# Mathematics needed for string theory

Hello,

like I said in the topic title I am interested in what mathematics is needed for string theory to be studied. I am in faculty of electrical engineering, sub-filed nanoelectronics. I have good knowledge of math but not good in comparison with what is needed for superstrings, which I would like to start to learn for about two years (perhaps PhD, since I have two more undergraduate years).

So, I would be grateful if someone could give me a guide what parts of math are required. I have most of books for linear algebra, analytic geometry, real and complex calculus, tensors and differential geometry and topology (surly, suggest some book if you think that it is good :) ). But the main problem is that I do not want to learn just pure math (it would take probably much more time for this), but those parts that are obligatory for strings (like we do not need to know how to solve all possible differential equations in order to learn classical mechanics).
If someone have some experience whit this things just write down :)
Thanks ;)

Try Leonard Susskinds lecture series. He covers the main areas of physics from the very basics to advanced modern theoretical physics, and covers the maths needed to understand the core principles therein. They are very much a stripped back approach and whilst I appreciate having the lectures to access, studying by yourself and not having anyone to bounce ideas off does make it tough. For example i'm currently following the Einsteins theory series and he's covering tensor calc which I have no experience of. Trying to follow the jump from scalars and vectors to n-dimensional tensors and deciphering the Kronecker Delta is hard work especially after a day in the office (i'm an undergrad engineer on work experience).

Also go to a bookshop and have a look at a copy of Road to Reality by Roger Penrose. The first half covers the mathmatical foundations from the ground up but very swiftly. It'll give you an idea of the breadth of matierial you have to learn!

The math of string theories is still a work in progress involving complicated topologies in high dimensions. For instance, I believe it has moved beyond the Calabi-Yau shapes.

http://www.physorg.com/news70021180.html

Well I also believe that string math is in continuous progress, and certainly the current math that is used in describing strings is very hard and difficult. But what I need is a little guide through math required for this theory, just at the beginners level (linear algebra required for tensors, and that sort of stuff), since I am going to learn all of this by myself :) .

Thanks for recommendation for Roger Penrose book :).

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Well I also believe that string math is in continuous progress, and certainly the current math that is used in describing strings is very hard and difficult. But what I need is a little guide through math required for this theory, just at the beginners level (linear algebra required for tensors, and that sort of stuff), since I am going to learn all of this by myself :) .

Thank for recommendation for Roger Penrose book :).

I've read through "Road to Reality" three times and will begin my fourth reading soon. It's not a textbook. Penrose summarizes different areas of math and is very good at showing how the different pieces fit together. Advanced algebras, including non-commutative algebras (and their related geometries) are probably the most important areas relevant to string theory and other cutting edge areas of theoretical physics. You can't learn all the math you need from reading Penrose. However, it is a good supplement to textbooks and coursework. It gives a certain perspective. The following link outlines a study program for the mathematics of string theory. It's in three parts.

http://superstringtheory.com/math/math1.html

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I don't want to sound snobbish, but I can't promise that you'll get a satisfactory understanding of string theory with an abbreviated understanding of the math behind it. Penrose's book is well-written, and it is good for obtaining an intuitive idea of what some mathematicians think of given definitions of objects, but it won't give you concise definitions that you can depend on to build your own interpretations. Also, it will be difficult for you to get all the way to understanding string theory if you can't get excited about the intermediate steps (this is also something that Penrose's book is good for).

Demystifier
Gold Member
Nemanja, I would also recommend the Penrose's book. (By the way, is Nemanja a Serbian name?)

haushofer
I would recommend Nakahara's "geometry and topology"-book. It is imho very accessible, and the perfect mid-road between mathematical rigour and physical relevance.

Besides that, you'll need quite some group theory, which can be found in e.g. Georgi's book.

HiggsBoson1
@Demystifier

Thank you for recommendation of Penrose book. Well yes it is :), it's an orthodox Serbian name, and it can be translated and have several meanings, few of them are: I'm the monster (in sense someone very powerful and the one who causes awe) [Nemanja= Neman(monster)+ja(I,me)], and the second one means ˝never to give up˝ from
ne(not to-negation)+maniti(give up). Serbian kings long time ago (1190 - ~1300) used that name as title, and not everyone could use that name. I know that this was a little extensive answer, but I'm interested how do you know for Serbia (it's very small country)?

@haushofer

Thank you also, for that group theory book did you mean on this one: https://www.amazon.com/dp/0738202339/?tag=pfamazon01-20

If you know some other physical orientated mathematical book, please let me know. :)

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Interesting thread. I too seek the math to understand modern physics.

bapowell
Nakahara's "Geometry, Topology, and Physics" covers in a very cursory way the topics needed to get started in strings. This is a good first stop to get a decent introduction to these ideas, and there are plenty of references that you can use to continue your studies.

Main topics include:

Group Theory (Lie groups, representation theory)
Differential Geometry (on real and complex manifolds)
Topology (Algebraic: homology and homotopy theory)
Characteristic Classes
Algebraic Geometry

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Demystifier
Gold Member
but I'm interested how do you know for Serbia (it's very small country)?
I live in an even smaller country in the neighborhood (Croatia).

HiggsBoson1
haushofer
@Demystifier

@haushofer

Thank you also, for that group theory book did you mean on this one: https://www.amazon.com/dp/0738202339/?tag=pfamazon01-20
:)

Yes. It has its drawbacks; I find the writing style sometimes terrible, but it covers a lot of useful stuff. For a more mathematical treatment, I like Brian Hall's book, but there is not a lot of physics in that one.

To be honest, I've never seen a book or lecture notes about group theory and Lie algebras which I found really satisfying.

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