- Source: Fukaya category
In symplectic topology, a Fukaya category of a symplectic manifold
(
X
,
ω
)
{\displaystyle (X,\omega )}
is a category
F
(
X
)
{\displaystyle {\mathcal {F}}(X)}
whose objects are Lagrangian submanifolds of
X
{\displaystyle X}
, and morphisms are Lagrangian Floer chain groups:
H
o
m
(
L
0
,
L
1
)
=
C
F
(
L
0
,
L
1
)
{\displaystyle \mathrm {Hom} (L_{0},L_{1})=CF(L_{0},L_{1})}
. Its finer structure can be described as an A∞-category.
They are named after Kenji Fukaya who introduced the
A
∞
{\displaystyle A_{\infty }}
language first in the context of Morse homology, 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. This conjecture has now been computationally verified for a number of examples.
Formal definition
Let
(
X
,
ω
)
{\displaystyle (X,\omega )}
be a symplectic manifold. For each pair of Lagrangian submanifolds
L
0
,
L
1
⊂
X
{\displaystyle L_{0},L_{1}\subset X}
that intersect transversely, one defines the Floer cochain complex
C
F
∗
(
L
0
,
L
1
)
{\displaystyle CF^{*}(L_{0},L_{1})}
which is a module generated by intersection points
L
0
∩
L
1
{\displaystyle L_{0}\cap L_{1}}
. The Floer cochain complex is viewed as the set of morphisms from
L
0
{\displaystyle L_{0}}
to
L
1
{\displaystyle L_{1}}
. The Fukaya category is an
A
∞
{\displaystyle A_{\infty }}
category, meaning that besides ordinary compositions, there are higher composition maps
μ
d
:
C
F
∗
(
L
d
−
1
,
L
d
)
⊗
C
F
∗
(
L
d
−
2
,
L
d
−
1
)
⊗
⋯
⊗
C
F
∗
(
L
1
,
L
2
)
⊗
C
F
∗
(
L
0
,
L
1
)
→
C
F
∗
(
L
0
,
L
d
)
.
{\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
J
{\displaystyle J}
on the symplectic manifold
(
X
,
ω
)
{\displaystyle (X,\omega )}
. For generators
p
d
−
1
,
d
∈
C
F
∗
(
L
d
−
1
,
L
d
)
,
…
,
p
0
,
1
∈
C
F
∗
(
L
0
,
L
1
)
{\displaystyle p_{d-1,d}\in CF^{*}(L_{d-1},L_{d}),\ldots ,p_{0,1}\in CF^{*}(L_{0},L_{1})}
and
q
0
,
d
∈
C
F
∗
(
L
0
,
L
d
)
{\displaystyle q_{0,d}\in CF^{*}(L_{0},L_{d})}
of the cochain complexes, the moduli space of
J
{\displaystyle J}
-holomorphic polygons with
d
+
1
{\displaystyle d+1}
faces with each face mapped into
L
0
,
L
1
,
…
,
L
d
{\displaystyle L_{0},L_{1},\ldots ,L_{d}}
has a count
n
(
p
d
−
1
,
d
,
…
,
p
0
,
1
;
q
0
,
d
)
{\displaystyle n(p_{d-1,d},\ldots ,p_{0,1};q_{0,d})}
in the coefficient ring. Then define
μ
d
(
p
d
−
1
,
d
,
…
,
p
0
,
1
)
=
∑
q
0
,
d
∈
L
0
∩
L
d
n
(
p
d
−
1
,
d
,
…
,
p
0
,
1
)
⋅
q
0
,
d
∈
C
F
∗
(
L
0
,
L
d
)
{\displaystyle \mu _{d}(p_{d-1,d},\ldots ,p_{0,1})=\sum _{q_{0,d}\in L_{0}\cap L_{d}}n(p_{d-1,d},\ldots ,p_{0,1})\cdot q_{0,d}\in CF^{*}(L_{0},L_{d})}
and extend
μ
d
{\displaystyle \mu _{d}}
in a multilinear way.
The sequence of higher compositions
μ
1
,
μ
2
,
…
,
{\displaystyle \mu _{1},\mu _{2},\ldots ,}
satisfy the
A
∞
{\displaystyle A_{\infty }}
relations because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.
This definition of Fukaya category for a general (compact) symplectic manifold has never been rigorously given. The main challenge is the transversality issue, which is essential in defining the counting of holomorphic disks.
See also
Homotopy associative algebra
References
Bibliography
Denis Auroux, A beginner's introduction to Fukaya categories.
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, vol. 46, 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, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, ISBN 978-0-8218-4837-1, MR 2548482
External links
The thread on MathOverflow 'Is the Fukaya category "defined"?'
Kata Kunci Pencarian:
- Fukaya category
- Kenji Fukaya
- Fukaya, Saitama
- Triangulated category
- Brane
- Mirror symmetry (string theory)
- Homological mirror symmetry
- Floer homology
- String theory
- Denis Auroux
