Geometrical and topological foundations of theoretical physics: from gauge theories to string program

structures known to mathematicians as connections in fibre bundles. The discovery of this equivalence has made it possible to understand why and how powerful mathematical concepts and structures are necessary and suitable for the description and explanation of physical reality. In a very important paper, Wu and Yang introduced the fundamental concept of nonintegrable—that is, path-dependent—phase factor as the basis of a description of electromagnetism [66]. Further this concept is made to correspond to the definition of a gauge field; to extend it to global problems, they analyzed, in relation with the original Dirac’s result, the field produced by a magnetic monopole. The monopole discussion leads to the recognition that in general the phase factor (and indeed the vector potential Aμ) can only be properly defined in each of many overlapping regions of spacetime. In the overlap of any two regions, there exists a gauge transformation relating the phase factors defined for the two regions. The concept of monopole leads to the definition of global gauges and global gauge transformations. A surprising result is that the monopole types are quite different for SU2 and SO3 gauge fields and for electromagnetism. The mathematics of these results is the fiber bundle theory. Furthermore gauge fields, including in particular the electromagnetic field, are fiber bundles, and all gauge fields are thus based on geometry. So maybe all of the fundamental interactions of the physical worlds could be based on these geometrical and topological objects. The exact formulation of the concept of a nonintegrable phase factor depends on the definition of global gauge transformations, that is, on the choice of the overlapping regions of R (where R is a region of spacetime, precisely, all spacetime minus the origin r = 0) and of the potentialAμ in this region. Through a certain kind of operations, called distorsion, one arrives at a large number of possibilities, each with a particular choice of overlapping regions and with a particular choice of gauge transformation from the original (Aμ)a or (Aμ)b to the new Aμ in each region. Each of such possibilities will be called a gauge (or global gauge). This definition is a natural generalization of the usual concept, extended to deal with the intricacies of the field of a magnetic monopole. Notice that the gauge transformation factor in the overlap between Ra and Rb does not refer to any specific Aμ . (The gauge transformation in the overlap of the two regions is S = Sab = exp(−iα)= exp(2ige/hc)φ.) Thus two different gauges may share the same characterizations (a) and (b). In the case of the monopole field, one can attach to the gauge any (Aμ)a and (Aμ)b provided they are gauge-transformed into each other in the region of overlap. Thus a gauge is a concept not tied to any specific vector potential. Wu and Yang called the process of distorsion leading from one gauge to another a global gauge transformation. It is also a concept not tied to any specific vector potential. The collection of gauges that can be globally gauge-transformed into each other will be said to belong to the same gauge type. The phase factor exp(ie/hc ∫ Aμdx) (which is nonintegrable, i.e., path-dependent) around a loop starts and ends at the same point in the same region. Thus it does not change under any global transformation, so that we have the following theorem for Abelian gauge fields. Theorem 5.1. The phase factor around any loop is invariant under a global gauge transformation. The next two theorems follow trivially from this by taking an infinitesimal loop. GEOMETRICAL AND TOPOLOGICAL FOUNDATIONS 1797 Theorem 5.2. The field strength fμν is invariant under a global gauge transformation. Theorem 5.3. Between two gauge fields defined on the same gauge there exists a continuous interpolating gauge field defined on the same gauge. Theorem 5.4. Consider gauge D and define any gauge field on it. The total magnetic flux through a sphere around the origin r = 0 is independent of the gauge field and depends on the gauge only: ∫∫ fμνdxdx =− ihc e ∫ ∂ ∂xμ ( lnSab ) dxμ, (5.9) where S is the gauge transformation defined by (5.8) for the gauge D in question, and the integral is taken around any loop around the origin r = 0 in the overlap between Ra and Rb, such as the equation on a sphere r = 1. As in the case of electromagnetism, in the non-Abelian gauge fields both the concept of a gauge and the concept of a global gauge transformation are not tied to any specific gauge potentials. The nonintegrable phase factor for a given path is now an element of the gauge group (see [41]). Since these phase factors do not in general commute with each other, Theorems 5.1 and 5.2 for the Abelian case need to be modified as follows. Theorem 5.5. Under a global gauge transformation, the phase factor around any loop remains in the same class. The class does not depend on which point is taken as the starting point around the loop. Theorem 5.6. The field strength fk μν is covariant under a global gauge transformation. Theorem 5.5 defines the class of a loop. This concept is a generalization of the phase factor for electromagnetism around a loop with the magnetic flux as the exponent. It is a gauge-invariant concept. Rigorously the mathematical structure of gauge theory is that of a vector bundle E with structure group G over a compact Riemannian manifold M . We assume that G ∈ O(m) and E carries an inner product compatible with G. Let E be the space of G-connections on E, and let be the space of G-automorphisms of E. Then acts on E as before, and we have a quotient space B ≡ E/ . To each connection ∇∈ E there is associated a curvature 2-form R∇, and at each point x, we can take its norm ∥∥R∇∥∥2x ≡ ∑ i<j ∥∥R∇ ei,ej ∥∥2 x, (5.10) where {e1, . . . ,en} is an orthonormal basis of TxM and the norm of R∇ ei,ej is the usual one on Hom(E,E)—namely, 〈A,B〉 ≡ trace(At ◦B). Given any g ∈ , we recall that Rg(∇) = g◦R∇◦g−1, so ∥∥Rg(∇)∥∥≡ ∥∥R∇∥∥ on M. (5.11) This says that the pointwise norm of the curvature is gauge-invariant.