Brane

D-branes

In string theory, 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, 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.

Categorical description

Mathematically, branes can be described using the notion of a category. 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. One can likewise consider categories where the objects are D-branes and the morphisms between two branes ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are states of open strings stretched between ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.

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. Intuitively, one can think of a submanifold as a surface embedded inside of a Calabi–Yau manifold, although submanifolds can also exist in dimensions different from two. In mathematical language, the category having these branes as its objects is known as the derived category of coherent sheaves on the Calabi–Yau. 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. Roughly speaking, they are what mathematicians call special Lagrangian submanifolds. 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. The category having these branes as its objects is called the Fukaya category.

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. On the other hand, the Fukaya category is constructed using symplectic geometry, a branch of mathematics that arose from studies of classical physics. Symplectic geometry studies spaces equipped with a symplectic form, a mathematical tool that can be used to compute area in two-dimensional examples.

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. This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.

General and cited references

• Aspinwall, Paul; Bridgeland, Tom; Craw, Alastair; Douglas, Michael; Gross, Mark; Kapustin, Anton; Moore, Gregory; Segal, Graeme; Szendröi, Balázs; Wilson, P.M.H., eds. (2009). Dirichlet Branes and Mirror Symmetry. Clay Mathematics Monographs . Vol. 4. American Mathematical Society. ISBN 978-0-8218-3848-8.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. ISBN 978-0-387-98403-2.
• Moore, Gregory (2005). "What is ... a Brane?" (PDF). Notices of the AMS. 52: 214. Retrieved June 7, 2018.
• Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. ISBN 978-0-465-02023-2.
• Zaslow, Eric (2008). "Mirror Symmetry". In Gowers, Timothy (ed.). The Princeton Companion to Mathematics. ISBN 978-0-691-11880-2.