Topics in Physical Mathematics

We suppose that the reader is familiar with basic structures of algebra such as groups, rings, fields, and vector spaces and their morphisms, as well as the elements of representation theory of groups. Theory of groups was discovered by Cauchy. He called it “theory of substitutions.” He found it so exotic that he is said to have remarked: “It is a beautiful toy, but it will not have any use in the mathematical sciences.” In fact, quite the opposite was revealed to be true. The concept of group has proved to be fundamental in all mathematical sciences. In particular, the theory of Lie groups enjoys wide applicability in theoretical physics. We will discuss Lie groups in Chapter 3. Springer has started to reissue the volumes originally published under the general title “Éléments de mathématique” by Nicholas Bourbaki (see the note in Appendix B). The volumes dealing with Lie groups and Lie algebras are [57, 56]. They can be consulted as standard reference works, even though they were written more than 20 years ago.

Several areas of research in modern mathematics have developed as a result of interaction between two or more specialized areas. For example, the subject of algebraic topology associates with topological spaces various algebraic structures and uses their properties to answer topological questions. An elegant proof of the theorem that R ^m and R ⁿ with their respective standard topologies, are not homeomorphic for m≠n is provided by computing the homology of the one point compactification of these spaces. Indeed, the problem of classifying topological spaces up to homeomorphism was fundamental in the creation of algebraic topology. In general, however, the knowledge of these algebraic structures is not enough to decide whether two topological spaces are homeomorphic. The equivalence of algebraic structures follows from a weaker relation among topological spaces, namely, that of homotopy equivalence. In fact, homotopy equivalent spaces have isomorphic homotopy and homology structures. Equivalence of algebraic structures associated to two topological spaces is a necessary but not sufficient condition for their homeomorphism. Thus, one may think of homotopy and homology as providing obstructions to the existence of homeomorphisms. As we impose further structure on a topological space such as piecewise linear, differentiable, or analytic structures other obstructions may arise.

The mathematical background required for the study of modern physical theories and, in particular, gauge theories, is rather extensive and may be divided roughly into the following parts: elements of differential geometry, fiber bundles and connections, and algebraic topology of a manifold. The first two of these parts are nowadays fairly standard background for research workers in mathematical physics. In any case, physicists are familiar with classical differential geometry, which forms the cornerstone of Einstein’s theory of gravitation (the general theory of relativity). Therefore, in this chapter we give only a summary of some results from differential geometry to establish notation and make the monograph essentially self-contained. Fiber bundles and connections are discussed in Chapter 4. Characteristic classes, which are fundamental in the algebraic topology of a manifold, are discussed in detail in Chapter 5. The study of various classical field theories is taken up in Chapter 6. The reader familiar with the mathematical material may want to start with Chapter 6 and refer back to the earlier chapters as needed.

In 1931 Hopf studied the set [S ³, S ²] of homotopy classes of maps of spheres in his computation of π₃(S ²).He showed that π₃(S ²) is generated by the class of a certain map that is now well known as the Hopf fibration (see Example 4.5). This fibration decomposes S ³ into subspaces homeomorphic to S ¹ and the space of these subsets is precisely the sphere S ². In 1933 Seifert introduced the term fiber space to describe this general situation. The product of two topological spaces is trivially a fiber space, but the example of the Hopf fibration shows that a fiber space need not be a global topological product. It continues to be a local product and is now referred to as a fiber bundle.

In 1827 Gauss published his classic book Disquisitiones generales circa superficies curvas. He defined the total curvature (now called the Gaussian curvature) \(\kappa \) as a function on the surface. In his famous theorema egregium Gauss proved that the total curvature \(\kappa \) of a surface S depends only on the first fundamental form (i.e., the metric) of S. Gauss defined the integral curvature \(\kappa (\Sigma )\) of a bounded surface Σ to be \({\int\nolimits \nolimits }_{\Sigma }\kappa \ d\sigma \). He computed \(\kappa (\Sigma )\) when Σ is a geodesic triangle to prove his celebrated theorem where A, B, C are the angles of the geodesic triangle Σ. Gauss was aware of the significance of equation (5.1) in the investigation of the Euclidean parallel postulate (see Appendix B for more information). He was interested in surfaces of constant curvature and mentions a surface of revolution of constant negative curvature, namely, a pseudosphere. The geometry of the pseudosphere turns out to be the non-Euclidean geometry of Lobačevski–Bolyai.

In recent years gauge theories have emerged as primary tools for research in elementary particle physics. Experimental as well as theoretical evidence of their utility has grown tremendously in the last two decades. The isospin gauge group SU(2) of Yang–Mills theory combined with the U(1) gauge group of electromagnentic theory has lead to a unified theory of weak interactions and electromagnetism. We give an account of this unified electroweak theory in Chapter 8. In this chapter we give a mathematical formulation of several important concepts and constructions used in classical field theories. We begin with a brief account of the physical background in Section 6.2. Gauge potential and gauge field on an arbitrary pseudo-Riemannian manifold are defined in Section 6.3. Three different ways of defining the group of gauge transformations and their natural equivalence is also considered there. The geometric structure of the space of gauge potentials is discussed in Section 6.4 and is then applied to the study of Gribov ambiguity in Section 6.5. A geometric formulation of matter fields is given in Section 6.6. Gravitational field equations and their generalization is discussed in Section 6.7. Finally, Section 6.8 gives a brief indication of Perelman’s work on the geometrization conjecture and its relation to gravity.

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.

The concept of moduli space was introduced by Riemann in his study of the conformal (or equivalently, complex) structures on a Riemann surface. Let us consider the simplest non-trivial case, namely, that of a Riemann surface of genus 1 or the torus T ². The set of all complex structures \(\mathcal{C}({T}^{2})\) on the torus is an infinite-dimensional space acted on by the infinite-dimensional group Diff(T ²). The quotient space is the moduli space of complex structures on T ². Since T ² with a given complex structure defines an elliptic curve, \(\mathcal{M}({T}^{2})\) is, in fact, the moduli space of elliptic curves. It is well known that a point ω = ω₁ + iω₂ in the upper half-plane H (ω₂ > 0) determines a complex structure and is called the modulus of the corresponding elliptic curve. The modular group SL(2, Z) acts on H by modular transformations and we can identify \(\mathcal{M}({T}^{2})\) with H ∕ SL(2, Z). This is the reason for calling \(\mathcal{M}({T}^{2})\) the space of moduli of elliptic curves or simply the moduli space. The topology and geometry of the moduli space has rich structure. The natural boundary ω₂ = 0 of the upper half plane corresponds to singular structures. Several important aspects of this classical example are also found in the moduli spaces of other geometric structures. Typically, there is an infinite-dimensional group acting on an infinite-dimensional space of geometric structures with quotient a “nice space” (for example, a finite-dimensional manifold with singularities). For a general discussion of moduli spaces arising in various applications see, for example, [193].

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].

In this chapter we make some historical observations and comment on some early work in knot theory. Invariants of knots and links are introduced in Section 11.2. Witten’s interpretation of the Jones polynomial via the Chern–Simons theory is discussed in Section 11.3. A new invariant of 3-manifolds is obtained as a byproduct of this work by an evaluation of a certain partition function of the theory. We already met this invariant, called the Witten–Reshetikhin–Turaev (or WRT) invariant in Chapter 10. In Section 11.4 we discuss the Vassiliev invariants of singular knots. Gauss’s formula for the linking number is the starting point of some more recent work on self-linking invariants of knots by Bott, Taubes, and Cattaneo. We will discuss their work in Section 11.5. The self-linking invariants were obtained earlier by physicists using Chern–Simons perturbation theory. This work now forms a small part of the program initiated by Kontsevich [235] to relate topology of low-dimensional manifolds, homotopical algebras, and non-commutative geometry with topological field theories and Feynman diagrams in physics. See also the book [176] by Guadagnini. Khovanov’s categorification of the Jones polynomial by Khovanov homology is the subject of Section 11.6. We would like to remark that in recent years many applications of knot theory have been made in chemistry and biology (for a brief of discussion of these and further references see, for example, [260]). Some of the material in this chapter is from my article [263]).