String theory and the geometry of the universes hidden dimensions shingtung yau and steve nadis i d like to talk about how mathematics and physics can come together to the benefit of both fields, particularly in the case of calabiyau spaces and string theory. String theory usually builds on this with at the very least a little algebraic geometry. The num ber of dimensions and the geometry fourdimensional minkowski space. String theory and the history of noneuclidean geometry. Complex geometry first appeared as a distinct area of study in the work of bernhard riemann in his study of riemann surfaces. If you take cues from people on the nlab, category theory can be big in string theory. Enumerative geometry and string theory cover image. One would therefore like to understand the space of all worldsheet qfts whose effective target space geometry and particle content is close to. Enumerative geometry is very closely tied to intersection theory. As a result, the theory is independent of backgrounds.
Mathematician trying to learn string theory mathoverflow. Nuclear physics b268 1986 253294 northholland publishing company noncommutative geometry and string field theory edward witten joseph henry laboratories, princeton university, princeton, new jersey 08540, usa received 2 december 1985 an attempt is made to interpret the interactions of bosonic open strings as defining a noncommutative, associative algebra, and to formulate the classical. Einsteins unified field theory with teleparallelism a translation by a. Pick any combination of differentialalgebraic and geometry topology and it will be useful in string theory. This says to pick a 2d superconformal field theory of central charge 15, collect its npoint functions into one formal power series, interpret this as a loopwise finite, hence normalized, scattering matrix known from qft, then study this. In particular, we hope to understand further the role that spectral curves and spectral data plays in string theory, and in particular for higgs bundles on higher dimensional spaces. Aside from being the best candidate for the theory of quantum gravity, string theory has branched o. Scientists in the department of theoretical physics and at the mathematical science institute are collaborating in understanding the mathematics behind string theory and quantum field theory. This, not coincidentally, is the subject of the new book. Moreover, string theory is very much a work in progress and certain aspects of the theory are far from understood. String theory is a rather new field of theoretical physics. 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. Nuclear physics b268 1986 253294 northholland publishing company noncommutative geometry and string field theory edward witten joseph henry laboratories, princeton university, princeton, new jersey 08540, usa received 2 december 1985 an attempt is made to interpret the interactions of bosonic open strings as defining a noncommutative, associative algebra, and to. Einsteins later later publications have been translated to many languages, but not to english yet maybe because there are people who get paid for doing that work slowly.
The 2002 clay school on geometry and string theory was held at the isaac. A general theme that runs through this thesis is to. It appeared only twenty years ago to describe phenomenology of strong interactions of elementary particles, and until recently, it has been developing rather slowly. The plan is for a systematic, focused activity with the dual aim of discussing new results and putting new insights gained from physics into the general. In this direction, we give some foundational results and propose.
An introduction to string theory kevin wray abstract. In this thesis we investigate several problems which have their roots in both topolog ical string theory and enumerative geometry. The research field number theory and geometry brings together people in the department with interests in arithmetic and various aspects of geometry, especially arithmetic and diophantine geometry. String theory is not, in contrast to general relativity and quantum field the. By sheldon katz enumerative geometry and string theory by 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. In section 3, we solve the equations of motion and obtain a string geometry solution that represents perturbative string. I would adopt an analytic point of view, such as in the book by griffiths and harris principles of algebraic geometry, since this is going to be closer to the language that. String geometry and nonperturbative formulation of string theory. Kostas skenderis in the spring of 2009 at the university of amsterdam. A course on noncommutative geometry in string theory.
The group organizes the number theory seminar and the annual number theory days, jointly with epf lausanne and university of basel. Introduction in topological string theory on calabiyau manifolds the gromovwitten invariants, which are symplectic invariants of. Number theory and geometry department of mathematics eth. String theory would not, of course, have seen its success without all the important concurrent developments in mathematics.
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. Penroses book is wellwritten, and it is good for obtaining an intuitive idea of what some mathematicians think of given definitions of objects, but it wont give you concise. String theory max planck institute for gravitational physics. During the academic year 200405, a yearlong program of activity will take place at the fields institute and perimeter institute on the general subject the geometry of string theory. Enumerative geometry and string theory ams bookstore. Zaslow, eric, northwestern algebraic geometry and string theory additional funding. Fourier mukai transforms and applications to string theory by b. Complex geometry and the theory of quantum strings jetp.
String theory is invariant underexchange of momentum and winding. In this note i will describe very basic and elementary examples of the kind of mathematical structure motivated either directly or indirectly by string theory. Jun 20, 2006 at the beginning of the last century, albert einstein posited a now famous theory that forever linked geometry and fundamental physics. In the past decade, there has been huge surge of interest in topological aspects of condensed matter physics. According to general relativity, spacetime is curved, and. Geometry and group theory abstract inthiscourse, wedevelopthebasicnotionsofmanifoldsandgeometry, withapplications in physics, and also we develop the basic notions of. This is the homepage for the ut geometry and string theory seminar.
The spacetime in string theory 1 2 is often described by means of a mathematical object called manifold 3. String theory a mathematical model that we will use often is that of. Manifolds are very important objects from the mathematical and the physics point of view, not only in string theory. Many reasons for doing string theory involve physics. Lifshitz geometries in string and mtheory jerome gauntlett aristomenis donos aristomenis donos, nakwoo kim, oscar varela to appear thursday, 9 september 2010. Berges hepph0409233, 1 pages introduction to yangian symmetry in integrable field theory by n. Ag0412328, 52 pages introduction to nonequilibrium quantum field theory by j. Eva silverstein abstract in this thesis we investigate several problems which have their roots in both topological string theory and enumerative geometry. This involves applying fairly recent mathematical disciplines, such as noncommutative geometry, but also involves developing and studying new mathematics. Mathematics needed for string theory physics stack exchange.
The mathematics of string theory and quantum field theory. Galois theory and projective geometry 5 projective space subject to pappus axiom is a projectivization of a vector space over a. We will then digress to introduce mtheory as a strongcoupling limit of the type iia string theory, and ftheory as a variant of the type iib string theory whoseexistenceismotivatedbytduality. String theory b ecame then a consistent candidate for a theory of quantum gravity, and later on, in the early 1980s, a candidate for a uni. Mathematics needed for string theory physics forums. Aug 26, 2011 i dont want to sound snobbish, but i cant promise that youll get a satisfactory understanding of string theory with an abbreviated understanding of the math behind it. Emanuel malek university of cambridge the od, d geometry of string theory 9th october 20 6 38. In the twentieth century algebraic geometry became a forbiddingly technical subject, wellinsulated from nonmathematical in uences. Topological string theory and enumerative geometry yun s.
Enumerative geometry of stable maps with lagrangian. Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces. Enumerative geometry and string theory sheldon katz publication year. For three of the previous workshops at the aspen center for physics in which dan freed was an organizer in 1987, 1989, and 2004 the nsf and clay mathematical. Destination page number search scope search text search scope search text. May 06, 2015 this is first of a series of notes on the geometry of string theory compactifications. An overview of the geometry of string theory, which sets the various contributions to this proceedings in this context. Ideas from string theory have an independent existence in the mathematical world, separate from any question about their relationship to reality. This is first of a series of notes on the geometry of string theory compactifications. The aim of this lecture course is to show how ideas from string theory and quantum. In principle perturbative string theory is well defined.
Singular geometry and higgs bundles in string theory. Certainly it is, mathematically, a structure that contains in various limits a plethora of quantum field theories. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime. The study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomology. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry.
Its interest for experimental high energy physics lies in the hypothesis that it provides a theory of everything in the sense of fundamental physics, but the jury on that is still out. 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. Elias kiritsis, string theory in a nutshell, princeton up 2007, 608 pp. Newton institute for mathematical sciences, cambridge, uk from 24 march 20. This set of notes is based on the course introduction to string theory which was taught by prof. The very rst beginnings of string theory go back to the late 1960s and early 1970s, with e.
Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables, and has found applications to string theory and mirror symmetry. Geometry and supersymmetry in type ii string theory. They also construct a lagrangian for the od,d metric that reproduces the usual one when there is no ya dependence but they also have an od. Pick any combination of differentialalgebraic and geometrytopology and it will be useful in string theory.
With this warning given, let me say that it seems to me that it would be near impossible to understand string theory without some understanding of algebraic geometry. String geometry and nonperturbative formulation of string. Noncommutative geometry and string field theory sciencedirect. There is no experimental evidence that string theory is the correct description of our world and scant hope that hard evidence will arise in the near future. This is a nice, informal, introduction to enumerative geometry and string theory. The study of quantum cohomology, gromovwitten invariants and mirror symmetry gave a significant progress in clemens conjecture.
An evaluation peter woit department of mathematics, columbia university january 29, 2001 for nearly seventeen years now most speculative and mathematical work in particle theory has centered around the idea of replacing quantum eld theory with something that used to be known as \superstring theory, but now goes under the name \m. At the organizational meeting we will flesh out the details of our plans for the semester. At the beginning of the last century, albert einstein posited a now famous theory that forever linked geometry and fundamental physics. We have also drawn on some ideas from the books string theory and. Algebraic geometry and string theory royal society. Manifolds are very important objects from the mathematical and the physics point of.
Pdf enumerative geometry and string theory download full. Enumerative geometry and string theory sheldon katz. Before string theory introduced the concept of extra dimensions, the fascination with strange warping of space in the 1800s was perhaps nowhere as clear as in the creation of noneuclidean geometry, where mathematicians began to explore new types of geometry that werent based on the rules laid out 2,000 years earlier by euclid. However strings have also had an important impact on mathematics. Over the last three decades, string theory has had a profound impact in pure mathematics connected to string theory, including generalized geometry, vertex algebras, topological tduality and related topics.
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