It has been a very popular idea among theoretical physicists, for a number of reasons, for several decades it was a hit back when i was a student, before physics was cool, and even well before. Martin department of physics, northern illinois university, dekalb il 60115 i provide a pedagogical introduction to supersymmetry. Perturbation theory is generalized beyond tree level to include radiative corrections loops. Morse theory the simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics. The renormalization procedure and renormalization group are thoroughly discussed, the asymptoticfreedomofnonabeliangaugetheoriesisderived,andapplicationsinquantum. Windey cern, geneva, switzerland we present a brief description of the use of supcrsymmetry in the theory of indices in geometry and physics. Instantons in wittens supersymmetry and morse theory.
What are the implications of supersymmetry being incorrect. If the lhc never sees such particles, then that shows that supersymmetry, if it is realized in na. On the work of edward witten university of california. If we include gravity, then the spin2 graviton with 2 helicity states, so nb 2 has a spin32 superpartner called the gravitino. Basic constituents of matter molecules are built up from the atom, which is the basic unit of any chemical element. On graphtheoretic identifications of adinkras, supersymmetry. The idea of morse theory is to extract information about the global topology of x from the critical points off,i. It implies that minkowski space is a stable ground state. In this last theorem of the initial chapter of our document, we combine the. If this theory is true, it would at least double the kinds of. Supersymmetry and the search for a unified theory of matter and force on free shipping on qualified orders. The level of discussion is aimed at readers who are familiar with the standard model and quantum.
One of the most powerful, and useful, of these theories is morse theory. I assume this question is referring to the fact that the lhc hasnt yet seen evidence of supersymmetric partners. Morse theory typically deals with more nontrivial examples, where the di erentiability of mand fplay a larger role. Since supersymmetry is a required component of superstring theory, any discovered supersymmetry would be. Threlfall, varlationsrechnung im grossen, published in the united states by chelsea, new york, 1951. In these lectures we describe an approach to differential topology and geometry rooted in supersymmetric quantum theory. We show how the basic concepts and notions of differential geometry emerge from concepts and notions of the quantum theory of nonrelativistic particles with spin, and how the classification of different types of differential geometry follows the classification of.
Supersymmetry is a scientific theory that says that when elementary particles such as photons, electrons, and quarks were formed at the beginning of the universe, matching kinds of theoretical superparticles were also created. Supersymmetry and noncommutative geometry springerlink. In this sequel to 6, we bring the proof of results announced in 7 and 8. Supersymmetry article about supersymmetry by the free. For a colorful history of morse theory and the events leading up to wittens paper, we highly recommend reading raoul botts recollections in 8. There are other possible combinations of particles with spins that can satisfy eq. This article tries to demonstrate that the fundamental structural and methodological differences.
It is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian. Although 6 might be a good introduction, the present paper is relatively self contained. Morse, the calculus of variations in the large, american mathematical society, new york, 1934. Help our scientists and scholars continue their fieldshaping work. Supersymmetry not only has played a most important role in the development of theoretical physics over the last three decades, but also has strongly in. As ordinary eld theories, supersymmetric theories describe particles and interactions between them. Michael dine errata some ancient wisdom on the subject of teachers and error. Smooth morse theory depends heavily on the notion of a critical point. Its not clear that string theory predicts supersymmetry but it is certainly a central ingredient and, symbiotically, supersymmetry has played a central role in string theory and its successes. Proving the morse homology theorem is highly nontrivial and goes beyond the scope of this thesis. Although there is no empirical evidence for string theory it is a very compelling framework to consider fundamental interactions.
The atom in turn is made from the proton, neutron, and electron. Supersymmetry of the morse oscillator sciencedirect. Susy manifests itself in the speci c particle spectrum a theory enjoys, and in. Supersymmetry is a conjectured symmetry of space and time and a unique one. 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. Here is a more detailed presentation of the contents. Quevedoa adepartment of applied mathematics and theoretical physics, centre for mathematical sciences, university of cambridge, wilberforce road, cambridge cb3 0wa, united kingdom. Supersymmetry and morse theory 665 the results of 2 have an analogue for complex manifolds, which will be discussed in a separate paper. As we will see in chapter 4, however, most smooth functions are morse. Thus in the hypothesis of the previous theorem, we could have said that fis a c. Friedan enrico fermi institute and department of physics, university of chicago, chicago, il 60637, usa and p.
The rst two are known as grand uni ed theories, or guts, because they are based on simple lie groups. Pdf supersymmetry and morse theory semantic scholar. If string theory is telling us something about nature, nature is supersymmetric at some energy scale, giving us good reason to regard supersymmetry as a genuine symmetry of nature. We combine these facts into the sequence of superfield. In this thesis, the profound connection between morse theory and supersymmetry will.
A course for mathematicians, volumes 1 and 2 edited by pierre deligne, pavel etingof, daniel s. If there are two teachers available for hire one who has studied extensively but is not meticulous about mistakes, and another who is meticulous but has not studied as much, we hire the one who. Introduction 25th anniversary edition twentyfive years ago, michel green, john schwarz, and edward witten wrote two volumes on string theory. Introduction to supersymmetry university of oxford. This result, also known as the morse homology theorem, is one of the main results in morse theory. Supersymmetry definition of supersymmetry by merriamwebster. Quantum field theory, and symmetries and particle physics. Supersymmetry definition is the correspondence between fermions and bosons of identical mass that is postulated to have existed during the opening moments of the big bang and that relates gravity to the other forces of nature.
Supersymmetry susy is a generaliza tion of the spacetime symmetries of quantum. Beyond the standard model on free shipping on qualified orders. A supersymmetric eld theory is a set of elds and a lagrangian which exhibit such a symmetry. Ask any credentialed nerd what the ultimate theory of physics is, and chances are theyll reply, string theory. The proof is based on a version of infinite dimensional morse theory which is due to floer. Existence of an infinite sequence of harmonic maps between spheres of certain dimensions was proven by bizon and chmaj. Georgi and glashows su5 theory, georgis theory based on the group spin10 physicists call this the so10 theory, and the patisalam model. According to the basic insights of marston morse, a typical differentiable function on a. For further information on morse theory, the following sources are extremely useful. Supersymmetry and morse theory school of natural sciences. The kinds of theorems we would like to prove in morse theory will typically only apply to morse functions. Field theory ii begins with the path integral formulation of quantum field theory.
Scientific realism in the age of string theory richard dawid string theory currently is the only viable candidate for a unified description of all known natural forces. The second part consists of applications of morse theory over the reals, while the last part describes the basics and some applications of complex morse theory, a. Physica 15d 1985 7174 northholland, amsterdam supersymmetry and index theorems d. The gravitino would be massless if supersymmetry were unbroken, and so it has nf 2 helicity states. This sequence shares many features of the bartnikmckinnon sequence of solutions to the einsteinyangmills equations as well as sequences of solutions that have arisen in other physical models. Division algebras and supersymmetry ii baez, john c. Morse theory for periodic solutions of hamiltonian systems. The book contains several problems, and passwordprotected solutions will be available to lecturers at. 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. Instanton solutions or tunneling paths in this theory would be extrema of this lagrangian, written with a euclidean. Morse theory arises from the recognition that the number of critical points of fis constrained by the topology of m. Morse complex, and that the corresponding homology is isomorphic to the singular homology. It is the morsetheoretic analogue of the hodge operator d o g where g is the fundamental solution for the laplacian the greens operator.
Finally we end with a brief look at supersymmetry in other dimensions, describing spinors and supersymmetry algebras in various dimensions, 5dimensional n1 and 2 theories, and 6dimensional n2,0 and 1,1 theories. Division algebras and supersymmetry iii huerta, john, advances in theoretical and mathematical physics, 2012. For instance, since mis compact, then fattains its maximum value, and hence, fhas at least one critical point. There is a very close relationship between the topology of a smooth manifold m and the. Morse theory is a method to determine the topology of a.
Wipf theoretischphysikalischesinstitut friedrichschilleruniversit at, max wien platz 1 07743 jena 4. By the osterwalderschrader theorem one can consider. We combine this with the index theorem of the previous. All of them, though, feature supersymmetric equations that treat matter and forces identically source. Published during a period of rapid progress in this subject, these volumes were highly influential for a generation of students and researchers. If experimental evidence confirms supersymmetry in the form of supersymmetric particles such as the neutralino that is often believed to be the lightest superpartner, some people believe this would be a major boost to superstring theory. An artistic impression depicts the formation of a galaxy cluster in the early universe, released on october, 2014 in a new. Type ii a string theory 201 type ii b theory 203 the massless spectrum of different sectors 203 summary 204 quiz 205 chapter 12 heterotic string theory 207 the action for so32 theory 208 quantization of so32 theory 209 the spectrum 214 compacti. Morse theory and floer homology p p p p p 1 1 f 2 p 1 2 3 f 2 4 fig. Supersymmetry is a complex mathematical framework based on. Bott, morse theory and its application to homotopy theory.
1450 563 322 386 1588 1580 849 521 1031 10 851 208 1153 239 576 474 1604 1138 1597 1109 705 1550 1160 153 1211 657 644 1376 307 984 1050 813 1130 362 301 119 1240 493