Let us also mention that if one follows the analogy of the deformation in b05 with the witten laplacian, then c 1t, so that t b. Pdf upper bounds on the witten index for supersymmetric. It implies that minkowski space is a stable ground state. I am trying to read the article supersymmetry and morse theory by e. Witten, \supersymmetry and morse theory, journal of di erential geometry 17, 661 1982. Witten writes down the relevant supersymmetric lagrangian and then states. Bibliography of edward witten 1 heavy quark contributions to deep inelastic scattering, nucl. Through hodge theory generalized to elliptic operators, witten proves that the dimension of. Morse theory is a method to determine the topology of a nite or in nite dimensional mani fold such as the space of paths or loops on a compact manifold from the critical points of. 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.
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. Kirwan, \ morse functions for which the stationary phase approximation is exact, topology 26, 37 1987. M\to\mathbbr and their associated gradient flows classical morse theory centered around simple statements like morse inequalities, concerning just the betti numbers. Witten has been honored with numerous awards including a macarthur grant 1982, the fields medal 1990, the nemmers prize in mathematics 2000. Related questions instantons in wittens supersymmetry and morse theory. A brief history of morse homology yanfeng chen abstract morse theory was originally due to marston morse 5. Edward witten american mathematical physicist britannica. Upper bounds on the witten index for supersymmetric lattice models by discrete morse theory article pdf available in european journal of combinatorics 302. Edward witten, the index of the dirac operator in loop space. In the last years, these functions have mostly been used only implicitly, and instead one constructs acyclic matchings on hasse diagrams of face posets. W1, witten developed the whole theory with the help of some perturbated generalized laplacian operators acting on the exterior bundle of di. Where does the supersymmetry in wittens proof of the morse. Instanton solutions or tunneling paths in this theory would be extrema.
Division algebras and supersymmetry ii baez, john c. We study representations of the super poincar e algebra in d 4. For superstring theory to be consistent, supersymmetry seems to be required at some level although it may be a strongly broken symmetry. In chapter 4, we step aside from milnors account of morse theory and take a look at edward wittens article supersymmetry and morse theory, published in 1982. It is shown that the morse inequalities can be obtained by a consideration of f certain supersymmetric quantum mechanics hamiltonian. Witten, \ supersymmetry and morse theory, journal of di erential geometry 17, 661 1982. Syz mirror symmetry from wittenmorse theory ma, ziming nikolas arxiv. Thanks for contributing an answer to physics stack exchange. Based on the same idea, morse homology was introduced by thom, smale, milnor, and witten. Morse theory and supersymmetry uvafnwi universiteit van. Lectures on chernweil theory and witten deformations.
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. On the work of edward witten university of california, berkeley. 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. Edward witten, the index of the dirac operator in loop. 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. String journal club an introduction to morse theory. Another reference is ed wittens supersymmetry and morse theory this classical aspects of this story are really dodgy because we have fermions, so we need odd. In this thesis, the profound connection between morse theory and supersymmetry will be explored along the lines of wittens 1982 paper. Wittens approach makes essential use of two ideas borrowed from quantum physics and hodge theory. Morse theory the simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics.
Here is a more detailed presentation of the contents. Using the results of the present article and combining them with the analytic result of 30, one can show that for an arbitrary antiradial morse function, the witten complex in. He made significant contributions to morse theory, supersymmetry, and knot theory. It is well known that the traditional approach to morse theory via an analysis of the sets ma gives meaningful results only if the. Schrodinger operators with applications to morse theory pdf. Physics stack exchange is a question and answer site for active researchers, academics and students of physics.
It is shown that the morse inequalities can be obtained by consideration of a certain supersymmet ric quantum mechanics. It is useful not only for studying manifolds, but also for studying infinite cwtype spaces. Witten hel 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. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of morse functions. Pdf supersymmetry and morse theory semantic scholar. Syz mirror symmetry from witten morse theory ma, ziming nikolas arxiv. Morse theory has already been applied in phml, so there is a good chance that phml can use the supersymmetric approach to morse theory. Related questions instantons in witten s supersymmetry and morse 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. According to the basic insights of marston morse, a typical differentiable function on a.
Witten was awarded the fields medal at the international congress of mathematicians in kyoto, japan, in 1990. Upper bounds on the witten index for supersymmetric. It is shown that the morse inequalities can be obtained by consideration of a certain. The main new result of the present paper is a path integral formula for the cal. Edward witten it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian. On the work of edward witten university of california. We begin with supersymmetric quantum mechanics, covering the witten index, supersymmetric localization, sqm on a. This well known article applies some tools developed by physicists e. We introduce super elds and construct supersymmetric actions for gauge and matter theories.
Pdf 2019 applications of supermathematics to machine. In another context, a morse function can also mean an anharmonic oscillator see morse potential in mathematics, specifically in differential topology, morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. The supersymmetry algebra in a quantum eld theory, we have a hilbert space hwhich consists of. Morse theory and wittens proof of the morse inequalities. Question on wittens paper supersymmetry and morse theory. Upper bounds on the witten index for supersymmetric lattice. A complex in morse theory computing intersection homology. For x,y e c, define the space m g x,y of trajectories of the gradient flow connecting x and y. 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. This paper presented a new proof of the morse inequalities by considering certain supersymmetric quantum mechanics hamiltonian. Supersymmetry is part of superstring theory, a string theory of quantum gravity, although it could in theory be a component of other quantum gravity theories as well, such as loop quantum gravity. Using the results of the present article and combining them with the analytic result of 30, one can show that for an arbitrary antiradial morse function, the witten complex in 30 convergestothedualofcu.
Some of the implications of modern ideas in mathematics for supersymmetric theories are discussed. 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. Transactions of the american mathematical society volume 365, number 2, february 20, pages 885909 s 000299472012056510 article electronically published on july 2, 2012 th. Edward witten born august 26, 1951 is an american theoretical physicist and the charles simonyi professor in the school of natural sciences at the institute for advanced study. Let us also mention that if one follows the analogy of the deformation in b05 with the witten laplacian, then c. Supersymmetry and morse theory 665 the results of 2 have an analogue for complex manifolds, which will be discussed in a separate paper. The topological origin of this model gives a natural 1. For a colorful history of morse theory and the events leading up to wittens paper, we highly recommend reading raoul botts recollections in 8. Welcome to inspire, the high energy physics information system.
Inspire, the high energy physics information system. Morse homology can also be formulated for morsebott functions. In fact using arguments similar to the ones he presents when proving the morse inequalities, witten was also able to derive the lefschetz fixed point theorem as well as provide lower bounds on the betti numbers. His early research interests were in electromagnetism, but he soon developed an interest in what is now known as superstring theory in mathematical physics. The positive mass conjecture in general relativity the positive mass conjecture asserts that under appropriate hypotheses the total energy of a gravitating system is positive and can only be zero for flat minkowski space. It was invented by forman, who used the concept of discrete morse functions. Some of the implications of modem ideas in mathematics for supersymmetric theories are disscused. One such an important episode involved a 1982 paper on morse theory and supersymmetry by the physicist edward witten 30. Frederic bourgeois sketched an approach in the course of his work on a morsebott version of symplectic field theory, but this work was never published due to substantial analytic difficulties.
A supersymmetry theory is by definition a theory in which there are hermitian symmetry operators q. Introduction 25th anniversary edition twentyfive years ago, michel green, john schwarz, and edward witten wrote two volumes on string theory. Kirwan, \morse functions for which the stationary phase approximation is exact, topology 26, 37 1987. The supersymmetry algebra in a quantum eld theory, we have a hilbert space hwhich consists of things called states, jai. Instantons in wittens supersymmetry and morse theory. 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. Our goal will be to establish the weak form of the morse inequalities, that is m p b p where b p is the pth betti number and m p is the number of critical points of a morse function hwith index p.
Morse theory is the method of studying the topology of a smooth manifold m m by the study of morse functions m. Gromovwitten invariants in genus zero, as is predicted by mirror symmetry. Edward witten usc dana and david dornsife college of. Help our scientists and scholars continue their fieldshaping work. In addition to his contributions to physics, wittens work has significantly. Where does the supersymmetry in wittens proof of the. In his 1982 paper w1, witten developed the whole theory with the help of some pertur. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. In particle physics, supersymmetry susy is a conjectured relationship between two basic classes of elementary particles. Published during a period of rapid progress in this subject, these volumes were highly influential for a generation of students and researchers. Supersymmetry and morse theory school of natural sciences.
Supersymmetry l16 david skinner this course provides an introduction to the role of supersymmetry in quantum eld theory, with the emphasis on mathematics rather than phenomenology. Discrete morse theory is a method for reducing the number of cells of a cwcomplex, without changing its homotopy type. Coupled supersymmetric quantum mechanics and ladder structures beyond the harmonic oscillator williams, cameron l. Witten deformation and morse theory jonieramaralantunes april25,2016 introduction in 1982 edward witten published the paper supersymmetry and morse the.
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