r/StringTheory • u/QuantumPhyZ • Mar 08 '26

Question What math should I learn for HEP-Th?

Hi! I know HEP-Th is extremely competitive but I’m not shy to challenges.

I’m in undergrad senior level (3rd year in Europe, where I’m located at) and here’s the math courses I have done (I’m doing a physics major now):

Algebra (A first course to Abstract Algebra), Computational Algebra, Topology (A first course), Complex Analysis (A first course), Functional Analysis (A first course) and Differential Geometry (A first course). (Linear Algebra and all the Real Analysis/Calculus are subtended, in Real Analysis/Calculus 3 we learnt about Differential Equations and Fourier Transforms).

After this, in my Masters, what math applied to physics should I learn and deepen my knowledge on? Should I learn Topology but in a physics approach now that I have a first course? Is there more subjects that I should learn such as Geometric Algebra?

Bonus questions, I’m also interested in Plasma physics, the same questions applies to this!

Thanks in advance for the responses!

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u/CandidateDirect5127 PhD student Mar 08 '26

as MUCH math as you possibly can! ideally a lot of differential geometry, algebraic topology, group theory, etc. but really anything would probably be helpful (algebraic geometry and category theory for example). same with physics; as much statistical physics, electromagnetism, QM, QFT, GR, string theory if you can, etc. :) The more the better

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u/HEPTheorist Mar 10 '26

Great answer! I second the focuses on algebraic geometry and category theory if you see yourself becoming even some what mathematical/formal.

I will also anti-recommend geometric algebra. It's a cute thing which is disproportionately represented on YouTube, Blogs, etc. but researchers use differential geometry concepts, not funny things from geometric algebra.

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u/I-AM-MA Mar 10 '26

is algebraic geometry usually the main deal if you're interested in formal parts of st and hep in general

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u/HEPTheorist Mar 11 '26 edited Mar 11 '26

In ancient times, like the 90's or early 00's, some minimal working knowledge of complex algebraic geometry a la Griffiths & Harris or the Clay Mirror Symmetry book was useful. And these were tied to very dominant subjects in HEPth at the time (less so now). Sociologically, formal HEPth/ST would often cross-list to arXiv:math.AG: you can see Witten used to do it all the time, Nekrasov used to post directly there, etc.

Currently, I don't see a huge use for deep ideas in AG at the research level (not as much as some cheap street category theory anyway). Yes, it's implicitly used in papers on 2d gravity, but it's not usually critical to know about sheaves or something. As a string theory subreddit, we should also still acknowledge it because Calabi-Yau's or whatever. But my friends currently working on BH microstate/index calculations or bootstrap definitely don't know or care.

Most places I see some operational knowledge of AG being useful is in very formal mathematical work, e.g. in quantization, resurgence, or twisted QFT/holography. This is not surprising when you realize where these ideas came from historically.

On the other hand, as a student, now is the chance to sit down quietly without overbearing publication pressure and learn things properly once and for all. Then you don't have to learn sketchy handwaving category theory or algebraic geometry when you need it!

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u/QuantumPhyZ Mar 11 '26

You got me kinda confused, should I deepen my knowledge in AG or Category theory or not? AG seems fun to learn but isn’t Category theory too abstract for physics?

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u/HEPTheorist Mar 11 '26 edited Mar 11 '26

Apologies. In the reply to I-AM-MA, I am saying that the role of AG is perhaps less crucial than something like differential geometry and group theory, which essentially all theoretical physicists should know.

Right now, category theory is ubiquitous in modern research level QFT, i.e. in mathematical QFT, high energy QFT and string theory, and condensed matter theory. Even people who study lattice models and many-body QM are using category theory. In fact, the "global categorical symmetries" Simons Collaboration is currently one of the booming subjects in modern HEPth, see e.g. the very popular lectures here https://inspirehep.net/literature/2663314 or here https://inspirehep.net/literature/2684592

If you wish to be very formal or mathematical (see the papers I referenced in previous post), you should learn introductory algebraic geometry (but also, you must pick a good reference for a physicist). E.g. does the idea of Calabi-Yaus or topological strings excite you? If yes, then learn some AG and definitely homological algebra. If you do not plan on being a formal theorist, then you probably do not need to do this. But, as a student with lots of time on your hands, why not learn them now while you still have time to read books and take classes?

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u/QuantumPhyZ Mar 11 '26

I do still have time and that’s what I’m planning to do! Thank you!

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u/I-AM-MA Mar 11 '26

thank you for the response, what is the current research landscape like for any hep theorist that wants to be formal/ rigorous. Maybe they're more referred to as mathematical physicists idk ive seen names vary department by department. I have a big passion for hep and for rigour, in what ways to people usually mix these. Sorry if im not being specific enough, im vague on purpose because i dont know much about the formal mathematicians that work in hep or adjacent fields

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u/HEPTheorist Mar 11 '26

It depends how mathematical you want to go. If you want to say mathematical enough that you might write papers with mathematicians and not have it be unusual, or mathematical enough you won't get a job, I would say some topics include things like:

  • studies of SQFTs, dualities, and specifically their modern connection to things like 4d/2d correspondence or related
  • twisted and cohomological QFT/string theory
  • resurgence, wall crossing, etc.
  • more formal aspects of categorical symmetries (but you can also see way more practically minded people here as well)
  • algebraic QFT and operator algebras have seen a resurgence of interest in a mathematical form (especially related to the last point)
  • constructive QFT (there is now a Simons Collaboration for this)

But also, these subjects are related, e.g. some folks might see the first 3 as the same things (and where does Langlands fit?). String Field Theory also fits somewhere in the unable to get a job category, I suppose.

Perhaps a better strategy is to see what people at String Math have done the last few years: https://indico.ictp.it/event/10482 and https://icms.ac.uk/activities/workshop/stringmath26/

If you are willing to be more mathematical than any of your experimentalist friends will care about, but still able to get a job, we could add way more things: operators algebras (but QG); categorical symmetries (but less math); swampland and string pheno adjacent stuff; bootstrap, CFT and S-matrix; amplitudes; holography and AdS/CFT; CFT and defects; and so on. Then you can just see what people have been doing at Strings the past few years.

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u/I-AM-MA Mar 11 '26

thanks a lot that helped clarify some things

my interests are a bit weird and specific in the sense that im looking for mathematical rigour and ideas that are directly adjacent to hep, meanwhile one the biggest problem of the field is that its mostly not mathematically rigorous.

constructive and algebraic qft seem really cool

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u/Lower-Oil-9324 Mar 12 '26

Can I DM you?

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u/HEPTheorist Mar 11 '26 edited Mar 11 '26

Fun piece of evidence for the categorical symmetries comment I made: this bridge/cluster forming here between math-phys, condensed matter, and high energy theory on Paperscape is basically the aforementioned "categorical symmetries" programme

https://imgur.com/a/gjhSHtH

Edit: For aficionados, the glowing points are Sakura Schafer-Nameki's papers. I am showing the region that was shared with other high profile categorical symmetries people like Thorngren, Shao, etc.

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u/Minimum-Dot5165 Mar 19 '26

If you really wanna master something first I'd recommend mastering smooth manifolds and differential geometry first before doing anything else mathy. Cuz these tools are needed in almost all fundamental theory now. Whereas algebraic geo mainly needed if you are doing kahler geo stuff related to calabi-yau or very math-ph heavy stuff.

For geometry/topology i cant recommend John Lee's trilogy enough, be it you a mathematician or a physicist or a nobody.

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u/QuantumPhyZ Mar 19 '26

Thank you!

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u/rafisics Mar 09 '26

You have done some good courses. Alongside those, learn group theory. If you are interested in hep-th, it is better to prioritize relevant advanced physics courses. For instance, when you take GR, QFT, CFT, or String Theory courses, you would be guided to the specific required topics of mathematics via standard introductory texts, course instructors, and online forums. Best wishes.