A quadrillion ways for string theory to make our universe. In a sense, the topological string is a natural locus where mathematics and physics meet. We describe a compact moduli space of diagrams, and show how the cellular chain complex of this space gives algebraic operations on the singular chains of the free loop space lm of m. At these sites in m the curves may be cut and reconnected in some way. Previous treatments of string topology describe algebraic structures on the homology of the free loop space of the manifold and operations parameterized by a noncompact space of graphs. Vast size of the landscape arises from the plethora of possible geometries of the extra compact dimensions and. Kate poirier, university of california berkeley string topology studies the algebraic topology of the free loop space of a closed, oriented manifold. Since this is a textbook on algebraic topology, details involving pointset topology are often treated lightly or skipped entirely in the body of the text. Davide gaiotto, and yuji tachikawa showed that by compactifying this theory on a surface, one obtains a fourdimensional quantum field theory, and there is a duality known. A homotopy theoretic realization of string topology.
String topology in dimensions two and three springerlink. A main result of compactified string topology is the. We describe a compact moduli space of diagrams, and show how the cellular chain complex of this space gives algebraic. String topology and unstable homology of moduli space. Let x be an oriented compact manifold of dimension d. However, a limited number of carefully selected survey or expository papers are also included.
The idea of the fundamental group 23 paths, which are allowed to pass through the same point many times. Algebraic geometry and string theory royal society. Notice that mm e is the one point compactification of the tubular. That is, garbage collection in which the storage in use is moved to a contiguous region of memory and all pointers are adjusted to reflect this movement. Closed string operators in topology leading to lie bialgebras and higher string algebra.
B asic t opology t opology, sometimes referred to as othe mathematics of continuityo, or orubber sheet geometryo, or othe theory of abstract topo logical spaceso, is all of these, but, abo ve all, it is a langua ge, used by mathematicians in practically all branches of our science. Examples of nontrivial higher string topology operations have been regrettably rare in the literature. A homotopy theoretic realization of string topology 1. Then x is compact if and only if x is a complete lattice i. However, since there are copious examples of important topological spaces very much unlike r1, we should keep in mind that not all topological spaces look like subsets of euclidean space.
We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. This subject was initiated in the beautiful paper of chas and sullivan 3. Thus, the role of topology is likely to be far more central, in particular in tendimensional physics. We assume no prior knowledge of category theory, proof theory or computer science. Analogously, open topological string theory can be used to compute superpotentials for type ii string on cy3 with d branes.
Compactifying string topology kate poirier and nathaniel rounds abstract. Let lm be the free loop space over a compact oriented d di. The fundamental interval in western music is the half tone or semitone, e. Topological strings and their physical applications. In string theory, compactification is a generalization of kaluzaklein theory. We study the string topology of a closed oriented riemannian manifold m. Let x be a simply ordered set endowed with the order topology. Voronov 1 school of mathematics university of minnesota 206 church st. In this expository paper, we make some of these analogies precise using the concept of closed symmetric monoidal category. Notes on string topology stanford mathematics stanford university. Jun 20, 2016 examples of nontrivial higher string topology operations have been regrettably rare in the literature. Introduction to topology 3 prime source of our topological intuition. Homotopy and geometric perspectives on string topology.
The string landscape there is a vast landscape of vacua in string theory. Topology and its applications is primarily concerned with publishing original research papers of moderate length. Furthermore, the study of the topological string yielded an unanticipated but very exciting bonus. String theory in a nutshell pdf request pdf on researchgate on jan 1, elias kiritsis and others published string theory in a nutshell. Sample exam, f10pc solutions, topology, autumn 2011 question 1.
String theory and the real world gordon kane university of michigan. Curved string topology and tangential fukaya categories. Different type of inverter topologies for pv transformerless standalone system. Pdf closed string operators in topology leading to lie. Compactifying string topology pr11, with nathaniel rounds. String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. Introduction to string compactification contents albert einstein. Consequently, there are no local degrees of freedom in topological string theory. We describe a compact moduli space of diagrams, called sd, and show how the cellular chain complex of this space gives algebraic operations on the singular chains of the free loop space lm of m. One obtains operators on the set of parametrized collections of closed curves in m. When twisted topological complex ktheory is applied to this prospectrum, the. Pdf string topology and cyclic homology researchgate. Oct 28, 2012 i will cover the topology of the real line and the definition of continuous.
Klein, dennis sullivan, the homotopy invariance of the string topology loop product and string bracket, j. Not included in this book is the important but somewhat more sophisticated topic of spectral sequences. The methods of compactification are various, but each is a way of controlling points from going off to infinity by in. By making the coincidences transversal and compactifying, the operators can be made to act in the algebraic topology of the free loop space of m when m is oriented. Abstract we study the string topology of a closed oriented riemannian manifold m. Kate poirier uc berkeley compactifying string topologyjuly 24, 2012. Consider two families of closed oriented curves in a dmanifold. Research statement page 6 of 12 katherine poirier question 2. Three applications of topology to physics dan freed university of texas at austin january 12, 2018.
In this paper, working in the context of string topology of classifying spaces, we provide explicit calculations of a wealth of nontrivial higher string topology operations associated to a number. Uvadare digital academic repository topological strings and. Introduction to string theory universiteit utrecht. This paper is an exposition of the new subject of string topology. The methods of compactification are various, but each is a way of controlling points from going off to infinity by in some way adding points at infinity or preventing. String topology studies the algebraic topology of the free loop space of a closed, oriented manifold. We describe a compact moduli space of diagrams, and show how the cellular chain complex of this space gives algebraic operations on the singular chains of the free loop space lm.
Frg conference on topology and field theories june 4 8. Homotopy and geometric perspectives on string topology ralph l. Other topics include an openclosed version of string topology, a morse theoretic interpretation, relation to gromovwitten invariants, and brane topology, which deals with sphere spaces. A compact space is a space in which every open cover of the space contains a finite subcover. In string topology one studies the bvalgebrastructure on the ordinary homology of the. Tduality via noncommutative topology 707 consider the 2 dimensional torus t2 r2z2. Compactifying string topology kate poirier uc berkeley algebraic topology. Ralph cohen, homotopy and geometric perspectives on string topology, pdf. At each point of intersecction of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Any finite topological space, including the empty set, is compact. Higher operations in string topology of classifying spaces.
Compactification, topology change and surgery theory. Lectures given at geometric and topological methods for quantum. Stemming from the ftheory branch of string theory, each solution replicates key features of the standard model of particle. The resulting list structure has the same topology as the old, so that reentrancy and sharing of common substructure are preserved. Attempts to link superstrings with the reality of the physical world in.
Seyed morteza hosseini extremization principles from geometry hosseini. In 1998 the authors constructed a bracket on higher dimensional manifolds which is part of string topology 2. In string topology one studies the bvalgebrastructure on the. Orbifolds were rst introduced into topology and di erential. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur.
Stanford university august 30, 2005 in these lecture notes i will try to summarize some recent advances in the new area of study known as string topology. We study the process of compactification as a topology change. This translates in two possible gauge groups for the heterotic string theory. As a result, the pv array is disconnected from the utility. The mathematical focus of the journal is that suggested by the title. Curved string topology and tangential fukaya categories by daniel michael pomerleano a dissertation submitted in partial satisfaction of the requirements for the degree of doctor of philosophy in mathematics in the graduate division of the university of california, berkeley committee in charge. Transverse string topology and the cord algebra request pdf. Topological invariants on a chain of oriented strings in interaction are discussed. Applications and new developments stanford university, july 24, 2012.
String topology is ultimately about the differential and algebraic topology of spaces of paths and loops in compact, oriented manifolds. The string topology perspective gives a topological derivation of the cord algebra also isomorphic to degree 0 knot contact homology and relates it to the knot group. It tries to reconcile the gap between the conception of our universe based on its four observable dimensions with the ten, eleven, or twentysix dimensions which theoretical equations lead us to suppose the universe is made with. Subsequent investigations revealed however severe di. This larger picture containing both string theory and 11d supergravity is. Basicnotions 004e the following is a list of basic notions in topology.
Mathematical framework we begin by explaining the precise mathematical framework in which we are working. Conversely, 11d supergravity contains a membrane solution, and one can obtain string theory from it by compactifying on a subplanckian circle, so that a membrane wound around the circle becomes a string. Topological string theory is obtained by a topological twist of the worldsheet description of ordinary string theory. As an application of these calculations, we obtain an abundance of. On string topology of classifying spaces request pdf. In the twentieth century algebraic geometry became a forbiddingly technical subject, wellinsulated from nonmathematical in uences. In this paper, working in the context of string topology of classifying spaces, we provide explicit calculations of a wealth of nontrivial higher string topology operations associated to a number of different lie groups. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Introduction to orbifolds april 25, 2011 1 introduction orbifolds lie at the intersection of many di erent areas of mathematics, including algebraic and di erential geometry, topology, algebra and string theory. I will cover the topology of the real line and the definition of continuous. The idea of the fundamental group cornell university. By making the coincidences transversal and compactifying, the operators can be made to act in the. Compactifying string theory over an internal space x geometrizes 4.
It is possible that string topology can provide information about moduli space as well. String topology of classifying spaces dedicated to. Compactification of extra dimensions as has already been mentioned, to have a chance of being realistic, the six extra space dimensions must curl up into a tiny geometrical space, whose size should be comparable to the string length l st since spacetime geometry is determined dynamically as in general relativity, only geometries that satisfy the dynamical equations are allowed. Although the topological properties of the compactification manifold x are re. May 21, 2009 if a and b are conjugacy classes, the bracket a. String topology is the study of algebraic and differential topological. In summary, the lowenergy physics of the bosonic string in d1 dimensions consists of a metric g. The paper is a joint account of the lecture series given by each of us at the 2003 summer school on string topology and hochschild homology in almeria, spain.
Tube algebras, excitations statistics and compactification in gauge. What is the compacti cation of moduli space that is appropriate for string topology. In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. It employs an extra switch on the dc side of inverter. So kkreduction of 10dimensional supergravities can for a suitable ansatz cure the old problem of moduli stabilization in kktheory this means that physical model building using the specific ansatz of kkreduction of type ii supergravities on calabiyau fibers reduces. Moduli space provides information about the string topology of m via the action of its homology on the homology of the free loop space of m.915 1174 354 489 908 154 1125 1232 228 655 363 380 1213 874 1377 44 1183 957 1114 63 783 1313 1410 1563 239 1314 1102 603 302 899 952 1225 491 294 761 511 208