Special Lagrangian fibrations and McLean's theorem

Intuition Beginner

A Calabi-Yau space carries two pieces of geometric furniture beyond its curved metric. One is the symplectic form, a way of measuring area inside the space. The other is a holomorphic volume form, a complex-valued gadget that measures oriented volume of half-dimensional pieces. A submanifold that sits perfectly with respect to both pieces at once is called special Lagrangian. It is the most balanced kind of half-dimensional surface a Calabi-Yau admits.

Two conditions express the balance. First, the area form must measure zero on the surface: in every direction the surface spreads, no symplectic area accumulates. This is the Lagrangian condition. Second, the surface must line up with the real part of the volume form, so that the volume form's real part reproduces the surface's own honest volume and its imaginary part reads zero. A surface meeting both demands is special Lagrangian, and a calibration argument shows it has the least possible volume among all nearby surfaces in its class.

The deep question is: once you have one such surface, how many others sit beside it? Can you wiggle it and stay special Lagrangian, or is it rigid and locked in place? McLean answered this in 1998. The answer is strikingly clean. The ways to wiggle a compact special Lagrangian surface form a smooth space, and the number of independent wiggle directions equals a topological count of the surface itself — the number of independent loops you can draw on it that do not bound.

For the case that drives mirror symmetry the surface is a torus. A torus has a first loop-count equal to its dimension, so the wiggle space has the same dimension as the torus. The wiggle directions assemble into a base, and the original tori, swept out over that base, build the whole Calabi-Yau as a family of tori. This is the fibration picture: a Calabi-Yau presented as tori stacked over a flat base, with the base itself being the McLean moduli space of the fibres. The mirror Calabi-Yau is the same base with each torus swapped for its dual.

Visual Beginner

A two-panel picture. Left panel: a single torus $L$ drawn as a doughnut sitting inside a softly shaded Calabi-Yau, with two highlighted loops on it — one around the hole, one through the hole — labelled "the loops that do not bound." A small arrow nudges the doughnut to a faint copy of itself nearby, labelled "a special Lagrangian wiggle." Right panel: the moduli space of all these wiggled tori, drawn as a small flat patch $B$ with coordinate axes; each point of the patch carries a tiny doughnut floating above it, so the whole Calabi-Yau appears as doughnuts stacked over the flat patch. A caption box reads "wiggle directions = harmonic loops = the base $B$."

The point of the picture: the dimension of the flat base patch on the right equals the number of highlighted loops on the left. McLean's theorem is the precise statement that these two counts agree, and the fibration picture is what you get when the wiggled tori sweep out the entire Calabi-Yau without gaps. The mirror replaces each floating doughnut by its dual doughnut over the same flat patch.

Worked example Beginner

Take the flat model: the space of two complex coordinates, with its standard area form and the volume form $d{z}_1\wedge d{z}_2$. Inside it sits a flat two-torus $L$ built from a square lattice — pairs of angles $(\alpha ,\beta )$ with the real coordinates real and the imaginary coordinates locked to the lattice. We check it is special Lagrangian and count its wiggle directions by hand.

Step 1. Lagrangian. The area form is a sum of two basic area pieces, one for each complex coordinate. On the flat torus the imaginary directions are held fixed, so each basic area piece pairs a moving direction with a frozen one and reads zero. The total area form measures zero on $L$, so $L$ is Lagrangian.

Step 2. Special. The volume form $d{z}_1\wedge d{z}_2$ restricted to the real directions of $L$ becomes $d{x}_1\wedge d{x}_2$, a purely real expression. Its imaginary part is zero and its real part is the honest area form of the flat torus. So $L$ is special of phase zero, meeting both demands.

Step 3. Count the wiggles. The torus $L$ has two independent loops — one for each angle — that do not bound. McLean's count says the number of special Lagrangian wiggle directions equals this loop-count, which is two. And indeed the torus can be slid in two independent transverse directions while staying flat and special: two real parameters, matching the count.

What this tells us: the wiggle space of the flat two-torus is two-dimensional, exactly its loop-count. Sweeping the torus through its two slide directions fills out a four-real-dimensional region of the ambient space as a family of two-tori over a two-dimensional base. This is the simplest special Lagrangian fibration, and the base is the McLean moduli space.

Check your understanding Beginner

Formal definition Intermediate+

Fix a Calabi-Yau $n$-fold $(X,J,\omega ,\Omega )$: a compact Kähler manifold of complex dimension $n$ with Ricci-flat Kähler form $\omega$ and a nowhere-vanishing parallel holomorphic $n$-form $\Omega$, normalised so that $$ \frac{\omega^n}{n!} = (-1)^{n(n-1)/2}\left(\frac{i}{2}\right)^n \Omega \wedge \overline{\Omega}. $$

Definition (special Lagrangian submanifold; Harvey-Lawson 1982). A real $n$-dimensional oriented submanifold $L\subset X$ is special Lagrangian of phase $\theta \in \mathbb{R}/2\pi \mathbb{Z}$ if $$ \omega|_L = 0 \qquad \text{and} \qquad \Im!\big(e^{-i\theta}\Omega\big)\big|_L = 0. $$ Equivalently $\Re (e^{-i\theta }\Omega ){|}_L={\mathrm{vol}}_L$ is the Riemannian volume form of $L$. The form $\Re (e^{-i\theta }\Omega )$ is a calibration: it is closed, its comass is one, and a submanifold on which it restricts to the volume form is calibrated, hence volume-minimising in its homology class (Harvey-Lawson 1982). We take $\theta =0$ unless stated; the phase only rotates $\Omega$.

Definition (the moduli space of special Lagrangian deformations). Let $L\subset X$ be compact and special Lagrangian. The moduli space $$ \mathcal{M}_L = {, L' \subset X : L' \text{ special Lagrangian, } L' \text{ a small deformation of } L ,} $$ is the set of nearby special Lagrangian submanifolds, topologised as a subspace of compact submanifolds near $L$ in the $C^{\infty }$ topology.

The deformation map. A normal vector field along $L$ deforms $L$ to a nearby submanifold $L^{\prime }$. Since $L$ is Lagrangian, the symplectic form gives a bundle isomorphism between the normal bundle $\nu (L)$ and the cotangent bundle $T^{\ast }L$, sending a normal field $V$ to the 1-form ${\iota }_V\omega$. Under this identification a deformation of $L$ corresponds to a 1-form $\alpha \in {\Omega }^1(L)$. The linearised special Lagrangian conditions at $L$ read $$ \frac{d}{dt}\Big|{0}\big(\omega|{L_t}\big) = d\alpha, \qquad \frac{d}{dt}\Big|{0}\big(\Im\Omega|{L_t}\big) = d(\alpha), $$ where $$istheHodgestaroftheinducedmetricon$L$.SothedeformationstaysspecialLagrangiantofirstorderexactlywhen$d\alpha = 0$and$d(*\alpha) = 0$,thatis,when$\alpha$ is closed and coclosed — a harmonic 1-form.

Definition (the McLean metric and the dual affine structures; Hitchin 1997). On ${\mathcal{M}}_L$ the tangent space at $[L]$ is the space ${\mathcal{H}}^1(L)$ of harmonic 1-forms. The McLean $L^2$ metric is $$ g_{\mathcal{M}}(\alpha, \beta) = \int_L \alpha \wedge *\beta, \qquad \alpha, \beta \in \mathcal{H}^1(L). $$ There are two distinguished flat (integral affine) coordinate systems on ${\mathcal{M}}_L$: the symplectic periods $b\mapsto {\int }_{{\gamma }_i}(\text{flux of }\omega )$ and the complex periods $b\mapsto {\int }_{{\gamma }_i}\Im \Omega$ over a basis $\{{\gamma }_i\}$ of $H_1(L;\mathbb{Z})$. These two affine structures are exchanged by Legendre transform, the differential-geometric origin of the dual affine structures on the SYZ base 04.12.10.

Definition (special Lagrangian fibration). A special Lagrangian fibration of $X$ is a map $\pi :X\to B$ onto a real $n$-manifold $B$ whose generic fibre $L_b={\pi }^{-1}(b)$ is a compact special Lagrangian submanifold diffeomorphic to a torus $T^n$, smooth away from a discriminant $\Delta \subset B$ of codimension at least two. By McLean's theorem the smooth locus $B^{\circ }=B\setminus \Delta$ is locally identified with the moduli space ${\mathcal{M}}_{L_b}$, so $B^{\circ }$ inherits the integral affine structure above.

Counterexamples to common slips

• "McLean's dimension is $n$, the dimension of $L$." The dimension of ${\mathcal{M}}_L$ is $b_1(L)$, the first Betti number, not $n$. For $L=T^n$ these agree because $b_1(T^n)=n$, but for a special Lagrangian sphere $L=S^n$ (which occurs as a vanishing cycle) $b_1(S^n)=0$ for $n\ge 2$: such an $L$ is rigid, with a zero-dimensional moduli space. The fibration picture needs torus fibres precisely so the moduli has full dimension $n$.

• "The linearised operator is the Laplacian." The linearisation is the first-order operator $\alpha \mapsto (d\alpha ,d\ast \alpha )$, whose kernel is the harmonic 1-forms. The Laplacian $\Delta =d{d}^{\ast }+d^{\ast }d$ appears only when one assembles $d\oplus d^{\ast }$ into a self-adjoint elliptic operator to invoke Hodge theory; the deformation equation itself is the first-order closed-and-coclosed system.

• "Special Lagrangian deformations are obstructed like complex ones." They are unobstructed: the cokernel of the linearisation vanishes (it is again ${\mathcal{H}}^1$ by self-adjointness), so the implicit function theorem applies and every harmonic 1-form integrates. Complex submanifolds, governed by the $\bar{\partial }$-operator with a generally non-vanishing $H^1$ of the normal bundle, are obstructed — the contrast is the heart of McLean's result.

Key theorem with proof Intermediate+

Theorem (McLean's deformation theorem; McLean 1998 Comm. Anal. Geom. 6). Let $L$ be a compact special Lagrangian submanifold of a Calabi-Yau $n$-fold $X$. The moduli space ${\mathcal{M}}_L$ of special Lagrangian deformations of $L$ is a smooth manifold of dimension $b_1(L)$, and the natural map sending a deformation to its infinitesimal data is an isomorphism of the tangent space $T_{[L]}{\mathcal{M}}_L$ with the space ${\mathcal{H}}^1(L)$ of harmonic 1-forms on $L$. The deformations are unobstructed.

Proof. The argument linearises the special Lagrangian conditions, identifies the kernel with harmonic 1-forms by Hodge theory, and shows the cokernel vanishes so that the implicit function theorem produces a smooth moduli space.

Step 1 (the deformation 1-form). Because $L$ is Lagrangian, the map $V\mapsto {\iota }_V\omega$ is a bundle isomorphism $\nu (L)\stackrel{\sim }{\to }{T}^{\ast }L$ from the normal bundle to the cotangent bundle. A nearby submanifold $L_{\alpha }$ is the image of $L$ under the normal field dual to a 1-form $\alpha \in {\Omega }^1(L)$ via the exponential map. The two forms restricted to the moving submanifold satisfy, with $\iota$ the contraction and Cartan's formula ${\mathcal{L}}_V=d{\iota }_V+{\iota }_{V}d$ together with $d\omega =0$ and $d\Im \Omega =0$, $$ \frac{d}{dt}\Big|{0}\big(\omega|{L_{t\alpha}}\big) = \big(\mathcal{L}V \omega\big)\big|L = d(\iota_V\omega)|L = d\alpha, $$ $$ \frac{d}{dt}\Big|{0}\big(\Im\Omega|{L{t\alpha}}\big) = \big(\mathcal{L}_V \Im\Omega\big)\big|_L = d(\iota_V,\Im\Omega)|_L = d(*\alpha). $$ The last equality uses the pointwise identity ${\iota }_V\Im \Omega {|}_L=\ast {\iota }_V\omega {|}_L=\ast \alpha$, which holds because on a special Lagrangian $L$ the contraction of $\Im \Omega$ with a normal field equals the Hodge dual of the contraction of $\omega$ — a computation in the special Lagrangian frame where $\omega$ and $\Re \Omega ,\Im \Omega$ take their flat normal forms (McLean 1998, Lemma; Joyce 2007, Ch. 8).

Step 2 (the linearised system and its kernel). The deformation $L_{\alpha }$ remains special Lagrangian to first order exactly when both derivatives vanish: $$ d\alpha = 0 \qquad \text{and} \qquad d(\alpha) = 0. $$ The second equation says $\alpha$isclosed,equivalently$d^\alpha = 0$where$d^ = -d$ is the codifferential. So the kernel of the linearisation is $$ {\alpha \in \Omega^1(L) : d\alpha = 0,\ d^*\alpha = 0} = \mathcal{H}^1(L), $$ the harmonic 1-forms. By the Hodge theorem on the compact Riemannian manifold $L$, the inclusion of harmonic forms into closed forms induces an isomorphism ${\mathcal{H}}^1(L)\cong H^1(L;\mathbb{R})$, so $\dim {\mathcal{H}}^1(L)=b_1(L)$.

Step 3 (unobstructedness). Assemble the linearisation into the operator $$ P : \Omega^1(L) \to \Omega^2(L) \oplus \Omega^0(L), \qquad P\alpha = (d\alpha,\ d^\alpha), $$ a first-order elliptic operator (its symbol is the symbol of $d \oplus d^$,whichisinjectiveoffthezerocovector).Itskernelis$\mathcal{H}^1(L)$,computedinStep2.Theformaladjoint$P^* = d^* \oplus d$haskernelthepairs$(\beta, f)$with$d^*\beta = 0$and$df = 0$thatalsolieintheimage-orthogonal-complement;restrictingtotherelevantcomponent,thecokernelof$P$ontheslicetransversetoreparametrisationsisagainisomorphicto$\mathcal{H}^1(L)$byself-adjointnessoftheHodgeLaplacian$\Delta = P^P$onthisslice.Thepointisthattheobstructionspace—thecokernelresponsiblefornon-integrability—vanishesonceonequotientsbythediffeomorphismsof$L$, because every harmonic 1-form is closed and hence is the differential of nothing it must cancel. Concretely, McLean shows that the nonlinear special Lagrangian operator $$ F : \alpha \mapsto \big(,d\alpha,\ d(\alpha) + Q(\alpha),\big), $$ with $Q$ the higher-order remainder, has surjective linearisation onto the closed-coclosed slice with kernel ${\mathcal{H}}^1(L)$, so by the implicit function theorem in suitable Hölder or Sobolev spaces the solution set is a smooth manifold of dimension $\dim {\mathcal{H}}^1(L)=b_1(L)$, with tangent space ${\mathcal{H}}^1(L)$. Elliptic regularity upgrades the solutions to smooth submanifolds. $\square$

Bridge. McLean's theorem builds toward the SYZ fibration picture of 04.12.10, where the base $B^{\circ }$ is exactly the McLean moduli space of the torus fibres, and it appears again in 04.12.16 as the differential-geometric foundation that makes the A-model and B-model live over a common affine base. The foundational reason the moduli is smooth is that the special Lagrangian linearisation is the harmonic-form system $d\alpha =0$, $d^{\ast }\alpha =0$, whose kernel and cokernel are both ${\mathcal{H}}^1(L)$ by Hodge self-adjointness, so the obstruction space is dual to the deformation space and they cancel; this is exactly the mechanism by which the dimension equals $b_1(L)$ rather than something larger. The McLean metric is dual to its own Legendre transform, and the two period systems are dual under it — putting these together, the central insight is that the same compact special Lagrangian $L$ carries two flat affine coordinate systems on its moduli, one from $\omega$-periods and one from $\Im \Omega$-periods, and the bridge to mirror symmetry is that these two structures generalise the symplectic-and-complex pairing into the dual SYZ tori, so that fibrewise T-duality is the exchange of the two affine structures McLean's moduli already carries.

Exercises Intermediate+

Advanced results Master

McLean's theorem is the local engine of the SYZ programme, and its consequences organise into the global fibration structure, the metric geometry of the base, and the singularity theory that the compactness of real fibres forces.

The first strand is the global fibration and the affine base. Granted McLean's local smoothness, a special Lagrangian fibration $\pi :X\to B$ presents $B^{\circ }=B\setminus \Delta$ as a manifold each of whose points is a special Lagrangian torus, with $T_b{B}^{\circ }\cong {\mathcal{H}}^1(L_b)$. The lattice $H_1(L_b;\mathbb{Z})\subset {\mathcal{H}}^1(L_b{)}^{\ast }$ varies covariantly, producing the integral affine structure with monodromy in $\mathrm{GL}(n,\mathbb{Z})$ around $\Delta$. The Hitchin 1997 ^{[Hitchin 1997]} semi-flat construction realises the total space, near the large-complex-structure limit, as $X^{\circ }\cong T^{\ast }{B}^{\circ }/(\text{period lattice})$ with its canonical symplectic form, and the dual torus bundle ${\check{X}}^{\circ }\cong T{B}^{\circ }/(\text{dual lattice})$ as the mirror. The McLean metric $\mathrm{Hess}\varphi$ and its Legendre dual $\mathrm{Hess}\check{\varphi }$ are the semi-flat Kähler metrics on the two sides, each solving the real Monge-Ampère equation $\det ({\partial }^2\varphi /\partial x_i\partial x_j)=\text{const}$ in the large-complex-structure approximation 04.12.10.

The second strand is rigidity versus deformability through topology. Because the dimension is $b_1(L)$, the topology of $L$ dictates everything. A special Lagrangian sphere $S^n$ ($n\ge 2$) has $b_1=0$ and is rigid — these are the vanishing cycles at nodes, and they sit over the discriminant $\Delta$. A special Lagrangian torus $T^n$ has $b_1=n$ and moves in a full $n$-parameter family — these are the smooth fibres. The interpolation between the two, as a torus fibre degenerates to pinch a cycle and acquire a singular point, is governed by the change in $b_1$ across $\Delta$, and McLean's count is what makes the discriminant have the expected codimension.

The third strand is singularity theory of the fibres, where the compactness of real special Lagrangians becomes a genuine constraint. McLean's theorem is a statement about smooth compact $L$; near $\Delta$ the fibres are singular and the deformation theory must be extended. Joyce's analysis 04.12.10 classifies the generic codimension-two singularities (focus-focus in dimension two, and a short list of model cones in dimension three), each carrying its own local deformation theory. The harmonic-form count of McLean is the regular-locus shadow of this finer picture, and the obstruction to a globally smooth special Lagrangian fibration — central to the precise modern form of SYZ — is exactly the obstruction theory of these singular fibres that McLean's smooth theorem does not see.

Synthesis. The central insight is that McLean's deformation theorem is the foundational reason the SYZ base exists as a smooth affine manifold: the moduli of a compact special Lagrangian torus is smooth of dimension $b_1(L)=n$ precisely because the linearised special Lagrangian operator $d\oplus d^{\ast }$ has kernel and cokernel both equal to ${\mathcal{H}}^1(L)$, so the obstruction is dual to the deformation and they cancel. This is exactly the mechanism, and putting these together with the Hodge theorem gives the dimension as $b_1$, the topological count that the fibration picture demands of its torus fibres. The McLean $L^2$ metric is dual to itself under Legendre transform, and the two period systems — $\omega$-periods and $\Im \Omega$-periods — generalise the symplectic-and-complex pairing into the two integral affine structures that fibrewise T-duality exchanges; the bridge from McLean's local theorem to the global SYZ conjecture 04.12.10 is precisely this pair of dual affine structures carried already by the moduli of a single fibre. The pattern recurs throughout the mirror-symmetry programme: the unobstructed real elliptic theory on the A-side stands against the obstructed complex theory on the B-side, and mirror symmetry 04.12.16 is the assertion that these two faces are exchanged, with McLean's harmonic-form moduli the differential-geometric heart of the exchange.

Full proof set Master

Proposition 1 (the linearised special Lagrangian operator is elliptic with kernel ${\mathcal{H}}^1(L)$). Let $L$ be a compact special Lagrangian submanifold of a Calabi-Yau $n$-fold. The operator $P:{\Omega }^1(L)\to {\Omega }^2(L)\oplus {\Omega }^0(L)$, $P\alpha = (d\alpha, d^\alpha)$,isoverdetermined-elliptic,and$\ker P = \mathcal{H}^1(L)$with$\dim\ker P = b_1(L)$.*

Proof. The principal symbol of $P$ at a covector $\xi \ne 0$ is ${\sigma }_{\xi }(\alpha )=(\xi \wedge \alpha ,-{\iota }_{{\xi }^{♯}}\alpha )$. If ${\sigma }_{\xi }(\alpha )=0$ then $\xi \wedge \alpha =0$ and ${\iota }_{{\xi }^{♯}}\alpha =0$. The first gives $\alpha =\xi \wedge \beta$ proportionality, i.e. $\alpha =c\xi$ for a scalar $c$ (since $\alpha$ is a 1-form), and the second then gives $c|\xi {|}^2=0$, so $c=0$ and $\alpha =0$. The symbol is injective, so $P$ is elliptic (overdetermined). The kernel consists of $\alpha$ with $d\alpha =0$ and $d^{\ast }\alpha =0$; these are the harmonic 1-forms ${\mathcal{H}}^1(L)$. By the Hodge theorem on compact $L$, ${\mathcal{H}}^1(L)\cong H_{\mathrm{dR}}^1(L)\cong H^1(L;\mathbb{R})$, of dimension $b_1(L)$. $\square$

Proposition 2 (unobstructedness via vanishing of the relevant cokernel). With $P$ as above, the special Lagrangian deformation problem is unobstructed: the nonlinear special Lagrangian operator has surjective linearisation onto the closed-coclosed image slice, so its zero set near $L$ is a smooth manifold of dimension $b_1(L)$.

Proof. Write the deformation of $L$ by a 1-form $\alpha$ in a tubular neighbourhood and let $F(\alpha )=(\omega {|}_{L_{\alpha }},\Im \Omega {|}_{L_{\alpha }})$, valued in ${\Omega }^2(L)\oplus {\Omega }^n(L)$, with $F(0)=0$ since $L$ is special Lagrangian. By Step 1 of the key theorem, $d{F}_0(\alpha )=(d\alpha ,d(\ast \alpha ))$. Both components are exact for every $\alpha$, so the image of $F$ lands in the closed forms; moreover ${\int }_L\omega {|}_{L_{\alpha }}$ over each 2-cycle and ${\int }_L\Im \Omega {|}_{L_{\alpha }}$ over the fundamental class are deformation invariants fixed by the homology class of $L_{\alpha }$, so $F$ in fact maps into the exact forms $\mathrm{im}d\oplus \mathrm{im}d$. On this target the linearisation $d{F}_0$ is surjective: given exact $(d\mu ,d\nu )$, the Hodge decomposition writes $\mu =d^{\ast }\eta +(\text{harmonic}+\text{exact})$ and one solves $d\alpha =d\mu$, $d(\ast \alpha )=d\nu$ for $\alpha$ in the coexact-plus-harmonic complement, using invertibility of the Laplacian on the non-harmonic part. The kernel is ${\mathcal{H}}^1(L)$ by Proposition 1. The implicit function theorem in Hölder spaces $C^{k,a}$ then presents $F^{-1}(0)$ near $0$ as a smooth manifold modelled on $\ker d{F}_0={\mathcal{H}}^1(L)$, of dimension $b_1(L)$. Elliptic regularity makes the solutions smooth. $\square$

Proposition 3 (the McLean metric is the Hessian of a convex potential in affine coordinates). On the smooth locus $B^{\circ }$ of a special Lagrangian torus fibration, in the integral affine coordinates $x_i={\int }_{{\gamma }_i}\Im \Omega$, the McLean metric has the form $g_{\mathcal{M}}={\sum }_{i,j}{\varphi }_{ij}d{x}_{i}d{x}_j$ with ${\varphi }_{ij}={\partial }^2\varphi /\partial x_i\partial x_j$ the Hessian of a strictly convex $\varphi :B^{\circ }\to \mathbb{R}$.

Proof. The tangent space $T_b{B}^{\circ }\cong {\mathcal{H}}^1(L_b)$ has the basis $\{{\theta }_i\}$ dual to $\{{\gamma }_i\}$, where ${\theta }_i$ is the harmonic representative with ${\int }_{{\gamma }_j}{\theta }_i={\delta }_{ij}$. The affine coordinate $x_i$ is the $\Im \Omega$-period, and its differential $d{x}_i$ pairs with the deformation field of ${\theta }_j$ to give ${\delta }_{ij}$, so $\{d{x}_i\}$ is the coframe dual to the deformation basis. The McLean metric in this coframe is the matrix $g_{ij}={\int }_{L_b}{\theta }_i\wedge \ast {\theta }_j$. Hitchin 1997 shows this matrix is the Hessian of the convex function $$ \phi(x) = \int_{L_b}!!\big(\text{Kähler potential / period generating function}\big), $$ explicitly $\partial \varphi /\partial x_i={\check{x}}_i={\int }_{{\Gamma }_i}\omega$, the conjugate symplectic period, whence ${\partial }^2\varphi /\partial x_i\partial x_j=\partial {\check{x}}_j/\partial x_i=g_{ij}$. Strict convexity is positive-definiteness of $g_{\mathcal{M}}$, established in Exercise 5. Thus $g_{\mathcal{M}}=\mathrm{Hess}\varphi$, and the conjugate coordinates ${\check{x}}_i=\partial \varphi /\partial x_i$ furnish the Legendre-dual affine structure with potential the Legendre transform $\check{\varphi }$. $\square$

Connections Master

• Calibrated geometries 03.09.19. The special Lagrangian condition is the calibration $\Re (e^{-i\theta }\Omega ){|}_L={\mathrm{vol}}_L$ of Harvey-Lawson, one of the four named calibrations of that unit. The present unit takes the special Lagrangian instance and develops its deformation theory: the calibration certifies volume-minimisation, and McLean's theorem then counts how the calibrated submanifold can move. The bridge is that calibrated submanifolds are the volume-minimisers, and special Lagrangians are the calibrated submanifolds whose moduli builds the SYZ base.

• Lagrangian submanifold 05.05.01. The first half of the special Lagrangian condition is the Lagrangian condition $\omega {|}_L=0$ of that unit, and the symplectic isomorphism $\nu (L)\cong T^{\ast }L$ it provides is exactly the device that turns a normal deformation field into the 1-form $\alpha$ at the start of McLean's proof. Without the Lagrangian structure the deformation 1-form would not exist; the special condition then constrains $\alpha$ to be harmonic.

• Strominger-Yau-Zaslow conjecture 04.12.10. McLean's theorem is the differential-geometric foundation that unit states without proving: the SYZ base $B^{\circ }$ is the McLean moduli space of the torus fibres, the dimension $b_1(T^n)=n$ is McLean's count, and the two integral affine structures the SYZ base carries are the $\omega$-period and $\Im \Omega$-period structures of the McLean moduli. This unit supplies the deformation theorem and the Hitchin metric that 04.12.10 quotes as a black box.

• The A-model, the B-model, and mirror symmetry 04.12.16. McLean's unobstructed real elliptic theory is the A-side counterpart to the obstructed complex deformation theory on the B-side, and mirror symmetry exchanges them. The dual affine structures on the McLean moduli are the local origin of the A-model / B-model exchange that unit frames, with fibrewise T-duality interchanging the symplectic-period and complex-period coordinates carried already by a single special Lagrangian's moduli.

Historical & philosophical context Master

The special Lagrangian condition was isolated by Harvey and Lawson in their 1982 Acta Mathematica paper ^{[Harvey-Lawson 1982]}, the founding document of calibrated geometry. They observed that on a Calabi-Yau the real part of the holomorphic volume form is a calibration, and they named the submanifolds it calibrates special Lagrangian, developing their local theory and many explicit examples. For more than a decade special Lagrangian geometry was a beautiful corner of Riemannian geometry with no external pull, until Strominger, Yau and Zaslow in 1996 ^{[Strominger-Yau-Zaslow 1996]} argued from string-theoretic D-brane charges that mirror symmetry should be carried by dual special Lagrangian torus fibrations. Their argument depended on knowing that the moduli of a special Lagrangian torus is smooth of the right dimension — a fact McLean was, at that moment, in the process of proving.

McLean's 1998 Communications in Analysis and Geometry paper ^{[McLean 1998]} supplied exactly the missing theorem: the deformations of a compact special Lagrangian are unobstructed, the moduli smooth of dimension $b_1(L)$, the tangent space the harmonic 1-forms. The cleanness of the result — a hard analytic deformation problem collapsing to elementary Hodge theory — was a surprise, and it is the contrast with the obstructed complex case that gives mirror symmetry its A-side / B-side asymmetry. Hitchin's 1997 paper ^{[Hitchin 1997]} then read off the metric geometry of the moduli, discovering the two Legendre-dual affine structures that became the integral affine base of the SYZ programme. The philosophical lesson is characteristic of mirror symmetry's history: a purely geometric theorem proved for its own interest (McLean's), and a metric refinement of it (Hitchin's), turned out to be the precise mathematical content a physical heuristic (SYZ) required, and the subject advanced by recognising that the moduli of a single calibrated submanifold already encodes, in its dual affine structures, the duality that mirror symmetry globalises.

Bibliography Master

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