Exotic $t$-structures on the product of two elliptic curves

In this talk I will recall the notion of a partially defined operation of a tensor product of $t$-structures on enhanced triangulated categories. In general, the existence of a product is very difficult to establish. Then I will sketch the proof of the following statement: such a tensor product exists in the case of two isogenous elliptic curves and $t$-structures associated with irrational slopes, which are related by a fractional-linear transform from $PGL^{-2}(Q)$.