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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.

Four-manifolds (Topology) --- Seiberg-Witten invariants. --- Mathematical physics. --- Physical mathematics --- Physics --- Invariants --- 4-dimensional manifolds (Topology) --- 4-manifolds (Topology) --- Four dimensional manifolds (Topology) --- Manifolds, Four dimensional --- Low-dimensional topology --- Topological manifolds --- Mathematics --- 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.

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This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Algebres de Clifford --- Clifford [Algebra's van ] --- Clifford algebras --- Fysica [Mathematische ] --- Fysica [Wiskundige ] --- Mathematische fysica --- Physics -- Mathematics --- Physics [Mathematical ] --- Physique -- Mathématiques --- Physique -- Méthodes mathématiques --- Wiskundige fysica --- Clifford, Algèbres de --- Spin, Nuclear --- Geometric algebras --- Clifford algebras. --- Spin geometry. --- Clifford, Algèbres de --- Spin geometry --- 514.76 --- Algebras, Linear --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Geometry of differentiable manifolds and of their submanifolds --- Global differential geometry --- Geometry --- Mathematical physics --- Topology --- Nuclear spin --- -Mathematics --- Géométrie --- Physique mathématique --- Spin nucléaire --- Topologie --- Mathematics --- Mathématiques --- Algebraic theory. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Binary icosahedral group. --- Binary octahedral group. --- Bundle metric. --- C*-algebra. --- Calabi conjecture. --- Calabi–Yau manifold. --- Cartesian product. --- Classification theorem. --- Clifford algebra. --- Cobordism. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Complete metric space. --- Complex manifold. --- Complex vector bundle. --- Complexification (Lie group). --- Covering space. --- Diffeomorphism. --- Differential topology. --- Dimension (vector space). --- Dimension. --- Dirac operator. --- Disk (mathematics). --- Dolbeault cohomology. --- Einstein field equations. --- Elliptic operator. --- Equivariant K-theory. --- Exterior algebra. --- Fiber bundle. --- Fixed-point theorem. --- Fourier inversion theorem. --- Fundamental group. --- Gauge theory. --- Geometry. --- Hilbert scheme. --- Holonomy. --- Homotopy sphere. --- Homotopy. --- Hyperbolic manifold. --- Induced homomorphism. --- Intersection form (4-manifold). --- Isomorphism class. --- J-invariant. --- K-theory. --- Kähler manifold. --- Laplace operator. --- Lie algebra. --- Lorentz covariance. --- Lorentz group. --- Manifold. --- Mathematical induction. --- Metric connection. --- Minkowski space. --- Module (mathematics). --- N-sphere. --- Operator (physics). --- Orthonormal basis. --- Principal bundle. --- Projective space. --- Pseudo-Riemannian manifold. --- Pseudo-differential operator. --- Quadratic form. --- Quaternion. --- Quaternionic projective space. --- Ricci curvature. --- Riemann curvature tensor. --- Riemannian geometry. --- Riemannian manifold. --- Ring homomorphism. --- Scalar curvature. --- Scalar multiplication. --- Sign (mathematics). --- Space form. --- Sphere theorem. --- Spin representation. --- Spin structure. --- Spinor bundle. --- Spinor field. --- Spinor. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Tangent bundle. --- Tangent space. --- Tensor calculus. --- Tensor product. --- Theorem. --- Topology. --- Unit disk. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Nuclear spin - - Mathematics --- -Clifford algebras.