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The Seiberg-Witten equations and applications to the topology of smooth four-manifolds / / John W. Morgan



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Autore: Morgan John W. <1946-> Visualizza persona
Titolo: The Seiberg-Witten equations and applications to the topology of smooth four-manifolds / / John W. Morgan Visualizza cluster
Pubblicazione: Princeton, New Jersey : , : Princeton University Press, , 1996
©1996
Descrizione fisica: 1 online resource (138 p.)
Disciplina: 514/.2
Soggetto topico: Four-manifolds (Topology)
Seiberg-Witten invariants
Mathematical physics
Soggetto non controllato: Affine space
Affine transformation
Algebra bundle
Algebraic surface
Almost complex manifold
Automorphism
Banach space
Clifford algebra
Cohomology
Cokernel
Complex dimension
Complex manifold
Complex plane
Complex projective space
Complex vector bundle
Complexification (Lie group)
Computation
Configuration space
Conjugate transpose
Covariant derivative
Curvature form
Curvature
Differentiable manifold
Differential topology
Dimension (vector space)
Dirac equation
Dirac operator
Division algebra
Donaldson theory
Duality (mathematics)
Eigenvalues and eigenvectors
Elliptic operator
Elliptic surface
Equation
Fiber bundle
Frenet–Serret formulas
Gauge fixing
Gauge theory
Gaussian curvature
Geometry
Group homomorphism
Hilbert space
Hodge index theorem
Homology (mathematics)
Homotopy
Identity (mathematics)
Implicit function theorem
Intersection form (4-manifold)
Inverse function theorem
Isomorphism class
K3 surface
Kähler manifold
Levi-Civita connection
Lie algebra
Line bundle
Linear map
Linear space (geometry)
Linearization
Manifold
Mathematical induction
Moduli space
Multiplication theorem
Neighbourhood (mathematics)
One-form
Open set
Orientability
Orthonormal basis
Parameter space
Parametric equation
Parity (mathematics)
Partial derivative
Principal bundle
Projection (linear algebra)
Pullback (category theory)
Quadratic form
Quaternion algebra
Quotient space (topology)
Riemann surface
Riemannian manifold
Sard's theorem
Sign (mathematics)
Sobolev space
Spin group
Spin representation
Spin structure
Spinor field
Subgroup
Submanifold
Surjective function
Symplectic geometry
Symplectic manifold
Tangent bundle
Tangent space
Tensor product
Theorem
Three-dimensional space (mathematics)
Trace (linear algebra)
Transversality (mathematics)
Two-form
Zariski tangent space
Note generali: Description based upon print version of record.
Nota di bibliografia: Includes bibliographical references.
Nota di contenuto: Front matter -- Contents -- 1. Introduction -- 2. Clifford Algebras and Spin Groups -- 3. Spin Bundles and the Dirac Operator -- 4. The Seiberg-Witten Moduli Space -- 5. Curvature Identities and Bounds -- 6. The Seiberg-Witten Invariant -- 7. Invariants of Kahler Surfaces -- Bibliography
Sommario/riassunto: The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.
Titolo autorizzato: Seiberg-Witten equations and applications to the topology of Smooth four-manifolds  Visualizza cluster
ISBN: 1-4008-6516-6
Formato: Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione: Inglese
Record Nr.: 9910828604403321
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Serie: Mathematical notes (Princeton University Press) ; ; 44.