That's usually the one, yes

If I pressed you what language would you use to describe the mathematical foundation of QFT? (or what such a foundation would/should look like)

@s.harp is there a single result related to that field

@s.harp Bit of a complicated question!
There's quite a few
and how they relate to each other, and also to QFT as it is practiced, isn't easy

@bolbteppa I know next to nothing about the theory, but here is one result: A state gives a time evolution which makes the state "thermal", ie an explicit version of the connecction between statistical physics and time usually (which is usually vaguely suggested by everybody)

5:42 PM
You can start from scratch and build the QFT yourself or you can take a classical theory and quantize it, you can define it from an operator theory, an integrator, functors, graphs, algebras, and also amidst all that you have to do some kind of regularization procedure because I think we gave up doing QFT without renormalization these days

Not sure what that means, I would bet a lot that algebraic qft doesn't show anything like that

also there's the lattice version
I don't know if there's a theory that's considered more modern QFT than the others
The true QFT is the friends we made along the way
4

I need to eat, but thanks for your comments I will come back later
But to say something about how I would answer that question: I always used to think that the mathematical foundation of QFT should be represenation theory - a QFT should be a projective representaiton of an appropriate symmetry group (ie wightman axioms)

I mean that's certainly part of it
but not all of it I'm afraid
According to nlab, this is how you define QFT :
$$\mathrm{PolyObs}(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \otimes \mathrm{PolyObs}(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \overset{\star_{\tfrac{i}{2}\Delta}}{\longrightarrow} \mathrm{PolyObs}(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]$$
Hope that helps

It's simply not possible to understand any qft without the BV-BRST polyobservables :p

5:53 PM
Well, you don't need it for a free theory without gauge, I suppose!

Luckily Bhabha had eaten his broccoli and learned higher topos theory before getting to scattering amplitudes!
@s.harp Folland and Ticciati are two books which are most likely to strike a balance between rigor and a normal qft book, but they are probably way too much if one hasn't learned it the normal way first

is perturbation theory the protagonist of quantum field theory?

@Bohemianrelativist I haven't studied QFT literary analysis

@s.harp ...would you say that an ordinary QM system with its Hamiltonian is "a projective representation of an appropriate symmetry group"?
Certainly the space of states is always a representation of the algebras and groups of operators acting on it, but the core of the physical theory, the time evolution generated by the Hamiltonian, doesn't really have all that much to do with representation theory

I mean, I think when people say "QFT", they mean "the standard model" specifically
Not a random QFT

6:07 PM
@Slereah It's like Coleman's preface says perturbation theory is the main contents of QFT.

Perturbation theory is pretty important, and on a practical level, that's mostly what people do

There is no reason the mathematical tools needed to explain a physical theory should correspond neatly to one of the subfields of mathematics, and indeed they often don't

I don't know if that makes it the protagonist
Or the love interest

@Bohemianrelativist perturbation theory is what many people do because its computationally tractable and gives excellent results for a large variety of situations
but e.g. strongly-coupled QCD isn't amenable to perturbation theory and certainly also an important part of applied QFT

also you could technically solve a QFT without ever using perturbation

6:09 PM
and I'm not sure how perturbative various condensed matter applications are

But that would be challenging

the "QFT is perturbation theory" viewpoint is usually something from people who mostly look at high-energy colliders :P

Those damn S-matrix fetishists

@Slereah I think the problem is that we actually don't have the techniques to do that :P

@ACuriousMind I didn't say it would be easy!

6:11 PM
just do the sum

The perturbative sum is divergent
So it's pretty easy
$$D(x, y) = \infty$$

incredible

6:23 PM
There is some notion that the perturbative expansion converges to the actual solution up to some specific order IIRC
And diverges after that
I'm not sure if there's an actual theorem for that or if it's just vague

6:44 PM
Not sure that a non-perturbative perspective makes sense tbh

It may or may not, depending on the theory

@NiharKarve I know
I don't know if there's a theorem as it applies to QFT, though
IIRC it's tied to the coupling constant

7:02 PM
of interest

@ACuriousMind I mean thats because global time is special for QM (vs QFT). Unlike (classical) field theories the way you talk about QM has kinematics and dynamics - here the representation of the observables (not the symmetry group, right?) is the kinematics.
I guess my second question is trying to get an answer to "what do you think a QFT is", regardless whether or not its possible to DO something with this "what"

You can have a QFT without any symmetries

QFT is quantum mechanics with the Galilean group replaced by the Poincare group

@Slereah You mean like you can have a completely inhomogeneous solution to Einstein equations for GR? Where the isometry group is trivial? Or do you mean something else?

yes
Although the Poincaré group still plays a role, 'course
since it's always a local symmetry

7:06 PM
I should say relativistic qft is...

I mean rly I think QFT is good enough to define what QFT is
it's a quantum theory of fields
Defining what that means is a bit tricky but that about encapsulate it

Describing an "inhomogenous" configuration really is something that a hypothetical grounding should be able to do - so I agree that moving away from the "representation of the Poincare group" viewpoint is good
unfortunately everything is so muddled in my head

@s.harp 1. QFT is not inherently relativistic, you can do QFT in Euclidean/Galilean space. 2. How do you think the dynamics of a QFT - its Lagrangian - are encoded in a representation?

there's a lot of things you could include or exclude under "QFT"

It's pretty much a shock to actually see how to go directly from the usual QM to non-relativistic qft

7:10 PM
@s.harp "A QFT" is what we do to get our usual predictions, really
i.e. some very non-rigorous muddling about with Feynman diagrams or path integrals that in the end results in some scattering amplitudes or expectation values

The problem with the viewpoint "QFT is that what works to describe it" is that its probably philosophically sound (and probably philosophically superior to Platonism which is for mental adolescents like me)
but I didnt get interested in physics for reasons like that - it doesn't satisfy the craving for order or truth that I had as a "kid"

If you phrase everything in terms of path integrals there aren't any "states" or "representations", there's just this strange procedure where you put in some polynomial in the fields and get out expectation values

if you want order and truth I'm not sure you're in the right field :p

well I'm a mathematician

it's a bit of a mess
If you want some fancy QFT you can try Reed & Simon

7:14 PM
wha? Simon & Reed didn't write a book on QFT did they? just operator theory I thought

If you read it you'd notice that there's a few chapters dedicated to QFT!
Book 2 has the theory for Klein Gordon QFT

@s.harp physics isn't about platonic truth, it's about providing working approximate models that can usefully describe reality

except GR, 'course

You wont get a platonic perspective from any math book on qft, you will from L&L :p

GR is perfect and beautiful

7:16 PM
The Course of Theoretical Physics is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s. It is said that Landau composed much of the series in his head while in an NKVD prison in 1938-1939. However, almost all of the actual writing of the early volumes was done by Lifshitz, giving rise to the witticism, "not a word of Landau and not a thought of Lifshitz". The first eight volumes were finished in the 1950s, written in Russian and translated into English in the late 1950s...

L&L is a bit old timey
They don't even talk about Hilbert spaces

Hilbert spaces are a lie

tell that to Hilbert

First off they need to be rigged, and few people care to be careful about that

"It is thus clear that such a theory cannot lead to healthy correlation functions,in other words for any negativeλthe power series (12) will diverge. From this w ecan conclude the radius of convergence being R= 0!"
noooo

7:18 PM
It's old timey in the sense that Yang-Mills and SSB were being worked out when those books came out, but up to QED it's all solid

there are dynamical formulations of (classical) electromagnetism but there are also non-dynamical ones that involve more machinery. Its much harder (and stupider) to use sections of fibre bundles to design a telegraph wire.
But the perspective that (classical) field theories are static and not dynamic is more common in theoretical and mathemtiacl circles - right?
so my distinction QM - QFT is not non-relativistic - relativistc | but finite dimensional phase space - infinite dimensional phase space

You still need a way to find the dynamics even in very fancy formulations
You need to define a Hamiltonian vector field or what have you on that bundle

@s.harp yes, finite vs. infinite d.o.f. is the correct distinction, but you said something about "global time", which is a relativistic/non-relativistic distinction usually

Simply because given the same representations or whatever else, you can define multiple theories

7:23 PM
there's nothing "static" about a field theory - the Euler-Lagrange equations are usually still thought about as answering the question "given an initial field configuration at an instant, what will it evolve to?"

you need a form on your
JET BUNDLE
dun dun

I need to think about the things you said before I reply further. But I also want to ask one more thing about the "practical" perspective ("QFT is anything that can usefully describe reality").
Because this perspective is fundamentally at odds with the very popular (among physicists) string theory! I don't want to be polemic, but that seems like a counter-example to any claim that people aren't interested in what QFT "is" and that the utilitarian perspective (if it works then thats QFT) is king

Well not everything that works is QFT
Otherwise Newtonian mechanics would be QFT

everything that works to describe a certain regime of observation

QFT is a specific field in physics
It's a quantum theory of fields, that's about the best I can describe it without getting more specific!
You have a classical theory of fields, you quantize it
Or you have a quantum theory from the wavefunctions on the space of fields
Or you have a C^* algebra for field operators

7:34 PM
Quantum mechanics/qft says classical mechanics doesn't exist, but it still needs to reduce to this non-existent theory in a limit that can only be defined by the existence of the theory that doesn't exist

I mean it does exist, but I'm still not 100% sure if there's a generic process for it?
The dequantization
We talked about it a few times here
How do you go back from the Hilbert space to the configuration space

somebody told me to do a Rieffel deformation for that direction, but I'll be quiet for a while now

The first thing QM says is that the variables upon which all of classical mechanics is founded on do not even exist. If they did we could theoretically repeat experiments exactly which is exactly what experiments do not show

I mean yeah, but [something something limit process]

(sorry I looked it up again, deformation is [classical] -> [quantum] not the otherway around)

7:39 PM
Although I'm not 100% sure this works for everything

So instantly we have absolutely nothing, no theory. The only way to define QM from then on is by the fact that classical mechanics has to exist in some vague sense [yeah limiting, which also has to be defined]

Like if there are correlations involved, is there a classical limit?

I'm not sure about that stuff yet but if there's a serious problem with it it would mean Bohr etc were wrong about QM and that's very unlikely

Classical mechanics cannot be contextual

@s.harp your perception of popularity of string theory among physicists is probably exaggerated - many physicists don't even know all that much about QFT!

7:42 PM
I know a physicist IRL
One time I showed him the stuff I was writing about and he burst into flames

that's a vampire, not a physicist
it's an easy mistake to make

He is a rheologist so he is not used to math beyond "a function"
mostly he inflicts punishment on fluids

8:23 PM
that paper actually explains quite nicely why perturbation doesn't work and also works
Although as usual, the warning
QFT is divergent in several different ways
Taming all these beasts is the main issue

9:04 PM
Classical field theory is also divergent and it's directly due to special relativity and point particles
QFT just exacerbates those problems

where does classical field theory become divergent?

Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.For example, an electron theory may begin by postulating an electron with an initial mass...
This, the self-reaction and I think some fluid example are basically what motivated Kramers etc... note the 'classical electron radius' violates special relativity so it's an incompleteness in classical physics

ah i see

2 hours later…
10:53 PM
regarding the induced metric on the worldsheet in string theory, along which map do we pull-back the background spacetime metric?

@Charlie along the coordinates of the string, usually denoted $X$

hmm ok

you don't sound convinced :P

11:10 PM
I guess it's odd to me that we pull back a flat metric onto flat spacetime

I'm not sure what that means - you're pulling back a flat spacetime metric onto a worldsheet that a priori has no metric

but is the worldsheet not just like, a submanifold of spacetime?

The $X$ makes it one

If we want to measure distances/areas on the worldsheet, why not just do it in a chart? it seems like the string is being treated as something that isn't just a sheet sitting in spacetime

The worldsheet is a 2d manifold $\Sigma$ and the coordinates $X : \Sigma \to\mathbb{R}^{26}$ are an embedding that identifies it as a submanifold
@Charlie I think you're trying to imbue the mathematics with a bit too much ontology here
Maybe an analogy helps: When we consider a curve in a manifold, the curve is not just "a submanifold of dimension 1", it is an embedding of the abstract manifold "the unit interval" into the manifold, i.e. an embedding $\gamma : I \to M$. Likewise, a worldsheet is an embedding of a 2d manifold into another manifold
you wouldn't ask "why are we treating the curve as an interval that's not sitting in spacetime", would you?

11:20 PM
oh, is the metric begin pulled all the way back to the 2d manifold itself?
that actually would make more sense
my problem was that it seemed trivial to give an submanifold a metric obtained by pulling back the metric of the surrounding space to it

since the 2d manifold is isomorphic to its image under the embedding, I'm not sure that's a meaningful question :P
The reason why "pulling back" a metric is the natural thing to do is precisely because the pulled back metric gives the same results as "restricting" the metric to the submanifold would

ohh
If it's the same as just restricting the metric that makes sense, both Tong's notes and my lecturer made a point of saying it's the pullback of the metric, which made me think there was something special about it
a bit like taking a circle, then considering a small part of the circle and measuring distances on it by just using the metric on the surrounding circle :p
that seemed like a strange point to labour

it's just standard differential geometry, but for some reason some physicists feel they suddenly need to pretend to do actual math when doing string theory (and then still not do it :P)

oh god wait the "coordinates" $X$ we're talking about here go from the 2d manifold into spacetime, right? We're not talking about manifold coordinates in the sense of charts
or hm

11:36 PM
@Charlie yes, they're maps $X: \Sigma \to M$ where $M$ is the spacetime.

you've written the coordinates as $X:\Sigma\rightarrow \Bbb R^{26}$, is $\Bbb R^{26}$ the Euclidean space that is the target space of charts on spacetime or is it spacetime itself?
ah
now it seems like a slightly less trivial thing to do, because $\Sigma$ is now a separate manifold

but when the physicists write $X^\mu(\sigma, \tau)$, that of course is $X$ "expressed in charts" - the $(\sigma,\tau)$ live in some $\mathbb{R}^2$ chart of $\Sigma$ and the $X^\mu$ values live in some $\mathbb{R}^{26}$ chart of $M$
it's common not just in physics to not care much about the distinction between expressions "in local coordinates" and the abstract map, don't worry about that too much

I seee
ok ty, that is clearer