In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and especially
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, the Bogomolny equation for
magnetic monopole
In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". ...
s is the equation
:
where
is the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
of a
connection
Connection may refer to:
Mathematics
*Connection (algebraic framework)
*Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold
* Connection (affine bundle)
*Connection (composite bun ...
on a
principal -bundle over a
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
,
is a section of the corresponding
adjoint bundle In mathematics, an adjoint bundle is a vector bundle naturally associated with any smooth principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bun ...
,
is the
exterior covariant derivative
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of ...
induced by
on the adjoint bundle, and
is the
Hodge star operator
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a Dimension (vector space), finite-dimensional orientation (vector space), oriented vector space endowed with a Degenerate bilinear form, nonde ...
on
. These equations are named after E. B. Bogomolny and were studied extensively by
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...
and
Nigel Hitchin
Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of O ...
.
The equations are a dimensional reduction of the
self-dual Yang–Mills equations
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dua ...
from four dimensions to three dimensions, and correspond to global minima of the appropriate action. If
is closed, there are only trivial (i.e. flat) solutions.
See also
*
Monopole moduli space
*
Ginzburg–Landau theory
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomen ...
*
Seiberg–Witten theory
In theoretical physics, Seiberg–Witten theory is an \mathcal = 2 supersymmetric gauge theory with an exact low-energy effective action (for massless degrees of freedom), of which the kinetic part coincides with the Kähler potential of the ...
*
Bogomol'nyi–Prasad–Sommerfield bound
References
External links
*
Bogomolny equation on
nLab
The ''n''Lab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory. The ''n''Lab ...
*
Differential geometry
Magnetic monopoles
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