Redpill me on Topological QFT

youtube.com/watch?v=Bo8GNfN-Xn4
he explains

thanks user

A n-dim. TQFT is a symmetric monoidal functor from the category of n-cobordisms to the category of vector spaces.

The category of vector spaces consists:

- vector spaces
- linear maps between them

The category of n-cobordisms consists of:

- closed (n-1)-manifolds
- cobordisms between them

So a n-dim. TQFT associated to every closed (n-1)-manifold, a vector space. And to every cobordism of (n-1)-manifolds, a linear map.

The "symmetric monoidal" condition means it maps disjoint unions to tensors products.

i.e. the image of a disjoint union of two (n-1)-manifolds, is the tensor product of the images of each manifold

woah...okay, i see...Thanks user! Can you tell me what background do i have to have to get into it? i'm thinking about doing my graduate thesis (a paper you need to right to get a degree) about it

are you a physicist or a mathematician ?

Yes, it is a cobordism from a circle to a disjoint union of 2 circles.

Just standard stuff on manifolds and algebra.

I like this as an introduction: arxiv.org/abs/math/0512103

i'm a physicist.. but i mostly study mathematics, thus im interested in such a field
thank you!!
Do you think its too muck for an undergrad-level student (i mean it for my paper before the degree)

ask your professors about it. they know you, and your background.

>Do you think its too muck for an undergrad-level student (i mean it for my paper before the degree)

I don't think so. That math really isn't that bad, and you can blackbox a lot of the physics used to motivate it.

well i'm too embarrassed to ask cause i'm only in my second year of undergrad and i know its too early for even thinking about it. But i like to have things planed beforehand, to have a rough idea of what i would like to study. Thats why i ask your opinion on it.

I'm not the other user, but if you're thinking about this you need the "standard stuff on manifolds and algebra" he mentioned. This includes, at the very least, courses in groups, rings and fields, in topology, in differential geometry. perhaps some algebraic / differential topology too.

don't be embarrassed. talk to a professor who teachers a class you like and participate in, and be honest. tell them you find this stuff interesting and would like to get there in time for your thesis project. you seem to have a lot of time left to get there.

Well i'm familiar with these subjects at least in an introductory level (currently taking diffgeo)... Well i guess you are right.. but heck i haven't even had my first QM course yet it would be foolish of me to discuss stuff like that, you see what i mean..

It would be a bit weird as a physics student if you haven't even taken QM, but not as a math student.

Maybe add math as a second major?

Thought of that already...i would love to learn all these subjects in a deeper level but i would lose at least 2-3 years in which i can do my masters degree im mathematical physics, thus covering both (and doing only the math i would like to)

dude, you're totally fine, don't overthink it
the worst case scenario: you will write a shitty thesis but guess what, NOBODY gives a fuck about your undergrad thesis

there's nothing weird about being interested in something you know jack shit about. your professors know the feeling well, and will help you out if you're up front about it

well i guess you are right...and i will learn much more in the upcoming two years left for my thesis...although, i've been told it matters if you want to proceed in academia..

Okay, i'll consider asking for help or talking about it at least!

what matters is (proven) drive to study hard and will to succeed. you're in your second year, and it sounds like you're doing quite well. talk to your professors so they can help you get involved with research projects that are right for you, and to get on the path to the things you want to learn.

I never thought I'd see someone actually use penrose notation

okay i hear you..Thanks anons, this really helped me!

Not quite the same thing.

The nlab entry talks about manifolds being cobordant or cobordism classes. But aren't any two manifolds of the same dimension going to be cobordant?

No because cobordisms are required to be compact.

ok. But if the two manifolds are compact then they'll be cobordant?

math.stackexchange.com/questions/1385708/examples-of-manifolds-that-are-not-boundaries

cool, thanks

I mean the n-manifold, determining whether 2 closed (n-1)-manifolds are cobordant, is compact.

cocks

10/10 comment

>can't describe reality because reality has geometric structure

TQFT BTFO

>im a fat faggot
kys retard

RIP Voevodsky, 2016-2017. We shall all unite under your HoTT-Coq (also known as UniMath).

It also needs to be modular and maps the empty cobordism to [math]\mathbb{C}[/math].
In general a TQFT is a tuple [math](\mathscr{T},\tau)[/math] where [math]\mathscr{T}[/math] is the functor you've mentioned and [math]\tau[/math] is the quantum invariant. [math]\mathscr{T}[/math] encodes all the categorical information (i.e. naturality and functoriality of glueing patterns) while [math]\tau[/math] makes sure the TQFT is a topological field theory (i.e. computes topological invariants with values in the R-module).
The category of 2-cobordisms can be finitely generated by cusps, pants and cylinders which satisfy certain algebraic relations, which is why 2D TQFTs are isomorphic to commutative Frobenius algebras. The quantum invariants is just the Euler characteristic.
In the case of 3D TQFTs more work is needed since the Euler characteristic is not a cobordism invariant. Knot theory and Dehn surgery theory has been used to embed ribbon graphs into [math]S^3[/math] and surgering 3-manifolds from it, which keeps track of 3-cobordisms in an invariant way. The quantum invariant in this case can then be defined via knot invariants.
4D TQFTs are yet harder, because there exists exotic 4-manifolds that are h-cobordant but does not admit diffeomorphic differential structures. This is a problem for studying 4D Yang-Mills from the TQFT/CFT point of view, and I'm my opinion presents one of the biggest roadblocks to the Yang-Mills gap problem. In fact it has been proven a few years back that 4D unitary TQFTs induce a non-positive definite inner product on [math]\mathbb{C}[/maths]-linear spaces, which is a fatal problem if you want your UTQFT to describe any physical theory. This is precisely due to the fact that the knot-based quantum invariants generalized from 3D TQFT is not a h-cobordism invariant on exotic Mazur 4-manifolds.

lol i only understand the words manifold and invariant

>doesn't know what pants are

dat thigh gap tho

hhhhhhhhhhhhhhhhhnnnnnnnnnnnnnnnnnnnnnnnnnggggggggggggggggggggh

What's a cusp and why does it have to map the empty cobordism to C?

Sorry I meant that the empty set maps to C. The cusp is a cobordism between a loop and the empty set so it's assigned a C-linear morphism from a C-space V to C.
The choice that [math]\mathscr{T}[/math] maps the empty set to the ground ring R (C in the case of unitary TQFTs) is just a normalization condition.

>just a normalization condition

meaning?

also, nice pic.

Sorry for the slow response, my phone battery died as I was at the gym.
To see why the functor maps the empty set to the ground ring R, note that [math]\mathscr{T}[/math] maps the manifold [math]\overline{M}[/math] with the reverse orientation of [math]{M}[/math] to the dual [math]\mathscr{T}(M)^*[/math] of the R-module [math]\mathscr{T}(M)[/math], so since [math]\overline{\emptyset} = \emptyset[/math], we have [math]\mathscr{T}(\emptyset)^* = \mathscr{T}(\emptyset)[/math] and hence by the non-degeneracy (prove this!) of the pairing [math]\mathscr{T}(\emptyset) \times \mathscr{T}(\emptyset)^* = \mathscr{T}(\emptyset) \times \mathscr{T}(\emptyset) \rightarrow R[/math] we see that there is an isomorphism [math] \mathscr{T}(\emptyset) \cong R[/math], since the pairing would just be a multiplication by an element in R. Hence for any cobordism [math]W[/math] that ends/starts with the empty set, we can pre/post-compose the R-linear homomorphism [math]\mathscr{T}(W)[/math] with the isomorphism [math] \mathscr{T}(\emptyset) \cong R[/math] such that the new [math]\mathscr{T}(W)[/math] maps whatever space to the empty set (or vice versa). This isomorphism is unique up to isotopy so this doesn't change the categorical structure of R-modules.
This "shifting" of morphisms can be considered as a "normalization".

Hey T(ouhou)QFT poster, what do you make of this

news.rice.edu/2017/12/18/rice-u-physicists-discover-new-type-of-quantum-material/
Forgot link

is the gym worth it?

Really interesting. I was actually studying topological superconductors before I switched to mathematical physics.
One of the most powerful tools for studying quantum cirticality is conformal field theory, since a condensed matter system has infinite correlation length at the critical point and hence achieves conformal symmetry. In 2D the symmetry group is the modular (Mobius) group of [math]\mathbb{C}[/math], which is so large that all observables in the theory can be described by an affine Lie algebra. This means that the conformal blocks satisfy a finite set of differential KZ equations and so the [math]n[/math]-point correlation functions of the CFT can be solved exactly at the critical point.
General mathematical structures of CFT have been outlined by Seiberg and Moore, and connections to category theory made by Nayak and others. The latter is manifestly important since this gives a way to frame CFTs in terms of certain types of TQFTs, and quantum criticality described by the CFT can be computed via topological methods. An example would be the fact that the non-trivial particle statistics arising from the fusion relations in a CFT can be computed via Vafa's theorem in the corresponding TQFT.
It was good. Did some oly lifts while flirting with my bf.

Do you think mathematics is unreasonably effective at describing reality?

I wouldn't say it's unreasonable. Math was created to serve the needs of physics so it's only natural that it finds application in it, and sometimes even the other way around. No matter how far the mathematicians abstract away from the original, physically motivated first principles physics will always progress to a point where that level of abstraction is needed (or can be used) to describe reality. Even Diophantine equations can be used to describe the duality of quantum Hall states.

user what is pic related?

Turaev.

Two holes

if I would take the disjoint union of 5 circles, which combinations remain after the operation?

Uhh please repeat the question.

The red pill on topological QFT is that Topological and QFT are both meme words

fuck off

Based role-player.

Based Lurie. Pls give him an Abel prize.
>do you think it's too much
Yes, and frankly it'd probably be too much for your supervisor too unless he's doing string theory, and even then he'd only be familiar with specific examples of TQFT like Chern-Simons, WZW or topological theta models in the best scenario. In any other cases he'll either direct you to other profs or (implicitly) tell you to stop worrying about things (specifically the categorical definition of a TQFT a la Atiyah) you won't ever use in physics.
Mathematical physics is actually part of mathematics, so you'd be better off asking math faculties about TQFT than physics faculties. The examples above are already rich enough for you to spend years studying if you want to continue to be a physicist.
This is not to say that you shouldn't pursue your interests, but I'm just laying out what I've went through to bare so that you won't waste 2 years for your masters like me doing something you're not as interested in.

hi hank pls halp

Let M be an even dimensional smooth manifold.
I want to find an example M such that "Kahler cone ≠ symplectic cone" with non-empty Kahler cone, satisfying the following conditions:
M admits a Kahler structure.
ω is a symplectic form on M.
There is no Kahler structure (M,ω,J) such that [ω]=[ω]∈H^2 (M;R).

>In 2D the symmetry group is the modular (Mobius) group of C, which is so large that all observables in the theory can be described by an affine Lie algebra.

Can you explain this a little more? Are you saying that at the critical point you can start describing things with affine Lie algebras? I don't really understand what this signifies.

oh i see... my prof is studying string but yet i dont know if he would know all that stuff... you say that cause you ended up not liking the subject?

In the presence of scaling symmetry near the critical point you can write the energy-momentum tensor as a holomorphic part and an antiholomorphic part, which can be evaluated using contour integrals. Then you can build up the generators of the conformal symmetry with this using Cauchy integral formula
[math]L_n(z) = \int_\gamma \frac{dw}{(z-w)^{n+1}}T(w)[/math] as well as its dual part [math]\overline{L}_n[/math], and from the commutation relations of [math]T[/math] you can deduce the commutation relations of the generators [math]L_n[/math], which turns out to be the Virasoro algebra [math][L_n,L_m] =
L_{n+m} + \delta_{n+m}\frac{c}{12}[/math]. Given a simple Lie algebra [math]\mathfrak{g}[/math], the generators form the basis for the affine Lie algebra [math]\mathfrak{g}_\mathbb{C} = (\mathfrak{g}^+ \otimes \mathfrak{g}^-) \oplus c \mathbb{C}[/math] of [math]\mathfrak{g}[/math], where [math]\mathfrak{g}^+[/math] is generated by [math](L_n,f), f \in \mathbb{C}((z))[/math] and [math]\mathfrak{g}^-[/math] by [math](\overline L_n,f), f \in \mathbb{C}((z))[/math], where [math]\mathbb{C}((z))[/math] is the algebra of Laurent series in [math]z[/math] (you can generalized this to sheaves of Laurent series [math]\mathcal{O}_R[/math] on a Riemann surface [math]R[/math]). The Verma module that describes the physical states are then defined to be the set of vectors [math]v \in V[/math] such that [math]U(\mathfrak{g}^+)v =
0[/math], and the "primary field operators" of the CFT lie within the loop subgroup [math]LG^+[/math] of the loop group [math]LG[/math] of the central extension [math]\hat{\mathfrak{g}}_\mathbb{C}[/math] of the affine Lie algebra [math]\mathfrak{g}_\mathbb{C}[/math] of the Lie algebra [math]\mathfrak{g}[/math].
It is then possible to define a Hitchin's connection on [math]LG^+[/math], the flatness condition of which gives rise to the KZ equations, and these are the equations that your conformal blocks satisfy.

If I recall correctly Kahler manifolds are symplectic manifolds with a Kahler potential 0-form [math]K[/math] such that locally [math]\omega = \frac{\partial^2 K}{\partial z\partial \overline{z}}dzd\overline{z}[/math]. You need to elaborate on what a "Kahler cone" is since as far as I know these cones of a 2-form [math]\omega[/math] are defined to be vector fields [math]\xi[/math] such that [math]\omega_\xi(\eta) = 0[/math] for all vector fields [math]\eta[/math].
It never hurts to ask, and it's always good to let your prof know where your interests lies. They don't want a student who's begrudgingly going through the motions on something they don't have interest in either.
>you say that cause you ended up not liking the subject?
I studied condensed matter theory focusing on topological defects, but it wasn't mathematical enough. I was mainly interested in the mathematics that came out of the physics instead of the physics itself, while theoretical physics is still very much tied to experimental results, meaning that what I say needs to be at least experimentally feasible and isn't just some interesting mathematical result. This fact pushed me over the edge while pursuing a PhD.
Again, this is just anecdotal and you should decide on what you want to do based on what your prof suggests, not what some random internet stranger says.

>I was mainly interested in the mathematics that came out of the physics instead of the physics itself
are you me? hahaha
i hear you user..thank for the advice, much appreciated. I will see if i can do my ungergrad thesis with a math prof. although i think those subjects tend more to the interest of physicists than mathematicians (at least i haven't found one math prof in our department interested at QFTs...the closest i got is applications of diff geo which is awesome bit not something i would want for my undergrad thesis)

t-thank you p-professor y-yukari

Thanks user. I don't know about some of the math you're using, and I had to look up stuff to try to follow, so let me just recap in simpleton terms and you tell me if this is correct:

We have an energy-momentum tensor, which can be described as a polynomial including all negative powers (which is what I think holomorphic and antiholomorphic is saying).

From this we get generators (of what, and how do you get generators from the energy-momentum tensor? I know how to do integrals in the complex plane, but is there some sort of procedural way to obtain generators I'm not aware of?).

The generators are that of a scale transformation, so that [math]mathcal{g}^+[/math] and [math]mathcal{g}^-[/math] are the spaces of the transformations for zooming in and out of your system.

The second half is still too confusing for me. I'd need some concrete examples of physical systems using these concepts and definitions before my brain accepts them as meaningful. Or I need more math classes.

Are you by any chance the Russian user from Lomonosov Moscow State University? If so, how rigorous was your gauge fields and fibred spaces course? I'm disappointed by my university's take on it and am thinking about transfering to either your uni or some uni affiliated with Max Planck Institute after this semester (currently third semester undergrad in physics).

>including all negative powers
All powers. That's what Laurent series mean.
>(which is what I think holomorphic and antiholomorphic is saying)
Not quite. A function [math]f[/math] is holomorphic if [math]\overline \partial f = 0[/math] and antiholomorphic if [math]\partial f =
0[/math].
>of what
The Virasoro algebra.
>and how do you get generators from the energy-momentum tensor?
With the formula up there.
>The generators are that of a scale transformation
No, the generators are the generators of the Virasoro algebra, which describes the algebra of the field operators, not the symmetry.
No, though I'm affiliated with Perimeter. Here the courses are taught from physical motivations but can become quite rigorous depending on what stream you're following. Since I'm following the mathematical physics stream (along Ed Witten), all courses are extremely rigorous.

>though I'm affiliated with Perimeter
You lucky son of a bitch. I'm applying there now (Waterloo too since the guy I work under knows Achim Kempf and Brian Forrest). From what a friend of mine said it's tough as all hell to get in but looking it over if definitely seems like it's worth it. I figure I might have a somewhat remote chance of getting in since I got a package from them with a letter saying to apply. You doing the PSI program or the PhD?

Thigh gap goals

PhD. Another option for you is to apply for other universities around Perimeter and transfer into it later through doing PSI and some research. It's pretty bothersome but, as you've said, it's worth it.
Good luck user.

You should see his thighs :hearteyes:

i was wondering if any user would be willing to answer a question i have regarding the nature of 4D topology

>nature of 4D topology
>4D
The most topologically non-trivial dimension there is. Shoot, maybe someone will know

>his

Alright, I am attempting to extrapolate a 3D argument.

Suppose you have a 2+1 space with a stationary particle tracing out a world line. Now imagine that over a period of time you take a line segment and form it into a circle around that particle. You then break the circle and remove the line segment. When one looks at the world volumes this will result in a 'threading' action where the the particle world-line is threaded through a hole in the line segment world-sheet.

Now my question is, when we upgrade to 4D. i.e. when we temporarily surround a stationary particle in 3D with a 2D surface shaped into a sphere, and then break that sphere and remove the 2D surface, do we obtain the same kind of 'threading' action as we saw in the previous case.

> No, the generators are the generators of the Virasoro algebra, which describes the algebra of the field operators, not the symmetry.

I was under the impression a Lie algebra describes *a* symmetry in the system. Is there a short answer for how the Virasoro algebra connects to field operators and their commutation relations?

Also, is there a generator for scale transformations that fit in all this?

Is there a generator for scale transformation?

What would you say is a priority in terms of fields of math for undergrad looking to get into mathematical physics?
I've been through most graduate courses, but again my university is teaching them in a pretty bad way. I've been taking as many courses as i could on math department, but now i'm hitting barriers of our legal system. For example i'm not allowed to take non-stable K theory because it has mandatory prerequisites which are only available to math students. I still go to lectures and do all asignments, i even took the exam and passed it with full score, but from legal standpoint it is as if i never took the class. This applies to most of the courses on math department i went through.
At this point i think i have pretty solid understanding of MSc-level math, my focus being on higher gauge theories using methods of supergeometry, simplical homotopy and homotopy algebras. This is a topic i'm very interested in, but unfortunately none of the proffessors here are. For this reason i'm thinking of switching into environment that is more suited to me. Upon reading it, i was thinking of extending arxiv.org/abs/1604.01639 which unfortunately is out of interest of all the proffessors here. The only people who were interested in this were from Perimeter Institute, Max Planck Institute and MIPT. The most realistic choice for me is either Max Planck-affiliated uni or MIPT-affiliated uni (pretty much just Lomonosov Moscow State is worth mentioning) as i can't, due to my criminal record, leave EU (i can get student visa to Russia, but certainly not to US or Canada).
Do you know of any worthwhile institute in EU or Russia (apart from MIPT to which i can't apply) that would allow me to focus on this topic while giving me a good foundation for mathematical physics? I'm mainly interested in specific people that share my interest, rather than a curriculum.

>I was under the impression a Lie algebra describes *a* symmetry in the system.
This isn't always true, and even if it were it wouldn't mean Lie algebras has to describe symmetries anyway. In fact the special thing about CFTs is that the entirety of its operator algebra can be described by an affine Lie algebra [math]given[/math] its symmetries.
>Is there a short answer for how the Virasoro algebra connects to field operators and their commutation relations?
The Virasoro algebra defines a Verma module on which the algebra relations gives rise to KZ equations, and the solutions are the correlation functions (i.e. conformal blocks) of the primary fields.
>Also, is there a generator for scale transformations that fit in all this?
No, it's something else entirely. The translation, rotation and scaling symmetries are symmetries of the Hamiltonian which gives you the sufficient (and necessary) conditions to decompose it into anti-/holomorphic parts. That's the extent to which these symmetries are significant.
>Is there a generator for scale transformation?
Yes, and it's easily derived by putting [math]x^\mu \rightarrow x^\mu + \partial^\mu \epsilon(x)[/math] into the variation of the Hamiltonian.
There are several good books on the basics of CFT like DiFrancesco, Henkel, Kohno or Ueno that you can look into.
>What would you say is a priority in terms of fields of math for undergrad looking to get into mathematical physics?
Diff. top./geo., alg. top., functional analysis, group/representation theory, cohomology theory, Seiberg-Witten/Donaldson theory, Deligne-Mumford compactification, etc.
I don't know much about the institutes in the EU so I can't help you there.

>do we obtain the same kind of 'threading' action as we saw in the previous case.
It would seem so, at least thinking about standard R^4, you should actually ask this on math or physics stack exchange.

Sorry for bothering, but i was more interested in the etc. part, could you please expand on that? I have nobody else to ask and as i am, i have trouble using my math knowledge in physics. As in i can comfortably read mathematical papers, but i have trouble reading hep-th papers that are more focused on physics. I think it's mostly because these assume some basic level of knowledge in physics which i lack as i've been focusing on the math. All i've done in physics is pretty much just reading Landau and an introductory string book by Polchinski.
To give you a better idea of the bad state i'm in, i read papers from mathematical journals and understand them just fine, but i can't get into string theory papers (unless it's by mathematician), it just seems arcane and obfuscated by layers upon layers of notation, convention and assumptions. Is it just a barrier that i'll get through with more energy, or do i lack some fundamental component to be useful in physics?

At some point the distinction vanishes. If you're looking for physics background I suggest you read Landau-Lifshitz.
>obfuscated by layers upon layers of notation, convention and assumptions
Example? I've never seen this happen.
>Is it just a barrier that i'll get through with more energy, or do i lack some fundamental component to be useful in physics?
Hard to say without an example.

>There are several good books on the basics of CFT like DiFrancesco, Henkel, Kohno or Ueno that you can look into.

Ok, cool. I have a lot on my reading list right now though. I've been trying to teach myself QFT this past semester after switching research from plasma-physics, but progress is slow.

I've been trying to read the "intro" books like Mark Srednicki, Pier Ramond, and Peskin and Schroeder, which is good for basic knowledge but seem much less rigorous or lacking in giving an intuition when explaining the basic mechanics and how you obtain any particular Hamiltonian, or why a certain way to calculate a probability amplitude works.

The one colleague of mine 'teaching' me QFT just has me draw Feynman diagrams and write down the integrals, (and not even normalized so that I can calculate a probability of a scattering process).

The most insight I've gotten into this subject has been from condensed matter concepts and thermo. It seems to motivate the subject more as you can have an oversimplified toy model of an interaction on a small scale, but get all these interesting phenomena after re-normalizing to a larger scale. I mean, this is essentially what's happening in QFT, right? We just don't know the toy model at the plank-scale. Otherwise it just seems to me like Hamiltonians and Lagrangians fall out of the sky.

Read Baez's Guage Fields, Knots and Gravity.
>have an oversimplified toy model of an interaction on a small scale, but get all these interesting phenomena after re-normalizing to a larger scale. I mean, this is essentially what's happening in QFT, right?
Not really.

If the physical information contained in a ket is not affected by multiplication by a non-zero complex number why isn't quantum theory built on rays and not vectors?

It is.
en.wikipedia.org/wiki/Wigner's_theorem
en.wikipedia.org/wiki/Gelfand–Naimark–Segal_construction

Oh thanks.

>Wigner early work laid the ground for what many physicists came to call the group theory disease[1] in quantum mechanics – or as Hermann Weyl (co-responsible) puts it in his The Theory of Groups and Quantum Mechanics (preface to 2nd ed.), "It has been rumored that the group pest is gradually being cut out from quantum mechanics. This is certainly not true..."
Bastards.

I'll check this one out.

Do you have any good overarching intuitions for what's happening then? I thought it was a reasonable base intuition with analogies I perceived between QFT and condensed matter.

>Do you have any good overarching intuitions
Not really, I just know that integrating out momentum shells to get renorm flows isn't a proper group operation since the Hubbard-Stratonovich transform doesn't have unique inverses, so this whole thing isn't very well understood mathematically (by myself at least) and I try not to have any impressions about what renormalization actually is before I do.

shut the fuck up hank

Well the way I've been thinking about renormalization as integrating/ averaging over smaller scale phenomena, I wouldn't expect there to be a unique inverse. The process of smoothing out/ integrating over variables throws out information.

post thighs