nLab
brane

Idea

The term brane referes in formal high energy physics and in particular in string theory to entities that one thinks of as physical objects that generalize the notion point particles to higher dimensional objects.

The term derives from the word membrane that was originally used to describe 2-dimensional “particles”. When the need was felt to speak also about 3-, 4- and higher dimensional such “particles” the usage “3-brane”, “4-brane” etc. was introduced. Ordinary particles would be 0-branes in this counting, the strings in string theory would be 1-branes and membranes themselves 2-branes.

There are however two fundamentally different concrete realizations of this somewhat vague notion.

• fundamental or $\sigma$-model branes

  A sigma-model quantum field theory with parameter space a $k+1$-dimensional manifold $\Sigma$ and target space $X$ may be thought of as describing the quantum mechanics of a $k$-brane object in $X$.

  In this context one speaks of $\Sigma$ as a fundamental $k$-brane propagating in $X$.

  • For $k=0$ this describes the ordinary quantum mechanics of a point particle on $X$. And such point particles are the fundamental particles for instance of the standard model of particle physics.

  • For $k=1$ this describes the quantum propagation of a string, and accordingly one speaks of the fundamental string of F1-brane (fundamental 1-brane).

  • For $k=2$ this describes the quantum propagation of a membrane.

  • There are good indications that there is a way to describe heterotic string theory not in terms of fundamental 1-branes but in terms of the sigma-model of a fundamental 5-brane – the magnetic dual of the 1-brane in 10-dimensions.

  • etc.

• boundary conditions or D-branes

  A related but a priori very different notion of brane is that properly called D-brane for Dirichlet-brane. The term arises from the characterization of boundary conditions for 1-brane sigma-models. Today one identifies more generally

  • a D-brane is by definition a boundary condition for a 1+1-dimensional sigma-model

  For instance in the description of rational conformal field theory in terms of Frobenius algebra objects $A$ inside a modular tensor category $C$, the category ${\mathrm{Mod}}_C(A)$ of $A$-module objects internal to $C$ is the category of D-branes for the CFT encoded by $A$.

  At least for 2-dimensional TQFTs analogous considerations yield not just categories but stable (infinity,1)-categories of boundary condition objects. For instance for what is called the B-model 2-d TQFT the category of D-branes is the derived category of coherent sheaves on some Calabi-Yau space.

  In simple special cases, however, bouandray conditions for sigma-models with target $X$ are encoded in terms of $k$-dimensional subspaces $QhoookrightarrowX$. It is those subspaces $Q$ that were originally identified as $k$-branes as extended objects. More general sigma-model boundary conditions that are not as geometrically encoded are therefore sometimes called non-geometic D-branes.

Starting with Kontsevich’s homological algebra reformulation of mirror symmetry the study of (derived) D-brane categories has become a field in its own right in pure mathematics.

… lots of further things to say …

References

… lots of references going here …

A classical text describing how the physics way to think of D-branes leads to seeing that they are objects in derived categories is

• Paul Aspinwall9, D-Branes on Calabi-Yau Manifolds (arXiv)

This can to a large extent be read as a dictionary from homological algebra terminology to that of D-brane physics.

More recent similar material with the emphasis on the K-theory aspects is

• Richard Szabo?, D-Branes and Bivariant K-Theory