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In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole. A magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence.
A topological quantum field theory is a quantum field theory which computes topological invariants.
In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension and holonomy group contained in Sp(k). Hyperkähler manifolds are special classes of Kähler manifolds. They can be thought of as quaternionic analogues of Kähler manifolds. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten.
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 a theory that determines an exact low-energy effective action of a supersymmetric gauge theory—namely the metric of the moduli space of vacua.
Peter Benedict Kronheimer is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University.
In mathematical physics, 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."
Richard Samuel Ward FRS is a British mathematical physicist. He is a Professor of Theoretical Physics at the University of Durham.
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, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994), using the Seiberg–Witten theory studied by Nathan Seiberg and Witten during their investigations of Seiberg–Witten gauge theory.
In mathematics, a stable vector bundle is a vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder-Narasimhan filtration. Stable bundles were defined by David Mumford in Mumford (1963) and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others.
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 geometric Langlands correspondence over the field of complex numbers; related to conformal field theory. A genus zero analogue of the Hitchin system arises as a certain limit of the Knizhnik–Zamolodchikov equations. Almost all integrable systems of classical mechanics can be obtained as particular cases of the Hitchin system.
S (Sundararaman) Ramanan is an Indian mathematician who works in the area of algebraic geometry, moduli spaces and Lie groups. He is one of India's leading mathematicians and recognised as an expert in algebraic geometry, especially in the area of moduli problems. He has also worked in differential geometry: his joint paper with MS Narasimhan on universal connections has been influential. It enabled SS Chern and B Simons to introduce what is known as the Chern-Simons invariant, which has proved useful in theoretical physics..
In mathematics, the Atiyah–Jones conjecture is a conjecture about the homology of the moduli space of instantons over a sphere. It was introduced by Michael Francis Atiyah and John D. S. Jones (1978) and proved by Charles P. Boyer, Jacques C. Hurtubise, and Benjamin M. Mann et al..
In differential geometry, the Atiyah–Hitchin–Singer theorem, introduced by Michael Atiyah, Nigel Hitchin, and Isadore Singer, states that the space of SU(2) anti self dual Yang–Mills fields on a 4-sphere with index k > 0 has dimension 8k – 3.
Jacques-Claude HurtubiseFRSC is a Canadian mathematician who works as a professor of mathematics and chair of the mathematics department at McGill University. His research interests include moduli spaces, integrable systems, and Riemann surfaces. Among other contributions, he is known for proving the Atiyah–Jones conjecture.
References
- Atiyah, Michael; Hitchin, Nigel (1988), The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, ISBN 978-0-691-08480-0, MR 0934202
Sir Michael Francis Atiyah was a British-Lebanese mathematician specialising in geometry.
Nigel James Hitchin FRS is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics. He is a professor emeritus of mathematics at Oxford University.
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.