Connection (principal bundle)

. Let \pi : P \to M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra \mathfrak g of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P. In other words, it is an element ω of \Omega^1(P,\mathfrak g)\cong C^\infty(P, T^*P\otimes\mathfrak g) such that • \hbox{Ad}_g(R_g^*\omega)=\omega where R_g denotes right multiplication by g, and \operatorname{Ad}_g is the adjoint representation on \mathfrak g (explicitly, \operatorname{Ad}_gX = \frac{d}{dt}g\exp(tX)g^{-1}\bigl|_{t=0}); • if \xi\in \mathfrak g and X_\xi is the vector field on P associated to ξ by differentiating the G action on P, then \omega(X_\xi)=\xi (identically on P). Sometimes the term principal G-connection refers to the pair (P,\omega) and \omega itself is called the connection form or connection 1-form of the principal connection. Computational remarks Most known non-trivial computations of principal G-connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let G \to H \to H/G, be a principal G-bundle over H/G.) This means that real-valued 1-forms on the total space H are canonically isomorphic to C^\infty(H,\mathfrak{h}^*), where \mathfrak{h}^* is the dual Lie algebra, hence G-connections are in bijection with C^\infty(H,\mathfrak{h}^*\otimes \mathfrak{g})^G. Relation to Ehresmann connections A principal G-connection \omega on P determines an Ehresmann connection on P in the following way. First note that the fundamental vector fields generating the G action on P provide a bundle isomorphism (covering the identity of P) from the bundle V to P\times\mathfrak g, where V=\ker(d\pi) is the kernel of the tangent mapping {\mathrm d}\pi\colon TP\to TM which is called the vertical bundle of P. It follows that \omega determines uniquely a bundle map v:TP\rightarrow V which is the identity on V. Such a projection v is uniquely determined by its kernel, which is a smooth subbundle H of TP (called the horizontal bundle) such that TP=V\oplus H. This is an Ehresmann connection. Conversely, an Ehresmann connection H\subset TP (or v:TP\rightarrow V) on P defines a principal G-connection \omega if and only if it is G-equivariant in the sense that H_{pg}=\mathrm d(R_g)_p(H_{p}). Pull back via trivializing section A trivializing section of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s*ω of a principal connection is a 1-form on U with values in \mathfrak g. If the section s is replaced by a new section sg, defined by (sg)(x) = s(x)g(x), where g:M→G is a smooth map, then (sg)^* \omega = \operatorname{Ad}(g)^{-1}s^* \omega + g^{-1} dg. The principal connection is uniquely determined by this family of \mathfrak g-valued 1-forms, and these 1-forms are also called connection forms or connection 1-forms, particularly in older or more physics-oriented literature. Bundle of principal connections The group G acts on the tangent bundle TP by right translation. The quotient space TP/G is also a manifold, and inherits the structure of a fibre bundle over TM which shall be denoted dπ:TP/G→TM. Let ρ:TP/G→M be the projection onto M. The fibres of the bundle TP/G under the projection ρ carry an additive structure. The bundle TP/G is called the bundle of principal connections . A section Γ of dπ:TP/G→TM such that Γ : TM → TP/G is a linear morphism of vector bundles over M, can be identified with a principal connection in P. Conversely, a principal connection as defined above gives rise to such a section Γ of TP/G. Finally, let Γ be a principal connection in this sense. Let q:TP→TP/G be the quotient map. The horizontal distribution of the connection is the bundle :H = q^{-1}\Gamma(TM) \subset TP. We see again the link to the horizontal bundle and thus Ehresmann connection. Affine property If ω and ω′ are principal connections on a principal bundle P, then the difference is a \mathfrak g-valued 1-form on P that is not only G-equivariant, but horizontal in the sense that it vanishes on any section of the vertical bundle V of P. Hence it is basic and so is determined by a 1-form on M with values in the adjoint bundle :\mathfrak g_P:=P\times^G\mathfrak g. Conversely, any such one form defines (via pullback) a G-equivariant horizontal 1-form on P, and the space of principal G-connections is an affine space for this space of 1-forms. == Examples ==

Maurer-Cartan connection For the trivial principal G-bundle \pi:E \to X where E = G\times X, there is a canonical connectionpg 49\omega_{MC} \in \Omega^1(E,\mathfrak{g})called the Maurer-Cartan connection. It is defined at a point (g,x) \in G\times X by(\omega_{MC})_{(g,x)} = (L_{g^{-1}}\circ \pi_1)_* for x \in X, g \in Gwhich is a compositionT_{(g,x)}E \xrightarrow{\pi_{1*}} T_gG \xrightarrow{(L_{g^{-1}})_*} T_eG = \mathfrak{g}defining the 1-form. Note that\omega_0 = (L_{g^{-1}})_*: T_gG \to T_eG = \mathfrak{g}is the Maurer-Cartan form on the Lie group G and \omega_{MC} = \pi_1^*\omega_0. Trivial bundle For a trivial principal G-bundle \pi:E \to X, the identity section i: X \to G\times X given by i(x) = (e,x) defines a 1-1 correspondencei^*:\Omega^1(E,\mathfrak{g}) \to \Omega^1(X,\mathfrak{g})between connections on E and \mathfrak{g}-valued 1-forms on Xpg 53. For a \mathfrak{g}-valued 1-form A on X, there is a unique 1-form \tilde{A} on E such that • \tilde{A}(X) = 0 for X \in T_xE a vertical vector • R_g^*\tilde{A} = \text{Ad}(g^{-1}) \circ \tilde{A} for any g \in G Then given this 1-form, a connection on E can be constructed by taking the sum\omega_{MC} + \tilde{A}giving an actual connection on E. This unique 1-form can be constructed by first looking at it restricted to (e,x) for x \in X. Then, \tilde{A}_{(e,x)} is determined by A because T_{(x,e)}E = ker(\pi_*)\oplus i_*T_xX and we can get \tilde{A}_{(g,x)}by taking\tilde{A}_{(g,x)} = R^*_g\tilde{A}_{(e,x)} = \text{Ad}(g^{-1})\circ \tilde{A}_{(e,x)}Similarly, the form\tilde{A}_{(x,g)} = \text{Ad}(g^{-1}) \circ A_x \circ \pi_*: T_{(x,g)}E \to \mathfrak{g} defines a 1-form giving the properties 1 and 2 listed above. Extending this to non-trivial bundles This statement can be refinedpg 55 even further for non-trivial bundles E \to X by considering an open covering \mathcal{U} = \{U_a\}_{a \in I} of X with trivializations \{\phi_a\}_{a \in I} and transition functions \{g_{ab}\}_{a,b\in I}. Then, there is a 1-1 correspondence between connections on E and collections of 1-forms\{A_a \in \Omega_1(U_a,\mathfrak{g}) \}_{a \in I}which satisfyA_b = Ad(g_{ab}^{-1})\circ A_a + g_{ab}^*\omega_0on the intersections U_{ab} for \omega_0 the Maurer-Cartan form on G, \omega_0 = g^{-1}dg in matrix form. Global reformulation of space of connections For a principal G bundle \pi: E \to M the set of connections in E is an affine spacepg 57 for the vector space \Omega^1(M,E_\mathfrak{g}) where E_\mathfrak{g} is the associated adjoint vector bundle. This implies for any two connections \omega_0, \omega_1 there exists a form A \in \Omega^1(M, E_\mathfrak{g}) such that\omega_0 = \omega_1 + AWe denote the set of connections as \mathcal{A}(E), or just \mathcal{A} if the context is clear. Connection on the complex Hopf-bundle Wepg 94 can construct \mathbb{CP}^n as a principal \mathbb{C}^*-bundle \gamma:H_\mathbb{C} \to \mathbb{CP}^n where H_\mathbb{C} = \mathbb{C}^{n+1}-\{0\} and \gamma is the projection map\gamma(z_0,\ldots,z_n) = [z_0,\ldots,z_n]Note the Lie algebra of \mathbb{C}^* = GL(1,\mathbb{C}) is just the complex plane. The 1-form \omega \in \Omega^1(H_\mathbb{C},\mathbb{C}) defined as\begin{align} \omega &= \frac{\overline{z}^tdz}{|z|^2} \\ &= \sum_{i=0}^n\frac{\overline{z}_i}{|z|^2}dz_i \end{align}forms a connection, which can be checked by verifying the definition. For any fixed \lambda \in \mathbb{C}^* we have\begin{align} R_\lambda^*\omega &= \frac{\overline{(z\lambda)}^td(z\lambda)}{|z\lambda|^2} \\ &= \frac{ \overline{\lambda}\lambda\overline{z}^tdz }{|\lambda|^2\cdot |z|^2} \end{align}and since |\lambda|^2 = \overline{\lambda}{\lambda}, we have \mathbb{C}^*-invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any z \in H_\mathbb{C} we have a short exact sequence0 \to \mathbb{C} \xrightarrow{v_z} T_zH_\mathbb{C} \xrightarrow{\gamma_*} T_{[z]}\mathbb{CP}^n \to 0where v_z is defined asv_z(\lambda) = z\cdot \lambdaso it acts as scaling in the fiber (which restricts to the corresponding \mathbb{C}^*-action). Taking \omega_z\circ v_z(\lambda) we get \begin{align} \omega_z\circ v_z(\lambda) &= \frac{\overline{z}dz}{|z|^2}(z\lambda) \\ &= \frac{\overline{z}z\lambda}{|z|^2} \\ &= \lambda \end{align} where the second equality follows because we are considering z\lambda a vertical tangent vector, and dz(z\lambda) = z\lambda. The notation is somewhat confusing, but if we expand out each term\begin{align} dz &= dz_0 + \cdots + dz_n \\ z &= a_0z_0 + \cdots +a_nz_n \\ dz(z) &= a_0 + \cdots + a_n \\ dz(\lambda z) &= \lambda\cdot (a_0 + \cdots + a_n) \\ \overline{z} &= \overline{a_0} + \cdots + \overline{a_n} \end{align}it becomes more clear (where a_i \in \mathbb{C}). ==Induced covariant and exterior derivatives==

For any linear representation W of G there is an associated vector bundle P\times^G W over M, and a principal connection induces a covariant derivative on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of P\times^G W over M is isomorphic to the space of G-equivariant W-valued functions on P. More generally, the space of k-forms with values in P\times^G W is identified with the space of G-equivariant and horizontal W-valued k-forms on P. If α is such a k-form, then its exterior derivative dα, although G-equivariant, is no longer horizontal. However, the combination dα+ωΛα is. This defines an exterior covariant derivative dω from P\times^G W-valued k-forms on M to P\times^G W-valued (k+1)-forms on M. In particular, when k=0, we obtain a covariant derivative on P\times^G W. ==Curvature form==

The curvature form of a principal G-connection ω is the \mathfrak g-valued 2-form Ω defined by :\Omega=d\omega +\tfrac12 [\omega\wedge\omega]. It is G-equivariant and horizontal, hence corresponds to a 2-form on M with values in \mathfrak g_P. The identification of the curvature with this quantity is sometimes called the ''(Cartan's) second structure equation''. Historically, the emergence of the structure equations are found in the development of the Cartan connection. When transposed into the context of Lie groups, the structure equations are known as the Maurer–Cartan equations: they are the same equations, but in a different setting and notation. Flat connections and characterization of bundles with flat connections We say that a connection \omega is flat if its curvature form \Omega = 0. There is a useful characterization of principal bundles with flat connections; that is, a principal G-bundle \pi: E \to X has a flat connectionpg 68 if and only if there exists an open covering \{U_a\}_{a\in I} with trivializations \left\{ \phi_a \right\}_{a \in I} such that all transition functionsg_{ab}: U_a\cap U_b \to Gare constant. This is useful because it gives a recipe for constructing flat principal G-bundles over smooth manifolds; namely taking an open cover and defining trivializations with constant transition functions. ==Connections on frame bundles and torsion==

If the principal bundle P is the frame bundle, or (more generally) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since the additional structure of the solder form θ, which is an equivariant Rn-valued 1-form on P, should be taken into account. In particular, the torsion form on P, is an Rn-valued 2-form Θ defined by : \Theta=\mathrm d\theta+\omega\wedge\theta. Θ is G-equivariant and horizontal, and so it descends to a tangent-valued 2-form on M, called the torsion. This equation is sometimes called the ''(Cartan's) first structure equation''. ==Definition in algebraic geometry==

If X is a scheme (or more generally, stack, derived stack, or even prestack), we can associate to it its so-called de Rham stack, denoted XdR. This has the property that a principal G bundle over XdR is the same thing as a G bundle with *flat* connection over X. ==References==