Brane

Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics.

[2] The corresponding object of arbitrary dimension p is called a p-brane, a term coined by M. J. Duff et al. in 1988.

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.

[5] 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.

[12] In another version of string theory called the topological A-model, the D-branes can again be viewed as submanifolds of a Calabi–Yau manifold.

[13] This means, among other things, that they have half the dimension of the space in which they sit, and they are length-, area-, or volume-minimizing.

[15] The derived category of coherent sheaves is constructed using tools from complex geometry, a branch of mathematics that describes geometric shapes in algebraic terms and solves geometric problems using algebraic equations.

[16] On the other hand, the Fukaya category is constructed using symplectic geometry, a branch of mathematics that arose from studies of classical physics.