Lecture notes Geometry, Topology and Physics II 136.004 summer term 2009 Maximilian Kreuzer Institut f¨ur Theoretische Physik, TU–Wien +43 – 1 – 58801-13621 Wiedner Hauptstraße 8–10, A-1040 Wien [email protected] http://hep.itp.tuwien.ac.at/˜kreuzer/inc/gtp2.pdf Contents 1 Vector bundles 1 1.1 Vielbein and Lorentz connection . ........ 1 1.2 Fiberbundles .................................... ... 4 1.3 LiegroupsandLiealgebras . ...... 9 1.4 Connectionsonfiberbundles . ...... 11 2 Lie algebras and representations 14 3 Complex manifolds 23 3.1 Vector bundles on complex manifolds . ......... 23 Chapter 1 Vector bundles 1.1 Vielbein and Lorentz connection So far we described the tangent space with local coordinates (xm,vn), where the components n m v of a vector at a point with coordinates x refer to the holonomous basis ∂ m of T . As a x M consequence the transition functions for vn on the overlap of two charts is given by the Jacobi ∂ym matrix ∂xl , which is a general linear transformation. But the general linear group only has tensor representations and therefor we cannot describe spinors in this formalism. This problem a m a can be solved by introducing an orthonormal basis e = dx em for the cotangent space and m the dual basis Ea = Ea ∂m with g(Ea,Eb)= ηab for the tangent space, so that m n a n n a b gmnEa Eb = ηab, em Ea = δm, gmn = ηabem en . (1.1) We will denote Lorentz indices by letters from the beginning of the alphabet and world indices by k,l,m,...; for a Riemannian metric we have ηab = δab and in the pseudo-Riemannian case a 1 the diagonal matrix ηab has entries 1. In physics em is called vielbein, and we can use it ±m a m to write down a Dirac operator ∂/ = γ ∂mγ Ea ∂m and an action e ψ(iD/ m)ψ for spinor a b − ab fields in curved space in terms of the usual γ-matrices satisfying γR,γ = 2η . The volume a { } element can be written as e = det(em ) = g . The x-dependent γ-matrices in curved space m | a m | | | a are linear combinations γ = γ Ea of constantp representation matrices γ of the Clifford m algebra, where Ea is determined only up to a local (i.e. x-dependent) Lorentz transformation on its Lorentz index a. In order to define the covariant derivative Dψ of spinor fields we introduce the Lorentz b m b connection (or spin connection) ωa = dx ωma for objects, like spinors and tensors with Lorentz indices, that transform in some representation under local Lorentz transformations. Metricity of 1 a m a In mathematics e = dx em is called soldering form since the vielbein provides a soldering of the (principal bundle of the) cotangent bundle with an orthonormal frame bunde (see below). Geometry / M.Kreuzer — 1 — version June 2, 2009 the connection tranlates into antisymmetry in the last two indices ωab +ωba = 0, which preserves orthonormality of tangent vectors under parallel transport. If we want to use holonomous and orthogonal bases (i.e. tensors with both types of indices) at the same time we may do so by defining the total covariant derivative as D = d +Γ m∆ l + 1 ω lab, l v = η v η v . (1.2) l m 2 ab ab c ac b − bc a n m i m i with D = dx Dn, Γl = dx Γil and ωab = dx ωiab. The constant flat metric ηab is invariant under lab so that upper and lower Lorentz indices transform with the same sign. On spinors the algebra [l , l ]= η l η l η l + η l (1.3) ab cd ac bd − bc ad − ad bc bd ac 1 of Lorentz transformations is represented as labψ = 4 [γa,γb]ψ. Cartan invented a very efficient calculus by introducing differential forms and orthonormal a a bases e of contangent space: On (co)vector fields v = e va the spin-connection term of the b covariant derivative acts by matrix multiplication Dva = dva + ωa vb. Curvature and torsion can then be defined by Cartan’s structure equations (CSEq) c c b c a a a b Ra = dωa + ωa ωb , T = de + ω be , (1.4) or, in an even more compact symbolic form,2 R = D2 and T = De. The Bianchi identities dT + ω T = R e, dR + ω R R ω = 0. (1.5) ∧ ∧ ∧ − ∧ are now a trivial consequence of d2 = 0. The total antisymmetrization that is implicit in these 3-forms replaces the cyclic sum over the respective indices. In order to relate the spin connection ω to the affine connection Γ we impose that parallel transport should not depend on which basis we use for tangent space. The vielbein thus has to be covariantly constant, D e a = ∂ e a Γ le a + ω a e b = ∂ e a Γ a ω a = 0, (1.6) m n m n − mn l m b n m n − mn − mn which provides a relation between Γ and the spin connection (here we use the vielbein and its inverse to convert the second index of ω and the third index of Γ to the appropriate basis). Contraction of this equation with dxmdxn leads to dea dxmdxnΓ a dxmdxnω ae b = − mn − mb n dea + ωa eb dxmdxnΓ a = 0, which shows that the definition of the torsion via the CSEq b − mn agrees with our previous definition. In order to establish the equivalence of the definitions of ω b b b the curvature we introduce the symbod (D )a = δad + ωa for the covariant derivative that only acts on Lorentz indices. Then Cartan’s definition amounts to Rω =(Dω)2 and ω b 1 i j ω ω 1 i j l l Γ j i (R )a vb = 2 dx dx (Di Dj v)a = 2 dx dx (DiDjv +ΓijDlv)a = Ea(R )i ∆j vl, (1.7) 2 Derivatives of the vector field components drop out in D2v = d2v + d(ωv) + ωdv + ω2v = (dω + ω2)v. Geometry / M.Kreuzer — 2 — version June 2, 2009 where [D ,D ] = T lD + RΓ l∆ k. In a sense, the torsion contribution to [D ,D ] is thus i j − ij l ijk l i j taken into account in Cartan’s calculus by contracting the form index of the spin connection with a differential and having the connection only act on the Lorentz indices. A formula for the spin connection as a function of vielbein and torsion can be obtained by b n b b a b using ωma = Ea ∂men Γma , gmn = em en ηab and our formula for Γ(g, T ). The same result − a a a b can also be obtained directly from the structure equation T = de +ω be if we use the formula 2ω = ω ω that allows to compute a tensor that is antisymmetric in its last [mn]r mnr mnr − rmn 3 2 indices fromP its antisymmetrization in the first two indices: ω = e ∂ e a + e ∂ e a + e ∂ e a + 1 (T + T T ) (1.8) − mnl ma [n l] la [n m] na [m l] 2 mnl lmn − nlm For fixed vielbein the components of spin connection and torsion are thus related by invertible linear equations, so that it is equivalent to use e a,ω b or e a, T a as independent of { m ma } { m mn } fields. Usually it is more convenient to work with the former set. Example: The vielbein calculus is very useful for the evaluation of the curvature tensor. As an example we derive the Schwarzschild metric for a spherically symmetric black hole. Parametrizing the time dilatation with T (r) and the length contraction in the radial direction with R(r), where 4πr2 is the surface of the sphere at fixed r: ds2 = T 2dt2 R−2dr2 r2(dθ2 + sin2 θ dϕ2). (1.9) − − A convenient choice for the vielbein is therefore e0 = Tdt, e1 = R−1 dr, e2 = rdθ, e3 = r sin θ dϕ. (1.10) de0 = T ′ drdt, de1 = 0, de2 = drdθ, de3 = sin θ dr dϕ + r cos θ dθ dϕ. (1.11) a a b a Vanishing torsion T = de + e ωb = 0 thus yields the nonvanishing connection coefficients ω 1 = T ′Rdt, ω 2 = Rdθ, ω 3 = R sin θ dϕ, ω 3 = cos θ dϕ. (1.12) 0 − 1 − 1 − 2 − and the curvatures 1 ′ ′ 2 ′ 2 3 ′ 2 R0 =(T R) dt dr, R0 = T R dt dθ, R0 = T R sin θ dt dϕ, (1.13) R 2 = R′ dr dθ, R 3 = R′ sin θ dr dϕ, R 3 = (1 R2) sin θ dθ dϕ. (1.14) 1 − 1 − 2 − The vacuum Einstein equations in the orthonormal basis are = R c = E mE nR c = 0. Rab acb a c mnb The Ricci tensor turns out to be diagonal with = R(T ′R)′/T + 2T ′R2/rT, = R(T ′R)′/T 2R′R/r, (1.15) R00 R11 − − = = T ′R2/rT R′R/r + (1 R2)/r2. (1.16) R22 R33 − − − 3 The cyclic sum gives the total antisymmetrization of such a tensor, so that both, ω[mn]r and mnr ωmnr, n r are functions of vielbein and torsion. We thus obtain ωmab = Ea Eb (ω[mn]r ω[nr]m + ω[rm]n) with ω[mn]r = a b 1 − P ηaber ∂ e Tmnr. [m n] − 2 Geometry / M.Kreuzer — 3 — version June 2, 2009 Summation of = 0 and = 0 implies that R and T are proportional and by a rescaling of R00 R11 t we can set T = cR, where c is the speed of light.