\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

constructing mirror manifolds and implications

The Hodge-number flip explains the name, but it does not yet explain where the mirror should come from. The next layer of the story is constructive: one tries to build the mirror family by degenerating \(X\) to a combinatorial, toric, or affine limit, dualizing the relevant data, and then smoothing the result. The constructions below are schematic, but the point is precise: mirror symmetry is usually a statement about matched families near boundary points of moduli, not just about two isolated manifolds.

1: constructing mirrors

The first examples of mirror symmetry involved Calabi--Yau hypersurfaces in projective space. Let \(H\subset \CP^n\) be a smooth hypersurface of degree \(d\), so \(H\) has complex dimension \(n-1\). We can compute its first Chern class using the exact sequence \[ 0 \to T^{1,0}H\to T^{1,0}\CP^n|_H\to N_{H/\CP^n}\to 0. \] This tells us that \[ c(T\CP^n)|_H=c(TH)\cdot c(N_{H/\CP^n}). \] Suppose that the hypersurface has degree \(d\), and let \(h=c_1(\mathcal O_{\CP^n}(1))|_H\). Then \[ c(T\CP^n)|_H=(1+h)^{n+1},\qquad c(N_{H/\CP^n})=1+dh. \] Therefore, \[ 1+(n+1)h+\cdots=(1+c_1(TH)+\cdots)(1+dh), \] so \[ c_1(TH)=(n+1-d)h. \] Equivalently, by adjunction, \[ K_H\simeq \mathcal O_H(d-n-1). \] Thus \(d=n+1\) is necessary and sufficient for \(H\) to have trivial canonical bundle. If \(H\) is smooth, a degree \(n+1\) hypersurface in \(\CP^n\) is a compact Kähler manifold with trivial canonical bundle, hence is Calabi--Yau in the sense of (Calabi--Yau manifold). The quintic threefold in \(\CP^4\) is the standard example.

definition 0.0.1

Let \[ \pi:\mathcal X^*\to \Delta^* \] be a one-parameter family of Calabi--Yau \(n\)-folds over a punctured disc. Parallel transport around the puncture gives a monodromy operator \[ T:H^n(X_t;\QQ)\to H^n(X_t;\QQ). \] After replacing the punctured disc by a finite cover if necessary, assume that the monodromy is unipotent, and write \(T\) for this unipotent monodromy. Then one can define \[ N:=\log T=(T-I)-\frac{(T-I)^2}{2}+\frac{(T-I)^3}{3}-\cdots . \] The series is finite because \(T-I\) is nilpotent. In this introductory convention, a boundary point \(t=0\) is called a large complex-structure limit, or a maximally unipotent monodromy point, if \[ N^n\neq 0 \] on \(H^n(X_t;\QQ)\). Equivalently, \(N\) has nilpotency index \(n+1\), the maximum possible for a weight-\(n\) variation of Hodge structure. More refined definitions also impose conditions on the limiting mixed Hodge structure; the maximally unipotent monodromy criterion is the part used in this introductory discussion. For the degree \(n+1\) hypersurface family, there is a particularly degenerate limit to the union of the \(n+1\) coordinate hyperplanes. The central fiber is a singular toric normal-crossings variety, not a smooth toric manifold. A general strategy for constructing mirrors is to do mirror symmetry in families. Let \(\mathcal X\to \Delta\) be a degeneration of Calabi--Yau manifolds whose central fiber \(X_0\) is toric or toroidal and possibly singular. Then the strategy is:
  1. find a toric or toroidal degeneration of the family to \(X_0\);
  2. construct a mirror degeneration \(\check X_0\) using the dual toric or affine-combinatorial data;
  3. deform or smooth \(\check X_0\) to obtain the mirror family \(\check{\mathcal X}\), and match the deformation parameters using the mirror map.
There are several motivating reasons for taking this path:
  • The SYZ proposal interprets mirror symmetry as fiberwise \(T\)-duality, which suggests looking at dual tori.
  • Homological mirror symmetry suggests that a Lagrangian torus fiber equipped with a flat unitary local system should correspond to a skyscraper sheaf at a point of the mirror. Thus torus fibrations are natural objects to examine.
  • The topological calculation below shows why dualizing torus fibers exchanges the two Hodge numbers.

definition 0.0.2

Let \((X,\omega,\Omega)\) be a Calabi--Yau \(n\)-fold with its Ricci-flat Kähler metric, and assume that \(\Omega\) is normalized with respect to this metric. For \(\theta\in \RR/2\pi\ZZ\), an oriented real \(n\)-dimensional submanifold \(L\subset X\) is special Lagrangian of phase \(\theta\) if \[ \omega|_L=0,\qquad \operatorname{Im}(e^{-i\theta}\Omega)|_L=0, \] and \[ \operatorname{Re}(e^{-i\theta}\Omega)|_L=\operatorname{vol}_L \] as positive volume forms on \(L\). A special Lagrangian torus fibration of phase \(\theta\) is a continuous map \[ f:X\to B \] to a real \(n\)-dimensional base such that, over a dense open subset \(B_0\subset B\), the restriction \[ f^{-1}(B_0)\to B_0 \] is a smooth fiber bundle whose fibers are \(n\)-tori that are special Lagrangian submanifolds of phase \(\theta\). In the SYZ setting, nontrivial compact examples are expected to have singular fibers over a discriminant locus \(\Delta=B\setminus B_0\).

definition 0.0.3

Let \[ f:X_0\to B_0 \] be a smooth torus fibration with fibers \(F_b\simeq V_b/\Lambda_b\), where \(V_b\) is a real vector space and \(\Lambda_b\subset V_b\) is a lattice. The dual torus fibration is \[ \check f:\check X_0\to B_0, \qquad \check F_b:=\check f^{-1}(b)=\operatorname{Hom}(\Lambda_b,U(1)) \simeq H^1(F_b;\RR)/H^1(F_b;\ZZ). \] Fiberwise, \[ H_1(\check F_b;\ZZ)\simeq H^1(F_b;\ZZ), \qquad H^1(\check F_b;\ZZ)\simeq H_1(F_b;\ZZ). \] If the three-torus fibers are oriented, then Poincar\'e duality, with respect to the chosen orientations, gives the useful identifications \[ H^1(\check F_b;\CC)\simeq H^2(F_b;\CC), \qquad H^2(\check F_b;\CC)\simeq H^1(F_b;\CC). \] These fiberwise identifications induce the corresponding dual local systems over \(B_0\). Here is a motivating calculation from [Gro01]. It deliberately ignores many details about singular fibers, which are essential in the actual theory, but it captures the reason dual tori produce the mirror flip. Suppose that we have a special Lagrangian torus fibration \[ f:X\to B \] of a Calabi--Yau threefold. Assume, for this heuristic calculation, that the fibration is smooth, that the fibers are three-tori \(F=f^{-1}(p)\), that \(X\) has the Hodge diamond shape from (Hodge Diamond and Deformation), and that the base \(B\) is a closed oriented three-manifold with \(H^1(B,\CC)=H^2(B,\CC)=0\). A globally smooth torus fibration with simply connected total space is not the actual situation for compact Calabi--Yau threefolds; singular fibers are necessary. The simplified model is still useful because it isolates the cohomological mechanism. Let \[ \mathcal H^q:=R^q f_*\CC \] denote the local system over \(B\) whose fiber at \(p\) is \(H^q(F,\CC)\). The special Lagrangian condition gives an orientation of the fibers: after choosing the phase, \(\operatorname{Re}\Omega\) restricts to a volume form on each fiber. Thus \[ \mathcal H^0\simeq \CC,\qquad \mathcal H^3\simeq \CC. \] The Leray--Serre spectral sequence has \[ E_2^{p,q}=H^p(B,\mathcal H^q)\Longrightarrow H^{p+q}(X,\CC). \] The \(E_2\) page has the form Leray--Serre \(E_2\) pageUsing \(\mathcal H^0\simeq \mathcal H^3\simeq \CC\) and \(H^1(B,\CC)=H^2(B,\CC)=0\), the top and bottom rows become Leray--Serre page with trivial top and bottom rowsAssume further, as part of this simplified model, that the spectral sequence degenerates at \(E_2\) and that the relevant invariant-cycle terms vanish: \[ H^0(B,\mathcal H^1)=H^0(B,\mathcal H^2)=0. \] By duality, this also gives \[ H^3(B,\mathcal H^1)=H^3(B,\mathcal H^2)=0. \] Then the relevant part of the spectral sequence is relevant terms in the Leray--Serre pageConsequently, \[ H^2(X,\CC)\simeq H^1(B,\mathcal H^1), \] and \[ H^3(X,\CC)\simeq \CC\oplus H^1(B,\mathcal H^2)\oplus H^2(B,\mathcal H^1)\oplus \CC. \] Since \(X\) is a Calabi--Yau threefold with \(h^{2,0}=0\), we have \[ \dim H^2(X,\CC)=h^{1,1}(X),\qquad \dim H^3(X,\CC)=2h^{2,1}(X)+2. \] Moreover, Poincar\'e duality on the base with local coefficients, together with fiberwise duality, gives \[ \dim H^1(B,\mathcal H^2)=\dim H^2(B,\mathcal H^1). \] Therefore, \[ h^{1,1}(X)=\dim H^1(B,\mathcal H^1),\qquad h^{2,1}(X)=\dim H^1(B,\mathcal H^2). \] Now look at the dual torus fibration \[ \check f:\check X\to B. \] By definition 0.0.3, fiberwise the dual torus satisfies \[ H^1(\check F,\CC)\simeq H_1(F,\CC)\simeq H^2(F,\CC), \] and similarly \[ H^2(\check F,\CC)\simeq H_2(F,\CC)\simeq H^1(F,\CC). \] Thus the local systems for the dual fibration are swapped: \[ \check{\mathcal H}^1\simeq \mathcal H^2, \qquad \check{\mathcal H}^2\simeq \mathcal H^1. \] Applying the same calculation to \(\check X\) gives \[ h^{1,1}(\check X)=\dim H^1(B,\check{\mathcal H}^1) =\dim H^1(B,\mathcal H^2)=h^{2,1}(X), \] and \[ h^{2,1}(\check X)=\dim H^1(B,\check{\mathcal H}^2) =\dim H^1(B,\mathcal H^1)=h^{1,1}(X). \] This is the topological shadow of the mirror flip of Hodge numbers.

Connections

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Used in

References

[Gro01]Mark Gross. Special Lagrangian fibrations i: Topology. AMS/IP Studies in Advanced Mathematics, 23:65--94, 2001.