Martin department of physics, northern illinois university, dekalb il 60115 i provide a pedagogical introduction to supersymmetry. Lectures on morse theory, old and new morse theory is. Introduction to morse theory let mand nbe smooth manifolds, and let f. This is the basic fact of morse theory, and we also see that this equality arises from cancellations between critical points achieved by the boundary operator.
A brief history of morse homology yanfeng chen abstract morse theory was originally due to marston morse 5. In particular, we find an example of a group that is of type fpz but is not finitely presented. Morse theory investigates the topology of a surface by looking at critical points of a function on that surface. 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. In section 2 we give an overview of conley index theory, focusing on gradient ows and the relation to morse theory. Morse theory and finiteness properties of groups springerlink. Morse homology also serves a simple model for floer homology, which is covered in the second part.
In this paper we construct an in nite dimensional extraordinary cohomology theory and a morse theory corresponding to it. Chapter iii introduces the theory discrete morse functions developed by robin forman 12 in the 1990s. We introduce super elds and construct supersymmetric actions for gauge and matter theories. On the work of edward witten university of california, berkeley. 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.
Stochastic calculus, using brownian paths which im plement. These theories have some special properties which make them useful in the study of criti. There is an analogue of the morse complex which counts gradient. 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. 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. On the instanton complex of holomorphic morse theory. The information of the e1terms already implies the holomorphic morse inequalities. Canonical brst quantization of the topological particle defined by a morse function h is described. As already mentioned, there is one other aspect to morse theory, namely. In this thesis, the profound connection between morse theory and supersymmetry will be explored. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of morse functions.
That question can then be answered using floer theory. An introduction to floer homology daniel ruberman floer homology is a beautiful theory introduced in 1985 by andreas floer 8. For a colorful history of morse theory and the events leading up to wittens paper, we highly recommend reading raoul botts recollections in 8. This theory extends many ideas from smooth morse theory to the context of cw complexes. Morse theory and the euler characteristic 3 the points x2xat which df xfails to have full rank are called critical points of f.
Supersymmetry and morse theory 665 the results of 2 have an analogue for complex manifolds, which will be discussed in a separate paper. The latter chapters are predominately focused on discussing discrete morse theory and its relationship with smooth morse theory. The level of discussion is aimed at readers who are familiar with the standard model and quantum. Morse homology for the yangmills gradient flow jan swoboda maxplanckinstitut fur mathematik, vivatsgasse 7, d53111 bonn, germany abstract we use the yangmills gradient. It implies that minkowski space is a stable ground state. Here is a more detailed presentation of the contents. Division algebras and supersymmetry ii baez, john c. Division algebras and supersymmetry iii huerta, john, advances in theoretical and mathematical physics, 2012.
The first part is a thorough introduction to morse theory, a fundamental tool of differential topology. It combined new ideas about morse theory, gauge theory, and cassons approach 1, 14 to homology 3spheres and the representations of their fundamental groups into lie groups such as su2 and so3. On the instanton complex of holomorphic morse theory 777 natural coboundary maps, are the couterparts in holomorphic morse theory of the instanton complex in ordinary morse theory. Aithchison, supersymmetry in particle physics, cambridge university press is super clear and basic. Morse theory is a beautiful and natural extension of the minimum principle for a continuous function on a compact space. It defines the morse complex and the morse homology, and develops some of their applications. Wess and bagger, supersymmetry and supergravity, princeton university pub. However, there have been important developments during the past few years which are not mentioned. Pdf supersymmetry and morse theory semantic scholar. A supersymmetric eld theory is a set of elds and a lagrangian which exhibit such a symmetry. An infinite dimensional morse theory with applications wojciech kryszewski and andrzej szulkin abstract. Bott, morse theory and its application to homotopy theory.
Susy manifests itself in the speci c particle spectrum a theory enjoys, and in the way particles interact between themselves. But unlike ordinary morse theory, the spectral sequences do not always degenerate. Help our scientists and scholars continue their fieldshaping work. As alexandrov spaces do not have as much structure as riemannian manifolds, our theory gives more detailed information on the nature of critical points and.
Novikov inequalities, in quite a similar way as in morse theory one obtains morse inequalities. Massey in honor of robert macpherson on his 60th birthday 1 introduction in 1974, mark goresky and robert macpherson began their development of intersection homology theory. We examine the finiteness properties of certain subgroups of right angled artin groups. Since supersymmetry is a required component of superstring theory, any discovered supersymmetry would be. Create an aipowered research feed to stay up to date with new papers like this posted to arxiv. In section 3 we describe the construction of the seibergwitten floer stable homotopy type for rational homology. Morse theory the simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics. This book gives a presentday account of marston morse s theory of the calculus of variations in the large. Morse theory and supersymmetry uvafnwi universiteit van.