# [PDF / Epub] ✎ Enumerative Geometry and String Theory (Student Mathematical Library) ☂ Sheldon Katz – Tactical-player.co.uk

A great introduction to the interaction between geometry and string theory at an elementary level. This is for those interested in algebraic geometry as well as its connections with physics Highly recommended. 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 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 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 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. It won t be easy, but the effort using it will pay off.Plan on spending a lot of time on it 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 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

## About the Author: Sheldon Katz

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