Wall-crossing
Encyclopedia
In algebraic geometry
and string theory
, the phenomenon of wall-crossing describes the discontinuous change of a certain quantity, such as an integer geometric invariant, an index
or a space of BPS state
, across a codimension
one wall in a space of stability conditions, a so-called wall of marginal stability.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, the phenomenon of wall-crossing describes the discontinuous change of a certain quantity, such as an integer geometric invariant, an index
Index (mathematics)
The word index is used in variety of senses in mathematics.- General :* In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a...
or a space of BPS state
BPS state
In theoretical physics, BPS states are massive representations of an extended supersymmetry algebra with mass equal to the supersymmetry central charge Z. Quantum mechanically, if the supersymmetry does not get broken, the mass is exactly equal to the modulus of Z...
, across a codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
one wall in a space of stability conditions, a so-called wall of marginal stability.