Name for construction on two vector bundles

Question

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

Answer 1

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.

Answer 2

We can actually define this in terms of hom, but first one needs to extend $A$ and $B$ to bundles on $X\times Y$.

Let $\pi_X:X\times Y \to X$ and $\pi_Y:X\times Y \to Y$ be the projection maps. The pullback $\pi_X^*(A)$ will be, in your notation, $\{((x,y),A_x)|(x,y)\in X\times Y\}$, and similarly for $\pi_Y^*(B)$. The bundle you are looking for is then $\mathcal{H}om(\pi_X^*(A),\pi_Y^*(B))$