The Fukaya Category and Homological Mirror Symmetry¶

Job: develop the $A_\infty$-Fukaya category (objects, Floer morphisms, the $m_k$ operations, bounding cochains / curved structure), the wrapped Fukaya category, the proven HMS cases, and the family / relative HMS / Auroux superpotential — with the elliptic curve Polishchuk–Zaslow computation as the worked example.

Core¶

The Fukaya category $\mathrm{Fuk}(X)$¶

For a symplectic manifold $(X^{2n},\omega)$, the (immersed / unobstructed) Fukaya category $\mathrm{Fuk}(X)$ is an $A_\infty$-category whose:

Objects are (unobstructed) Lagrangian submanifolds $L\subset X$ ($\dim_\mathbb{R}L=n$, $\omega|_L=0$) equipped with:
a grading $\alpha_L:L\to\mathbb{R}$, a lift of the phase map $L\to S^1$, $x\mapsto e^{i\alpha_L(x)}=\mathrm{arg}(\Omega|_{T_xL})$ (the Maslov class $\mu_L\in H^1(L,\mathbb{Z})$ must vanish for a grading to exist);
a flat $U(1)$ local system / Novikov-field coefficients;
(optionally) a bounding cochain $b\in CF^1(L,L)$.
Morphisms $\mathrm{Hom}(L_0,L_1)=CF^\bullet(L_0,L_1)$ are the Lagrangian Floer complexes: the chain complex generated by the intersection points $L_0\cap\phi(L_1)$ (for a generic Hamiltonian perturbation $\phi$ of $L_1$, making the intersection transverse), graded by the Maslov index, with the Floer differential $$ \partial x=\sum_y n(x,y)\,y,\qquad n(x,y)=#{u:\mathbb{R}\times[0,1]\to X\text{ rigid holomorphic strip}}, $$ where $u$ solves $\partial_s u+J(\partial_t u)=0$ (the Cauchy–Riemann equation for the compatible almost-complex structure $J$), with boundary $u(\cdot,0)\in L_0$, $u(\cdot,1)\in\phi(L_1)$, and limits $u(-\infty)=x$, $u(+\infty)=y$.
$A_\infty$-structure: the higher products $$ m_k:\mathrm{Hom}(L_0,L_1)\otimes\cdots\otimes\mathrm{Hom}(L_{k-1},L_k)\to\mathrm{Hom}(L_0,L_k)[2-k] $$ count pseudo-holomorphic polygons: rigid $J$-holomorphic maps $u:\Sigma\to X$ from the $k$-pointed disk $\Sigma$ (the Stasheff associahedron) with boundary on $L_0,\dots,L_k$ cyclically and corners mapping to the intersection points. The $m_k$ satisfy the $A_\infty$-relations $$ \sum_{i,j}(-1)^{|x_1|+\cdots+|x_i|-i}m_{j-i+1}(x_1,\dots,m_i(x_{i+1},\dots,x_{i+j}),\dots,x_k)=0, $$ the Stasheff identities; in particular $m_1=\partial$ with $m_1^2=0$ (the Floer differential), $m_2$ the Floer product, and $m_k$ the higher operations.

Fukaya data	Definition
Object	Lagrangian $L$ + grading $\alpha_L$ + flat $U(1)$ local system + bounding cochain $b$
Morphism $CF(L_0,L_1)$	$\mathbb{C}$-span of $L_0\cap\phi L_1$, graded by Maslov index
$m_1=\partial$	counts rigid holomorphic strips $u:\mathbb{R}\times[0,1]\to X$
$m_2$	counts rigid holomorphic bigons (the Floer product)
$m_k$	counts rigid holomorphic $(k+1)$-gons (the Stasheff associahedra)

Obstructions and bounding cochains¶

The Floer differential $\partial=m_1$ satisfies $\partial^2\ne0$ in general — the obstruction is the count of holomorphic disks bounding $L$ (the $m_0$ term, the "curvature"). A Lagrangian is unobstructed if there exists a bounding cochain $b\in CF^1(L,L)$ satisfying the Maurer–Cartan equation $$ m_0 + m_1(b)+m_2(b,b)+m_3(b,b,b)+\cdots=0\qquad\text{(the curved $A_\infty$ MC equation)}. $$ A Lagrangian with such $b$ is (weakly) unobstructed; the deformed differential $\partial_b=m_1+m_2(b,\cdot)+m_2(\cdot,b)+\cdots$ squares to zero, giving a genuine complex. The curved $A_\infty$-category has $m_0\ne0$ as a curvature term; the bounding cochains deform away the curvature.

The split-closed derived Fukaya category $D^\pi\mathrm{Fuk}(X)$ is the category obtained by taking the twisted complexes (the formal cones / shifts / direct sums) and then the idempotent completion (the Karoubi envelope). The HMS conjecture is stated for $D^\pi\mathrm{Fuk}$, not the bare $A_\infty$-category.

Homological mirror symmetry (HMS)¶

Conjecture (Kontsevich HMS). For a mirror pair $(X,\check X)$ of Calabi–Yau manifolds, $$ D^\pi\mathrm{Fuk}(X)\simeq D^b\mathrm{Coh}(\check X) $$ (an equivalence of triangulated / $A_\infty$-categories). For Fano $X$, the mirror is a Landau–Ginzburg model $(\check Y,W)$ with $D^\pi\mathrm{Fuk}(X)\simeq D^b\mathrm{MF}(\check Y,W)$ (matrix factorizations). For Stein / Weinstein $X$, $D^\pi\mathcal{W}(X)\simeq D^b\mathrm{Coh}(\check Y)$ (wrapped Fukaya).

Mirror type	A-side (symplectic)	B-side (algebraic)	Proven cases
Calabi–Yau	$\mathrm{Fuk}(X)$	$D^b\mathrm{Coh}(\check X)$	elliptic curve (Polishchuk–Zaslow); abelian variety (Polishchuk); quartic / quintic (Sheridan); toric CY hypersurfaces
Fano	$\mathrm{Fuk}(X)$	$D^b\mathrm{MF}(\check Y,W)$ (LG model)	toric Fano (Abouzaid, via the coherent-constructible correspondence); $\mathbb{P}^n$ (Auroux–Katzarkov–Orlov)
Stein / Weinstein	$\mathcal{W}(X)$ (wrapped)	$D^b\mathrm{Coh}(\check Y)$ (affine / categorical mirror)	cotangent bundles (Nadler–Zaslow, Abouzaid); Milnor fibers (Bee–Tonks); Weinstein tubular neighborhoods

The wrapped Fukaya category¶

For a Liouville / Weinstein manifold $(X,\omega=d\lambda)$ (the Liouville vector field $Z$ with $\omega=d\lambda$, $i_Z\omega=\lambda$, expanding at infinity), the wrapped Fukaya category $\mathcal{W}(X)$ has:

objects: asymptotically non-compact Lagrangians (the cocores of the critical handles, or the Lagrangians going to infinity) — not the compact Lagrangians of $\mathrm{Fuk}(X)$;
morphisms: the wrapped Floer complexes $CF(L_0,L_1)$, generated by Reeb-flow chords at infinity (the Hamiltonian flow with the quadratic / linear-at-infinity Hamiltonian, the wrapping), with the wrapped differential counting pseudo-holomorphic polygons in the wrapping direction;
$A_\infty$-structure: the wrapped $m_k$ counts polygons with the wrapping; the continuation maps / the Reeb flow encode the wrapping.

Theorem (Abouzaid–Seidel generation). The wrapped Fukaya category $\mathcal{W}(W)$ of a Weinstein manifold $W$ (obtained by attaching critical handles to a subcritical skeleton) is generated by the cocores of the critical handles — the categorical mirror of a Lefschetz fibration / the Orlov / Knörrer periodicity. In particular, $\mathcal{W}(W)$ is generated by finitely many objects (one per critical handle).

Compact $\mathrm{Fuk}(X)$	Wrapped $\mathcal{W}(X)$
compact Lagrangians	non-compact / asymptotic Lagrangians (cocores, stops)
morphisms: intersections $L_0\cap L_1$	morphisms: Reeb chords at infinity (wrapped Floer)
finite-dimensional complexes	typically infinite-dimensional (the wrapping)
generation: by Lagrangian spheres / tori	generation: by cocores of the critical handles (Abouzaid–Seidel)
HMS target: $D^b\mathrm{Coh}(\check X)$ for CY	HMS target: $D^b\mathrm{Coh}(\check Y)$ for affine / Stein mirror

Family / relative HMS and the SYZ connection¶

Family HMS: the SYZ mirror as a family of tori, the Fukaya category fiberwise; the family HMS equates the Fukaya category of $X$ with the (derived) category of a family of coherent sheaves on the mirror base.
Relative HMS: the wrapped Fukaya category of the complement $X\setminus D$ of an ample divisor $D$ is mirror to a Landau–Ginzburg model $(\check Y,W)$.

Theorem (Auroux). For $X$ Fano (or CY) with an anticanonical / ample divisor $D$, the mirror of $X\setminus D$ is a Landau–Ginzburg model $(\check Y,W)$ where $\check Y=(\mathbb{C}^*)^n$ (the SYZ mirror of the complement) and the superpotential $$ W=\sum_i c_i z^{\beta_i} $$ is the sum over the holomorphic disk classes $\beta_i$ bounding the Lagrangian torus fibers, with coefficients $c_i$ the disk counts — the Auroux superpotential from the count of holomorphic disks.

Wall-crossing / scattering: the instanton corrections to the SYZ mirror (the holomorphic disks bounding the torus fibers) produce the wall-crossing phenomena — the superpotential / the Floer products change across walls (the codimension-1 loci where disks appear), governed by the Gross–Siebert scattering diagrams (see SYZ & Gross–Siebert).

HMS variant	A-side	B-side
Compact CY	$\mathrm{Fuk}(X)$	$D^b\mathrm{Coh}(\check X)$
Fano	$\mathrm{Fuk}(X)$	$D^b\mathrm{MF}(\check Y,W)$ (LG / matrix factorizations)
Stein / Weinstein	$\mathcal{W}(X)$ (wrapped)	$D^b\mathrm{Coh}(\check Y)$ (affine)
Complement of divisor	$\mathcal{W}(X\setminus D)$	LG $(\check Y,W)$ (Auroux superpotential)
Family / SYZ	$\mathrm{Fuk}(X)$ fiberwise	family of sheaves on the SYZ base

See it¶

Elliptic curve HMS (Polishchuk–Zaslow). Let $E_\tau=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ be the elliptic curve with complex parameter $\tau$ (the B-side) and $\check E_\rho$ the symplectic torus $T^2$ with complexified area $\rho$ (the A-side, $\mathrm{Im}\,\rho$ the symplectic area, $\mathrm{Re}\,\rho$ the B-field). The Fukaya category $\mathrm{Fuk}(\check E_\rho)$ has objects the simple closed curves of rational slope $p/q\in\mathbb{Q}\cup\{\infty\}$ (the Lagrangian circles $L_{p/q}$, with the grading + a flat $U(1)$ local system on $L_{p/q}$). The Floer complex $CF(L_{p/q},L_{p'/q'})$ is nonzero iff the curves are transverse, i.e. $pq'-p'q=\pm1$ (the determinant of the slope matrix is $\pm1$), in which case $L_{p/q}\cap L_{p'/q'}$ is one point and $CF=\mathbb{C}\cdot\mathrm{pt}$. The Floer product $m_2$ of two such circles $L_{p/q},L_{p'/q'}$ (with $pq'-p'q=1$) with a third $L_{p''/q''}$ (with the mediant condition $(p+p')/(q+q')=p''/q''$, the Farey sum) counts the unique triangle bounded by the three curves, giving $$ m_2(L_{p/q},L_{p'/q'})\propto L_{(p+p')/(q+q')}. $$ This reproduces the theta-line multiplication: the line bundle $\mathcal{O}(D_{p/q})$ on $E_\tau$ (the mirror) corresponds to the Lagrangian circle $L_{p/q}$ (a fiber circle / a rank-1 local system), and the multiplication of theta functions (the structure sheaf / line-bundle algebra) matches the Floer product. The HMS equivalence $$ D^\pi\mathrm{Fuk}(\check E_\rho)\xrightarrow{\ \sim\ }D^b\mathrm{Coh}(E_\tau) $$ sends the slope $p/q$ circle to the line bundle / skyscraper sheaf $\mathcal{O}(D_{p/q})$ (or $\mathcal{O}_{(p/q)}$ the skyscraper), with the Ext-algebras $\mathrm{Ext}^*(\mathcal{O}_{p/q},\mathcal{O}_{p'/q'})$ matching the Floer complexes $CF^*(L_{p/q},L_{p'/q'})$ — the prototypical HMS computation, where the slope / Farey structure of the Fukaya category reproduces the categorical structure of coherent sheaves on the elliptic curve.

Check¶

Why does the Floer differential $\partial^2\ne0$ in general, and what is the geometric meaning of the $m_0$ obstruction?

$\partial^2$ counts pairs of rigid holomorphic strips $u_1,u_2$ forming a broken trajectory, but the compactification of the 1-dimensional moduli of strips includes boundary components that are broken strips (canceling $\partial^2$) and components that are rigid holomorphic disks bounding $L$ (the "boundary at infinity" of the strip moduli). The count of these disks is $m_0\in CF^2(L,L)$ (the curvature), and $\partial^2=m_0$ as an operator (the $A_\infty$-relation $m_1^2=m_0$ with the multiplication by $m_0$). A bounding cochain $b$ with $m_0+m_1(b)+m_2(b,b)+\cdots=0$ cancels the curvature, making $\partial_b^2=0$. The geometric meaning: $m_0$ counts the holomorphic disks bounding $L$, the obstruction to $L$ defining an object of the (uncurved) Fukaya category.

Show that for the wrapped Fukaya category of the cotangent bundle $T^*M$ of a closed manifold $M$, the cocores (the zero-section fibers / the conormal bundles) generate, and identify the mirror.

By Nadler–Zaslow (and Abouzaid's refinement), $\mathcal{W}(T^*M)$ is equivalent to the category of local systems (constructible sheaves) on $M$ — the cotangent zero-section $M\subset T^*M$ (a cocore / the core of the cotangent bundle) maps to the constant local system, and the conormal bundles $\nu^*S$ to submanifolds $S\subset M$ map to the local systems on $S$. The generation by cocores (Abouzaid–Seidel) says $M$ itself (the zero-section, the core of $T^*M$) generates $\mathcal{W}(T^*M)$; the mirror is the algebra of differential operators / the constructible-sheaf category on $M$ (the microlocal mirror).

Pitfalls¶

HMS is proven only for specific cases (elliptic curve, toric Fano, the quintic, abelian varieties) — NOT for general CY. Do not state "HMS is a theorem"; state the proven cases explicitly.
The Fukaya category is curved (the bounding cochains are essential, not a technicality) — without bounding cochains, $\partial^2\ne0$ and the category is not triangulated. The "deformation by $b$" is the definition of the object, not an afterthought.
The split-closed derived category $D^\pi\mathrm{Fuk}$ is the HMS target, not the bare $A_\infty$-category — the twisted complexes and the idempotent completion are required for the triangulated structure.
The wrapped Fukaya category is for non-compact / Weinstein manifolds; the compact Fukaya $\mathrm{Fuk}(X)$ and the wrapped $\mathcal{W}(X)$ are different (different objects, different morphisms) — do not conflate.
The Auroux superpotential is the sum over holomorphic disk classes; the wall-crossing is the change of the superpotential / the Floer products across walls (the disks appear/disappear), governed by the scattering diagrams.
The Maslov class / grading: a Lagrangian must have vanishing Maslov class $\mu_L=0\in H^1(L,\mathbb{Z})$ to admit a grading; ungraded Lagrangians live in a $\mathbb{Z}/2$-graded category (and the HMS with the bounded derived category requires the $\mathbb{Z}$-grading).

Refs¶

Fukaya–Oh–Ohta–Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction (AMS 2009); Seidel, Fukaya Categories and Picard–Lefschetz Theory (EMS 2008)
Polishchuk–Zaslow, "Categorical mirror symmetry: the elliptic curve" (Adv. Theor. Math. Phys. 1998)
Sheridan, "Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space" (Invent. 2015)
Abouzaid, "A geometric criterion for generating the Fukaya category" (Publ. Math. IHÉS 2010); Abouzaid–Seidel, "An open string analogue of Vafa–Witten theory" (2010)
Auroux, "Mirror symmetry and $T$-duality in the complement of an anticanonical divisor" (J. Gökova Geom. Topol. 2007); Nadler–Zaslow, "Constructible sheaves and the Fukaya category" (J. Symplectic Geom. 2009)

Next¶

MS index · Foundations · Gromov–Witten & Quantum Cohomology · SYZ & Gross–Siebert · Hitchin & Geometric Langlands · Symplectic / Floer · Homological algebra

Fukaya data	Definition
Object	Lagrangian \(L\) + grading \(\alpha_L\) + flat \(U(1)\) local system + bounding cochain \(b\)
Morphism \(CF(L_0,L_1)\)	\(\mathbb{C}\)-span of \(L_0\cap\phi L_1\), graded by Maslov index
\(m_1=\partial\)	counts rigid holomorphic strips \(u:\mathbb{R}\times[0,1]\to X\)
\(m_2\)	counts rigid holomorphic bigons (the Floer product)
\(m_k\)	counts rigid holomorphic \((k+1)\)-gons (the Stasheff associahedra)