Fukaya category

In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold ${\displaystyle (M,\omega )}$ is a category ${\displaystyle {\mathcal {F}}(M)}$ whose objects are Lagrangian submanifolds of ${\displaystyle M}$, and morphisms are Floer chain groups: ${\displaystyle \mathrm {Hom} (L_{0},L_{1})=FC(L_{0},L_{1})}$. Its finer structure can be described in the language of quasi categories as an A-category.

They are named after Kenji Fukaya who introduced the ${\displaystyle A_{\infty }}$ language first in the context of Morse homology[1], and exist in a number of variants. As Fukaya categories are A-categories, they have associated derived categories, which are the subject of the celebrated homological mirror symmetry conjecture of Maxim Kontsevich[2]. This conjecture has been computationally verified for a number of comparatively simple examples.

Formal Definition

Let ${\displaystyle (X,\omega )}$ be a symplectic manifold. For each pair of Lagrangian submanifolds ${\displaystyle L_{0},L_{1}\subset X}$, suppose they intersect transversely, then define the Floer cochain complex ${\displaystyle CF^{*}(L_{0},L_{1})}$ which is a module generated by intersection points ${\displaystyle L_{0}\cap L_{1}}$. The Floer cochain complex is viewed as the set of morphisms from ${\displaystyle L_{0}}$ to ${\displaystyle L_{1}}$. The Fukaya category is an ${\displaystyle A_{\infty }}$ category, meaning that besides ordinary compositions, there are higher composition maps

${\displaystyle \mu _{d}:CF^{*}(L_{d-1},L_{d})\otimes CF^{*}(L_{d-2},L_{d-1})\otimes \cdots \otimes CF^{*}(L_{1},L_{2})\otimes CF^{*}(L_{0},L_{1})\to CF^{*}(L_{0},L_{d}).}$

It is defined as follows. Choose a compatible almost complex structure ${\displaystyle J}$ on the symplectic manifold ${\displaystyle (X,\omega )}$. For generators ${\displaystyle p_{d-1d},\ldots ,p_{01}}$ of the cochain complexes on the left, and any generator ${\displaystyle q_{0d}}$ of the cochain complex on the right, the moduli space of ${\displaystyle J}$-holomorphic polygons with ${\displaystyle d+1}$ faces with each face mapped into ${\displaystyle L_{0},L_{1},\ldots ,L_{d}}$ has a count

${\displaystyle n(p_{d-1d},\ldots ,p_{01};q_{0d})}$

in the coefficient ring. Then define

${\displaystyle \mu _{d}(p_{d-1d},\ldots ,p_{01})=\sum _{q_{0d}\in L_{0}\cap L_{d}}n(p_{d-1d},\ldots ,p_{01})\cdot q_{0d}\in CF^{*}(L_{0},L_{d})}$

and extend ${\displaystyle \mu _{d}}$ in a multilinear way.

The sequence of higher compositions ${\displaystyle \mu _{1},\mu _{2},\ldots ,}$ satisfy the ${\displaystyle A_{\infty }}$ relation because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.

References

1. ^ Kenji Fukaya, Morse homotopy, ${\displaystyle A_{\infty }}$ category and Floer homologies, MSRI preprint No. 020-94 (1993)
2. ^ Kontsevich, Maxim, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel, 1995.
• Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich lectures in Advanced Mathematics
• Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, 46.1, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4836-4, MR 2553465
• Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, 46.2, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4837-1, MR 2548482
• The thread on MathOverflow 'Is the Fukaya category "defined"?'