Geometric Construction of Quiver Tensor Products

By a classic theorem of Beilinson, the perfect derived category $\operatorname{Perf}(\mathbb{P}^n)$ of projective space is equivalent to the category of derived representations of a certain quiver with relations. The vertex-wise tensor product of quiver representations corresponds to a symmetric monoidal structure $\otimes_{\mathsf{Q}}$ on $\operatorname{Perf}(\mathbb{P}^n)$. We prove that, for a certain choice of equivalence, the symmetric monoidal structure $\otimes_{\mathsf{Q}}$ may be described geometrically as an \emph{extended convolution product} in the sense that the Fourier--Mukai kernel is given by the closure of the torus multiplication map in $(\mathbb{P}^n)^3$. We also set up a general framework for such problems, allowing us to generalize the extended convolution description of quiver tensor products to the case where $\mathbb{P}^n$ is replaced by any smooth complete toric variety of Bondal--Ruan type. Under toric mirror symmetry, this extended convolution product corresponds to the tensor product of constructible sheaves on a real torus. As another generalization of our results for $\mathbb{P}^n$, we show that any finite-dimensional algebra $A$ gives rise to a monoidal structure $\star_A'$ on $\operatorname{Perf}(\mathbb{P}(A))$, providing insights into the moduli of monoidal structures on $\operatorname{Perf}(\mathbb{P}^n)$.