Monday, February 16, 2009

An invitation to algebraic geometry




I have mentioned previously in this blog that I was studying algebraic geometry. O.K. I have concluded my first phase of study and it is time to tell something about it.

Algebraic geometry is a very broad topic with different possible approaches and epochs of development. Most traditional courses/books of initiation on the subject go through the algebraic approach. As the name suggests there is a lot of algebra involved. It is expected that the reader would be familiar with basic abstract algebra (groups, rings and the basics of ideals of rings), possibly field (in the mathematical sense nothing to do with a physic field, of course) theory and the very related subject of Galois theory. This is the standard content of undergraduate courses in alge3bra. In addition it is needed what is known under the name of "commutative algebra" which may, or maybe not, be covered in undergraduate courses. Commutative algebra consists mainly of a deeper analysis of rings and ideals. In particular noetherian rings, Hilbert nullentensantz theorem and things like that.

A bibliography of all this could be DORRONSORO, J. : NUMEROS GRUPOS Y ANILLOS (in Spanish, for sure there are lot of books covering the same topic in the English literature)for basic abstract algebra. For field/Galois theory a very canonical reference could be Ian Stewart: Galois Theory.

About commutative algebra a classical reference is the book of atiyah with precisely that name. Actually a friend of me, disr3ecomended me that book and considered a much better choice Miles Reid: Undergraduate Commutative Algebra.

Well, that are the expected prerequisites. Quite a lot if you are a physicist and not a mathematician. Later I´ll suggest other paths, but by now I´ll follow the traditional way.

With That algebraic basic you could try to bit one of the hard books, but it would be a better option to begin with an introductory book. One of the traditional introductory books is Fulton´s one algebraic curves available on line for free (info posted by Peter Woit in his blog, not even wrong). The books begins with a introductory chapter which serves a remaindering of the necessary algebra, but if you don´t previously know it it will not be too helpful. I must say that this summer I followed a course on that subject. The teacher was a physicist reconverted in mathematician. He has a paper, in collaboration, about string theory. It was that background of the teacher which aimed me to follow the course. I must say that it was a great course. To begin not all the audience were mathematicians so the teacher took a lot of time to reminder, and exemplify, the algebraic machinery. It also explained very clearly the geometric aspects. Later in this post ll say actually something about algebraist geometry and not only about the bibliography, by now I´ll just say that a (regular) algebraic curve can be identified with a Riemann surface. Every string theorists is well aware of the fact that the theory of Riemann surfaces is the basic of the Polyakov path integral. But, in fact, a traditional course in algebraic curves doesn't include the material specifically necessary for that purpose. Fortunately it includes some concepts, mainly the theory of divisor, which has become important for the subject of compactifications. Later more about it. It also tells about Blowing ups, which are actually of interest for compactifications.

After such a book, or similar ones (for example, "undergraduate algebraic geometry", also by Miles Reid one could try more serous books. For example Shafarevich: basic algebraic geometry, which covers most general algebraic spaces, aspects of the analytic approach as well as abstract geometry (schemes and all that, more on it later), or the book of Robin Hartsone which is still harder (and cover more topics), or, much better, both books.

Certainly that is a lot for a physicist. And he will not go into the most necessary goals until the last books. It is time to offer alternatives.

Recently it has appeared totally recommendable book, that gives name to this post, An Invitation to Algebraic Geometry by various authors, mainly Karen Smith.



The algebraic prerequisites of the book are only linear algebra. It gives a concise, and worked, intro to the necessary extra algebra. It does a very good job in explaining the flavour of the traditional algebraic geometry (which is not present in analytic geometry). The geometric aspects, and it´s relations to geometry are very well explained. It covers most of the topics necessary for string theory, the previously mentioned, (divisor,blowing ups), it gives some hints on abstract algebraic geometry (schemes, which is related to sheaves and the analytic side of algebraic geometry) and also, and this is important, families of algebraic surfaces. Under that name probably is not familiar for an sting theorist. The important fact is that is the concept which is behind the idea of moduli space. Loosely speaking a moduli space is an algebraic space which has the property that any of its point can be identified with other algebraic space. The most basic example is the projective plane. Any point of the projective plane can be identified with the rect that pass through it.

The concept of moduli space is a very recent one. The book of Joe Harris: Algebraic Geometry, a first course, cover more details about it than the book I was talking previously. The precise definition of the concept is very hard. The problem is that a moduli space usually is going to have singular points. To work in a proper way with that point you must go to abstract algebraic varieties (schemes), and, further, orbifolded schemes and things like that. The first introduction of the idea of moduli space dates back to the work of grothendieck which goes beyond the previous works of Teichmuller. There the re talking about families of algebraic curves (Riemann surfaces) and their complex structures. The teichmuller space of Riemann surface S of a given genus (intuitively, the genus is the number of holds) is the quotient of Conf(S)/Diff(S). Here Conf(S) is Met(S)/Cinf(S) where Met(s) is the set of possible metrics on S and Cinf is the group al infinite differentiable functions on S, which acts in the group o f metrics generating conformal transformations. -this is well known material for string theorists. It is covered in the chapters about the Polyakov integral. There one is explained how to get the moduli space form the teichmuller space and that the beltrami differentials form a basic of the tangent space to the moduli space and many other things. Is one is familiarized with differential geometry on manifolds, some basic group theory and complex analysis(and who ins´t nowadays) he can follow the ideas and accept that space3s as "abstract" spaces whose dimension can be determined (with the help of the Riemann Roch theorem) and can work with them to get the desired result, expressions for the cross sections. But if one reads books on algebraic geometry one becomes aware of the fact that those topic have a more geometric, and not so abstract, nature. Of course, as I said, the precise formulation of the idea requires heavier machinery that the one covered in string theory books. The pre3cise concept of moduli space of a Riemann surfaces is due to David Mumford and he got a Fields medal for it. And it is a very recent work, it is dated in the last sixtie3s and former seventies. It is somewhat incredible how soon such an abstract and difficult concept, combined with the also very recent, for the date, theory of conformal field theory, were combined in a baby string theory in a time where the math basic of more physicists was essentially the book of Arfken (or Mathews Walker) plus some tensor analysis and pedestrian Lie groups theory. I guess that there3 is a history in that achievement that deserves to e told, pity I have idea of that.

Well, we have a recommended book in the algebraic part. Lets go for a book in the analytic part. My choice 8freely available online) is U. Bruzzo. INTRODUCTION TO. ALGEBRAIC TOPOLOGY AND. ALGEBRAIC GEOMETRY.

It is a short book. the first part covers algebraic topology,almost form the beginning. It, later, introduce the very important concept of presheaves and sheaves. Them he introduces Cech co-homology. He treats fibered spaces, de rham theory, characteristic classes, the very hard topic of spectral sequences and, in general it covers a good part the material covered in canonical books of algebraic topology such as the one by Spanier, or the famous "differential forms in algebraic topology" of Bott and Tue (see this entry int he U-duality blog for more info in that book and in the general subject of math for string theory).

The second part of the book covers, from the analytic viewpoint, the same material on divisors (and the related concept of line bundles,ample and very ample bundles) that is mentioned in the book y Karen Smith and others. It also covers algebraic curves and the Riemann Roch theorem. The last chapter has a somewhat misleading title "nodal curves". The "nodal" makes reference to a type of singularity in an algebraic curve. An algebraic curve (it is time to say at last something about them xD) is, roughly speaking the set of zeros of a polynomials (enough polynomials to get a one dimensional space in the field of definition of the polynomials-note, a complex one dimensional space is a two dimensional real space-). A singularity is a point where the tangent space is not defined. That can e due to a self-intersection of the curve (so we have "two tangent spaces) or a point where the tangent space is not defined (the curve has not a derivative). The former is a nodal point, and can be resolved by the technique of blowing-up. This consist of cutting the singular point and to replace it by the projective space over it (see the books for the details). Well, the question is under the name of "nodal curves" is hidden the most well known concept of blowing-up. I would mention that I have followed this semester a course in the UCM which covered this analytic viewpoint of algebraic geometry. The official teacher was a mathematician who also has a publication on string theory, actually, a good amount of the course was finally imparted by a different teacher. I previously knew sheave theory, but certainly my knowledge on the subject has greatly growth ;-).


Well, with that two books one has a decent basic in algebraic geometry. He could go from there directly to specific reviews relating this to string theory. Soon ill comment something about it. Firs another suggestions. In the blog entry of U-duality it is recommended the Nakaharas book. I would add another book which I have already mentioned here. Topology and quantum field theory, by Charles Nash. It covers Riemann surfaces, and it relation to string theory. It also cover many other subjects. For example the theory of elliptic operators and its relation to topology, which is central to many results in the analytic part of algebraic geometry. In fact the Nash book covers that aspects (I suspect that it follows closely the book of Weill on analytic manifolds). Certainly a great book.

ON the pure math side one could go with the previously mentioned books of shafarevich and Hartsone (I have read some chapters of the former and I have been teach ed the first chapter of the second). An additional recommendation is the encyclopedic Principles of Algebraic Geometry by Griffith & Harris. Of those books the only one that covers moduli spaces is the one of Hartsone.

OK, what about not pure math books? Where is the string theory?. Lets go to it.

The canonical reference would be the book Mirror symmetries freely available on-line form the claymath institute (yes,the same claymath institute of the millenium prizes). Really one could begin with that book and to forget about all the above ones. It is a very extensive one, and covers almost all the mentioned subjects, together with the new, ones that I´ll speak now, toric varieties. The first part is reasonably accessible to a physicist with a decent basic on modern math (without need of abstract algebra). I guess that the book somewhat fails in providing the geometric-geometric viewpoint of the subject, something that , I insist, is very well covered in the Karen Smith book in a very acce3sible way.

OK, last references, arxiv reviews of the application of all this to string theory.

I would begin by TASI LECTURES ON COMPACTIFICATION AND DUALITY by DAVID R. MORRISON. It dates in nov 2004 and the arxiv signature is: hep-th/0411120v1
The third and fourth part are specifically related to the subject of algebraic geometry in string theory. It explains how Calaby -Yau manifolds are good compactifications of string theory and it explains how the theory of divisors and ample line bundles are good tools to get kahler classes of Calabi-Yau manifolds and why that is a good thing.

A different use of algebraic geometry,and specially the blowing up techniques, is for resolution of singularities in orbifolds and also, of conifolds. Resolving singularities of conifolds one can describe process where by varying the moduli space of a calabi-yau one gos trought transitions where some topological aspects of the calabi yau change.You can describe that changing aspects in terms of intersection theory. And you can describe intersection theory in terms of divisors. The first papers on topological change are due to Brian Green and Aspinwell (who, B.T.W. also has a book named Mirror Symmetry which covers those topics) on one side and Witten on the other side. Perhaps the most closed paper to the algebraic geometric techniques is this. the paper is form 1993. Later the topic got a different, more complete, treatment by placing D-branes in the conifold singular point (sorry, I have not time now to search the arxiv reference for the paper).

For papers explaining toric geometry , kind of generalization of projective spaces, very useful in F-theory construction you can read the3 recommended papers of the string wiki:

Toric geometry and calabbi yau compactifications y Maximilian Kreuzer (arXiv:hep-th/0612307v2)
Lectures on complex geometry, Calabi–Yau manifolds and toric geometry by Vincent Bouchard (arXiv:hep-th/0702063v1) or, also:

The Geometer’s Toolkit to String Compactifications (arXiv:0706.1310v1)

Certainly I don´t find them the more funny papers in math and I have not, still, readed them completely.

Once you have all this math background you could go without fear to study string compactifications, specifically F-theory ones. If you want a complete and detailed physical guide on the field of aplication you could use this paper:

LES HOUCHES LECTURES ON CONSTRUCTING STRING VACUA by Frederik Denef (arXiv:0803.1194v1)
I apologyce by not saying nothingn about the subject of elliptic curves (a very special kind of algebraic curves) and the related topics of eliptic integrals and aliptics function and of its generalizations (modular forms), maybe in other post. In fact from now to that point I´ll probaly have a better knowledge of that subjects and it will be a better post anyway ;-).

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