CHAPTER Principal bundles and connections Motivation Gauge theory The simplest example of a gauge theory in physics is electromagnetism Recall Maxwells equations for an electromagentic eld B homogeneous equations r B r E t and r E r B E j inhomogeneous equations t Here we have chosen units so that and c are equal to E is the electric eld B the magnetic eld j the current density and the electric charge density All of these elds are functions on spacetime R with co ordinates x t and x x x x One can rewrite Maxwells equations in more concise co ordinate free form Let the electromagnetic eld strength F b e the following form on R F dx E dx E dx E dx B dx dx B dx dx B dx dx and dene a chargecurrent density form by J d x dx dx dx j dx dx j dx dx j dx d x k k Also R R b e the star op erator with resp ect to the standard ori entation and the Minkowski metric on R Then Maxwells equations are equivalent to dF homogeneous equations and d F J inhomogeneous equations The inhomogeneous equations require the integrability condition dJ which is easily recognized as charge conservation Note that the Equations are not quite as natural as the homogenous Maxwellequations The homogeneous Maxwell equations make sense 1 k For any pseudoRiemannian manifold M g of dimension n g extends to a metric on T M 2 If M is oriented one therefore obtains a unique oriented volume form such that jjjj The k nk 4 Ho dge star op erator M M is dened by the equation For R with 0 1 2 3 0 1 2 3 Minkowski metric one has dx dx dx dx thus eg dx dx dx dx PRINCIPAL BUNDLES AND CONNECTIONS on any manifold M while the inhomogeneous ones involve the choice of a pseudo Riemannian metric g More precisely since multiplying g by a p ositive function do es not change the star op erator on forms in R they dep end on the conformal structure of g The homogeneous Maxwell equations just say that F is a closed form Of course on R any closed form is exact so F dA for some form A R called the electromagnetic p otential The form A is dened up to a closed form On R any closed form is exact so that any two p otentials A A for F are related by A A df for some function f R One says that f denes a gauge transformation of the p otential A At this stage the p otential A on R itself do es not have physical meaning Only F dA ie the gauge equivalence class of A can b e interpreted in terms of B and E and can b e measured in exp eriments Using A the inhomogeneous Maxwell equations can b e obtained as EulerLagrange equations for a Lagrange density a form L d A dA A J The rst term in this expression can also b e written jjF jj dx dx dx dx by A denition of the star op erator In terms of B and E jjF jj jjB jj jjE jj The A functional R A dA dA 4 R dened on p otentials A which decrease suciently fast at innity is a sp ecial case of the YangMil ls functional Gauge transformations b ecome more interesting if the electromagnetic eld is coupled to some particle eld Consider for example an electron describ ed according to Diracs theory by a wave function R C Here C accomo dates b oth the electron particle and its antiparticle the p ositron Diracs equation for a free as a spin P m where the gammamatrices are matrices electron reads i giving a representation of the Cliord algebra of R The equation for in a non P A dx are obtained by minimal coupling quantized electromagnetic eld A ie replacing derivatives by covariant derivatives eA i i X i eA m This equation is invariant under the gauge transformations if A e A df 2 0 Note that on a nonsimply connected space time M it is p ossibly for two forms A A to have the same dierential without b eing gauge equivalent Thus gauge equivalence of A is a ner invariant than eld strength F dA The famous AharanovBohm exp eriment shows that the gauge equivalence A class does have physical meaning PRINCIPAL BUNDLES AND CONNECTIONS if for f R R Note that only g e R U really enters the gauge transfor mations In terms of g A g A idg g In this sense the theory of an electromagnetic eld is a gauge theory with gauge group G fg R U Note that the theory of an electron without an EM eld only has a global U gauge symmetry Electromagnetism is necessary to turn the global U gauge symmetry into a lo cal gauge symmetry The idea of YangMills theory is to replace the ab elian gauge group U by non commutative Lie groups G The gauge elds A are now forms with values in the Lie algebra g of G Again there is a notion of curvature form F which is interpreted as A a eld strength The coupling of this gauge eld to a particle eld with values in some complex inner pro duct space V dep ends on the choice of a unitary representation of G on V and replacing derivatives by covariant derivatives in such a way that the coupled theory b ecomes gauge invariant Principal bundles and connections In this section we review basic material on principal bundles connections curvature and parallel transp ort Fib er bundles Let F b e a given manifold A smo oth b er bundle with standard ber F is a smo oth map E B from a manifold E the total space to a manifold B called base with the following prop erty called local triviality There exists an op en covering U of B and dieomorphisms U U F such that pr x x That is intertwines with pro jection to the rst U factor We will denote the b ers by E b b A morphism of b er bundles E B with b ers F is a smo oth map j j j j E E taking b ers to b ers In particular induces a map B B on the base Conversely supp ose E B is a b er bundle with b er F and Y B is a smo oth map Then one can dene a pullback bundle E Y as a b ered pro duct E fy x Y E j y xg By construction one has a bundle map E E The bundle E is called trivializable if there exists a bundle map E B F the choice of is called a trivialization A sp ecial case of this construction is the bered product of two b er bundles E E B over the same base Let E E B B b e the direct pro duct and diag B B B B the diagonal map One denes B E E diag E E B B Thus E E B has b ers B E E E E b b b PRINCIPAL BUNDLES AND CONNECTIONS Additional structures on F give rise to sp ecial typ es of b er bundles If F V is a vector space one denes a vector bund le with standard b er V to b e a b er bundle E B where all b ers b are vector spaces and the lo cal trivializations can b e chosen to b e b erwise linear A homomorphism of two vector bundles is a b er bundle homomorphism that is b erwise linear The b ered pro duct of vector bundles E E is a vector bundle also called Whitney sum and denoted E E One has natural bundle maps E E E b erwise addition and a map R E E b erwise scalar multiplication If F G has the structure of a Lie group one denes a group bund le G B with standard b er G to b e a b er bundle where all b ers carry group structures and the lo cal trivializations can b e chosen to b e b erwise group homomorphisms A group bundle homomorphism is a b er bundle homomorphism which is b erwise a group homomorphism The b ered pro duct of group bundles is a group bundle B One has natural bundle maps G G G b erwise group multiplication and G G b erwise inversion Similarly one denes algebra bundles Lie algebra bundles An action of a group B bundle G B on a b er bundle E B is a smo oth map G E E preserving b ers and dening a group action b erwise Similarly one can dene b erwise linear actions of group or algebra bundles on vector bundles Example Given a vector bundle E B one obtains a group bundle GL E B with b er over b the invertible transformations automorphisms of the vector space E b and an algebra bundle EndE B with b ers the en b B domorphisms of E There are natural b erwise linear maps GL E E E and b B End E E E Principal bundles Let G b e a Lie group A principal homogeneous Gspace is a manifold X together with a transitive free Gaction For example G itself with action g a ag is a principal homogeneous space We could also use the leftaction of course Any principal homogeneous Gspace X is isomorphic to this example Given x X there is a unique Gequivariant isomorphism X G taking x to e The only reason for intro ducing the p edantic notion of principal homogeneous Gspace is that G in general there is no distinguished choice of x Note that the group Di X of G equivariant dieomorphisms of X to itself is dieomorphic to G but not canonically G so and that the space of Gequivariant dieomorphisms Di X G is a principal homogeneous Gspace with G acting on the target G by left multiplication which is canonically isomorphic to X itself Example If V is a real vector space of dimension n the space FrV of linear n isomorphisms frames V R is a principal homogeneous GLn Rspace The abstract vector space V can b e recovered from X as a quotient n V FrV R GL n R PRINCIPAL BUNDLES AND CONNECTIONS by the diagonal action In fact all the natural that is GL n Requivariant con n structions with R give rise to the corresp onding constructions of V by a similar quo n tient For instance P V FrV RP GL n R is the pro jectivization of V k k n V FrV R GL n R is the exterior algebra and so on A principal Gbund le is a b er bundle P B where each b er P carries the b structure of a principal homogeneous