Mirror symmetry conjecture

Originally, the construction of mirror manifolds was discovered through an ad-hoc procedure. Essentially, to a generic quintic threefold $X\; \backslash subset\; \backslash mathbb\{CP\}^4$ there should be associated a one-parameter family of Calabi-Yau manifolds $X\_\backslash psi$ which has multiple singularities. After blowing up these singularities, they are resolved and a new Calabi-Yau manifold $X^\backslash vee$ was constructed. which had a flipped Hodge diamond. In particular, there are isomorphisms

$$H^q(X,\backslash Omega\_X^p)\; \backslash cong\; H^q(X^\backslash vee,\; \backslash Omega\_\{X^\backslash vee\}^\{3-p\})$$

but most importantly, there is an isomorphism

$$H^1(X,\backslash Omega\_X^1)\; \backslash cong\; H^1(X^\backslash vee,\; \backslash Omega\_\{X^\backslash vee\}^\{2\})$$

where the string theory (the A-model of $X$) for states in $H^1(X,\backslash Omega\_X^1)$ is interchanged with the string theory (the B-model of $X^\backslash vee$) having states in $H^1(X^\backslash vee,\; \backslash Omega\_\{X^\backslash vee\}^\{2\})$. The string theory in the A-model only depended upon the Kahler or symplectic structure on $X$ while the B-model only depends upon the complex structure on $X^\backslash vee$. Here we outline the original construction of mirror manifolds, and consider the string-theoretic background and conjecture with the mirror manifolds in a later section of this article.

Recall that a generic quintic threefoldfor example, as a set, a Calabi-Yau manifold is the subset of complex projective space$\backslash \{[x\_0:x\_1:x\_2:x\_3:x\_4]\; \backslash in\; \backslash mathbb\{CP\}^4\; :\; x\_0^5\; +\; x\_1^5\; +\; x\_2^5\; +\; x\_3^5\; +\; x\_4^5\; =\; 0\; \backslash \}$ $X$ in $\backslash mathbb\{P\}^4$ is defined by a homogeneous polynomial of degree $5$. This polynomial is equivalently described as a global section of the line bundle $f\; \backslash in\; \backslash Gamma(\backslash mathbb\{P\}^4,\backslash mathcal\{O\}\_\{\backslash mathbb\{P\}^4\}(5))$.{{cite journal|last=Morrison|first=David R.|year=1993 | title=Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians|journal=J. Amer. Math. Soc. | volume=6|pages=223–247|doi=10.1090/S0894-0347-1993-1179538-2|arxiv=alg-geom/9202004|s2cid=9228037}} Notice the vector space of global sections has dimension$$\backslash dim\; \{\; \backslash Gamma\; (\backslash mathbb\; \{P\}\; ^\{4\},\{\backslash mathcal\; \{O\}\}\_\{\backslash mathbb\; \{P\}\; ^\{4\}\}(5))\}\; =\; 126$$ but there are two equivalences of these polynomials. First, polynomials under scaling by the algebraic torus $\backslash mathbb\{G\}\_m$Which can be thought of as the $\backslash mathbb\{C\}^*$-action on $\backslash mathbb\{C\}^5\; -\; \backslash \{\; 0\backslash \}$ constructing the complex projective space $\backslash mathbb\{CP\}^4$ (non-zero scalers of the base field) given equivalent spaces. Second, projective equivalence is given by the automorphism group of $\backslash mathbb\{P\}^4$, $\backslash text\{PGL\}(5)$ which is $24$ dimensional. This gives a $101$ dimensional parameter space$$U\_\backslash text\{smooth\}\; \backslash subset\; \backslash mathbb\{P\}(\backslash Gamma(\backslash mathbb\{P\}^4,\backslash mathcal\{O\}\_\{\backslash mathbb\{P\}^4\}(5)))/PGL(5)$$ since $126\; -\; 24\; -\; 1\; =\; 101$, which can be constructed using Geometric invariant theory. The set $U\_\{\backslash text\{smooth\}\}$ corresponds to the equivalence classes of polynomials which define smooth Calabi-Yau quintic threefolds in $\backslash mathbb\{P\}^4$, giving a moduli space of Calabi-Yau quintics.More generally, such moduli spaces are constructed using projective equivalence of schemes in a fixed projective space on a fixed Hilbert scheme Now, using Serre duality and the fact each Calabi-Yau manifold has trivial canonical bundle $\backslash omega\_X$, the space of deformations has an isomorphism$$H^1(X,T\_X)\; \backslash cong\; H^2(X,\backslash Omega\_X)$$ with the $(2,1)$ part of the Hodge structure on $H^3(X)$. Using the Lefschetz hyperplane theorem the only non-trivial cohomology group is $H^3(X)$ since the others are isomorphic to $H^i(\backslash mathbb\{P\}^4)$. Using the Euler characteristic and the Euler class, which is the top Chern class, the dimension of this group is $204$. This is because

$$\backslash begin\{align\}$$

\chi(X) &= -200 \\

&= h^0 + h^2 - h^3 +h^4 + h^6 \\

&= 1 + 1 - \dim H^3(X) + 1 + 1

\end{align}

Using the Hodge structure we can find the dimensions of each of the components. First, because $X$ is Calabi-Yau, $\backslash omega\_X\; \backslash cong\; \backslash mathcal\{O\}\_X$ so$$H^0(X,\backslash Omega\_X^3)\; \backslash cong\; H^0(X,\backslash mathcal\{O\}\_X)$$ giving the Hodge numbers $h^\{0,3\}\; =\; h^\{3,0\}\; =\; 1$, hence $$\backslash dim\; H^2(X,\backslash Omega\_X)\; =\; h^\{1,2\}\; =\; 101$$ 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 $U\_\backslash text\{smooth\}$ 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.

There is a distinguished family of Calabi-Yau manifolds $X\_\backslash psi$ called the Dwork family. It is the projective family

$$X\_\backslash psi\; =\; \backslash text\{Proj\}\; \backslash left($$

\frac{\mathbb{C}[\psi][x_0,\ldots, x_4]}{(x_0^5 + \cdots + x_4^5 - 5\psi x_0x_1x_2x_3x_4)}

\right)

over the complex plane $\backslash text\{Spec\}(\backslash mathbb\{C\}[\backslash psi])$. Now, notice there is only a single dimension of complex deformations of this family, coming from $\backslash psi$ having varying values. This is important because the Hodge diamond of the mirror manifold $\backslash check\{X\}$ has $$\backslash dim\; H^\{2,1\}(\backslash check\{X\})\; =\; 1.$$The family $X\_\backslash psi$ has symmetry group $$G\; =\; \backslash left\backslash \{\; (a\_0,\backslash ldots,\; a\_4)\; \backslash in\; (\backslash mathbb\{Z\}/5)^5\; :\; \backslash sum\; a\_i\; =\; 0\; \backslash right\backslash \}$$ acting by $$(a\_0,\backslash ldots,a\_4)\backslash cdot\; [x\_0:\backslash cdots:x\_4]\; =\; [e^\{\; a\_0\backslash cdot\; 2\backslash pi\; i/5\}x\_0:\backslash cdots\; :\; e^\{\; a\_4\; \backslash cdot\; 2\backslash pi\; i/5\}x\_4]$$ Notice the projectivity of $X\_\backslash psi$ is the reason for the condition $$\backslash sum\_i\; a\_i\; =\; 0.$$ The associated quotient variety $X\_\backslash psi\; /\; G$ has a crepant resolution given by blowing up the $100$ singularities $$\backslash check\{X\}\; \backslash to\; X\_\backslash psi\; /\; G$$ giving a new Calabi-Yau manifold $\backslash check\{X\}$ with $101$ parameters in $H^\{1,1\}(\backslash check\{X\})$. This is the mirror manifold and has $H^3(\backslash check\{X\})\; =\; 4$ where each Hodge number is $1$.

{{See also|Mirror symmetry (string theory)}}

In string theory there is a class of models called non-linear sigma models which study families of maps $\backslash phi:\; \backslash Sigma\; \backslash to\; X$ where $\backslash Sigma$ is a genus $g$ algebraic curve and $X$ is Calabi-Yau. These curves $\backslash Sigma$ are called world-sheets and represent the birth and death of a particle as a closed string. Since a string could split over time into two strings, or more, and eventually these strings will come together and collapse at the end of the lifetime of the particle, an algebraic curve mathematically represents this string lifetime. For simplicity, only genus 0 curves were considered originally, and many of the results popularized in mathematics focused only on this case.

Also, in physics terminology, these theories are $(2,2)$ heterotic string theories because they have $N=2$ supersymmetry that comes in a pair, so really there are four supersymmetries. This is important because it implies there is a pair of operators $$(Q,\backslash overline\{Q\})$$ acting on the Hilbert space of states, but only defined up to a sign. This ambiguity is what originally suggested to physicists there should exist a pair of Calabi-Yau manifolds which have dual string theories, one's that exchange this ambiguity between one another.

The space $X$ has a complex structure, which is an integrable almost-complex structure $J\; \backslash in\; \backslash text\{End\}(TX)$, and because it is a Kähler manifold it necessarily has a symplectic structure $\backslash omega$ called the Kähler form which can be complexified to a complexified Kähler form $$\backslash omega^\backslash mathbb\{C\}\; =\; B\; +\; i\backslash omega$$ which is a closed $(1,1)$-form, hence its cohomology class is in $$[\backslash omega^\backslash mathbb\{C\}]\; \backslash in\; H^1(X,\backslash Omega\_X^1)$$ The main idea behind the Mirror Symmetry conjectures is to study the deformations, or moduli, of the complex structure $J$ and the complexified symplectic structure $\backslash omega^\backslash mathbb\{C\}$ in a way that makes these two dual to each other. In particular, from a physics perspective,{{Cite book| last1=Cox | first1 = David A. | last2 = Katz | first2 = Sheldon | title=Mirror symmetry and algebraic geometry|date=1999 | publisher=American Mathematical Society|isbn=978-0-8218-2127-5|oclc=903477225}}{{rp|pp=1–2}} the super conformal field theory of a Calabi-Yau manifold $X$ should be equivalent to the dual super conformal field theory of the mirror manifold $X^\backslash vee$. Here conformal means conformal equivalence which is the same as an equivalence class of complex structures on the curve $\backslash Sigma$.

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

Given a Calabi-Yau manifold $X$ with complexified Kähler class $[\backslash omega^\backslash mathbb\{C\}]\; \backslash in\; H^1(X,\backslash Omega\_X^1)$ the nonlinear sigma model of the string theory should contain the three generations of particles, plus the electromagnetic, weak, and strong forces.{{cite arXiv|last=Hamilton|first=M. J. D.|date=2020-07-24|title=The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking|class=math.DG|eprint=1512.02632}}{{rp|p=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 $H^1(X,\backslash Omega^1\_X)$. Note this space is the eigenspace of an operator $Q$ on the Hilbert space of states for the string theory.{{rp|pp=3–5}} This three point function is "computed" as

$$\backslash begin\{align\}$$

\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{align}

using Feynman path-integral techniques where the $n\_\backslash beta$ are the naive number of rational curves with homology class $\backslash beta\; \backslash in\; H\_2(X;\backslash mathbb\{Z\})$, and $\backslash omega\_i\; \backslash in\; H^1(X,\backslash Omega\_X)$. Defining these instanton numbers $n\_\backslash beta$ 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.

In the A-model the corresponding moduli space are the moduli of pseudoholomorphic curves{{Cite book|last=McDuff | first = Dusa|title=J-holomorphic curves and symplectic topology|date=2012|publisher=American Mathematical Society | others=Salamon, D. (Dietmar)|isbn=978-0-8218-8746-2|edition=2nd|location=Providence, R.I.|oclc=794640223}}{{rp|p=153}}

$$\backslash overline\{\backslash mathcal\{M\}\}\_\{g,k\}(X,J,\backslash beta)\; =\; \backslash \{$$

(u:\Sigma \to X, j, z_1,\ldots, z_k) : u_*[\Sigma] = \beta, \overline{\partial}_Ju = 0

\}

or the Kontsevich moduli spaces{{Cite journal|last1=Kontsevich|first1=M.|last2=Manin|first2=Yu|date=1994|title=Gromov-Witten classes, quantum cohomology, and enumerative geometry|url=https://projecteuclid.org/euclid.cmp/1104270948|journal=Communications in Mathematical Physics|language=en|volume=164|issue=3|pages=525–562|doi=10.1007/BF02101490|arxiv=hep-th/9402147| bibcode=1994CMaPh.164..525K| s2cid=18626455|issn=0010-3616}}

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

These moduli spaces can be equipped with a virtual fundamental class

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

which is represented as the vanishing locus of a section $\backslash pi\_\{Coker\}(v)$ of a sheaf called the Obstruction sheaf $\backslash underline\{\backslash text\{Obs\}\}$ over the moduli space. This section comes from the differential equation$$\backslash overline\{\backslash partial\}\_J(u)\; =\; v$$ which can be viewed as a perturbation of the map $u$. It can also be viewed as the Poincaré dual of the Euler class of $\backslash underline\{\backslash text\{Obs\}\}$ if it is a Vector bundle.

With the original construction, the A-model considered was on a generic quintic threefold in $\backslash mathbb\{P\}^4$.{{Cite book|title=Mirror symmetry|date=2003|publisher=American Mathematical Society | last1 = Pandharipande | first1 = Rahul |last2 = Hori | first2 = Kentaro |isbn=0-8218-2955-6|location=Providence, RI|oclc=52374327}}

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

$$\backslash langle\; \backslash theta\_1,\backslash theta\_2,\backslash theta\_3\; \backslash rangle\; =$$

\int_X\Omega \wedge (\nabla_{\theta_1}\nabla_{\theta_2}\nabla_{\theta_3}\Omega)

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

$$\backslash Omega\; \backslash wedge\; (\backslash nabla\_\{\backslash theta\_1\}\backslash nabla\_\{\backslash theta\_2\}\backslash nabla\_\{\backslash theta\_3\}\backslash Omega)\; \backslash in\; H^3(X,\backslash Omega\_X^3)$$

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

The action of the cohomology classes $\backslash theta\; \backslash in\; H^1(X,T\_X)$ on the $\backslash Omega\; \backslash in\; H^0(X,\backslash Omega\_X^3)$ can also be understood as a cohomological variant of the interior product. Locally, the class $\backslash theta$ corresponds to a Cech cocycle $[\backslash theta\_\{i\}]\_\{i\; \backslash in\; I\}$ for some nice enough cover $\backslash \{U\_i\; \backslash \}\_\{i\; \backslash in\; I\}$ giving a section $\backslash theta\_i\; \backslash in\; T\_X(U\_i)$. Then, the insertion product gives an element $$\backslash iota\_\{\backslash theta\_i\}(\backslash Omega|\_\{U\_i\})\; \backslash in\; H^0(U\_i,\backslash Omega\_X^2|\_\{U\_i\})$$ which can be glued back into an element $\backslash iota\_\backslash theta(\backslash Omega)$ of $H^1(X,\backslash Omega\_X^2)$. This is because on the overlaps $$U\_i\backslash cap\; U\_j\; =\; U\_\{ij\},$$ $$\backslash theta\_\{i\}|\_\{ij\}\; =\; \backslash theta\_\{j\}|\_\{ij\}$$ giving

$$\backslash begin\{align\}$$

(\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{align}

hence it defines a 1-cocycle. Repeating this process gives a 3-cocycle $$\backslash iota\_\{\backslash theta\_1\}\backslash iota\_\{\backslash theta\_2\}\backslash iota\_\{\backslash theta\_3\}\backslash Omega\; \backslash in\; H^3(X,\backslash mathcal\{O\}\_X)$$ which is equal to $\backslash nabla\_\{\backslash theta\_1\}\backslash nabla\_\{\backslash theta\_2\}\backslash nabla\_\{\backslash theta\_3\}\backslash Omega$. This is because locally the Gauss-Manin connection acts as the interior product.

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

Relating these two models of string theory by resolving the ambiguity of sign for the operators $(Q,\backslash overline\{Q\})$ led physicists to the following conjecture:{{rp|p=22}} for a Calabi-Yau manifold $X$ there should exist a mirror Calabi-Yau manifold $X^\backslash vee$ such that there exists a mirror isomorphism $$H^1(X,\backslash Omega\_X)\; \backslash cong\; H^1(X^\backslash vee,\; T\_\{X^\backslash vee\})$$ giving the compatibility of the associated A-model and B-model. This means given $H\; \backslash in\; H^1(X,\backslash Omega\_X)$ and $\backslash theta\; \backslash in\; H^1(X^\backslash vee,T\_\{X^\backslash vee\})$ such that $H\; \backslash mapsto\; \backslash theta$ under the mirror map, there is the equality of correlation functions$$\backslash langle\; H,H,H\backslash rangle\; =\; \backslash langle\; \backslash theta,\backslash theta,\backslash theta\backslash rangle$$ This is significant because it relates the number of degree $d$ genus $0$ curves on a quintic threefold $X$ in $\backslash mathbb\{P\}^4$ (so $H^\{1,1\}\backslash cong\; \backslash mathbb\{Z\}$) to integrals in a variation of Hodge structures. Moreover, these integrals are actually computable!

• Cotangent complex
• Homotopy associative algebra
• Kuranishi structure
• Mirror symmetry (string theory)
• Moduli of algebraic curves
• Kontsevich moduli space

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

• [http://www.claymath.org/library/monographs/cmim01c.pdf Mirror Symmetry] - Clay Mathematics Institute ebook
• [https://bookstore.ams.org/surv-68-s Mirror Symmetry and Algebraic Geometry] - Cox, Katz
• On the work of Givental relative to mirror symmetry

• 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

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

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

• 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 $(\backslash mathbb\{C\}^*)^n$

Category:Mathematical physics

Category:Conjectures

Category:String theory

Category:Algebraic geometry