用户: Solution/ 试卷: 规范场

1Example Sheet

  Question 1. Let $\mathbb{K}=\mathbb{R},\mathbb{C}$ and identify the Lie algebra of an embedded Lie subgroup of $\mathrm{GL}(n,\mathbb{K})$ with a Lie subalgebra of $\mathrm{Mat}(n\times n,\mathbb{K})$. Prove that

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The Lie bracket on the Lie subalgebra of $\mathrm{Mat}(n\times n,\mathbb{K})$ is the standard commutator of matrices.

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The Lie algebras of classical groups are the following Lie subalgebras of $\mathrm{Mat}(n\times n,\mathbb{K})$. - The Lie algebras of the special linear groups $\mathrm{SL}(n,\mathbb{K})$ read$$\mathfrak{sl}(n,\mathbb{K})=\{M\in \mathrm{Mat}(n\times n,\mathbb{K}):\mathrm{tr}(M)=0\}.$$- The Lie algebras of the orthogonal groups $\mathrm{O}(n)$ read$$\mathfrak{o}(n)=\left\{M\in \mathrm{Mat}(n\times n,\mathbb{R}):M+M^T=0\right\}.$$- The Lie algebras of the unitary groups $\mathrm{U}(n)$ read$$\mathfrak{u}(n)=\left\{M\in \mathrm{Mat}(n\times n,\mathbb{C}):M+M^{†}=0\right\}.$$- The Lie algebras of the special orthogonal groups $\mathrm{SO}(n)$ read$$\mathfrak{so}(n)=\mathfrak{o}(n).$$- The Lie algebras of the special unitary groups $\mathrm{SU}(n)$ read$$\mathfrak{su}(n)=\left\{M\in \mathrm{Mat}(n\times n,\mathbb{C}):M+M^{†}=0,\mathrm{tr}(M)=0\right\}.$$

[Hint: Use the fact that ‘If $H$ is an embedded Lie subgroup of $G$, then $\mathfrak{h}$ is an embedded Lie subalgebra of $\mathfrak{g}$’ and compute the dimension of each matrix group.]

  Question 2. Let $(E,\pi ,M;F)$ be a fibre bundle. Prove that the bundle is isomorphic to a trivial bundle if and only if there exists a smooth map $\tau :E\to F$ such that the restrictions$$\tau {|}_{E_x}:E_x⟶F$$are diffeomorphisms for all $x\in M$.

  Question 3. Let $s_i:U_i\to P$ and $s_j:U_j\to P$ be local gauges with $U_i\cap U_j\ne \varnothing$ and $g_{ij}:U_i\cap U_j\to G$ transition functions. Prove that the local curvature $2$-forms $F_{s_i},F_{s_j}$ transform as$$F_{s_i}={\mathrm{Ad}}_{g_{ji}^{-1}}\circ F_{s_j}\text{ on }{U}_i\cap U_j.$$In particular, when $G$ is a matrix Lie group, prove that$$F_{s_i}=g_{ji}^{-1}\cdot F_{s_j}\cdot g_{ji}.$$[Hint: Adapt the proof of the analogous theorem for local connection $1$-forms.]

  Question 4. Show that the exterior covariant derivative $d_A$ satisfies$$\begin{array}{rlrl}{d}_A(\omega +{\omega }^{\prime })& =d_A\omega +d_A{\omega }^{\prime }& & \forall \omega ,{\omega }^{\prime }\in {\Omega }^k(M,E)\\ d_A(\sigma \otimes e)& =d\sigma \otimes e+(-1{)}^k\sigma \wedge {\nabla }^{A}e& & \forall \sigma \in {\Omega }^k(M),e\in \Gamma (E)\\ d_A(\sigma \wedge \omega )& =d\sigma \wedge \omega +(-1{)}^k\sigma \wedge d_A\omega & & \forall \sigma \in {\Omega }^k(M),\omega \in {\Omega }^l(M,E).\end{array}$$[Hint: Invoke the local expression of the exterior covariant derivative.]

  Question 5. We live in the following geometry.

•	$({\mathbb{R}}^4,g)$: the Minkowski spacetime with signature $(+,-,-,-)$,
•	$G$: a compact Lie group (say $\mathrm{U}(1)\times \mathrm{SU}(2)\times \mathrm{SU}(3)$) with a Lie algebra $\mathfrak{g}$,
•	$P\to M$: a principal $G$-bundle,
•	$\mathrm{Ad}(P)$: the adjoint bundle with bundle metric $⟨\cdot ,\cdot {⟩}_{\mathrm{Ad}(P)}$,
•	$E\to M$: an associated vector bundle with representation $\rho$ and Hermitian bundle metric $⟨\cdot ,\cdot {⟩}_E$.

Denote

•	$A$: connection $1$-forms on $P$,
•	$F_A$: curvature $2$-forms on $P$ with respect to $A$,
•	$\Phi$: smooth sections of $E$,
•	$d_A$: the exterior covariant derivative on the adjoint bundle $\mathrm{Ad}(P)$,
•	$d_{A,\rho }$: the exterior covariant derivative on the associated vector bundle $E$ with representation $\rho$,$$d_{A,\rho }:{\Omega }^k(M,E)⟶{\Omega }^{k+1}(M,E).$$

Consider the Yang–Mills–Higgs Lagrangian$${\mathcal{L}}_{\mathrm{YMH}}[\Phi ,A]=⟨d_{A,\rho }\Phi ,d_{A,\rho }\Phi {⟩}_E-V(⟨\Phi ,\Phi {⟩}_E)-\frac{1}{2}⟨F_A,F_A{⟩}_{\mathrm{Ad}(P)}$$with a function $V(x)=x^2/2-1$ on $\mathbb{R}$.

•	Prove that variation of $\Phi$ leads to the field equation$$d_{A,\rho }^{\ast }{d}_{A,\rho }\Phi =V^{\prime }(⟨\Phi ,\Phi {⟩}_E).\text{\hspace{1em}(1)}$$
•	Prove that variation of $A$ leads to the field equation$$d_A^{\ast }{F}_A\Phi =J_{\rho }(A,\Phi ),$$where $J_{\rho }(A,\Phi )\in {\Omega }^1({\mathbb{R}}^4,\mathrm{Ad}(P))$ is a twisted $1$-form defined by$$⟨\alpha ,J_{\rho }(A,\Phi ){⟩}_{\mathrm{Ad}(P)}=2\Re (⟨d_{A,\rho }\Phi ,\alpha \cdot \Phi {⟩}_E),\forall \alpha \in {\Omega }^1({\mathbb{R}}^4,\mathrm{Ad}(P)).\text{\hspace{1em}(2)}$$

[Background in physics: The system (1)-(2) is called the Yang–Mills–Higgs equation. It is used in Higgs mechanism to explain how gauge bosons $A$ gain mass through interactions with Higgs bosons $\Phi$. The Higgs bosons were proposed in 1960s and finally detected at CERN in 2012.]