# General wedge-product for vector bundle valued forms

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.

• 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.
– mme
Jul 31, 2021 at 16:15

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.