Mirror symmetry conjecture

In mathematics, Mirror symmetry (string theory) is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold" which allows one to relate Gromov–Witten invariants to period integrals on a variation of Hodge structures. In short, this means there is a relation between the number of genus ${\displaystyle g}$ algebraic curves of degree ${\displaystyle d}$ on a Calabi-Yau variety ${\displaystyle X}$ and integrals on a dual variety ${\displaystyle {\check {X}}}$. These relations were original discovered by Candelas, De la Ossa, Green, and Schwarts[1] in a paper studying a generic quintic threefold in ${\displaystyle \mathbb {P} ^{4}}$ as the variety ${\displaystyle X}$ and a construction[2] from the quintic Dwork family ${\displaystyle X_{\psi }}$ giving ${\displaystyle {\check {X}}={\tilde {X}}_{\psi }}$. Shortly after, Sheldon Katz wrote a summary paper[3] outlining part of their construction and conjectures what the rigorous mathematical interpretation could be.

For other uses, see Mirror symmetry.

Constructing the mirror of a quintic threefold

Complex moduli

Recall that a generic quintic threefold[2] ${\displaystyle X}$ 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][4]. Notice the vector space of global sections has dimension

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

but there are two equivalences of these polynomials. First, polynomials under scaling by ${\displaystyle \mathbb {G} _{m}}$ (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

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

since ${\displaystyle 126-24-1=101}$, which can be constructed using Geometric invariant theory. 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

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

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 ${\displaystyle U_{smooth}}$ 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

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

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

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

The associated quotient variety ${\displaystyle X_{\psi }/G}$ has a crepant resolution given[2][4] 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 ${\displaystyle 1}$.

Ideas from string theory

In string theory there is a class of models called non-linear sigma models which study families of maps ${\displaystyle \phi$ :\Sigma \to X} where ${\displaystyle \Sigma }$ is a genus ${\displaystyle g}$ algebraic curve and ${\displaystyle X}$ is Calabi-yau. This space ${\displaystyle X}$ has a complex structure, which is an integrable almost-complex structure ${\displaystyle J\in {\text{End}}(TX)}$, and because it is a Kähler manifold it necessarily has a symplectic structure ${\displaystyle \omega }$ called the Kähler form which can be complexified to a complexified Kähler form

${\displaystyle \omega ^{\mathbb {C} }=B+i\omega }$

which is a closed ${\displaystyle (1,1)}$-form, hence its cohomology class is in

${\displaystyle [\omega ^{\mathbb {C} }]\in H^{1}(X,\Omega _{X}^{1})}$

The main idea behind the Mirror Symmetry conjectures is to study the deformations, or moduli, of the complex structure ${\displaystyle J}$ and the complexified symplectic structure ${\displaystyle \omega ^{\mathbb {C} }}$ in a way that makes these two dual to each other.

There are two variants of the non-linear sigma models called the A-model and the B-model which consider the pairs ${\displaystyle (X,\omega ^{\mathbb {C} })}$ and ${\displaystyle (X,J)}$ and their moduli[5]^{ch 38 pg 729}.

A-model

In the A-model the corresponding moduli space are the moduli of pseudoholomorphic curves[6]^{pg 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[7]

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

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 a generic quintic threefold in ${\displaystyle \mathbb {P} ^{4}}$.[5]

B-model

Mathematically, the B-model is a variation of hodge structures which was originally given by the construction from the Dwork family.

See also

• Cotangent complex
• Homotopy associative algebra
• Moduli of algebraic curves
• Kuranishi structure

External links

• https://ocw.mit.edu/courses/mathematics/18-969-topics-in-geometry-mirror-symmetry-spring-2009/lecture-notes/

References

1. Candelas, Philip; De La Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991-07-29). "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory". Nuclear Physics B. 359 (1): 21–74. doi:10.1016/0550-3213(91)90292-6. ISSN 0550-3213.
2. Auroux, Dennis. "The Quintic 3-fold and Its Mirror" (PDF).
3. Katz, Sheldon (1993-12-29). "Rational curves on Calabi-Yau threefolds". arXiv:alg-geom/9312009.
4. Morrison, David R. (1992-02-10). "Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians". arXiv:alg-geom/9202004.
5. Mirror symmetry. Hori, Kentaro. Providence, RI: American Mathematical Society. 2003. ISBN 0-8218-2955-6. OCLC 52374327.CS1 maint: others (link)
6. McDuff, Dusa, 1945- (2012). J-holomorphic curves and symplectic topology. Salamon, D. (Dietmar) (2nd ed.). Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-8746-2. OCLC 794640223.CS1 maint: multiple names: authors list (link)
7. Kontsevich, M.; Manin, Yu (1994). "Gromov-Witten classes, quantum cohomology, and enumerative geometry". Communications in Mathematical Physics. 164 (3): 525–562. ISSN 0010-3616.

Books/Notes

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

Research

• 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 ${\displaystyle (\mathbb {C} ^{*})^{n}}$