Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Crash course on algebraic geometry with view to applications in physics

+ 9 like - 0 dislike
7142 views

Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)?

What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references.

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user anonymous
asked Jan 5, 2011 in Resources and References by Anonymous (40 points) [ no revision ]
retagged May 4, 2014

10 Answers

+ 7 like - 0 dislike

Griffiths and Harris' "Principles of Algebraic Geometry" (Wiley) is the best for your purposes (read only the parts on Kahler geometry). The sections on algebraic geometry in "Mirror Symmetry" (Clay/AMS) are essentially a Crib Notes version of that paper and some of the classic CY and special geometry papers referred to above.

What you should keep in mind going in is the following:

Kahler manifolds are complex manifolds with a hermitian inner product on tangent vectors which have a metric that is determined (locally) by a single function. It is the geometry in which the metric and the complex structure "get along very nicely." This simplifies lots of calculations and adds new symmetries. That's why we know so much about them.

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user Eric Zaslow
answered Jan 6, 2011 by Eric Zaslow (385 points) [ no revision ]
+ 6 like - 0 dislike

One text that immediately comes to mind is "lectures on complex geometry" by Philip Candelas, an accessible introduction covering the very basics. I think it is hidden in some Trieste proceedings or some such.

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user user566
answered Jan 5, 2011 by user566 (545 points) [ no revision ]
do you happen to have an electronic copy of these lectures? I couldn't find one online.

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user user346
I have a paper copy from a course I took, but I am sure you can email Philip and ask. I also second the recommendation of Brian Greene's lectures (from a TASI course I was lucky enough to take) but they go in different directions beyond just the basics.

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user user566
+ 5 like - 0 dislike

I am not sure about generic applications to physics but when I hear about algebraic geometry I immediately think of string theory. And when I hear about Kähler manifolds it's pretty hard not to think of Calabi-Yaus :-)

So if you don't mind this kind of applications, you could find useful a paper by Brian Greene: String Theory on Calabi-Yau Manifolds. It contains also general talk about differential geometry and string theory. But most importantly, you'll find there the basics of Kähler and Calabi-Yau manifolds as well as lots of applications like mirror symmetry and investigation of moduli spaces.

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user Marek
answered Jan 5, 2011 by Marek (635 points) [ no revision ]
+ 3 like - 0 dislike

Though not particularly aimed at physicist, you could take a look at An Invitation to Algebraic Geometry by Smith, Kahanpää, Kekäläinen and Traves. It's very short and avoids many technicalities and proof. Instead it give a bird's-eye view and succeeds well at transmitting some of the basics.

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user Michael
answered Jan 6, 2011 by Michael (5 points) [ no revision ]
+ 3 like - 0 dislike

A new and more popular exposition directly from fields medalist Shing-Tung Yau: The shape of inner space
Find a review here: http://plus.maths.org/content/node/5389
Find a very gentle and short intro here: http://plus.maths.org/content/node/5388
Find a preview here: http://books.google.de/books?id=M40Ytp8Os_gC&lpg=PP1&ots=3dHRt8v3KI&dq=The%20shape%20of%20inner%20space&hl=en&pg=PP1#v=onepage&q&f=false
Find the webpage here: http://www.shapeofinnerspace.com/

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user vonjd
answered Jan 10, 2011 by vonjd (0 points) [ no revision ]
+ 3 like - 0 dislike

Try this one too http://arxiv.org/abs/hep-th/0702063, by Vincent Bouchard of Alberta who I think was student of Candellas.

answered Jan 5, 2015 by conformal_gk (3,625 points) [ revision history ]
+ 2 like - 0 dislike

One good source my undergraduate adviser recommended to me are the lecture notes of Candelas on Complex Geometry. They are written with string theory in mind and cover a lot of basic ground. I am not sure, if they are available online. Griffiths and Harris is very good, but probably not suitable as your only source for self-study. Just to get an idea what ideas were needed in string theory 25 years ago a look at chapter 12,14,15,16 in the second volume of Green, Schwarz, Witten might be helpful. Especially 14 and 15 should be interesting to you, even if you did not take a course in string theory yet.

By now there are of course a lot of other applications of ideas from algebraic geometry to the study of string theory beyond those ordinarily found in textbooks. For example model building in $F$-theory requires among other things to the study of singularities of elliptic fibrations and the approximate dynamics of certain branes is determined by variations of hodge structure. To actually find interesting examples knowledge of toric varieties is helpful. Most of those topics are actually not discussed in introductory texts.

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user orbifold
answered May 30, 2012 by orbifold (195 points) [ no revision ]
+ 2 like - 0 dislike

Mirror Symmetry, especially the first two chapters give a brief introduction to algebric geometry. URL:http://www.claymath.org/library/monographs/cmim01.pdf

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user Craig Thone
answered May 30, 2012 by Craig Thone (40 points) [ no revision ]
+ 1 like - 0 dislike

Geometry , Topology and physics by nakahara and Geometry of physics are the best introduction for geometry and topology for physicists

This post imported from StackExchange Physics at 2014-05-04 14:08 (UCT), posted by SE-user med_student
answered May 30, 2012 by med_student (0 points) [ no revision ]
+ 1 like - 0 dislike

I found 'Lectures on Advanced Mathematical Methods for Physicists' by Sunil-Mukhi and N. Mukunda very helpful for Topology and Lie-Group.

http://www.amazon.com/Lectures-Advanced-Mathematical-Methods-Physicists/dp/9814299731/ref=la_B004150E8S_1_1?s=books&ie=UTF8&qid=1419968062&sr=1-1

answered Dec 30, 2014 by bhautik_shastra (0 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$y$\varnothing$icsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...