In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
, the Bogomolny equation for
magnetic monopole
In particle physics, a magnetic monopole is a hypothetical elementary 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 "magneti ...
s is the equation
:
where
is the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of a
connection on a
principal -bundle over a
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
,
is a section of the corresponding
adjoint bundle In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles ha ...
,
is the
exterior covariant derivative
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.
Definition
Let ''G' ...
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 finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
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 ...
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 phenomenol ...
*
Seiberg–Witten theory
In theoretical physics, Seiberg–Witten theory is a theory that determines an exact low-energy effective action (for massless degrees of freedom) of a \mathcal = 2 supersymmetric gauge theory—namely the metric of the moduli space of vacua.
S ...
*
Bogomol'nyi–Prasad–Sommerfield bound
The Bogomol'nyi–Prasad–Sommerfield bound (named after Evgeny Bogomolny, M.K. Prasad, and Charles Sommerfield) is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infi ...
References
*
Differential geometry
Magnetic monopoles
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