I have a construction on two vector bundles and I would like to give it a name and a symbol but I can't find anything.


For two vector bundles $A=\{(x,A\_x) : x \in X\}$ and $B=\{(y,B\_y):y\in Y\}$ there is a vector bundle on $X\times Y$:


$$
A \mathop{?} B = \{((x,y),\text{L}(A\_x,B\_y)) : (x,A\_x) \in A, (y,B\_y) \in B\}
$$


$\text{L}(V,W)$ denotes the linear mappings from $V$ to $W$. 


Does anyone know this construction? Has it a name?


For $X = Y$ I know the Hom bundle but this would be a vector bundle on $X$ instead of $X^2$ and this makes it incompatible with the construction above.


References appreciated, thanks!


imix



Name for construction on two vector bundles
Some references, e.g., Berline–Getzler–Vergne, *Heat kernels and Dirac operators*, will define, for convenience, a modified tensor product $A \boxtimes B \to X \times Y$ by $$A \boxtimes B := \operatorname{proj}\_1^\ast A \otimes \operatorname{proj}\_2^\ast B,$$ where $\operatorname{proj}\_1 : X \times Y \to X$ and $\operatorname{proj}\_2 : X \times Y \to Y$ are the canonical maps defined by $$\operatorname{proj}\_1(x,y) := x, \quad \operatorname{proj}\_2(x,y) :=y;$$ in particular, if $X=Y$, then one should have $A \otimes B = \Delta^\ast(A \boxtimes B)$, where $\Delta : X \to X \times X$ is the diagonal map $\Delta(x) := (x,x)$. The vector bundle you want can be constructed as $A^\ast \boxtimes B \to X \times Y$, where $A^\ast \to X$ denotes the dual bundle to $A \to X$, and indeed, the construction of this sort of modified homomomorphism bundle (in the case of $X=Y$) is exactly why Berline–Getzler–Vergne bother to introduce such notation.


