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Sergei Novikov generalized this construction to a homology theory associated to a closed one-form on a manifold. Morse homology is a special case for the one-form df. A special case of Novikov's theory is circle-valued Morse theory, which Michael Hutchings and Yi-Jen Lee have connected to Reidemeister torsion and Seiberg–Witten theory.
Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse ...
A third area mentioned in Atiyah's address is Witten's work relating supersymmetry and Morse theory, [29] a branch of mathematics that studies the topology of manifolds using the concept of a differentiable function.
Supersymmetry is an integral part of string theory, a possible theory of everything. There are two types of string theory, supersymmetric string theory or superstring theory, and non-supersymmetric string theory. By definition of superstring theory, supersymmetry is required in superstring theory at some level.
Because of this, the Witten index is independent of the temperature and gives the number of zero energy bosonic vacuum states minus the number of zero energy fermionic vacuum states. In particular, if supersymmetry is spontaneously broken then there are no zero energy ground states and so the Witten index is equal to zero.
A gauge theory is a field theory with gauge symmetry. Roughly, there are two types of symmetries, global and local. Roughly, there are two types of symmetries, global and local. A global symmetry is a symmetry applied uniformly (in some sense) to each point of a manifold .
In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics.
In supersymmetry, 4D = global supersymmetry is the theory of global supersymmetry in four dimensions with a single supercharge.It consists of an arbitrary number of chiral and vector supermultiplets whose possible interactions are strongly constrained by supersymmetry, with the theory primarily fixed by three functions: the Kähler potential, the superpotential, and the gauge kinetic matrix.