Enumerative Geometry and String Theory by Sheldon Katz (PDF)

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Perhaps the most famous example of how ideas from modern physics have revolutionized mathematics is the way string theory has led to an overhaul of enumerative geometry, an area of mathematics that started in the eighteen hundreds. Century-old problems of enumerating geometric configurations have now been solved using new and deep mathematical techniques inspired by physics! The book begins with an insightful introduction to enumerative geometry. From there, the goal becomes explaining the more advanced elements of enumerative algebraic geometry. Along the way, there are some crash courses on intermediate topics which are essential tools for the student of modern mathematics, such as cohomology and other topics in geometry. The physics content assumes nothing beyond a first undergraduate course. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry. Once these topics are in place, the connection between physics and enumerative geometry is made with the introduction of topological quantum field theory and quantum cohomology.

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Editorial Reviews: Review “The most accessible portal into very exciting recent material.” —- CHOICE Magazine”The book contains a lot of extra material that was not included in the original fifteen lectures. It is a nicely and intuitively written remarkable little booklet covering a huge amount of interesting material describing a beautiful area, where modern mathematics and theoretical physics meet. It can give inspiration to teachers for a lecture series on the topic as well as a chance for self-study by students.” —- EMS Newsletter”It is a welcome addition to the spectrum of available references on the topic and ideal for someone between undergraduate and beginning graduate education who wants to know more about this exciting field or for more advanced students who would like to see how the pieces of the puzzle fit together.” —- Mathematical Reviews

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⭐Enumerative geometry can be viewed in the non-rigorous “classical” setting of the Italian geometers, in the rigorous modern setting of sheaf theory and algebraic geometry, or in the non-rigorous setting of high-energy physics and string theory. Modern mathematics insists upon rigorous formulation for all of its constructions, so it has appropriately rejected the Italian and physicist setting for enumerative geometry. But the move to put the results of the Italian geometers on a rigorous basis resulted in much of the current field of algebraic geometry, esoteric as it may be. A similar movement is now occurring in the attempt to make rigorous some highly interesting predictions in enumerative geometry coming from physics. This has proven to be a challenge, since anyone involved in it must understand not only the mathematics behind enumerative geometry but also the physics behind string theory. The author of this book is one of the few that does have this understanding, and he has passed on some of his insights in this short but illuminating book.The main issue in the learning of advanced mathematics, particularly an esoteric subject like enumerative geometry, has centered on the proper method by which to motivate the central concepts. To better appreciate these concepts, it is better to present many examples of them, preferably in an historical context, and then illustrate the properties that these examples have in common. One can then show how the concepts arose from abstracting or generalizing over these concepts. The process then should be to present concrete examples preferably with diagrams and pictures, explain the historical reasons for the interest in these examples and the mathematical tools that were used for dealing with them, and finally present the current theories that subsume these examples.The author follows this process to a large degree in this book, presenting for example the ‘stable map’ as being a generalization of the intersection of conics, and viewing projective space as the compactification of complex n-space by “adding a point at infinity”. The book is based on a series of lectures that were directed to an audience of advanced undergraduates, so the author realizes that he must remain as concrete as possible initially. To explain Gromov-Witten theory, topological quantum field theory, and quantum cohomology to such an audience in a way that would make it understandable to them is a tremendous challenge. Without any assessment of his audience it is impossible to judge whether he succeeded in increasing their understanding, but no doubt they benefited greatly from the insights and examples at least as they are presented in this book. The author cautions the reader that the book is not self-contained, but given its size this is no surprise. If all the prerequisites were included this would swell the size of the book into many thousands of pages. The most pleasant feature of this book goes along with what was said above, namely that he motivates the subject of enumerative geometry from the “classical” viewpoint. Linear and quadratic equations are easily dealt with by the intended audience, who also has no difficulty in dealing with intersections of lines and conic sections.Central to “classical” enumerative geometry is Bezout’s theorem, which says that the number of points in the intersection of two plane curves is equal to the product of their degrees. The generalization of this theorem to varieties in projective n-space P(n) involves a generalization of the notion of degree, which for a k-dimensional variety is the number of points in its intersection with a (n-k)-dimensional linear subspace of P(n). Bezout’s theorem in P(n) states that the number of points in the intersection of a collection of hypersurfaces is the product of their degrees. The author initially studies the case where these hypersurfaces are (smooth) conics in P(2), and asks for the number of conics that pass through four distinct points. To use Bezout’s theorem to answer this question, one must compactify the space of smooth conics, which is done by realizing that the space of all conics is parametrized by P(5). However this strategy fails to get the right number of conics, as the author shows with a few examples, due to what he calls `excess intersection’, i.e. there are more intersections than expected from what is predicted by Bezout’s theorem. The excess intersection is familiar from classical differential geometry as an “osculating” or degenerate intersection, i.e. the dimension of the intersection of two curves in the plane is positive. In the area of differential topology this case is taken care of by imposing “transversality.” The author shows however that intersection theory is subtler even for cases where the intersection is transversal. He illustrates this for the case of the intersection of two plane conics that have a line in common. This example also shows the power of line and vector bundles in enumerative geometry, and the accompanying notion of characteristic classes, such as Chern classes.This leads the author to consider a different compactification of the space of smooth conics. This is the famous space of `stable maps’, which the author motivates by considering first a construction that involves attaching pairs of P(1) together in a manner that does not introduce any cycles. This is called a `tree” and the no-cycle condition is imposed since otherwise one can have an algebraic curve that does not arise as a limit of curves isomorphic to P(1). A `morphism’ from a tree to P(n) is then defined, with any parametrized rational curve being a morphism from the tree P(1) to P(n).Readers familiar with the notion of ‘transversal intersection’ from differential topology and have worked with characteristic classes will understand fully the role that cohomology plays in “counting’ the number of intersections of geometric objects. Loosely speaking, homology theory, and its dual, cohomology, can be viewed as “linear” theories since the “boundary of a boundary is zero” (and similarly the “coboundary of a coboundary is zero”). This is especially true in the context of de Rham cohomology, which the author briefly discusses but does not really use in the book. Thus when viewing the intersection of geometric objects from the standpoint of cohomology, one is looking at the intersection of linearized approximations to this object (the tangent or cotangent vector). The author introduces and uses a particular and very familiar notion of cohomology in this book, namely that of `singular cohomology’, which is given a very rapid review. One can still speak of the transversal intersection of two submanifolds but in this case in terms of local coordinates instead of tangent spaces as in the case of differential topology. This intersection defines the `intersection product’ in singular cohomology which for complex manifolds, which are the objects of interest in enumerative geometry, one counts the number of points in the singular cohomology class of the intersection. This whole project assumes that the cohomology of the manifold of which the submanifolds are a part is known. Once the submanifolds are characterized explicitly and their cohomology classes identified, their intersection products are calculated and then “integrated” over the entire manifold by using the “pairing” between cohomology and homology. The author shows how this goes through for the case of P(n) and how `cellular’ homology and cohomology, another version of homology and cohomology theory, can be used in enumerative geometry. The author illustrates the utility of this version for the case of the `Grassmannian’ of lines in P(3). It is in this discussion that the author introduces, via an example, the famous `Schubert calculus’ in order to study the cellular cohomology of the Grassmannian. It was the goal of making the Schubert calculus, which dates from the nineteenth century, rigorous that drove much of the research in modern algebraic geometry. It is the `Schubert cycles’ that allow the author to find the number of lines in P(3) that intersect four given lines. The Schubert cycles are the closures of the cells in the cellular decomposition of the Grassmannian.It is the predictions from string theory that have motivated many researchers in enumerative geometry to look in more detail at this complex but fascinating branch of physics. For the typical mathematician, the learning of string theory can be a formidable project. The author attempts to make it somewhat more palatable by including a few chapters on physics in the book, these chapters being couched in the language of modern mathematics as much as possible. The reader will see the origin of the very controversial formula for the number of rational curves on a quintic threefold, and understand the role of the Gromov-Witten theory in giving this formula a rigorous foundation.Fundamental to this discussion, as it was in the rest of the book and in the nineteenth century, is the role of projective space, it having the important properties of being compact and non-singular. It is also a complex manifold, an algebraic variety, and its homology and cohomology can be computed straightforwardly. The details of the Gromov-Witten theory can be formidable for both the mathematician who must deal with its motivation from string theory, and the physicist who must digest not only what a variety is but also a `stack’, which is a kind of generalization of an algebraic variety. The author does not define rigorously what a stack is, but instead begins with a compact complex submanifold X of a projective space and considers the collection of n-pointed stable maps to X. This is a generalization of the notion of the stable map defined earlier in the book and maps a tree of rational curves with distinct “marked points” to X. This map must represent a two-dimensional (integer) homology class in X and if constant when restricted to a component of the tree, must contain a node or one of the marked points. The author then defines an evaluation map on the marked points and the Gromov-Witten invariants for X are defined as an integral over the stack of the product of the pullbacks of these evaluation maps (in the de Rham cohomology). They can be computed by the `3-point correlation functions’ that give the connection between these invariants and the now ubiquitous field of `quantum cohomology.’

⭐This is a nice, informal, introduction to enumerative geometry and string theory. The first three chapters give a flavor of the former, indicating connections between algebra and geometry and motivating the use of complex numbers and projective spaces. The concrete question “How many lines in the plane pass through k fixed points and are tangent to 5-k fixed general lines?” is raised and answered here. Chapters 4-6 introduce standard tools of algebraic topology (general topology, manifolds, basic algebraic topology); more enumerative geometry, including an introduction to excess intersection theory, follows in Chapters 7-9. For example, the author introduces rudiments of Schubert calculus (intersection theory of cycles on Grassmannians) and uses it to determine the number of lines that lie on a typical quintic 3-fold in P^4. Chapters 10-13 concern physics, and the final chapter is meant to indicate relations between the two subjects.The book certainly achieves its aim of giving a feeling for enumerative geometry and string theory, but I do not feel it indicates the connections between the two in a meaningful way; perhaps, this would be too much to expect from such a short book. Overall, the discussion is very geometric, though for some reason the author chooses to introduce vector bundles in a rather formal way. Another, minor, drawback of the book is that the author references entire books; more specific references would often be helpful.The book would make a great text for a one-semester advanced undergraduate seminar, with each chapter taking up about a week for discussion. While graduate students (and beyond) are likely to be familiar with the material covered in Chapters 4-6, they will likely learn many new interesting things in the rest of the book, without putting in much effort.

⭐A great introduction to the interaction between geometry and string theory at an elementary level.

⭐It won’t be easy, but the effort using it will pay off.Plan on spending a lot of time on it

⭐This is for those interested in algebraic geometry as well as its connections with physics. Highly recommended.

⭐It has a very nice introduction to enumerative geometry and its connection with physics. You don’t require a lot of background.

⭐Nice book, but need to have a very solid background in Mathematics to read this book and that is expected.

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