# Mirror of the autoequivalences of the derived category of del Pezzo surface?

One version of the homological mirror symmetry conjecture states that for every Fano variety $X$ there exists a Landau--Ginzburg model $W$ such that the category of B-branes on $X$ (i.e. the bounded derived category of coherent sheaves) is equivalent to the category of A-branes of $W$ (i.e. the bounded derived category of Lagrangian vanishing cycles) $$D^bCoh(X)\simeq D^b Lag_{vc}(W).$$

This conjecture has been proven for del Pezzo surfaces, for example.

A theorem of Bondal--Orlov states that for a Fano variety $X$ the autoequivalence group of $D^bCoh(X)$ is equal to to $Aut(X)\ltimes(Pic(X)\oplus \mathbb{Z})$ where the first factor is the group of automorphisms of $X$, the second is the Picard group (acting by twisting) and the third factor is the shift functor.

My question is: on the Landau--Ginzburg side, what do the autoequivalences corresponding to $Aut(X)$ and $Pic(X)$ respectively look like? Do they correspond to some geometric operation, e.g. Dehn twist on the fiber? I am particularly interested in the case of LG models corresponding to del Pezzo surfaces.

Of course, for the question to make sense one needs to fix a particular identification of the categories on 2 sides of mirror symmetry (and I am not sure if there is a canonical choice) --- let us fix the identification provided by Auroux--Katzarkov--Orlov for this question.

• These are most easily seen, perhaps, in a mirror version where the Lagrangian branes are in (C*)^n \cong R^n x T^n --> R^n. Then line bundles are mirror to sections, and tensor is mirror to convolution --- or simply addition in the torus fibers. – Eric Zaslow Aug 12 '18 at 3:09