Complex moduli

Recall that a generic quintic threefold^[2][4] ${\displaystyle X}$ in ${\displaystyle \mathbb {P} ^{4}}$ is defined by a homogeneous polynomial of degree ${\displaystyle 5}$. This polynomial is equivalently described as a global section of the line bundle ${\displaystyle f\in \Gamma (\mathbb {P} ^{4},{\mathcal {O}}_{\mathbb {P} ^{4}}(5))}$.^[1][5] Notice the vector space of global sections has dimension

$${\displaystyle \dim {\Gamma (\mathbb {P} ^{4},{\mathcal {O}}_{\mathbb {P} ^{4}}(5))}=126}$$

but there are two equivalences of these polynomials. First, polynomials under scaling by the algebraic torus ${\displaystyle \mathbb {G} _{m}}$^[6] (non-zero scalers of the base field) given equivalent spaces. Second, projective equivalence is given by the automorphism group of ${\displaystyle \mathbb {P} ^{4}}$, ${\displaystyle {\text{PGL}}(5)}$ which is ${\displaystyle 24}$ dimensional. This gives a ${\displaystyle 101}$ dimensional parameter space

$${\displaystyle U_{\text{smooth}}\subset \mathbb {P} (\Gamma (\mathbb {P} ^{4},{\mathcal {O}}_{\mathbb {P} ^{4}}(5)))/PGL(5)}$$

since ${\displaystyle 126-24-1=101}$, which can be constructed using Geometric invariant theory. The set ${\displaystyle U_{\text{smooth}}}$ corresponds to the equivalence classes of polynomials which define smooth Calabi-Yau quintic threefolds in ${\displaystyle \mathbb {P} ^{4}}$, giving a moduli space of Calabi-Yau quintics.^[7] Now, using Serre duality and the fact each Calabi-Yau manifold has trivial canonical bundle ${\displaystyle \omega _{X}}$, the space of deformations has an isomorphism

$${\displaystyle H^{1}(X,T_{X})\cong H^{2}(X,\Omega _{X})}$$

with the ${\displaystyle (2,1)}$ part of the Hodge structure on ${\displaystyle H^{3}(X)}$. Using the Lefschetz hyperplane theorem the only non-trivial cohomology group is ${\displaystyle H^{3}(X)}$ since the others are isomorphic to ${\displaystyle H^{i}(\mathbb {P} ^{4})}$. Using the Euler characteristic and the Euler class, which is the top Chern class, the dimension of this group is ${\displaystyle 204}$. This is because

$${\displaystyle {\begin{aligned}\chi (X)&=-200\\&=h^{0}+h^{2}-h^{3}+h^{4}+h^{6}\\&=1+1-\dim H^{3}(X)+1+1\end{aligned}}}$$

Using the Hodge structure we can find the dimensions of each of the components. First, because ${\displaystyle X}$ is Calabi-Yau, ${\displaystyle \omega _{X}\cong {\mathcal {O}}_{X}}$ so

$${\displaystyle H^{0}(X,\Omega _{X}^{3})\cong H^{0}(X,{\mathcal {O}}_{X})}$$

giving the Hodge numbers ${\displaystyle h^{0,3}=h^{3,0}=1}$, hence

$${\displaystyle \dim H^{2}(X,\Omega _{X})=h^{1,2}=101}$$

giving the dimension of the moduli space of Calabi-Yau manifolds. Because of the Bogomolev-Tian-Todorov theorem, all such deformations are unobstructed, so the smooth space ${\displaystyle U_{\text{smooth}}}$ is in fact the moduli space of quintic threefolds. The whole point of this construction is to show how the complex parameters in this moduli space are converted into Kähler parameters of the mirror manifold.

Mirror manifold

There is a distinguished family of Calabi-Yau manifolds ${\displaystyle X_{\psi }}$ called the Dwork family. It is the projective family

$${\displaystyle X_{\psi }={\text{Proj}}\left({\frac {\mathbb {C} [\psi ][x_{0},\ldots ,x_{4}]}{(x_{0}^{5}+\cdots +x_{4}^{5}-5\psi x_{0}x_{1}x_{2}x_{3}x_{4})}}\right)}$$

over the complex plane ${\displaystyle {\text{Spec}}(\mathbb {C} [\psi ])}$. Now, notice there is only a single dimension of complex deformations of this family, coming from ${\displaystyle \psi }$ having varying values. This is important because the Hodge diamond of the mirror manifold ${\displaystyle {\check {X}}}$ has

$${\displaystyle \dim H^{2,1}({\check {X}})=1.}$$

Anyway, the family ${\displaystyle X_{\psi }}$ has symmetry group

$${\displaystyle G=\left\{(a_{0},\ldots ,a_{4})\in (\mathbb {Z} /5)^{5}:\sum a_{i}=0\right\}}$$

acting by

$${\displaystyle (a_{0},\ldots ,a_{4})\cdot [x_{0}:\cdots :x_{4}]=[e^{a_{0}\cdot 2\pi i/5}x_{0}:\cdots :e^{a_{4}\cdot 2\pi i/5}x_{4}]}$$

Notice the projectivity of ${\displaystyle X_{\psi }}$ is the reason for the condition

$${\displaystyle \sum _{i}a_{i}=0.}$$

The associated quotient variety ${\displaystyle X_{\psi }/G}$ has a crepant resolution given^[2][5] by blowing up the ${\displaystyle 100}$ singularities

$${\displaystyle {\check {X}}\to X_{\psi }/G}$$

giving a new Calabi-Yau manifold ${\displaystyle {\check {X}}}$ with ${\displaystyle 101}$ parameters in ${\displaystyle H^{1,1}({\check {X}})}$. This is the mirror manifold and has ${\displaystyle H^{3}({\check {X}})=4}$ where each Hodge number is ${\displaystyle 1}$.

A-model

Correlation functions from String theory

Given a Calabi-Yau manifold ${\displaystyle X}$ with complexified Kähler class ${\displaystyle [\omega ^{\mathbb {C} }]\in H^{1}(X,\Omega _{X}^{1})}$ the nonlinear sigma model of the string theory should contain the three generations of particles, plus the electromagnetic, weak, and strong forces.^[10]: 27 In order to understand how these forces interact, a three-point function called the Yukawa coupling is introduced which acts as the correlation function for states in ${\displaystyle H^{1}(X,\Omega _{X}^{1})}$. Note this space is the eigenspace of an operator ${\displaystyle Q}$ on the Hilbert space of states for the string theory.^[8]: 3–5 This three point function is "computed" as

$${\displaystyle {\begin{aligned}\langle \omega _{1},\omega _{2},\omega _{3}\rangle =&\int _{X}\omega _{1}\wedge \omega _{2}\wedge \omega _{3}+\sum _{\beta \neq 0}n_{\beta }\int _{\beta }\omega _{1}\int _{\beta }\omega _{2}\int _{\beta }\omega _{2}{\frac {e^{2\pi i\int _{\beta }\omega ^{\mathbb {C} }}}{1-e^{2\pi i\int _{\beta }\omega ^{\mathbb {C} }}}}\end{aligned}}}$$

using Feynman path-integral techniques where the ${\displaystyle n_{\beta }}$ are the naive number of rational curves with homology class ${\displaystyle \beta \in H_{2}(X;\mathbb {Z} )}$, and ${\displaystyle \omega _{i}\in H^{1}(X,\Omega _{X})}$. Defining these instanton numbers ${\displaystyle n_{\beta }}$ is the subject matter of Gromov–Witten theory. Note that in the definition of this correlation function, it only depends on the Kahler class. This inspired some mathematicians to study hypothetical moduli spaces of Kahler structures on a manifold.

Mathematical interpretation of A-model correlation functions

In the A-model the corresponding moduli space are the moduli of pseudoholomorphic curves^[11]: 153

$${\displaystyle {\overline {\mathcal {M}}}_{g,k}(X,J,\beta )=\{(u:\Sigma \to X,j,z_{1},\ldots ,z_{k}):u_{*}[\Sigma ]=\beta ,{\overline {\partial }}_{J}u=0\}}$$

or the Kontsevich moduli spaces^[12]

$${\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )=\{u:\Sigma \to X:u{\text{ is stable and }}u_{*}([\Sigma ])=\beta \}}$$

These moduli spaces can be equipped with a virtual fundamental class

$${\displaystyle [{\overline {\mathcal {M}}}_{g,k}(X,J,\beta )]^{virt}}$$

or

$${\displaystyle [{\overline {\mathcal {M}}}_{g,n}(X,\beta )]^{virt}}$$

which is represented as the vanishing locus of a section ${\displaystyle \pi _{Coker}(v)}$ of a sheaf called the Obstruction sheaf ${\displaystyle {\underline {\text{Obs}}}}$ over the moduli space. This section comes from the differential equation

$${\displaystyle {\overline {\partial }}_{J}(u)=v}$$

which can be viewed as a perturbation of the map ${\displaystyle u}$. It can also be viewed as the Poincaré dual of the Euler class of ${\displaystyle {\underline {\text{Obs}}}}$ if it is a Vector bundle.

With the original construction, the A-model considered was on a generic quintic threefold in ${\displaystyle \mathbb {P} ^{4}}$.^[9]

B-model

Correlation functions from String theory

For the same Calabi-Yau manifold ${\displaystyle X}$ in the A-model subsection, there is a dual superconformal field theory which has states in the eigenspace ${\displaystyle H^{1}(X,T_{X})}$ of the operator ${\displaystyle {\overline {Q}}}$. Its three-point correlation function is defined as

$${\displaystyle \langle \theta _{1},\theta _{2},\theta _{3}\rangle =\int _{X}\Omega \wedge (\nabla _{\theta _{1}}\nabla _{\theta _{2}}\nabla _{\theta _{3}}\Omega )}$$

where ${\displaystyle \Omega \in H^{0}(X,\Omega _{X}^{3})}$ is a holomorphic 3-form on ${\displaystyle X}$ and for an infinitesimal deformation ${\displaystyle \theta }$ (since ${\displaystyle H^{1}(X,T_{X})}$ is the tangent space of the moduli space of Calabi-Yau manifolds containing ${\displaystyle X}$, by the Kodaira–Spencer map and the Bogomolev-Tian-Todorov theorem) there is the Gauss-Manin connection ${\displaystyle \nabla _{\theta }}$ taking a ${\displaystyle (p,q)}$ class to a ${\displaystyle (p+1,q-1)}$ class, hence

$${\displaystyle \Omega \wedge (\nabla _{\theta _{1}}\nabla _{\theta _{2}}\nabla _{\theta _{3}}\Omega )\in H^{3}(X,\Omega _{X}^{3})}$$

can be integrated on ${\displaystyle X}$. Note that this correlation function only depends on the complex structure of ${\displaystyle X}$.

Another formulation of Gauss-Manin connection

The action of the cohomology classes ${\displaystyle \theta \in H^{1}(X,T_{X})}$ on the ${\displaystyle \Omega \in H^{0}(X,\Omega _{X}^{3})}$ can also be understood as a cohomological variant of the interior product. Locally, the class ${\displaystyle \theta }$ corresponds to a Cech cocycle ${\displaystyle [\theta _{i}]_{i\in I}}$ for some nice enough cover ${\displaystyle \{U_{i}\}_{i\in I}}$ giving a section ${\displaystyle \theta _{i}\in T_{X}(U_{i})}$. Then, the insertion product gives an element

$${\displaystyle \iota _{\theta _{i}}(\Omega |_{U_{i}})\in H^{0}(U_{i},\Omega _{X}^{2}|_{U_{i}})}$$

which can be glued back into an element ${\displaystyle \iota _{\theta }(\Omega )}$ of ${\displaystyle H^{1}(X,\Omega _{X}^{2})}$. This is because on the overlaps

$${\displaystyle U_{i}\cap U_{j}=U_{ij},}$$

$${\displaystyle \theta _{i}|_{ij}=\theta _{j}|_{ij}}$$

giving

$${\displaystyle {\begin{aligned}(\iota _{\theta _{i}}\Omega |_{U_{i}})|_{U_{ij}}&=\iota _{\theta _{i}|_{U_{ij}}}(\Omega |_{U_{ij}})\\&=\iota _{\theta _{j}|_{U_{ij}}}(\Omega |_{U_{ij}})\\&=(\iota _{\theta _{j}}\Omega |_{U_{j}})|_{U_{ij}}\end{aligned}}}$$

hence it defines a 1-cocycle. Repeating this process gives a 3-cocycle

$${\displaystyle \iota _{\theta _{1}}\iota _{\theta _{2}}\iota _{\theta _{3}}\Omega \in H^{3}(X,{\mathcal {O}}_{X})}$$

which is equal to ${\displaystyle \nabla _{\theta _{1}}\nabla _{\theta _{2}}\nabla _{\theta _{3}}\Omega }$. This is because locally the Gauss-Manin connection acts as the interior product.

Mirror conjecture

Relating these two models of string theory by resolving the ambiguity of sign for the operators ${\displaystyle (Q,{\overline {Q}})}$ led physicists to the following conjecture:^[8]: 22 for a Calabi-Yau manifold ${\displaystyle X}$ there should exist a mirror Calabi-Yau manifold ${\displaystyle X^{\vee }}$ such that there exists a mirror isomorphism

$${\displaystyle H^{1}(X,\Omega _{X})\cong H^{1}(X^{\vee },T_{X^{\vee }})}$$

giving the compatibility of the associated A-model and B-model. This means given ${\displaystyle H\in H^{1}(X,\Omega _{X})}$ and ${\displaystyle \theta \in H^{1}(X^{\vee },T_{X^{\vee }})}$ such that ${\displaystyle H\mapsto \theta }$ under the mirror map, there is the equality of correlation functions

$${\displaystyle \langle H,H,H\rangle =\langle \theta ,\theta ,\theta \rangle }$$

This is significant because it relates the number of degree ${\displaystyle d}$ genus ${\displaystyle 0}$ curves on a quintic threefold ${\displaystyle X}$ in ${\displaystyle \mathbb {P} ^{4}}$ (so ${\displaystyle H^{1,1}\cong \mathbb {Z} }$) to integrals in a variation of Hodge structures. Moreover, these integrals are actually computable!

Books/Notes

• Mirror Symmetry - Clay Mathematics Institute ebook
• Mirror Symmetry and Algebraic Geometry - Cox, Katz
• On the work of Givental relative to mirror symmetry

First proofs

• Equivariant Gromov - Witten Invariants - Givental's original proof for projective complete intersections
• The mirror formula for quintic threefolds
• Rational curves on hypersurfaces (after A. Givental) - an explanation of Givental's proof
• Mirror Principle I - Lian, Liu, Yau's proof closing gaps in Givental's proof. His proof required the undeveloped theory of Floer homology
• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties - first general construction of mirror varieties for Calabi-Yau's in toric varieties
• Mirror symmetry for abelian varieties

Derived geometry in Mirror symmetry

• Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4

Research

• Mirror symmetry: from categories to curve counts - relation between homological mirror symmetry and classical mirror symmetry
• Intrinsic mirror symmetry and punctured Gromov-Witten invariants

Homological mirror symmetry

• Categorical Mirror Symmetry: The Elliptic Curve
• An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves
• Homological mirror symmetry for the genus two curve
• Homological mirror symmetry for the quintic 3-fold
• Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space
• Speculations on homological mirror symmetry for hypersurfaces in ${\displaystyle (\mathbb {C} ^{*})^{n}}$