# Mirror to the dualizing sheaf

I wonder - is there a general characterization of the object in the Fukaya category that is mirror to the dualizing sheaf on the other side?

This question has two cases: 1. CY 2. Non-CY

In 1. what I know is that by Polischuk-Zaslow the mirror of the dualizing sheaf in case of the 2-torus is a linear Lagrangian torus of type (1,0).

In 2. there is a subquestion - what is the analogue of the dualizing sheaf for the category of matrix factorizations of a Landau-Ginzburg model?

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I'll comment on the related question "what is the Serre functor for the Fukaya category?"

Calabi-Yau setting

The Serre functor $S$, by definition, satisfies $\mathsf{Hom}(X,SY) \cong \mathsf{Hom}(Y,X)^\vee$; since it's characterized categorically, it's preserved by the derived equivalences which arise in mirror symmetry. For the derived category $D^b\mathsf{Coh}(X)$ of a non-singular projective $n$-variety, $S= \cdot \otimes \omega_X [n]$, where $\omega_X$ is the dualizing sheaf. So, when $X$ is Calabi-Yau, it's simply a shift.

For a Fukaya category of compact Lagrangians (of dimension $n$) in a symplectic manifold, $S$ is just a shift by $n$, because of the Floer-theoretic Poincare duality $HF^\ast(X,Y) \cong HF^{n-\ast}(Y,X)^\vee$. At the fully precise $A_\infty$-level, the claim that the Serre functor is a shift is partly conjectural.

The mirror to $\mathcal{O}$ is a section $\sigma$ of the SYZ fibration, so the mirror to the canonical sheaf is $\sigma[n]$.

LG models

To get a more interesting answer, consider Fukaya categories of Landau-Ginzburg models, a.k.a. Fukaya-Seidel categories. These arise as mirrors to Fano manifolds. Out of caution, I'll assume that the L-G model is a symplectic Lefschetz fibration $E\to \mathbb{C}$. The objects of the category are Lagrangian submanifolds which map to eventually-horizontal paths in $\mathbb{C}$ (for instance, Lefschetz thimbles). Kontsevich proposed that the Serre functor should then be the "wrapping" or "monodromy" functor. This has been proved (at least at the level of objects, probably more), by Seidel (cf. his Symplectic homology as Hochschild homology and Vanishing cycles and mutation).

The wrapping functor is defined as follows. Take a circle of large radius $R$ in $\mathbb{C}$, and consider the Dehn twist $\delta$ along this circle. So $\delta(z)=e^{i\rho(|z|)}z$, where the angle $\rho(|z|)$ runs from $0$ when $|z| < R-1$ to $2\pi$ when $|z| > R+1$. There's a symplectomorphism $\Phi$ of $E$, covering $\delta$, given by symplectic parallel transport of the fibration over the arc from $z$ to $\delta(z)$. The wrapping functor takes a Lagrangian $L$ to $\Phi(L)$. It takes a standard Lefschetz thimble (fibering over a ray) to a "once-wrapped thimble", i.e. a thimble for a path that wraps once around the circle.

In the case of the LG mirror to $\mathbb{CP}^2$, the mirror to $\mathcal{O}$ (which is one of the Beilinson generators of $D^b \mathsf{Coh}(\mathbb{CP}^2)$) is a thimble, so the mirror to the canonical sheaf is a once-wrapped thimble.

The proof that the wrapping functor is the Serre functor invokes a general characterization of the Serre functor in triangulated categories with full exceptional collections in terms of the algebraic process of "mutation". The thimbles associated with a collection of vanishing paths form a full exceptional collection, and mutation corresponds to Hurwitz moves on vanishing paths.

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Tim - thank you! This is very helpful and informative. A followup question - is it known what the Serre functor is for a component of the Fukaya category of \mathbb{C}P^2 corresponding to a given critical value of the superpotential? –  user21485 Feb 20 '12 at 13:19
You're welcome. You're referring, I take it, to the mirror symmetry between the Fukaya category of a Fano and matrix factorizations of its LG model. Since these Fanos are compact symplectic manifolds, their Fukaya categories are Calabi-Yau, i.e., the Serre functor is a shift w.r.t. the cyclic grading. Matrix factorizations also form a Calabi-Yau category. –  Tim Perutz Feb 20 '12 at 15:27