# Introduction to Algebraic Geometry in String Theory?

I'm a beginning graduate student in string theory and I'm in the process of teaching myself algebraic geometry. I'm using lecture notes that go without mention of physics. I'm curious if there is a introductory book, or paper, or set of lecture notes that describes the application of alg. geometry to physics and string theory, specifically?

On a related note, I know algebraic topology and complex manifold theory also need to be on my to-do list; are there nice references for these in relation to physics? All I could find on the internet were papers on the arxiv that seem to assume one already knows most of the mathematics. Thanks for any help.

ShayanJ
Gold Member
I'm myself a beginner but it seems to me that Mikio Nakahara's Geometry, Topology and physics is the book you want and I think its good enough.

marcus and David Horgan
great, thanks! does that include algebraic geometry, or is it more for topology and differential geometry?

ShayanJ
Gold Member
great, thanks! does that include algebraic geometry, or is it more for topology and differential geometry?
It explicitly states that it discusses algebraic topology but doesn't make any statement about algebraic geometry. But there are books on algebraic geometry in its references section. Also I don't think a book covering that much things and also discussing bosonic string theory, omits algebraic geometry. Also the members who liked my post know about the subject much more than me so it seems they think it gives you what you want too.
I give you the contents list so you can see yourself.

Quantum Physics
Mathematical Preliminaries
Homology Groups
Homotopy Groups
Manifolds
DeRham Cohomology Groups
Riemannian Geometry
Complex Manifolds
Fibre Bundles
Connections on Fibre Bundles
Characteristic Classes
Index Theorems
Anomalies in Gauge Field Theories
Bosonic String

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I've briefly looked through a soft copy of Nakahara's text and it looks like in Chapter 2 and onward, there isn't much discussion of physics. Is this more of an introduction to the mathematics necessary to understand the physics, or is there good physics in there that I missed when I skimmed it?

ShayanJ
Gold Member
I've briefly looked through a soft copy of Nakahara's text and it looks like in Chapter 2 and onward, there isn't much discussion of physics. Is this more of an introduction to the mathematics necessary to understand the physics, or is there good physics in there that I missed when I skimmed it?
Actually when he's talking about math, he's just talking about math. But the fact that made me introduce you this book, was that its written by a physicist and so knows the right amount of rigour appropriate for a physics student. Also it sometimes talks about physics in separate sections. For example in the chapter for Riemannian Geometry, it discusses GR and Bosonic string theory in the last two sections. But now that I think, it seems to me there is no book that meets your requirement. I don't think any book aimed at teaching math, discusses that much physics that you have in mind. It seems to me you should find a book on string theory that contains some algebraic geometry, like GR books that contain some differential geometry. But I can't help here because I don't know about books on string theory.

Haelfix
I'm a beginning graduate student in string theory and I'm in the process of teaching myself algebraic geometry. I'm using lecture notes that go without mention of physics. I'm curious if there is a introductory book, or paper, or set of lecture notes that describes the application of alg. geometry to physics and string theory, specifically?
It would help if you told us what you want to learn in string theory specifically. Algebraic geometry is quite esoteric for most physicists and the places where it intersects with string theory is generally considered extremely sophisticated and challenging (at least for mere mortals). I would venture to guess that this is more of an active area of research in mathematican circles and hence the language and formalism is going to be found more from mathematician textbooks.

Of course it's always a good idea to read up on Edward Witten papers from the 80s and early 90s. He often spends entire sections of his papers reviewing the formalism and there is simply no better pedagogue anywhere in the world for this sort of thing.

Algebraic topology and differential geometry of course are much closer to working physicists and Nakahara is a standard reference and you can consider it a prerequisite before you start venturing into crazy elliptic curve counting.

ShayanJ
Gold Member
I would recommend looking at Principles of Algebraic Geometry', by Griffiths and Harris, which may also cover complex manifolds sufficiently. This book is rather long, but you probably don't need to read it cover to cover. If you want a deep appreciation of algebraic geometry, I would recommend starting with Shafarevich volume 1. The textbook by Joe Harris (algebraic geometry, a first course) may be a good source of intuition, e.g. for blow-ups. Applications in string theory seem to use varieties most of the time instead of schemes, so Hartshorne's book might be overkill for the time being.

For algebraic topology, Allen Hatcher's book is free, and does a good job motivating the material as it's presented. Hawking and Young is also excellent, as is the book (on algebraic topology) by Munkres. Unfortunately, I don't know off hand any books that combine specifically algebraic topology and physics. The closest books that come to mind are closer to differential topology: 'Gauge fields, knots, and gravity', and 'Differential topology and quantum field theory'. These books might also have references that are interesting to you.

In Green, Schwarz, Witten volume 2, they have a chapter on Algebraic Geometry that I found to be very accessible. There is a nice introductory chapter to algebraic geometry in Dummit and Foote too. As far as full books on the subject are concerned, I agree with the the recommendation of "Principles of Algebraic Geometry" by Griffiths and Harris. This is an extremely well written book that even physicists can learn things from. I think Hartshorne is still the canonical text on the subject, but is one of the most daunting books I've ever looked at.

Thanks a lot. Yes, Hartshorne is very frightening...I've heard some of his exercises are open problems . I'm not looking for anything that rigorous. Do you know in which chapter in GWS they cover alg. geo?

It's chapter fifteen of volume II. I should note that Green, Schwarz, Witten and Griffiths and Harris come at the subject from more of a topology of complex manifolds standpoint, which in imo is more useful to a physicist, and the Chapter in Dummit and Foote and Hartsthorne come at it from more of Grothendieck standpoint, but I found the chapter and in Dummit and Foote to still be very understandable and helped me grasp the concept of a scheme.

There is also Mirror symmetry' by Hori et. al. that discusses applications of algebraic geometry in string theory.