弦理论
- What is a particle?
- Is a particle the same as a concept?
Kasper Olsen, Richard Szabo. Constructing D-branes from K-theory Discusses Bott periodicity.
Bott-periodicity and string theory
- Edward Witten. D-Branes And K-Theory. 1998 "one can see Bott periodicity in the brane spectrum of Type IIB, Type IIA, and Type I superstrings. (For Type II, one has unitary gauge groups in every even or every odd dimension, and for Type I, one flips from SO to Sp and back to SO in adding four to the brane dimension.)
- Brane A physical object that generalizes the notion of a point particle to higher dimensions. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge. A point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one.
- In addition to point particles and strings, it is possible to consider higher-dimensional branes. A p-dimensional brane is generally called "p-brane". A p-brane sweeps out a (p+1)-dimensional volume in spacetime called its worldvolume. Physicists often study fields analogous to the electromagnetic field, which live on the worldvolume of a brane.
- A string may be open (forming a segment with two endpoints) or closed (forming a closed loop). D-branes are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to the Dirichlet boundary condition (the fixed boundary condition) which the D-brane satisfies.
- One crucial point about D-branes is that the dynamics on the D-brane worldvolume is described by a gauge theory, a kind of highly symmetric physical theory which is also used to describe the behavior of elementary particles in the standard model of particle physics. This connection has led to important insights into gauge theory and quantum field theory. For example, it led to the discovery of the AdS/CFT correspondence, a theoretical tool that physicists use to translate difficult problems in gauge theory into more mathematically tractable problems in string theory.
- Categories where the objects are D-branes and the morphisms between two branes α and β are states of open strings stretched between α and β.
- In one version of string theory known as the topological B-model, the D-branes are complex submanifolds of certain six-dimensional shapes called Calabi–Yau manifolds, together with additional data that arise physically from having charges at the endpoints of strings.
- In the topological B-model version of string theory, the D-branes are complex submanifolds of six-dimensional shapes called Calabi–Yau manifolds, together with additional data that arise physically from having charges at the endpoints of strings. The derived category of coherent sheaves on the Calabi–Yau manifold has these branes as its objects. In the topological A-model version of string theory, the D-branes can again be viewed as submanifolds of a Calabi–Yau manifold, namely, special Lagrangian submanifolds. They have half the dimension of the space in which they sit, and they are length-, area-, or volume-minimizing. The Fukaya category has these branes as its objects and is constructed using symplectic geometry.
- The homological mirror symmetry conjecture of Maxim Kontsevich states that the derived category of coherent sheaves on one Calabi–Yau manifold is equivalent in a certain sense to the Fukaya category of a completely different Calabi–Yau manifold, thus linking complex geometry and symplectic gometry.
String theory