I've some question about not(at)ions i've found on this article https://arxiv.org/pdf/hep-th/0403048.pdf
Let $(M,g)$ be a $n$-dimensional ortientable compact riemannian manifold with boundary $\Gamma$.
- A "gauge field $A$ defined in a Hermitian vector bundle $E$ over $M$ of rank $N$" (page 3, beginning of the section 2) is just a connection on the vector bundle?
- If the answer to the previous question is positive, $d_A\colon C^{\infty}(M,E)\to \Omega^1(M,E)$ (end of page 3) acts like the connection $A$?
- If the answer to the previous question is positive, how can i consider $\star d_A\psi$ (eq (2.5)), where $\star$ is the Hodge star operator of $M$ and $\psi\in C^\infty(M,E)$.
I'm a bit confused because the Hodge star operator works with $p$-forms of $M$ and not $E$-valued $p$-forms (i.e. elements of $\Omega^p(M,E)$).
Is there a generalization of this operator that I ignore?
Thanks in advance!
Best Answer
From my understanding, gauge fields are essentially physics terminology (probably originating from electrodynamics) for connections in some bundle of interest (here a vector bundle over $M$).
Next, note that $C^{\infty}(M,E)$ typically denotes all smooth maps from $M$ into $E$, so I find it odd that the paper uses it to denote the space of sections. If we wish to speak of only the smooth sections, the correct notation is something like $\Gamma(E)$ or $\Omega^0(M,E)$ ($E$-valued $0$-forms on $M$). The object $d_{A}$ is known as the exterior covariant derivative (which is typically denoted as $d_{\nabla}$ or $d^{\nabla}$). Also, yes, for a smooth section $\psi\in \Omega^0(M,E)$, we define $d_{\nabla}\psi:= \nabla\psi$. The reason it becomes an $E$-valued $1$-form on $M$ is because $\nabla_{(\cdot)}\psi$ has a slot which can be filled in by an element of $TM$, and if you feed it a $\xi_x\in T_xM$ then the output is $\nabla_{\xi_x}\psi\in E_x$.
Finally, regarding Hodge stars, let us first consider the case of vector spaces; we can then generalize to vector bundles by doing things fiber-by-fiber. Let $(V,\langle\cdot,\cdot\rangle)$ be an $n$-dimensional oriented pseudo-inner product space and let $E$ be a vector space. Note that an $E$-valued $k$-form on $V$ can be defined in several equivalent ways:
We shall use the third description. Let $\star:\bigwedge^k(V^*)\to \bigwedge^{n-k}(V^*)$ be the usual Hodge-star defined by the orientation and pseudo-inner product of $V$. This allows us to consider a bilinear map $\bigwedge^k(V^*)\times E\to \bigwedge^{n-k}(V^*)\otimes E$ defined as $(\alpha,\xi)\mapsto (\star\alpha)\otimes \xi$. Since this is a bilinear map, the universal property of tensor products tells us that we get a corresponding linear map $\star_E:\bigwedge^k(V^*)\otimes E\to \bigwedge^{n-k}(V^*)\otimes E$ such that for all "pure tensors" $\alpha\otimes \xi\in \bigwedge^{k}(V^*)\otimes E$, we have \begin{align} \star_E(\alpha\otimes \xi)&=(\star \alpha)\otimes \xi. \end{align} Hence, $\star_E$ is the desired mapping. It takes $E$-valued $k$-forms on $V$ to $E$-valued $(n-k)$-forms on $V$, and this definition is such that in the special case $E=\Bbb{R}$ is the underlying field, it coincides with the usual definition of $\star$ (after identifying a vector space of the form $W\otimes \Bbb{R}$ with $W$ in the natural way).
Now that we know how Hodge-star works at the level of vector spaces, you can do it at the vector-bundle level by doing things fiber-by-fiber to get a vector bundle morphism $\star_E:\bigwedge^k(T^*M)\otimes E\to \bigwedge^{n-k}(T^*M)\otimes E$, and hence by applying this to forms pointwise, we obtain a mapping $\Omega^k(M,E)\to \Omega^{n-k}(M,E)$ for each value of $k$. In particular, if $\psi$ is a smooth section, then $d_{\nabla}\psi$ is an $E$-valued 1-form on $M$, so its Hodge star is an $E$-valued $(n-1)$-form on $M$.