Morse theory and wittens proof of the morse inequalities. Supersymmetry, part i theory revised september 2015 by howard e. Dissociation of the no molecule article pdf available in epl europhysics letters 614. Wood, banach algebras and bott periodicity, topology, 4 196566, pp. 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. Wipf theoretischphysikalischesinstitut friedrichschilleruniversit at, max wien platz 1 07743 jena 4. In addition to his contributions to physics, wittens work has significantly. This correspondence is actively discussedinliterature. In a theory where supersymmetry is realized, each oneparticle state has at least a superpartner. His paper 2 on supersymmetry and morse theory is obligatory reading for geometers interested in understanding modern quantum field theory. Instanton solutions or tunneling paths in this theory would be extrema of this lagrangian, written with a euclidean. 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. We address some aspects of four dimensional chiral n1 supersymmetric theories on which the scalar manifold is described by k\ahler geometry and can further be viewed as k\ahlerricci soliton generating a oneparameter family of.
The idea of morse theory is to extract information about the global topology of x from the critical points off,i. 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. Morse theory and floer homology p p p p p 1 1 f 2 p 1 2 3 f 2 4 fig. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. It is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian.
Supersymmetry, sometimes abbreviated susy, is a fairly new area of theoretical physics that promises to answer several outstanding questions relating to particle physics, string theory and cosmology. Supersymmetry and morse theory school of natural sciences. Supersymmetry is known to aleviate some of the uv divergences of quantum. Morse, the calculus of variations in the large, american mathematical society, new york, 1934. In particle physics, supersymmetry susy is a conjectured relationship between two basic classes of elementary particles. 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.
Supersymmetry susy is a generaliza tion of the spacetime symmetries of quantum. This issue is handled in more generality by the introduction of the boundary operator. Therefore, in a susy world, instead of single particle states, one has to deal with supermultiplets of particle states. 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. According to the basic insights of marston morse, a typical differentiable function on a. The supersymmetry algebra in a quantum eld theory, we have a hilbert space hwhich consists of. Morse theory is a method to determine the topology of a. Syz mirror symmetry from wittenmorse theory ma, ziming nikolas arxiv. 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. Some of the implications of modern ideas in mathematics for supersymmetric theories are discussed. A complex in morse theory computing intersection homology. It also contains a brilliant proof of the classic morse inequalities, relating critical points to homology.
As we will see in chapter 4, however, most smooth functions are morse. Supersymmetry protected topological phases of isostatic lattices and kagome antiferromagnets lawler, michael j. Morse theory has provided the inspiration for exciting developments. 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. Related questions instantons in wittens supersymmetry and morse theory. Its phenomenologically minded and is reasonably easy to read and dip in and out of.
Division algebras and supersymmetry ii baez, john c. Bott, on the periodicity theorem for complex vector bundles, acts, mathematica, vol. 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. Time evolution in the morse potential using supersymmetry. Witten deformation and morse theory jonieramaralantunes april25,2016 introduction in 1982 edward witten published the paper supersymmetry and morse the. According to string theory, all particles in the universe can be divided into two types. What does it mean for string theory that the lhc has not. Morse theory the simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics. If the universe has extra dimensions mtheory by edward witten predicts up.
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. If this theory is true, it would at least double the kinds of particles in the universe. String theory predicts that a type of connection, called supersymmetry, exists between these two particle types. String theory requires supersymmetry, but it makes no requirements about the masses of supersymmetric particles either. On the work of edward witten university of california. Under supersymmetry, a fermion must exist for every boson and a boson for every fermion. Division algebras and supersymmetry iii huerta, john, advances in theoretical and mathematical physics, 2012. Help our scientists and scholars continue their fieldshaping work. In a new blow for the futuristic supersymmetry theory of the universes basic anatomy, experts reported fresh evidence monday of subatomic activity consistent with. The classic example of a gauge theory is maxwells theory of electromagnetism.
Threlfall, varlationsrechnung im grossen, published in the united states by chelsea, new york, 1951. Supersymmetry is a complex mathematical framework based on the theory of group transformations. Yes, i know the headlines said the lhc would probe string theory, and the lhc would probe supersymmetry. The kinds of theorems we would like to prove in morse theory will typically only apply to morse functions.
Supersymmetry and morse theory 665 the results of 2 have an analogue for complex manifolds, which will be discussed in a separate paper. 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. Here is a more detailed presentation of the contents. It was discovered in 1971 by gelfand and likhtman, raymond, and neveu and schwartz. Thus in the hypothesis of the previous theorem, we could have said that fis a c. 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. One such an important episode involved a 1982 paper on morse theory and supersymmetry by the physicist edward witten 30. Supersymmetry generators have speci c commutation properties with other generators. In all generality, symmetries both internal and spacetime implement constraints on the structure of the theory, and on its quantum corrections. The superpotentials in the supersymmetric theory are found to be equivalent to morse functions. Morse theory and floer homology university of texas. Rparity and the lightest supersymmetric particle i. Supersymmetry, or susy for short, is an extension of the classical symmetries of.
Supersymmetry, in particle physics, a symmetry between fermions subatomic particles with halfinteger values of intrinsic angular momentum, or spin and bosons particles with integer values of spin. Coupled supersymmetric quantum mechanics and ladder structures beyond the harmonic oscillator williams, cameron l. Instantons in wittens supersymmetry and morse theory. For a colorful history of morse theory and the events leading up to wittens paper, we highly recommend reading raoul botts recollections in 8. For further information on morse theory, the following sources are extremely useful. In this thesis, the profound connection between morse theory and supersymmetry will. This is also the reason why the noshow of supersymmetry has no consequences for string theory. Inspire, the high energy physics information system. Hidden sectors and the structure of supersymmetrybreaking i. 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. Theory and applications, hepph0312378, chung et al. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of morse functions. In fact, by going to an even larger theory, string theory, the nonrenormalizability issue of quantum gravity is resolved. Pdf supersymmetry and morse theory semantic scholar.
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