Algebraic and geometric methods in enumerative combinatorics. Shingtung yau, geometric structures in field and string theory postdocts under my mentorship 2018 2020dennis borisov harvard university, cmsa scienti cacademic honors and awards. The structure we have described is roughly speaking the socalled. String theory seems to be a consistent theory of quantum gravity. 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. 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. For any given enumerative question three problems naturally arise. Today, we will talk about donaldsonthomas theory, which is an enumerative theory of curves in smooth algebraic 3folds x, like the projective space p3. Translate the enumerative problem into an intersection problem. 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.
Mathematician trying to learn string theory mathoverflow. The next step is to understand the effects due to the. Enumerative geometry and string theory pdf free download. In practice, geometric questions are usually reduced to computations in intersection theory k theory. Enumerative invariants in algebraic geometry and string theory lectures given at the c. 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. This is a nice, informal, introduction to enumerative geometry and string theory. The enumerative geometry of rational and elliptic curves in projective space ravi vakil abstract. 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. From the june 2016 issue the fall and rise of string theory. There is no need to assume x is calabiyau, or anything like this, for the problem to be interesting and relevant. For instance an extensive research activity carried out during the past years was to prove the.
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. Curriculum vitae artan sheshmani harvard university. Enumerative geometry and string theory sheldon katz publication year. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry. The typical question is to nd the number of objects with a given set of properties. Enumerative geometry, intersection theory and moduli spaces. This has deep mathematical consequences, which are still far from being completely understood.
For instance, mirror symmetry and gromovwitten theory are two hot. 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. 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 of stable maps with lagrangian boundary. The di culties lie in solving the rst two problems. Quantum field theory, supersymmetry, and enumerative geometry. In addition to its applications in enumerative geometry, mirror symmetry is a fundamental tool for doing calculations in string theory.
The study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomology. Classical enumerative geometry and quantum cohomology. Topological string theory and enumerative geometry yun s. More on the localization of the supersymmetric integral is available as a pdf. Algebraic and geometric methods in enumerative combinatorics federico ardila 0 introduction enumerative combinatorics is about counting. 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. Enumerative geometry is very closely tied to intersection theory. We compute intersection numbers on these spaces that correspond to the number of degree dcurves incident to various general linear. The first chapters are pretty basic, but the end of the book is relatively advanced. The study of quantum cohomology, gromovwitten invariants and mirror symmetry gave a significant progress in clemens conjecture.
However, enumerative combinatorics is not just about counting. Summer school held in cetraro, italy june 611, 2005. So in a very concrete way the line and the hyperbola are tangent at in. In this thesis we investigate several problems which have their roots in both topolog ical string theory and enumerative geometry. 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. String theory is much more technically demanding than loop quantum gravity and thus less accessible to most philosophers. Eva silverstein abstract in this thesis we investigate several problems which have their roots in both topolog. 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.
Enumerative geometry from string theory sciencedirect. Enumerative invariants in algebraic geometry and string theory. Nov 07, 2011 enumerative geometry and string theory sheldon katz. But there has certainly been remarkable progress, in particular in the realm of applications to enumerative geometry. Things started with the discovery of mirror pairs of calabiyau manifolds 1 with a subsequent application 2 to superstring compacti cations. Destination page number search scope search text search scope search text.
Enumerative geometry and knot theory have bene tted considerably from the insights and results in string theory and topological eld theory. The aim of this lecture course is to show how ideas from string theory and quantum. Written for the 2015 algebraic geometry summer institute. Enumerative geometry and string theory sheldon katz. Enumerative geometry and string theory cover image. Introduction to string theory chapter 0 eth zurich, hs prof. Enumerative geometry is also currently one of the most active areas of research in algebraic geometry, mainly due to a recent in. Ordinary geometry is probed by point particles here we learn about a generalization. Pdf enumerative geometry and string theory download full. Chernsimons theory, matrix models, and topological strings. A first course in string theory pdf epub download cause of.