In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold (encoded as Gromov–Witten invariants) to integrals from a family of varieties (encoded as period integrals on a variation of Hodge structures). In short, this means there is a relation between the number of genus algebraic curves of degree on a Calabi-Yau variety and integrals on a dual variety . These relations were original discovered by Candelas, de la Ossa, Green, and Parkes[1] in a paper studying a generic quintic threefold in as the variety and a construction[2] from the quintic Dwork family giving . Shortly after, Sheldon Katz wrote a summary paper[3] outlining part of their construction and conjectures what the rigorous mathematical interpretation could be.

Constructing the mirror of a quintic threefold

Originally, the construction of mirror manifolds was discovered through an ad-hoc procedure. Essentially, to a generic quintic threefold there should be associated a one-parameter family of Calabi-Yau manifolds which has multiple singularities. After blowing up these singularities, they are resolved and a new Calabi-Yau manifold was constructed. which had a flipped Hodge diamond. In particular, there are isomorphisms

but most importantly, there is an isomorphism

where the string theory (the A-model of ) for states in is interchanged with the string theory (the B-model of ) having states in . The string theory in the A-model only depended upon the Kahler or symplectic structure on while the B-model only depends upon the complex structure on . 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.

Complex moduli

Recall that a generic quintic threefold[2][4] in is defined by a homogeneous polynomial of degree . This polynomial is equivalently described as a global section of the line bundle .[1][5] Notice the vector space of global sections has dimension

but there are two equivalences of these polynomials. First, polynomials under scaling by the algebraic torus [6] (non-zero scalers of the base field) given equivalent spaces. Second, projective equivalence is given by the automorphism group of , which is dimensional. This gives a dimensional parameter space

since , which can be constructed using Geometric invariant theory. The set corresponds to the equivalence classes of polynomials which define smooth Calabi-Yau quintic threefolds in , giving a moduli space of Calabi-Yau quintics.[7] Now, using Serre duality and the fact each Calabi-Yau manifold has trivial canonical bundle , the space of deformations has an isomorphism

with the part of the Hodge structure on . Using the Lefschetz hyperplane theorem the only non-trivial cohomology group is since the others are isomorphic to . Using the Euler characteristic and the Euler class, which is the top Chern class, the dimension of this group is . This is because

Using the Hodge structure we can find the dimensions of each of the components. First, because is Calabi-Yau, so

giving the Hodge numbers , hence

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 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 called the Dwork family. It is the projective family

over the complex plane . Now, notice there is only a single dimension of complex deformations of this family, coming from having varying values. This is important because the Hodge diamond of the mirror manifold has

Anyway, the family has symmetry group

acting by

Notice the projectivity of is the reason for the condition

The associated quotient variety has a crepant resolution given[2][5] by blowing up the singularities

giving a new Calabi-Yau manifold with parameters in . This is the mirror manifold and has where each Hodge number is .

Ideas from string theory

In string theory there is a class of models called non-linear sigma models which study families of maps where is a genus algebraic curve and is Calabi-Yau. These curves 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 heterotic string theories because they have supersymmetry that comes in a pair, so really there are four supersymmetries. This is important because it implies there is a pair of operators

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 has a complex structure, which is an integrable almost-complex structure , and because it is a Kähler manifold it necessarily has a symplectic structure called the Kähler form which can be complexified to a complexified Kähler form

which is a closed -form, hence its cohomology class is in

The main idea behind the Mirror Symmetry conjectures is to study the deformations, or moduli, of the complex structure and the complexified symplectic structure in a way that makes these two dual to each other. In particular, from a physics perspective,[8]:1–2 the super conformal field theory of a Calabi-Yau manifold should be equivalent to the dual super conformal field theory of the mirror manifold . Here conformal means conformal equivalence which is the same as and equivalence class of complex structures on the curve .

There are two variants of the non-linear sigma models called the A-model and the B-model which consider the pairs and and their moduli.[9]:ch 38 pg 729

A-model

Correlation functions from String theory

Given a Calabi-Yau manifold with complexified Kähler class 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 . Note this space is the eigenspace of an operator on the Hilbert space of states for the string theory.[8]:3–5 This three point function is "computed" as

using Feynman path-integral techniques where the are the naive number of rational curves with homology class , and . Defining these instanton numbers 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

or the Kontsevich moduli spaces[12]

These moduli spaces can be equipped with a virtual fundamental class

or

which is represented as the vanishing locus of a section of a sheaf called the Obstruction sheaf over the moduli space. This section comes from the differential equation

which can be viewed as a perturbation of the map . It can also be viewed as the Poincaré dual of the Euler class of if it is a Vector bundle.

With the original construction, the A-model considered was on a generic quintic threefold in .[9]

B-model

Correlation functions from String theory

For the same Calabi-Yau manifold in the A-model subsection, there is a dual superconformal field theory which has states in the eigenspace of the operator . Its three-point correlation function is defined as

where is a holomorphic 3-form on and for an infinitesimal deformation (since is the tangent space of the moduli space of Calabi-Yau manifolds containing , by the Kodaira–Spencer map and the Bogomolev-Tian-Todorov theorem) there is the Gauss-Manin connection taking a class to a class, hence

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

Another formulation of Gauss-Manin connection

The action of the cohomology classes on the can also be understood as a cohomological variant of the interior product. Locally, the class corresponds to a Cech cocycle for some nice enough cover giving a section . Then, the insertion product gives an element

which can be glued back into an element of . This is because on the overlaps

giving

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

which is equal to . This is because locally the Gauss-Manin connection acts as the interior product.

Mathematical interpretation of B-model correlation functions

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

Mirror conjecture

Relating these two models of string theory by resolving the ambiguity of sign for the operators led physicists to the following conjecture:[8]:22 for a Calabi-Yau manifold there should exist a mirror Calabi-Yau manifold such that there exists a mirror isomorphism

giving the compatibility of the associated A-model and B-model. This means given and such that under the mirror map, there is the equality of correlation functions

This is significant because it relates the number of degree genus curves on a quintic threefold in (so ) to integrals in a variation of Hodge structures. Moreover, these integrals are actually computable!

See also

References

  1. 1 2 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. Bibcode:1991NuPhB.359...21C. doi:10.1016/0550-3213(91)90292-6. ISSN 0550-3213.
  2. 1 2 3 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. for example, as a set, a Calabi-Yau manifold is the subset of complex projective space
  5. 1 2 Morrison, David R. (1993). "Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians". J. Amer. Math. Soc. 6: 223–247. arXiv:alg-geom/9202004. doi:10.1090/S0894-0347-1993-1179538-2. S2CID 9228037.
  6. Which can be thought of as the -action on constructing the complex projective space
  7. More generally, such moduli spaces are constructed using projective equivalence of schemes in a fixed projective space on a fixed Hilbert scheme
  8. 1 2 3 Cox, David A.; Katz, Sheldon (1999). Mirror symmetry and algebraic geometry. American Mathematical Society. ISBN 978-0-8218-2127-5. OCLC 903477225.
  9. 1 2 Pandharipande, Rahul; Hori, Kentaro (2003). Mirror symmetry. Providence, RI: American Mathematical Society. ISBN 0-8218-2955-6. OCLC 52374327.
  10. Hamilton, M. J. D. (2020-07-24). "The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking". arXiv:1512.02632 [math.DG].
  11. McDuff, Dusa (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.
  12. Kontsevich, M.; Manin, Yu (1994). "Gromov-Witten classes, quantum cohomology, and enumerative geometry". Communications in Mathematical Physics. 164 (3): 525–562. arXiv:hep-th/9402147. Bibcode:1994CMaPh.164..525K. doi:10.1007/BF02101490. ISSN 0010-3616. S2CID 18626455.

Books/Notes

First proofs

Derived geometry in Mirror symmetry

Research

Homological mirror symmetry

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