Compactifying string topology pdf

String topology of classifying spaces dedicated to. This translates in two possible gauge groups for the heterotic string theory. Kate poirier, university of california berkeley string topology studies the algebraic topology of the free loop space of a closed, oriented manifold. Compactifying string topology kate poirier uc berkeley algebraic topology. Consider two families of closed oriented curves in a dmanifold.

Generalizes results of abbaspourcohengruher 05 and vaintrob 06 when m kg,1, so cm kg. Oct 28, 2012 i will cover the topology of the real line and the definition of continuous. Introduction to string compactification contents albert einstein. Examples of nontrivial higher string topology operations have been regrettably rare in the literature. Let x be a simply ordered set endowed with the order topology. In summary, the lowenergy physics of the bosonic string in d1 dimensions consists of a metric g. This larger picture containing both string theory and 11d supergravity is. Compactifying string topology pr11, with nathaniel rounds. 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. We assume no prior knowledge of category theory, proof theory or computer science. In string topology one studies the bvalgebrastructure on the ordinary homology of the.

Vacuum selection from cosmology on networks of string. A compact space is a space in which every open cover of the space contains a finite subcover. Transverse string topology and the cord algebra request pdf. Algebraic geometry and string theory royal society. 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. What is the compacti cation of moduli space that is appropriate for string topology. Notes on string topology stanford mathematics stanford university. At these sites in m the curves may be cut and reconnected in some way.

So if one is thinking of a loop as something made of stretchable string, one has to give the string the magical power of being able to pass through itself unharmed. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. A main result of compactified string topology is the. Topology and its applications is primarily concerned with publishing original research papers of moderate length. In a sense, the topological string is a natural locus where mathematics and physics meet. By considering modular invariance of the genus one string partition function, the possible shapes of the torus are restricted up to two. Orbifolds were rst introduced into topology and di erential. Three applications of topology to physics dan freed university of texas at austin january 12, 2018. Topological strings and their physical applications. Abstract we study the string topology of a closed oriented riemannian manifold m. Pdf closed string operators in topology leading to lie. 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. This paper is an exposition of the new subject of string topology.

String topology is ultimately about the differential and algebraic topology of spaces of paths and loops in compact, oriented manifolds. Consequently, there are no local degrees of freedom in topological string theory. This subject was initiated in the beautiful paper of chas and sullivan 3. Not included in this book is the important but somewhat more sophisticated topic of spectral sequences. Let lm be the free loop space over a compact oriented d di. It is possible that string topology can provide information about moduli space as well. 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. Thus, the role of topology is likely to be far more central, in particular in tendimensional physics. 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. 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. 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. Closed string operators in topology leading to lie bialgebras and higher string algebra.

Seyed morteza hosseini extremization principles from geometry hosseini. A homotopy theoretic realization of string topology. Curved string topology and tangential fukaya categories. Pdf string topology and cyclic homology researchgate. Introduction to topology 3 prime source of our topological intuition. Mathematical framework we begin by explaining the precise mathematical framework in which we are working. Ralph cohen, homotopy and geometric perspectives on string topology, pdf. Analogously, open topological string theory can be used to compute superpotentials for type ii string on cy3 with d branes. Basicnotions 004e the following is a list of basic notions in topology. 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. 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.

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. Jun 20, 2016 examples of nontrivial higher string topology operations have been regrettably rare in the literature. 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. In 1998 the authors constructed a bracket on higher dimensional manifolds which is part of string topology 2. Uvadare digital academic repository topological strings and. Compactification, topology change and surgery theory.

Klein, dennis sullivan, the homotopy invariance of the string topology loop product and string bracket, j. Voronov 1 school of mathematics university of minnesota 206 church st. Homotopy and geometric perspectives on string topology ralph l. 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. Lectures given at geometric and topological methods for quantum. 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. When twisted topological complex ktheory is applied to this prospectrum, the. Let x be an oriented compact manifold of dimension d. Then x is compact if and only if x is a complete lattice i. String topology and unstable homology of moduli space. String theory in a nutshell pdf request pdf on researchgate on jan 1, elias kiritsis and others published string theory in a nutshell. 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. In string topology one studies the bvalgebrastructure on the. A nonempty compact subset of the real numbers has a greatest element and a least element.

The idea of the fundamental group cornell university. The mathematical focus of the journal is that suggested by the title. We study the process of compactification as a topology change. By making the coincidences transversal and compactifying, the operators can be made to act in the. 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. String theory and the real world gordon kane university of michigan. As a result, the pv array is disconnected from the utility. In string theory, compactification is a generalization of kaluzaklein theory. Different type of inverter topologies for pv transformerless standalone system. Sample exam, f10pc solutions, topology, autumn 2011 question 1.

Compactifying string topology kate poirier and nathaniel rounds abstract. Homotopy and geometric perspectives on string topology. The book is well founded on three decades of compacti. Tube algebras, excitations statistics and compactification in gauge. Pdf compactifying string topology kate poirier academia. 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.

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. Kate poirier uc berkeley compactifying string topologyjuly 24, 2012. We describe a compact moduli space of diagrams, and show how the cellular chain complex of this space gives algebraic. The resulting list structure has the same topology as the old, so that reentrancy and sharing of common substructure are preserved. A homotopy theoretic realization of string topology 1. It employs an extra switch on the dc side of inverter. 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. Notice that mm e is the one point compactification of the tubular. The methods of compactification are various, but each is a way of controlling points from going off to infinity by in. In this expository paper, we make some of these analogies precise using the concept of closed symmetric monoidal category. String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. Although the topological properties of the compactification manifold x are re. In the twentieth century algebraic geometry became a forbiddingly technical subject, wellinsulated from nonmathematical in uences.

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. 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. Furthermore, the study of the topological string yielded an unanticipated but very exciting bonus. One obtains operators on the set of parametrized collections of closed curves in m. Applications and new developments stanford university, july 24, 2012. 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. Compactifying string theory over an internal space x geometrizes 4. Introduction to string theory universiteit utrecht. Stemming from the ftheory branch of string theory, each solution replicates key features of the standard model of particle. Tduality via noncommutative topology 707 consider the 2 dimensional torus t2 r2z2.

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. However, a limited number of carefully selected survey or expository papers are also included. Research statement page 6 of 12 katherine poirier question 2. String topology is the study of algebraic and differential topological. 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. 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. Attempts to link superstrings with the reality of the physical world in. Subsequent investigations revealed however severe di.

The fundamental interval in western music is the half tone or semitone, e. Vast size of the landscape arises from the plethora of possible geometries of the extra compact dimensions and. On string topology of classifying spaces request pdf. May 21, 2009 if a and b are conjugacy classes, the bracket a. Higher operations in string topology of classifying spaces. The operation is fully analogous to the construction of topological field theory which is a related concept. The idea of the fundamental group 23 paths, which are allowed to pass through the same point many times. Topological invariants on a chain of oriented strings in interaction are discussed. As an application of these calculations, we obtain an abundance of. String topology studies the algebraic topology of the free loop space of a closed, oriented manifold. Topological string theory is obtained by a topological twist of the worldsheet description of ordinary string theory. The string landscape there is a vast landscape of vacua in string theory. A quadrillion ways for string theory to make our universe. 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.

Any finite topological space, including the empty set, is compact. We study the string topology of a closed oriented riemannian manifold m. In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. 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.