Branes

[share]

Mathematically,
branes can be described using the notion of a category.[6]
This is a mathematical structure consisting of objects, and for any pair of objects,
a set of morphisms between them.

In most examples, the objects are mathematical structures
(such as sets, vector spaces, or topological spaces) and
the morphisms are functions between these structures.[7]

One can also consider categories where the objects are D-branes and
the morphisms between two branes alpha and beta are states of open strings
stretched between alpha and beta.[8]

Wiki

[share]

In string theory and related theories such as supergravity theories,
a brane is 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.

Mathematically, branes can be represented within categories,
and are studied in pure mathematics for insight into homological mirror symmetry
and noncommutative geometry.

[share]

In string theory,
D-branes, short for Dirichlet membrane, are a class of extended objects upon which
open strings can end with Dirichlet boundary conditions, after which they are named.

D-branes were discovered by Dai, Leigh and Polchinski,[1] and
independently by Hořava,[2] in 1989.

In 1995, Polchinski identified D-branes with black p-brane solutions of supergravity,
a discovery that triggered the Second Superstring Revolution and
led to both holographic and M-theory dualities.

[share]

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

Mathematically, branes can be represented within categories, and
are studied in pure mathematics for insight into homological mirror symmetry and noncommutative geometry.

[share]

p-branes

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".

The term "p-brane" was coined by M. J. Duff et al. in 1988;[1]
"brane" comes from the word "membrane" which refers to a two-dimensional brane.[2]

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.[3]

[share]

Mathematical Aspects of D-branes

In these notes we review recent work on describing D-branes
with nonzero Higgs vevs in terms of sheaves,
which gives a physical on-shell D-brane interpretation to more sheaves than
previously understood as describing D-branes.

The mathematical ansatz for this encoding is checked by
comparing open string spectra between D-branes with nonzero Higgs vevs
to Ext groups between the corresponding sheaves.

https://arxiv.org/abs/hep-th/0402023