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SYZ conjecture, the Glossary

The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics.^[1]

27 relations: Abelian variety, Andrew Strominger, Bogomol'nyi–Prasad–Sommerfield state, Coherent sheaf, Dual abelian variety, Elliptic curve, Enumerative geometry, Eric Zaslow, Fibration, Gromov–Witten invariant, Homological algebra, Homological mirror symmetry, Jacobian variety, K3 surface, Line bundle, Maxim Kontsevich, Mirror symmetry (string theory), Moduli space, Riemannian manifold, Sheaf (mathematics), Shing-Tung Yau, String theory, Supersymmetry, Symplectic manifold, T-duality, Torsion sheaf, Type II string theory.
String theory stubs

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.

See SYZ conjecture and Abelian variety

Andrew Strominger

Andrew Eben Strominger (born 1955) is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature.

See SYZ conjecture and Andrew Strominger

Bogomol'nyi–Prasad–Sommerfield state

In theoretical physics, massive representations of an extended supersymmetry algebra called BPS states have mass equal to the supersymmetry central charge Z. Quantum mechanically, if the supersymmetry remains unbroken, exact equality to the modulus of Z exists.

See SYZ conjecture and Bogomol'nyi–Prasad–Sommerfield state

Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space.

See SYZ conjecture and Coherent sheaf

Dual abelian variety

In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field k. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this fails for higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions. SYZ conjecture and dual abelian variety are duality theories.

See SYZ conjecture and Dual abelian variety

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point.

See SYZ conjecture and Elliptic curve

Enumerative geometry

In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.

See SYZ conjecture and Enumerative geometry

Eric Zaslow

Eric Zaslow is an American mathematical physicist at Northwestern University.

See SYZ conjecture and Eric Zaslow

Fibration

The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.

See SYZ conjecture and Fibration

Gromov–Witten invariant

In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. SYZ conjecture and Gromov–Witten invariant are string theory.

See SYZ conjecture and Gromov–Witten invariant

Homological algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.

See SYZ conjecture and Homological algebra

Homological mirror symmetry

Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. SYZ conjecture and Homological mirror symmetry are conjectures, duality theories, string theory and symmetry.

See SYZ conjecture and Homological mirror symmetry

Jacobian variety

In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles.

See SYZ conjecture and Jacobian variety

K3 surface

In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. SYZ conjecture and K3 surface are string theory.

See SYZ conjecture and K3 surface

Line bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space.

See SYZ conjecture and Line bundle

Maxim Kontsevich

Maxim Lvovich Kontsevich (Макси́м Льво́вич Конце́вич,; born 25 August 1964) is a Russian and French mathematician and mathematical physicist.

See SYZ conjecture and Maxim Kontsevich

Mirror symmetry (string theory)

In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. SYZ conjecture and mirror symmetry (string theory) are string theory.

See SYZ conjecture and Mirror symmetry (string theory)

Moduli space

In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.

See SYZ conjecture and Moduli space

Riemannian manifold

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined.

See SYZ conjecture and Riemannian manifold

Sheaf (mathematics)

In mathematics, a sheaf (sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them.

See SYZ conjecture and Sheaf (mathematics)

Shing-Tung Yau

Shing-Tung Yau (born April 4, 1949) is a Chinese-American mathematician.

See SYZ conjecture and Shing-Tung Yau

String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

See SYZ conjecture and String theory

Supersymmetry

Supersymmetry is a theoretical framework in physics that suggests the existence of a symmetry between particles with integer spin (bosons) and particles with half-integer spin (fermions). SYZ conjecture and Supersymmetry are symmetry.

See SYZ conjecture and Supersymmetry

Symplectic manifold

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form \omega, called the symplectic form.

See SYZ conjecture and Symplectic manifold

T-duality

T-duality (short for target-space duality) in theoretical physics is an equivalence of two physical theories, which may be either quantum field theories or string theories. SYZ conjecture and t-duality are string theory.

See SYZ conjecture and T-duality

Torsion sheaf

In mathematics, a torsion sheaf is a sheaf of abelian groups \mathcal on a site for which, for every object U, the space of sections \Gamma(U, \mathcal) is a torsion abelian group.

See SYZ conjecture and Torsion sheaf

Type II string theory

In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. SYZ conjecture and type II string theory are string theory and string theory stubs.

See SYZ conjecture and Type II string theory

References

[1] https://en.wikipedia.org/wiki/SYZ_conjecture

Also known as Strominger-Yau-Zaslow program, Strominger–Yau–Zaslow conjecture.