diffgeo.tex (5255B)
1 \subsection{Excurse} 2 \textbf{Manifold:} A topological space that is locally Euclidean. 3 \newline 4 \textbf{Riemannian Manifold:}A Manifold equipped with a Riemannian 5 Metric, a 6 symmetric bilinear form on Vector Fields $\Gamma(TM)$ 7 \begin{align} 8 &g: \Gamma(TM) \times \Gamma(TM) \rightarrow C(M) \\ 9 \text{with}& \nonumber\\ 10 &g(X, Y) \in \mathbb{R} \;\;\; \text{if $X, Y \in \mathbb{R}$}\\ 11 &\text{$g$ is $C(M)$-bilinear } \forall f\in C(M):\;\; g(fX, Y) = 12 g(X, 13 fY) = fg(X,Y)\\ 14 &g(X,X) \begin{cases}\geq 0 \;\;\; \forall X \\ = 0 \;\;\; \forall X 15 =0 16 \end{cases} 17 \end{align} 18 $g$ on $M$ gives rise to a distance function on $M$ 19 \begin{align} 20 d_g(x, y) = \inf_\gamma \left\{\int_0^1(\dot{\gamma}(t), 21 \dot{\gamma}(t))dt;\;\; \gamma(0) = x, \gamma(1) = y \right\} 22 \end{align} 23 Riemannian Manifold is called spin$^c$ if there exists a vector bundle $S 24 \rightarrow M$ with an algebra bundle isomorphism 25 \begin{align} 26 \mathbb{C}\text{I}(TM) &\simeq \text{End}(S)\;\;\; &\text{($dim(M)$ 27 even)}\\ 28 \mathbb{C}\text{I}(TM)^\circ &\simeq \text{End}(S)\;\;\; 29 &\text{($dim(M)$ odd)}\\ 30 \end{align} 31 $(M,S)$ is called the \textbf{spin$^c$ structure on $M$}. 32 \newline 33 $S$ is called the \textbf{spinor Bundle}. 34 \newline 35 $\Gamma(S)$ are the \textbf{spinors}. 36 37 Riemannian spin$^c$ Manifold is called spin if there exists an 38 anti-unitary 39 operator $J_M:\Gamma(S) \rightarrow \Gamma(S)$ such that: 40 \begin{enumerate} 41 \item $J_M$ commutes with the action of real-valued continuous 42 functions 43 on $\Gamma(S)$. 44 \item $J_M$ commutes with $\text{Cliff}^-(M)$ (even case)\\ 45 $J_M$ commutes with $\text{Cliff}^-(M)^\circ$ (odd case) 46 \end{enumerate} 47 $(S, J_M)$ is called the \textbf{spin Structure on $M$} 48 \newline 49 $J_M$ is called the \textbf{charge conjugation}. 50 51 \subsection{Operators of Laplace Type} 52 Let $M$ be a $n$ dimensional compact Riemannian manifold with $\partial M = 0$. 53 Then consider a vector bundle $V$ over $M$ (i.e. there is a vector space to 54 each point on $M$), so we can define smooth functions. We want to look at 55 arbitrary differential operators $D$ of Laplace type on $V$, they have the general 56 from 57 \begin{align} 58 D = -(g^{\mu\nu} \partial_\mu\partial_\nu + a^\sigma\partial_\sigma +b) 59 \end{align} 60 where $a^\sigma, b$ are matrix valued functions on $M$ and $g^{\mu\nu}$ is the 61 inverse metric on $M$. There is a unique connection on $V$ and a unique 62 endomorphism (matrix valued function) $E$ on $V$, then we can rewrite $D$ in 63 terms of $E$ and covariant derivatives 64 \begin{align} 65 D = -(g^{\mu\nu} \nabla_\mu \nabla_\nu +E) 66 \end{align} 67 Where the covariant derivative consists of $\nabla = \nabla^{[R]} +\omega$ the 68 standard Riemannian covariant derivative $\nabla^{[R]}$ and a "gauge" bundle 69 $\omega$ (fluctuations). WE can write $E$ and $\omega$ in terms of geometrical 70 identities 71 \begin{align} 72 \omega_\delta &= \frac{1}{2}g_{\nu\delta}(a^\nu 73 +g^{\mu\sigma}\Gamma^\nu_{\mu\sigma}I_V)\\ 74 E &= b - g^{\nu\mu}(\partial_\mu \omega_\nu + \omega_\nu \omega_\mu - 75 \omega_\sigma \Gamma^\sigma_{\nu\mu}) 76 \end{align} 77 where $I_V$ is the identity in $V$ and the Christoffel symbol 78 \begin{align} 79 \Gamma^\sigma_{\mu\nu} = g^{\sigma\varrho} \frac{1}{2} (\partial_\mu 80 g_{\nu\varrho} + \partial_\nu g_{\mu\varrho} - \partial_\varrho g_{\mu\nu}) 81 \end{align} 82 Furthermore we remind ourselves of the Riemmanian curvature tensor, Ricci 83 Tensor and the Scalar curavture. 84 \begin{align} 85 R^\mu_{\nu\varrho\sigma} &= \partial_\sigma \Gamma^{\mu}_{\nu\varrho} 86 -\partial_\varrho \Gamma^\mu_{\nu\sigma} 87 \Gamma^{\lambda}_{\nu\varrho}\Gamma^{\mu}_{\lambda\sigma} 88 \Gamma^{\lambda}_{\nu\sigma}\Gamma^{\mu}_{\lambda\varrho}\\ 89 R_{\mu\nu} &:= R^{\sigma}_{\mu\nu\sigma}\\ 90 R &:= R^\mu_{\ \mu} 91 \end{align} 92 93 The we let $\{e_1, \dots, e_n\}$ be the local orthonormal frame of 94 $TM$(tangent bundle $M$), which will be noted with flat indices $i,j,k,l 95 \in\{1,\dots, n\}$, we use $e^k_\mu, e^\nu_j$ to transform between flat indices 96 and curved indices $\mu, \nu, \varrho$. 97 \begin{align} 98 e^\mu_j e^\nu_k g_{\mu\nu} &= \delta_{jk}\\ 99 e^\mu_j e^\nu_k \delta^{jk} &= g^{\mu\nu} \\ 100 e^j_\mu e^\mu_k &= \delta^j_k 101 \end{align} 102 103 The Riemannian part of the covariant derivative contains the standard 104 Levi-Civita connection, so that for a $v_\nu$ we write 105 \begin{align} 106 \nabla_\mu^{[R]} v_\nu = \partial_\mu v_\nu - 107 \Gamma^{\varrho}_{\mu\nu}v_\varrho. 108 \end{align} 109 The extended covariant derivative reads then 110 \begin{align} 111 \nabla_\mu v^j = \partial_\mu v^j + \sigma^{jk}_\mu v_k. 112 \end{align} 113 the condition $\nabla_\mu e^k_\nu = 0$ gives us the general connection 114 \begin{align} 115 \sigma^{kl}_\mu = e^\nu_l\Gamma^{\varrho}_{\mu\nu}e^k_\varrho - e^\nu_l 116 \partial_\mu e^k_\nu 117 \end{align} 118 The we may define the field strength $\Omega_{\mu\nu}$ of the connection $\omega$ 119 \begin{align} 120 \Omega_{\mu\nu} = \partial_\mu \omega_\nu -\partial_\nu \omega_\mu 121 +\omega_\mu \omega_\nu -\omega_\nu\omega_\mu. 122 \end{align} 123 If we apply the covariant derivative on $\Omega$ we get 124 \begin{align} 125 \nabla_\varrho\Omega_{\mu\nu} = \partial_\varrho \Omega_{\mu\nu} - 126 \Gamma^{\sigma}_{\varrho \mu} \Omega_{\sigma\mu} + [\omega_\varrho, 127 \Omega_{\mu\nu}] 128 \end{align}