Brane
String theory |
---|
Fundamental objects |
Perturbative theory |
Non-perturbative results |
Phenomenology |
Mathematics |
In string theory and related theories (such as supergravity theories), a brane is a physical object that generalizes the notion of a zero-dimensional point particle, a one-dimensional string, or a two-dimensional membrane to higher-dimensional objects. 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.
The word "brane" originated in 1987 as a contraction of "membrane".[1]
p-branes
A point particle is a 0-brane, of dimension zero; a string, named after vibrating musical strings, is a 1-brane; a membrane, named after vibrating membranes such as drumheads, is a 2-brane.[2] The corresponding object of arbitrary dimension p is called a p-brane, a term coined by M. J. Duff et al. in 1988.[3]
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.[4]
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.[5]
One crucial point about D-branes is that the dynamics on the D-brane worldvolume is described by a
Categorical description
Mathematically, branes can be described using the notion of a
One can likewise consider categories where the objects are D-branes and the morphisms between two branes and areIn one version of string theory known as the
The derived category of coherent sheaves is constructed using tools from
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.[18] This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.[19]
See also
- Black brane
- Brane cosmology
- Dirac membrane
- Lagrangian submanifold
- M2-brane
- M5-brane
- NS5-brane
Citations
- ^ "brane". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
- ^ Moore 2005, p. 214
- ^ M. J. Duff, T. Inami, C. N. Pope, E. Sezgin , and K. S. Stelle, "Semiclassical quantization of the supermembrane", Nucl. Phys. B297 (1988), 515.
- ^ Moore 2005, p. 214
- ^ Moore 2005, p. 215
- ^ Moore 2005, p. 215
- ^ Aspinwall et al. 2009
- ^ A basic reference on category theory is Mac Lane 1998.
- ^ Zaslow 2008, p. 536
- ^ Zaslow 2008, p. 536
- ^ Yau and Nadis 2010, p. 165
- ^ Aspinwal et al. 2009, p. 575
- ^ Aspinwal et al. 2009, p. 575
- ^ Yau and Nadis 2010, p. 175
- ^ Aspinwal et al. 2009, p. 575
- ^ Yau and Nadis 2010, pp. 180–1
- ^ Zaslow 2008, p. 531
- ^ Aspinwall et al. 2009, p. 616
- ^ Yau and Nadis 2010, p. 181
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. 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. ISBN 978-0-465-02023-2.
- Zaslow, Eric (2008). "Mirror Symmetry". In Gowers, Timothy (ed.). ISBN 978-0-691-11880-2.