Enumerative geometry and string theory pdf files

The di culties lie in solving the rst two problems. New insights into gauge theory and string theory, in particular, holography and issues in the black hole information puzzle solution of a long standing problem in field theory. A first course in string theory pdf epub download cause of. There is no need to assume x is calabiyau, or anything like this, for the problem to be interesting and relevant. Shingtung yau, geometric structures in field and string theory postdocts under my mentorship 2018 2020dennis borisov harvard university, cmsa scienti cacademic honors and awards. For instance, mirror symmetry and gromovwitten theory are two hot. The line through p is represented by its slope, that is the ratio z yx. Enumerative geometry and geometric representation theory.

The typical question is to nd the number of objects with a given set of properties. Quantum field theory, supersymmetry, and enumerative geometry. After an introduction to matrix models and chernsimons theory, the book describes in detail the topological string theories that correspond to these gauge theories and develops the mathematical implications of this duality for the enumerative geometry of calabiyau manifolds and knot theory. More on the localization of the supersymmetric integral is available as a pdf.

We study the geometry of moduli spaces of genus 0 and 1 curves in pn with speci ed contact with a hyperplane h. Enumerative geometry, intersection theory and moduli spaces. Enumerative geometry and string theory sheldon katz. In this thesis we investigate several problems which have their roots in both topolog ical string theory and enumerative geometry. The next step is to understand the effects due to the. So in a very concrete way the line and the hyperbola are tangent at in. This is a nice, informal, introduction to enumerative geometry and string theory. In my opinion the ideas and structures that have come out of string theory are deeper, richer and in the long run will teach us more about the fundamental structure of reality. Proper treatment of the subject would be quite technical, instead we will attemt to develop a feeling of what a modern enumerative geometry is by working out some examples. Topological string theory and enumerative geometry yun s. String theory seems to be a consistent theory of quantum gravity. Written for the 2015 algebraic geometry summer institute. Topological string theory and enumerative geometry a dissertation submitted to the department of physics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy yun s. Enumerative geometry and string theory cover image.

The structure we have described is roughly speaking the socalled. String theory is much more technically demanding than loop quantum gravity and thus less accessible to most philosophers. 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. This is a recurrent theme in string theory, where partition functions have often an enumerative interpretation as counting objects instantons, etc. Curriculum vitae artan sheshmani harvard university. However, enumerative combinatorics is not just about counting. Today, we will talk about donaldsonthomas theory, which is an enumerative theory of curves in smooth algebraic 3folds x, like the projective space p3. Ordinary geometry is probed by point particles here we learn about a generalization. Pdf enumerative geometry and string theory download full. 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. In practice, geometric questions are usually reduced to computations in intersection theory k theory. This volume presents three weeks of lectures given at the summer school on quantum field theory, supersymmetry, and enumerative geometry, three very active research areas in mathematics and theoretical physics. Some recent mathematical progress in enumerative geometry on a variety v is presented in the light of string theory, in its formulation as a topological field theory with target space v.

Algebraic and geometric methods in enumerative combinatorics. Destination page number search scope search text search scope search text. Classical enumerative geometry and quantum cohomology. From the june 2016 issue the fall and rise of string theory. The theory of gromovwitten invariants has emerged mostly from the consideration of topological sigma models and topological strings, and mirror symmetry has provided a surprising point of view with. Enumerative geometry from string theory sciencedirect. Translate the enumerative problem into an intersection problem.

Enumerative geometry and string theory sheldon katz publication year. One takes a ver sion of string theory or m theory, or f theory and compactifies the theory on a calabiyau manifold, obtaining an effective physical theory. It combines many interesting topics of quantum eld theory in two and higher dimensions. Enumerative geometry of stable maps with lagrangian boundary. For any given enumerative question three problems naturally arise. Enumerative geometry and knot theory have bene tted considerably from the insights and results in string theory and topological eld theory. The enumerative geometry of rational and elliptic curves in projective space ravi vakil abstract.

Mathematician trying to learn string theory mathoverflow. Introduction to string theory chapter 0 eth zurich, hs prof. Enumerative invariants in algebraic geometry and string theory. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry. But there has certainly been remarkable progress, in particular in the realm of applications to enumerative geometry. Enumerative geometry and string theory ams bookstore. The aim of this lecture course is to show how ideas from string theory and quantum.

Enumerative geometry and string theory sheldon katz 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. String theory has led to new ideas in enumerative geometry, and it has provided concrete conjectural answers to many enumerative questions, particularly through mirror symmetry. This has deep mathematical consequences, which are still far from being completely understood. Things started with the discovery of mirror pairs of calabiyau manifolds 1 with a subsequent application 2 to superstring compacti cations. Summer school held in cetraro, italy june 611, 2005. For instance an extensive research activity carried out during the past years was to prove the. Eva silverstein abstract in this thesis we investigate several problems which have their roots in both topolog.

Enumerative geometry and string theory pdf free download. It had an immediate impact on enumerative geometry and motivated various mathematical investigations including the formulation of gromovwitten invariants. In addition to its applications in enumerative geometry, mirror symmetry is a fundamental tool for doing calculations in string theory. Algebraic and geometric methods in enumerative combinatorics federico ardila 0 introduction enumerative combinatorics is about counting. The study of quantum cohomology, gromovwitten invariants and mirror symmetry gave a significant progress in clemens conjecture. In the amodel of topological string theory, physically interesting quantities are expressed in terms of infinitely many numbers called gromovwitten invariants, which are. The second is topological string theory, which is related to the enumerative geometry of calabiyau threefolds and computes, for example, gromovwitten invariants, donaldsonthomas invariants, gopakumarvafa bps invariants, and key aspects of kontsevichs homological mirror symmetry conjecture. The first chapters are pretty basic, but the end of the book is relatively advanced. Enumerative geometry is also currently one of the most active areas of research in algebraic geometry, mainly due to a recent in. Nov 07, 2011 enumerative geometry and string theory sheldon katz. Enumerative invariants in algebraic geometry and string theory lectures given at the c.