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The category having these branes as its objects is called the Fukaya category. [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]
When two D-branes approach each other the interaction is captured by the one loop annulus amplitude of strings between the two branes. The scenario of two parallel branes approaching each other at a constant velocity can be mapped to the problem of two stationary branes that are rotated relative to each other by some angle.
[6] [7] Gogberashvili also showed that the four-dimensionality of the Universe is the result of the stability requirement found in mathematics since the extra component of the Einstein field equations giving the confined solution for matter fields coincides with one of the conditions of stability. [8]
It is also possible to consider higher-dimensional branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane. [46] 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 ...
The M2-brane solution can be found [1] by requiring () symmetry of the solution and solving the supergravity equations of motion with the p-brane ansatz. The solution is given by a metric and three-form gauge field which, in isotropic coordinates, can be written as
For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p, these are called p-branes. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics.
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Rigor is a cornerstone quality of mathematics, and can play an important role in preventing mathematics from degenerating into fallacies. well-behaved An object is well-behaved (in contrast with being Pathological ) if it satisfies certain prevailing regularity properties, or if it conforms to mathematical intuition (even though intuition can ...