UPTEC F12 018 Examensarbete 30 hp Juni 2012 Bundles & Gauges, a Math-Physics Duality the case of Gravity David Mendes Abstract Bundles & Gauges, a Math-Physics Duality - the case of Gravity - David Mendes Teknisk- naturvetenskaplig fakultet UTH-enheten A modern and straight forward summary of the necessary tools and concepts needed to Besöksadress: understand and work with gauge theory in a Ångströmlaboratoriet Lägerhyddsvägen 1 fibre bundle formalism. Due to the aim of Hus 4, Plan 0 being a quick but thorough introduction full derivations are rarely included, but Postadress: references to such are given where they Box 536 751 21 Uppsala have been omitted. General Relativity, although being a geometric theory, in the Telefon: sense that the gravitational force is 018 – 471 30 03 described by the curvature of space-time, Telefax: may not be derived from geometry like the 018 – 471 30 00 other fundamental forces as in Yang-Mills theory. Thus, a possibility of unification Hemsida: lies in a geometrical derivation of http://www.teknat.uu.se/student gravity from gauge principles. By applying the presented formalism to the case of Gravity such a derivation is pursued along the lines of nonlinear realizations of the gauge group. Handledare: Ulf Lindström Ämnesgranskare: Maxim Zabzine Examinator: Tomas Nyberg ISSN: 1401-5757, UPTEC F12 018 Bundles and Gauges, a Math-Physics Duality - the case of Gravity - David Mendes June 5, 2012 Master thesis Supervisor: Ulf Lindström Department of Physics and Astronomy Division of Theoretical Physics University of Uppsala Abstract A modern and straight forward summary of the necessary tools and concepts needed to understand and work with gauge theory in a bre bun- dle formalism. Due to the aim of being a quick but thorough introduction full derivations are rarely included, but references to such are given where they have been omitted. General Relativity, although being a geometric theory, in the sense that the gravitational force is described by the cur- vature of space-time, may not be derived from geometry like the other fundamental forces as in Yang-Mills theory. Thus, a possibility of unica- tion lies in a geometrical derivation of gravity from gauge principles. By applying the presented formalism to the case of Gravity such a derivation is pursued along the lines of nonlinear realizations of the gauge group. Contents 1 introduction 4 1.1 manifold . 4 1.2 bre bundle . 5 1.3 gauge theory . 6 2 bre bundles & gauge theory 8 2.1 vector bundles & elds . 9 2.2 principal bundles & gauge transformations . 9 2.3 associated bundles & eld transformations . 10 2.4 connection one-forms & gauge potentials . 11 2.5 eld strengths & curvatures . 13 2.6 parallel transport & horizontal lifts . 14 2.7 connection coecients & representations . 16 2.8 frames, frame elds & frame bundles . 16 3 the case of Gravity 19 3.1 tensor bundles in general . 19 3.2 tensor bundles in general relativity . 20 3.3 general bundles in general relativity . 22 3.4 orthonormal frame & vielbein . 22 3.5 Lorentz bundle & spin connection . 23 3.6 Poincare gauge theory . 25 3.7 Einstein-Cartan action . 26 3.8 de Sitter gauge theory . 27 4 epilogue & outlook 32 4.1 extended tangent bundle . 32 4.2 symmetry reduction of SO(2; 4) ................... 34 4.3 generalized Cartan geometry . 34 3 1 introduction Before any proper introduction and historical background are given we rst introduce the two main areas which will harmonize in this thesis, namely the theory of bre bundles and the concept of gauge theories. Since bre bundles is an entirely mathematical framework and the sole purpose of gauge theories is to describe reality, the two subjects are introduced as the separate entities they really are. The title of this thesis, indicating their correspondence, will become clear as the reader moves on. Firstly the concept of manifolds is briey mentioned because bre bundles will be introduced as a natural generalization of that concept. Throughout this thesis all manifolds are assumed to be dierentiable and Haussdorf and all maps are assumed to be smooth. 1.1 manifold From a geometrical point of view we may start the mathematical representation of our physical world with the notion of a manifold M. This manifold repre- sents space, or at least the part of space we wish to include in our description. Examples of 2-dimensional manifolds are the ordinary cylinder and the 2-sphere S2. Although the manifold may have a very complicated topology globally, its dening property is to be locally homeomorphic to Rm, m being the dimension of our manifold. The homeomorphism 'i from M to an open subspace Ui of Rm in called a chart m (1.1) 'i : M! Ui ⊂ R and allows us to assign coordinates to the manifold by using those of Ui, which we may choose freely. If the manifold globally diers in topology from Rm we need to apply a set of charts fUi;'ig, called an open covering of M, such that the whole of M is covered and transitions between any two charts Ui and Uj is carried out by dierentiable, or smooth, transition functions ij dened by −1 (1.2) ij = 'i ◦ 'j : Uj ! Ui The transition functions need to be dened wherever there is an overlap between two charts Ui \ Uj 6= ; and need to obey the following consistency conditions ii = idUi −1 (1.3) ij = ji ik = ij ◦ jk where is the identity map on . The third and most restrictive of the idUi Ui consistency conditions, called the cocycle condition, apply if there is an overlap between three charts Ui \ Uj \ Uk 6= ;. Considering our example-manifolds, one 4 chart is enough to cover the cylinder while the sphere needs two charts to avoid coordinate singularities at the poles. With a satisfying representation of space we now need a framework with additional complexity to describe forces, particles and other things we wish to incorporate in our description. Just like a plane may contain more structure than a line one might argue that the most natural way to achieve additional structure is to assign an additional manifold to every point of the manifold M. The mathematics of such a structure is called the theory of bre bundles. Fibre bundles provide a framework to naturally incorporate gauge theories, possibly covering all the complexity of our physical world. 1.2 bre bundle The manifold representing space M is called the base while the manifold de- ned at each point of M are identical copies of a manifold F, called the bre. Furthermore, all these manifolds together are made to form a total space which is also a manifold, denoted by U. To make the total space a well dened manifold several maps have to be introduced, one of which is a surjection π : U!M called the projection. The subset of elements fug 2 U which are projected down to a specic point p 2 M is called the bre at p, denoted Fp −1 π (p) = Fp ⊂ U (1.4) Note that although the bres at dierent points are all isomorphic to the typical bre F they consist of elements u 2 U wheres the typical bre is a manifold of its own with elements f 2 F. Exactly like we needed charts fUi;'ig to perform calculations on M we need a set of dieomorphisms to locally map U onto the direct product Ui ×F. This is accomplished by the inverse of the local trivializations, the local trivialization is dened the other way around: −1 φi : Ui × F ! π (Ui) ⊂ U (1.5) −1 so that φi;p(f) 7! π (p) where φi;p is the restriction of the local trivialization to p 2 Ui. In addition, exactly like we needed smooth coordinate transformations from Ui to Uj where they overlap Ui \Uj we now introduce the transition functions −1 (1.6) tij(p) ≡ φi;p ◦ φj;p to smoothly paste the direct products fUi × Fg together, forming a covering of the total space U. The transition functions are elements of a Lie group G called 5 the structure group which acts on F on the left. The given denitions form a bre bundle (U; π; M; F; G), also denoted U −!π M. In addition to the denition of the bre bundle and its constituents we also want to dene a local map called a section si from Ui ⊂ M to the total space such that (1.7) si : Ui !U π ◦ si = idUi The set of sections on Ui is denoted Γ(Ui; U), in some cases the section may be extended to the whole manifold and is then referred to as a global section s 2 Γ(M; U), but in general there are topological obstructions to this extension. Note that by we refer to both the subset of m and the corresponding subset Ui R of the manifold M. 1.3 gauge theory Consider the simplest action for a charged complex scalar eld Z 4 µ 2 S = d x(@µΨ@ Ψ¯ − m ΨΨ)¯ (1.8) This action remains invariant under multiplication of the elds by a complex constant since Ψ and Ψ¯ occurs in pairs in the Lagrangian.A Lie group whose action leaves the Lagrangian invariant is called a symmetry group. In the example considered the symmetry group is U(1), causing the transformations Ψ ! eiΛΨ ; Ψ¯ ! e−iΛΨ¯ (1.9) where Λ is the transformation parameter of the Lie group. So far this is a global symmetry of the action since the group operation is performed identically everywhere. If the action remains invariant under a space-time dependent group operation the symmetry is promoted to a local symmetry.