3
$\begingroup$

In mathematics and physics, especially gauge theory, there are many different but related notions of wedge products when discussing vector space- and vector bundle-valued differential forms. For example, if $U,V,W$ are finite-dimensional $\mathbb{R}$-vector spaces and $\mu:U\times V\to W$ is a bilinear map, then we always can define a wedge product of the type

$$\wedge_{\mu}:\Omega^{k}(\mathcal{M},U)\times\Omega^{l}(\mathcal{M},V)\to\Omega^{k+l}(\mathcal{M},W)$$

via the usual detinition, i.e.

$$\alpha\wedge_{\mu}\beta:=\sum_{i=1}^{\mathrm{dim}_{\mathbb{R}}(U)}\sum_{j=1}^{\mathrm{dim}_{\mathbb{R}}(V)}(\alpha^{i}\wedge\beta^{j})\mu(e_{i},f_{j})$$

where $(e_{i})_{i=1}^{\mathrm{dim}_{\mathbb{R}}(V)}$ is a basis of $V$ and $(f_{j})_{j=1}^{\mathrm{dim}_{\mathbb{R}}(U)}$ is a basis of $U$ and where $\alpha^{i}\in\Omega^{k}(\mathcal{M})$ and $\beta^{j}\in\Omega^{l}(\mathcal{M})$ are the coordinate forms with respect to this basis. There are many different examples where this appears in the literature: Choosing the vector space scalar product $\mu:\mathbb{R}\times V\to V$ leads to a wedge product between real-valued and $V$-values forms, choosing an $\mathrm{Ad}$-invariant inner product on a Lie algebra $\mathfrak{g}$ leads to the "trace" of $\mathfrak{g}$-valued forms, which is for example used in the definition of Chern-Simons forms, and when taking a Lie-bracket, we recover the well-known wedge product $[\cdot\wedge\cdot]$.

Now, in many applications, one also has to deal with bundle-valued forms and in this context, there are also many different type of wedge products. I would imagine that every product in this context can be defined in the following unified framework:

Let $E,F,G$ be three smooth $\mathbb{R}$-vector bundles. Furthermore, let $\mu\in\Gamma^{\infty}(E^{\ast}\otimes F^{\ast}\otimes G)$, which we can view as a smooth assignment $\mathcal{M}\ni p\mapsto\mu_{p}$ such that $\mu_{p}:E_{p}\times F_{p}\to G_{p}$ is a bilinear map. Then we define a product of the type $$\wedge_{\mu}:\Omega^{k}(\mathcal{M},E)\times\Omega^{l}(\mathcal{M},F)\to\Omega^{k+l}(\mathcal{M},G).$$ For this, we take a local frame $\{e_{a}\}_{a=1}^{\mathrm{rank}(E)}\subset\Gamma^{\infty}(U,E)$ of $E$ and a local frame $\{f_{b}\}_{b=1}^{\mathrm{rank}(F)}\subset\Gamma^{\infty}(V,F)$ of $F$ on some open subsets $U,V\in\mathcal{M}$. Then we can write any $\alpha\in\Omega^{k}(\mathcal{M},E)$ and every $\beta\in\Omega^{l}(\mathcal{M},F)$ as $$\alpha\vert_{U}=\sum_{a=1}^{\mathrm{rank}(E)}\alpha^{a}e_{a}\hspace{1cm}\text{and}\hspace{1cm}\beta\vert_{V}=\sum_{b=1}^{\mathrm{rank}(F)}\beta^{b}f_{b}$$ for some local coordinate forms $\alpha^{a}\in\Omega^{k}(U)$ and $\beta^{b}\in\Omega^{l}(V)$. We then define the wedge-product $\wedge_{\mu}$ locally as $$(\alpha\wedge_{\mu}\beta)\vert_{U\cap V}:=\sum_{a=1}^{\mathrm{rank(E)}}\sum_{b=1}^{\mathrm{rank(F)}}(\alpha^{a}\wedge\beta^{b})\mu(e_{a},f_{b}),$$ where $\mu(e_{a},f_{b})\in \Gamma^{\infty}(U\cap V,G)$ has to be understood as $\mu(e_{a},f_{b})(p):=\mu_{p}(e_{a}(p),f_{b}(p))$ for all $p\in U\cap V$. This is well-defined and independent of the choice of local frames.

A particular example of this would be a bundle metric $\langle\cdot,\cdot\rangle\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$. When choosing the adjoint bundle $\mathrm{Ad}(P):=P\times_{\mathrm{Ad}}\mathfrak{g}$ of some principal bundle $P$, we recover for example the wedge-product used in the definition of the Yang-Mills action, when using the bundle metric on $\mathrm{Ad}(P)$ induced by an $\mathrm{Ad}$-invariant inner product on $\mathfrak{g}$.

Are such general type of wedge-products discussed in the literature and does anyone have some references for them?

Context: I am preparing some text on mathematical gauge theory and for this, I include some discussion of preliminaries including vector bundle valued differential forms and I would really like to treat them in a very general framework.

$\endgroup$
1
  • 2
    $\begingroup$ I don't think the bilinear map is an important part of the setup. What you want is a wedge product $\Omega^*(M,E) \otimes \Omega^*(M,F) \to \Omega^*(M, E \otimes F)$. Then you can simply apply $\mu$, because a linear bundle map E -> E' induces a map from E-valued forms to E'-valued forms. Phrased like this I believe there's a discussion of bundle-valued wedge products in Sharpe's differential geometry textbook, but I don't have access to a copy to check right now. $\endgroup$
    – mme
    Jul 31 at 16:15
4
$\begingroup$

The most general definition I know is the following. Every fiberwise bilinear form $\eta: V_1 \times V_2 \to W$ of vector bundles $V_1, V_2, W$ over $M$ gives rise to the wedge product of vector-bundle-valued differential forms by $$(\alpha \wedge_\eta \beta)_m (X_1, \dots X_{p+q}) = \frac{1}{p!q!} \sum_{\sigma \in S(p+q)} \mathrm{sign}(\sigma) \eta_m\bigl(\alpha_m(X_\sigma(1), \dots, X_{\sigma(p)}), \beta_m(X_{\sigma(p+1)}, \dots, X_{\sigma(p+q)})\bigr)$$ where $\alpha \in \Omega^p(M; V_1), \beta \in \Omega^q(M, V_2), m \in M$ and $X_i \in T_m M$. No choice of basis or local frame needed.

A discussion of wedge products is this sense can be found e.g. in the Monastir Lecture Notes by Neeb.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.