The results are applied to an infinite dimensional problem involving the symplect. Help our scientists and scholars continue their fieldshaping work. The focus of the book is on the interplay of mathematical and physical aspects of the theory. In his paper, witten showed that supersymmetry, the fundamental connection between fermions and bosons, gave an elegant way of deriving im portant results in morse theory. In this thesis, the profound connection between morse theory and supersymmetry will be explored along the lines of wittens 1982 paper. Part 5 is devoted to advanced topics in mirror symmetry, including the role of dbranes in the context of mirror symmetry, and some of their applications in physics and mathematics. This book gives a presentday account of marston morses theory of the calculus of variations in the large. Witten dynamical breaking of supersymmetry 535 try is unbroken at the tree level, then any particle massless at the tree level remains massless to all finite orders of perturbation theory, even if its masslessness resulted from arbitrary adjustment of parameters. Published during a period of rapid progress in this subject, these volumes were highly influential for a generation of students and researchers. Question on wittens paper supersymmetry and morse theory. Supersymmetric field theory and wittenfloer cohomology. In the sevendimensional case, the picture appears to be more conventional and in tune with wittens version of morse theory 25. The strong morse inequalities are derived using related ideas from supersymmetry, but mention is made of the techniques used by witten to get at the strong morse inequalities.
The real aim of the paper is however to prepare the ground for supersymmetric quantum field theory as the hodgede rham theory of infinitedimensional manifolds. Introduction 25th anniversary edition twentyfive years ago, michel green, john schwarz, and edward witten wrote two volumes on string theory. Professor edward wittens talk at strings 20 held at the sogang university, june2429, 20. Edward witten it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian. Chapter 12 is all about susy qm, though let me warn that its likely not readable on its own.
Morse theory the simplest example of supersymmetric quantum mechanics is. Morse originally applied his theory to geodesics critical points of the energy functional on paths. On the work of edward witten university of california. A striking connection between morse theory and the semiclassical limit in quantum mechanics was laid bare by edward witten in the groundbreaking paper 15, where question 1 was solved under the assumption that f is morse.
Seibergwitten theory and duality in n 2 supersymmetric. Ideal zerodivisor complex milosevic, nela and petrovic, zoran z. His paper 2 on supersymmetry and morse theory is obligatory reading for geometers interested in understanding modern quantum field theory. Wittenhel ersj ostrand theory is an addition to morse theory and hodgede rham theory for riemannian manifolds and considerably improves on them by injecting some spectral theory of elliptic operators. Edward witten born august 26, 1951 is an american theoretical physicist and professor of mathematical physics at the institute for advanced study in princeton, new jersey. Appendix 4 of chapter 12 gives a brief resummation of wittens paper on morse theory that you cite. It essentially specializes the symmetries, fields, equations viewpoint developed in chapters 9, 10, 11 to this example. On the equivalence of multiplicative structures in floer. Three approaches to morsebott homology hurtubise, d. We will also prove that gwinvariants in floer homology and quantum homology are equivalent. These techniques were used in raoul botts proof of his periodicity theorem. Wittens approach makes essential use of two ideas borrowed from quantum physics and hodge theory. Here is a more detailed presentation of the contents. Upper bounds on the witten index for supersymmetric.
We then develop some of these anomaly free theories into phenomenological models featuring a heavy z. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. Pdf supersymmetric field theory and wittenfloer cohomology. Topological effects in particle physics phenomenology. The resulting equations are the haydyswitten equations. Related questions instantons in wittens supersymmetry and morse theory. Witten deformation and morse theory jonieramaralantunes april25,2016 introduction in 1982 edward witten published the paper supersymmetry and morse the. Hilbert space of this vedimensional theory is the khovanov homology.
In chapter 4, we step aside from milnors account of morse theory and take a look at edward witten s article supersymmetry and morse theory, published in 1982. Supersymmetry and morse theory 665 the results of 2 have an analogue for complex manifolds, which will be discussed in a separate paper. Im reading wittens paper on supersymmetry and morse theory and am confused about the details of the instanton calculation which he uses to define a morse complex beginning at page 11 of the pdf witten writes down the relevant supersymmetric lagrangian and then states. Inspire, the high energy physics information system.
Gromovwitten invariants in genus zero, as is predicted by mirror symmetry. Euclid, pdf, spire which showed that morse theory may be equivalently interpreted as the study of supersymmetric vacua in supersymmetric quantum mechanics, and which was part of what gained witten the fields medal 1990. Supersymmetric noether currents and seibergwitten theory by alfredo iorio a thesis submitted to the school of mathematics, trinity college, university of dublin. In their seminal work, seiberg and witten derived the wilsonian lowenergy e ective action of the n 2 su2 gauge theory by encoding the problem in a twodimensional holomorphic curve. A brief history of morse homology yanfeng chen abstract morse theory was originally due to marston morse 5. The existence of nontrivial critical point for a class of.
Supersymmetric noether currents and seibergwitten theory. Supersymmetry and string theory will enable readers to develop models for new physics, and to consider their implications for accelerator experiments. Gregory moore, rutgers university collaboration with. For a colorful history of morse theory and the events leading up to wittens paper, we highly recommend reading raoul botts recollections in 8. Pdf upper bounds on the witten index for supersymmetric. Wittens complex and infinite dimensional morse theory andreas floer abstract we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space. I do not know if this is true in all supersymmetric theories. Supersymmetry and morse theory school of natural sciences. Division algebras and supersymmetry ii baez, john c. Instantons in wittens supersymmetry and morse theory. Based on the same idea, morse homology was introduced by thom, smale, milnor, and witten.
Initially i discovered higher category theory and higher homotopy theory for myself in my phd work, in the course of analyzing the supersymmetric quantum mechanics of the superstring on loop space. I edward witten, supersymmetry and morse theory, j. A morse theory of critical points suggests some atiyahsinger. Instanton solutions or tunneling paths in this theory would be extrema of this lagrangian, written with a euclidean. In particle physics, supersymmetry susy is a conjectured relationship between two basic classes of elementary particles. Supersymmetry and morse theory edward witten abstract. Morse theory the simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics. Free fermions on an algebraic curve, the mathematical heritage of hermann weyl. Edward witten how supersymmetry is different youtube.
Before morse, arthur cayley and james clerk maxwell had developed some of the ideas of morse theory in the context of topography. Witten explains that supersymmetric quantum mechanics is just hodgede rham theory. Upper bounds on the witten index for supersymmetric lattice models by discrete morse theory. Upper bounds on the witten index for supersymmetric lattice models by discrete morse theory article pdf available in european journal of combinatorics 302. Wittens complex and infinitedimensional morse theory. String theory and the universe lecture by edward witten. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of morse functions.
However, there have been important developments during the past few years which are not mentioned. A type of spacetime symmetry, supersymmetry is a possible candidate for undiscovered particle physics, and seen by some physicists as an elegant solution to many current problems in. Edward witten delivers his lecture string theory and the universe to an audience at the institute of physics in london. Bibliography of edward witten 1 heavy quark contributions to deep inelastic scattering, nucl. Bibliography of edward witten school of natural sciences. Pdf supersymmetry and morse theory semantic scholar. In this paper, we will prove that floer homology equipped with either the intrinsic or exterior product is isomorphic to quantum homology as a ring. Appendix 4 of chapter 12 gives a brief resummation of witten s paper on morse theory that you cite. One such an important episode involved a 1982 paper on morse theory and supersymmetry by the physicist edward witten 30.
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