Kishore Marathe

The geometry of gauge fields

Abstract The aim of this work is to give a self-contained development of a differential geometric formulation of gauge theories and their interactions with the theories of fundamental particles and in particular, of the theory of Yang-Mills and Yang-Mills-Higgs fields. We discuss in detail principal and associated bundles and develop the theory of connections in a principal fiber bundle and the theory of characteristic classes. These are applied to give a general formulation of gauge theories. The special cases of the theory of Yang-Mills fields and the theory of instantons and their moduli spaces are discussed separately.

Spaces admitting gravitational fields

We start with a modern version of Einsteins definition of a gravitational field. Tensors of curvature type and the curvature product of symmetric tensors are defined. The interaction tensor is defined as the curvature product of the fundamental tensor and the energy‐momentum tensor. The tensor W obtained by coupling the Riemann tensor and the interaction tensor is used to obtain a characterization of gravitational fields. The linear transformation of the space of second‐order differential forms, induced by W, is used to give a new definition of a gravitational field. The field equations are expressed in terms of the gravitational sectional curvature function f. Thorpes theorem characterizing Einstein spaces is obtained as a corollary. New formulations of the field equations are used to solve the problem of classification of gravitational fields. The mathematical foundations of the theory of classification are examined and a geometric interpretation of classification is obtained by using the critical poi...

Topics in physical mathematics

Algebra.- Topology.- Manifolds.- Bundles and Connections.- Characteristic Classes.- Theory of Fields, I: Classical.- Theory of Fields, II: Quantum and Topological.- Yang-Mills-Higgs Fields.- 4-Manifold Invariants.- 3-Manifold Invariants.- Knot and Link Invariants.

The mean curvature of gravitational fields

The mean curvature of a gravitational field is defined as a generalization of the average curvature in a given direction by using the gravitational sectional curvature function on non-degenerate tangent 2-planes to the space-time manifold. We find that the mean curvature of a gravitational field is independent of direction as determined by a unit vector. The converse of this result provides a new characterization of spaces admitting gravitational fields. We also define the average curvature (or bending) of a non-degenerate k-plane in a gravitational field and show that it is independent of the choice of non-degenerate k-plane.

Geometric Topology and Field Theory on 3-Manifolds

In recent years the interaction between geometric topology and classical and quantum field theories has attracted a great deal of attention from both the mathematicians and physicists. This interaction has been especially fruitful in low dimensional topology. In this article We discuss some topics from the geometric topology of 3-manifolds with or without links where this has led to new viewpoints as well as new results. They include in addition to the early work of Witten, Casson, Bott, Taubes and others, the categorification of knot polynomials by Khovanov. Rozansky, Bar-Natan and Garofouladis and a special case of the gauge theory to string theory correspondence in the Euclidean version of the theories, where exact results are available. We show how the Witten-Reshetikhin-Turaev invariant in SU(n) Chern-Simons theory on S 3 is related via conifold transition to the all-genus generating function of the topological string amplitudes on a Calabi-Yau manifold. This result can be thought of as an interpretation of TQFT as TQG (Topological Quantum Gravity). A brief discussion of Perelman’s work on the geometrization conjecture and its relation to gravity is also included.

Quantization on V -manifolds

SummaryV-manifolds are spaces which generalize the notion of differential manifold with a certain type of singularities. They arise naturally as orbit spaces of Hamiltonian group actions on symplectic manifolds, when the action is not free. The quantization of the nonisotropic harmonic oscillator, whose orbit space is aV-manifold, is discussed by using Maslov’s quantization condition.RiassuntoLeV-varietà sono spazi che generalizzano la struttura di varietà differenziabile con particolari tipi di singolarità. Ne sono un esempio gli spazi delle orbite di gruppi di azione hamiltoniana su varietà simplettiche nel caso in cui l’azione non è libera. Si studia, usando la condizione di quantizzazione di Maslov, la quantizzazione dell’oscillatore armonico non isotropo, il cui spazio delle orbite è unaV-varietà.РезюмеV-множества представляют пространства, которые обобщают понятие дифференциального множества с некоторым типом сингулярностей. Эти множества возникают естественным образом, как пространства орбит для групп гамильтоновых действий на симплектических множествах, когда действия не являются свободными. Используя условие кватования Маслова, обсуждается квантование неизотропного гармонического осциллятора, пространство орбит которого представляетV-множество.

3-Manifold Invariants

In Chapter 9 we discussed the geometry and topology of moduli spaces of gauge fields on a manifold. In recent years these moduli spaces have been extensively studied for manifolds of dimensions 2, 3, and 4 (collectively referred to as low-dimensional manifolds). This study was initiated for the 2-dimensional case in [17]. Even in this classical case, the gauge theory perspective provided fresh insights as well as new results and links with physical theories. We make only a passing reference to this case in the context of Chern–Simons theory. In this chapter, we mainly study various instanton invariants of 3-manifolds. The material of this chapter is based in part on [263]. The basic ideas come from Witten’s work on supersymmetric Morse theory. We discuss this work in Section 10.2. In Section 10.3 we consider gauge fields on a 3-dimensional manifold. The field equations are obtained from the Chern–Simons action functional and correspond to flat connections. Casson invariant is discussed in Section 10.4. In Section 10.5 we discuss the Z 8-graded instanton homology theory due to Floer and its relation to the Casson invariant. Floer’s theory was extended to arbitrary closed oriented 3-manifolds by Fukaya. When the first homology of such a manifold is torsion-free, but not necessarily zero, Fukaya also defines a class of invariants indexed by the integer s, 0 ≤ s < 3, where s is the rank of the first integral homology group of the manifold. These invariants include, in particular, the Floer homology groups in the case s = 0. The construction of these invariants is closely related to that of Donaldson polynomials of 4-manifolds, which we considered in Chapter 9. As with the definition of Donaldson polynomials a careful analysis of the singular locus (the set of reducible connections) is required in defining the Fukaya invariants. Section 10.6 is a brief introduction to an extension of Floer homology to a Z-graded homology theory, due to Fintushel and Stern, for homology 3-spheres. Floer also defined a homology theory for symplectic manifolds using Lagrangian submanifolds and used it in his proof of the Arnold conjecture. We do not discuss this theory. For general information on various Morse homologies, see, for example, [29]. The WRT invariants, which arise as a byproduct of Witten’s TQFT interpretation of the Jones polynomial are discussed in Section 10.7. Section 10.8 is devoted to a special case of the question of relating gauge theory and string theory where exact results are available. Geometric transition that is used to interpolate between these theories is also considered here. Some of the material of this chapter is taken from [263].

Gravitational instantons with source

We discuss various geometric formulations of the equations of gravitational instantons with sources. We show that these equations include as a special case the classical Einstein equations with or without the cosmological constant. We discuss the notion of an Einstein pair to clarify the distinction between the Palatini and the metric variational equations. Our approach naturally leads to the introduction of a cosmological function and a generalized conservation law for the energy-momentum tensor. We also discuss the physical significance of the new conservation condition.

Generalized products and associated structures on Euclidean spaces

We define a generalized product of vectors in an arbitrary finite dimensional, inner product space which depends on a finite number of real parameters and which includes as special cases the usual cross‐product in R 3 and the product of n— 1 vectors in Rn . The concepts developed in this part are used to define a generalized determinant function of an arbitrary mx nmatrix. This determinant function allows us to define a general notion of non‐singularity for a rectangular matrix which turns out to be necessary and sufficient for the existence of certain one‐sided inverses. We obtain families of one‐sided inverses of rectangular matrices and use them to construct reflexive generalized inverses which include as a special case the well‐known Moore‐Penrose inverse.

Yang–Mills–Higgs Fields

Yang–Mills equations, originally derived for the isospin gauge group SU(2), provide the first example of gauge field equations for a non-Abelian gauge group. This gauge group appears as an internal or local symmetry group of the theory. In fact, the theory can be extended easily to include the other classical Lie groups as gauge groups. Historically, the classical Lie groups appeared in physical theories, mainly in the form of global symmetry groups of dynamical systems. Noether’s theorem established an important relation between symmetry and conservation laws of classical dynamical systems. It turns out that this relationship also extends to quantum mechanical systems. Weyl made fundamental contributions to the theory of representations of the classical groups [401] and to their application to quantum mechanics. The Lorentz group also appears first as the global symmetry group of the Minkowski space in the special theory of relativity. It then reappears as the structure group of the principal bundle of orthonormal frames (or the inertial frames) on a space-time manifold M in Einstein’s general theory of relativity. In general relativity a gravitational field is defined in terms of the Lorentz metric of M and the corresponding Levi-Civita connection on M. Thus, a gravitational field is essentially determined by geometrical quantities intrinsically associated with the space-time manifold subject to the gravitational field equations. This geometrization of gravity must be considered one of the greatest events in the history of mathematical physics.