I am a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason I have been thinking about this, is that I am doing a computation involving a torus with parameter $\tau \in \mathbb{H}$, and my answer in invariant under $\tau \to \tau+1$, but not $\tau \to -1/\tau$. So I am thinking that maybe I am only using the Kähler structure, not the complex structure.

Of course, the complex structure is given by $\tau \in \mathbb{H}/\rm{PSL}(2, \mathbb{Z})$. I think for the Kähler structure, we choose a class $[\omega] \in H^{2}(X,\mathbb{C})$, which we can parameterize by Kähler parameter $t=t_{1} + i \, t_{2}$: $$t=\frac{1}{2\pi i } \int_{X} [\omega].$$ We identify $t_{2}>0$ with the area of the torus. So unlike the complex structure, Kähler structures related by $\rm{PSL}(2,\mathbb{Z})$, are not necessarily identical, correct? After all, one will have small area, the other large.

So I am confused about how the mirror symmetry acts. If mirror symmetry is some mysterious equivalence of the torus under interchange of the complex and Kähler moduli, does that not imply that the space of equivalent Kähler structures is also $\mathbb{H}/\rm{PSL}(2, \mathbb{Z})$. Is this correct?


This is related to the fundamental question about how to define the Kähler moduli space. The Kähler moduli space is often not as naive as one thinks. A solution for a genus 1 curve is given by Bridgeland in the last section of this article, where the "extended" Kähler moduli is identified with the space of Bridgeland stability conditions modulo autoequivalences of derived category. After taking quotient by the natural $\mathbb{C}^\times$-action, the space is biholomorphic to $\mathbb{H}/\mathrm{PSL}_2(\mathbb{Z})$.

The key idea is, under mirror symmetry, the monodromy (symplectomorphism) group corresponds to the group of autoequivalences. The appearance of the $\mathrm{SL}_2(\mathbb{Z})$-action on the derived category of an elliptic curve was first observed by Mukai in his article on the Fourier-Mukai transformations. This action is not geometric as it mixes the rank and degree of objects.


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