In physics, Kaluza–Klein theory is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.
Sir Michael Francis Atiyah 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 Fields Medal in 1966 and the Abel Prize in 2004.
The Bogomol'nyi–Prasad–Sommerfield bound is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infinity. This set of inequalities is very useful for solving soliton equations. Often, by insisting that the bound be satisfied, one can come up with a simpler set of partial differential equations to solve the Bogomolny equations. Solutions saturating the bound are called "BPS states" and play an important role in field theory and string theory.
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.
In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.
In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle.
In theoretical physics, Seiberg–Witten theory is an supersymmetric gauge theory with an exact low-energy effective action, of which the kinetic part coincides with the Kähler potential of the moduli space of vacua. Before taking the low-energy effective action, the theory is known as supersymmetric Yang–Mills theory, as the field content is a single vector supermultiplet, analogous to the field content of Yang–Mills theory being a single vector gauge field or connection.
In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construction of Instantons."
In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators.
In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten, using the Seiberg–Witten theory studied by Nathan Seiberg and Witten during their investigations of Seiberg–Witten gauge theory.
In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the theory of Lie algebras and integrable system theory. It also plays an important role in the geometric Langlands correspondence over the field of complex numbers through conformal field theory.
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, do not change under local transformations according to certain smooth families of operations. Formally, the Lagrangian is invariant under these transformations.
In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle. These equations are
In mathematics, the monopole moduli space is a space parametrizing monopoles. Atiyah and Hitchin studied the moduli space for 2 monopoles in detail and used it to describe the scattering of monopoles.
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. They have also found significant use in mathematics.
Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.
In mathematics and theoretical physics, and especially gauge theory, the deformed Hermitian Yang–Mills (dHYM) equation is a differential equation describing the equations of motion for a D-brane in the B-model of string theory. The equation was derived by Mariño-Minasian-Moore-Strominger in the case of Abelian gauge group, and by Leung–Yau–Zaslow using mirror symmetry from the corresponding equations of motion for D-branes in the A-model of string theory.
In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon.
In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, written down by Nigel Hitchin in 1987. Hitchin's equations are locally equivalent to the harmonic map equation for a surface into the symmetric space dual to the structure group. They also appear as a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to two dimensions, and solutions to Hitchin's equations give examples of Higgs bundles and of holomorphic connections. The existence of solutions to Hitchin's equations on a compact Riemann surface follows from the stability of the corresponding Higgs bundle or the corresponding holomorphic connection, and this is the simplest form of the Nonabelian Hodge correspondence.
