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Merlin Christ
Merlin Christ
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Functorial Natural equivalence of Dehn and spherical twist of Fukaya category

We consider the setup of Seidel's book. Let $(M,\omega)$ be an exact symplectic manifold with $2c_1=0$. Seidel defines the Fukaya category $\text{Fuk}(M)$ of $M$.

A Lagrangian sphere $L\subset M$, equipped with a brane structure, defines an object of $\text{Fuk}(M)$. There are two twist functors associated to $L$. The first is the autoequivalence of the Fukaya category associated to the Dehn twist on $L$. The second is the spherical twist functor of the spherical object $L\in \text{Fuk}(M)$. It is shown by Seidel, see corollary III.17.17, that pointwise both twist functors are equivalent.

Question: Has it been shown in general that thethere is an equivalence of functors between both twist functors is functorial? Have counterexamples been found?

What I know so far: In remark III.17.22, Seidel writes that ongoing work of Wehrheim-Woodward on Lagrangian correspondences is supposed to show that this equivalence is functorialnatural. I don't know if this indeed follows from their papers, at least it is not explicitly mentioned. Using very different ideas, this functorialthe equivalence of functors is, as far as I understand, established in some examples of complex dimension 1 by S. Opper in arXiv:1904.04859.

Functorial equivalence of Dehn and spherical twist of Fukaya category

We consider the setup of Seidel's book. Let $(M,\omega)$ be an exact symplectic manifold with $2c_1=0$. Seidel defines the Fukaya category $\text{Fuk}(M)$ of $M$.

A Lagrangian sphere $L\subset M$, equipped with a brane structure, defines an object of $\text{Fuk}(M)$. There are two twist functors associated to $L$. The first is the autoequivalence of the Fukaya category associated to the Dehn twist on $L$. The second is the spherical twist functor of the spherical object $L\in \text{Fuk}(M)$. It is shown by Seidel, see corollary III.17.17, that pointwise both twist functors are equivalent.

Question: Has it been shown in general that the equivalence between both twist functors is functorial? Have counterexamples been found?

What I know so far: In remark III.17.22, Seidel writes that ongoing work of Wehrheim-Woodward on Lagrangian correspondences is supposed to show that this equivalence is functorial. I don't know if this indeed follows from their papers, at least it is not explicitly mentioned. Using very different ideas, this functorial equivalence is, as far as I understand, established in some examples of complex dimension 1 by S. Opper in arXiv:1904.04859.

Natural equivalence of Dehn and spherical twist of Fukaya category

We consider the setup of Seidel's book. Let $(M,\omega)$ be an exact symplectic manifold with $2c_1=0$. Seidel defines the Fukaya category $\text{Fuk}(M)$ of $M$.

A Lagrangian sphere $L\subset M$, equipped with a brane structure, defines an object of $\text{Fuk}(M)$. There are two twist functors associated to $L$. The first is the autoequivalence of the Fukaya category associated to the Dehn twist on $L$. The second is the spherical twist functor of the spherical object $L\in \text{Fuk}(M)$. It is shown by Seidel, see corollary III.17.17, that pointwise both twist functors are equivalent.

Question: Has it been shown in general that there is an equivalence of functors between both twist functors? Have counterexamples been found?

What I know so far: In remark III.17.22, Seidel writes that ongoing work of Wehrheim-Woodward on Lagrangian correspondences is supposed to show that this equivalence is natural. I don't know if this indeed follows from their papers, at least it is not explicitly mentioned. Using very different ideas, the equivalence of functors is, as far as I understand, established in some examples of complex dimension 1 by S. Opper in arXiv:1904.04859.

Source Link
Merlin Christ
Merlin Christ
  • 93
  • 6

Functorial equivalence of Dehn and spherical twist of Fukaya category

We consider the setup of Seidel's book. Let $(M,\omega)$ be an exact symplectic manifold with $2c_1=0$. Seidel defines the Fukaya category $\text{Fuk}(M)$ of $M$.

A Lagrangian sphere $L\subset M$, equipped with a brane structure, defines an object of $\text{Fuk}(M)$. There are two twist functors associated to $L$. The first is the autoequivalence of the Fukaya category associated to the Dehn twist on $L$. The second is the spherical twist functor of the spherical object $L\in \text{Fuk}(M)$. It is shown by Seidel, see corollary III.17.17, that pointwise both twist functors are equivalent.

Question: Has it been shown in general that the equivalence between both twist functors is functorial? Have counterexamples been found?

What I know so far: In remark III.17.22, Seidel writes that ongoing work of Wehrheim-Woodward on Lagrangian correspondences is supposed to show that this equivalence is functorial. I don't know if this indeed follows from their papers, at least it is not explicitly mentioned. Using very different ideas, this functorial equivalence is, as far as I understand, established in some examples of complex dimension 1 by S. Opper in arXiv:1904.04859.