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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 ...
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.
Given a Lie subgroup , the / gauged WZW model (or coset model) is a nonlinear sigma model whose target space is the quotient / for the adjoint action of on . This gauged WZW model is a conformal field theory, whose symmetry algebra is a quotient of the two affine Lie algebras of the G {\displaystyle G} and H {\displaystyle H} WZW models, and ...
Supersymmetric theory of stochastic dynamics (STS) is a multidisciplinary approach to stochastic dynamics on the intersection of dynamical systems theory, statistical physics, stochastic differential equations (SDE), topological field theories, and the theory of pseudo-Hermitian operators. The theory can be viewed as a generalization of the ...
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.
The standard paradigm for incorporating supersymmetry into a realistic theory is to have the underlying dynamics of the theory be supersymmetric, but the ground state of the theory does not respect the symmetry and supersymmetry is broken spontaneously. The supersymmetry break can not be done permanently by the particles of the MSSM as they ...
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.
Julius Erich Wess (5 December 1934 – 8 August 2007) was an Austrian theoretical physicist noted as the co-inventor of the Wess–Zumino model and Wess–Zumino–Witten model in the field of supersymmetry and conformal field theory.