9.5 Tropical mirror symmetry
4 min read•august 20, 2024
connects tropical and complex geometry, offering a combinatorial approach to study mirror symmetry. It explores relationships between and their mirror partners, providing insights into and of .
This framework allows for deeper understanding of in tropical settings. It examines Fukaya categories of tropical manifolds, derived categories of coherent sheaves, and , offering new perspectives on complex geometric structures.
Tropical mirror symmetry overview
- Tropical mirror symmetry establishes a correspondence between tropical geometry and complex geometry, providing a framework to study mirror symmetry in a combinatorial setting
- Investigates the relationship between tropical manifolds and their mirror partners, which are derived categories of coherent sheaves on algebraic varieties
- Offers new insights into the structure of Fukaya categories and derived categories, leading to a deeper understanding of homological mirror symmetry
Homological mirror symmetry in tropical setting
Fukaya category of tropical manifolds
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- Defines a category whose objects are Lagrangian submanifolds of a tropical manifold and morphisms are Floer cohomology groups
- Studies the symplectic geometry of tropical manifolds using the tools of Floer theory
- Provides a combinatorial description of the Fukaya category, making it more amenable to explicit computations
- Relates the Fukaya category of a tropical manifold to the derived category of its mirror partner
Derived category of coherent sheaves
- Considers the category of coherent sheaves on an algebraic variety, with morphisms given by derived functors
- Encodes important geometric information about the variety, such as its cohomology and intersection theory
- Serves as the mirror partner to the Fukaya category in the tropical setting
- Allows for the study of homological properties of algebraic varieties using the language of derived categories
Tropical Landau-Ginzburg models
Tropical superpotentials
- Defines a tropical analogue of the superpotential in the Landau-Ginzburg model, a function on a tropical affine space
- Encodes information about the critical points and gradient flow of the superpotential
- Relates the tropical superpotential to the mirror partner via the tropical SYZ fibration
- Provides a combinatorial description of the Landau-Ginzburg model in the tropical setting
Tropical Morse theory
- Develops a Morse theory for tropical manifolds using the tropical superpotential as a Morse function
- Studies the topology of tropical manifolds by analyzing the critical points and gradient flow of the superpotential
- Relates the tropical Morse complex to the Fukaya category of the tropical manifold
- Offers a combinatorial approach to understanding the topology of tropical manifolds
SYZ conjecture for tropical toric varieties
Tropical SYZ fibrations
- Constructs a special Lagrangian fibration on a tropical toric variety, analogous to the Strominger-Yau-Zaslow (SYZ) fibration in the complex setting
- Relates the tropical toric variety to its mirror partner via the tropical SYZ fibration
- Provides a geometric interpretation of the mirror symmetry between and their Landau-Ginzburg models
- Allows for the study of tropical mirror symmetry using the language of special Lagrangian fibrations
Tropical Lagrangian sections
- Defines Lagrangian sections of the tropical SYZ fibration, which are tropical subvarieties that intersect each fiber in a single point
- Relates to holomorphic vector bundles on the mirror partner
- Provides a correspondence between tropical geometry and complex geometry via the tropical SYZ fibration
- Offers a way to construct explicit examples of mirror pairs in the tropical setting
Real tropical curves vs complex curves
Tropical Welschinger invariants
- Defines invariants that count real rational curves on a tropical surface satisfying certain conditions
- Provides a tropical analogue of the Welschinger invariants in real algebraic geometry
- Relates the to the of the mirror partner
- Offers a combinatorial approach to studying real algebraic curves using tropical geometry
Mikhalkin's correspondence theorem
- Establishes a correspondence between complex algebraic curves and tropical curves satisfying certain conditions
- Allows for the computation of Gromov-Witten invariants of complex toric surfaces using tropical geometry
- Provides a tool for studying the enumerative geometry of complex algebraic curves using combinatorial methods
- Relates the tropical Welschinger invariants to the Gromov-Witten invariants of complex toric surfaces
Tropical quantum cohomology
Tropical Gromov-Witten invariants
- Defines invariants that count tropical curves on a tropical toric variety satisfying certain conditions
- Provides a tropical analogue of the Gromov-Witten invariants in complex geometry
- Relates the to the quantum cohomology of the mirror partner
- Offers a combinatorial approach to studying the enumerative geometry of tropical toric varieties
Tropical descendant invariants
- Defines invariants that generalize the tropical Gromov-Witten invariants by incorporating psi-classes on the moduli space of tropical curves
- Provides a tropical analogue of the descendant invariants in complex geometry
- Relates the to the higher genus Gromov-Witten invariants of the mirror partner
- Allows for the study of more refined enumerative information using tropical geometry
Applications of tropical mirror symmetry
Tropical enumerative geometry
- Applies the tools of tropical mirror symmetry to solve problems in enumerative geometry, such as counting curves satisfying certain conditions
- Uses the correspondence between tropical geometry and complex geometry to compute Gromov-Witten invariants and other enumerative invariants
- Provides a combinatorial approach to studying the enumerative geometry of complex algebraic varieties
- Offers new insights into the structure of the moduli space of curves and its intersection theory
Homological mirror symmetry in toric setting
- Applies the ideas of tropical mirror symmetry to the study of
- Relates the Fukaya category of a toric manifold to the derived category of coherent sheaves on its mirror partner
- Provides explicit computations of the Fukaya category and derived category in the toric setting using tropical geometry
- Offers a new perspective on the homological mirror symmetry conjecture and its relation to tropical geometry
Key Terms to Review (22)
Coherent Sheaves: Coherent sheaves are a class of sheaves that satisfy certain finiteness conditions, making them essential in algebraic geometry and related areas. They allow for the study of local properties of varieties and schemes, particularly in the context of coherent cohomology and their applications in mirror symmetry, where the relationships between different geometric objects can be explored through their associated sheaf structures.
Complex curves: Complex curves are one-dimensional algebraic varieties defined over the complex numbers, representing the solution set of polynomial equations in two variables. These curves can exhibit intricate geometric properties and play a crucial role in understanding various phenomena in algebraic geometry and tropical geometry, particularly when discussing mirror symmetry.
Derived Categories: Derived categories are a fundamental concept in modern mathematics, particularly in the fields of algebraic geometry and homological algebra. They provide a way to systematically handle complexes of objects, allowing for the study of their morphisms and cohomological properties. This framework is crucial for establishing equivalences between categories, facilitating deep connections between seemingly disparate areas, such as mirror symmetry and tropical geometry.
Fukaya Categories: Fukaya categories are mathematical structures used in symplectic geometry and mirror symmetry, focusing on the study of Lagrangian submanifolds and their morphisms. They encapsulate the rich algebraic and geometric data associated with these Lagrangians, providing a framework to understand the relationships between different symplectic manifolds and their dual structures through mirror symmetry.
Gromov-Witten invariants: Gromov-Witten invariants are mathematical objects that count the number of curves of a certain class on a given algebraic variety, taking into account their interactions with the geometry of the space. These invariants are crucial in enumerative geometry, linking the world of algebraic geometry with physical theories, especially in string theory. They provide a way to study the geometry of moduli spaces and can be extended to tropical geometry, where they help understand the combinatorial aspects of curves and their deformations.
Homological Mirror Symmetry: Homological mirror symmetry is a conjectural relationship between the categories of coherent sheaves on a complex manifold and the Fukaya category of a symplectic manifold. This principle suggests that certain geometric properties of these spaces are mirrored through their respective algebraic structures, creating profound links between algebraic geometry and symplectic geometry.
Homological Mirror Symmetry for Toric Varieties: Homological mirror symmetry for toric varieties is a conjectural framework that relates the derived category of coherent sheaves on a toric variety to the Fukaya category of its mirror dual. This concept ties together algebraic geometry and symplectic geometry, suggesting that these two seemingly different mathematical structures can be understood in a unified way through homological algebra.
Mikhalkin's Correspondence Theorem: Mikhalkin's Correspondence Theorem establishes a deep connection between tropical geometry and classical algebraic geometry, particularly focusing on stable intersections of tropical curves. It asserts that the count of certain combinatorial types of tropical curves, known as stable curves, corresponds to enumerative invariants of classical algebraic curves. This theorem highlights the interplay between the tropical and classical worlds, revealing how problems in one realm can be translated into the other.
Real tropical curves: Real tropical curves are geometric objects defined over the real numbers, arising in the field of tropical geometry. They can be thought of as piecewise linear objects that generalize classical algebraic curves, allowing for a new way to study their properties and relationships, particularly in relation to mirror symmetry and algebraic geometry. Real tropical curves are often visualized as graphs in the tropical setting, enabling researchers to connect complex algebraic structures with more intuitive geometric interpretations.
Syz Conjecture: The Syz Conjecture is a hypothesis in the field of algebraic geometry that proposes a relationship between certain algebraic cycles and syzygies in a projective variety. This conjecture posits that every projective variety has a finite number of syzygies, implying that the algebraic properties of the variety can be understood through these relationships, significantly impacting tropical mirror symmetry by suggesting connections between classical and tropical geometric properties.
Tropical descendant invariants: Tropical descendant invariants are algebraic tools used in tropical geometry that generalize classical intersection numbers and count curves in a tropical setting, incorporating contributions from various descendant classes. These invariants connect the counting of curves to the geometry of tropical varieties and facilitate the study of enumerative geometry. They also play a critical role in mirror symmetry, linking families of tropical curves to their dual mirror pairs.
Tropical Enumerative Geometry: Tropical enumerative geometry studies the solutions to geometric counting problems in the framework of tropical mathematics, which uses piecewise linear structures instead of classical algebraic varieties. This field connects tropical geometry to classical enumerative problems, allowing for new interpretations and computations involving counts of curves, intersection numbers, and more, using tropical methods.
Tropical Gromov-Witten Invariants: Tropical Gromov-Witten invariants are combinatorial invariants that count the number of certain types of tropical curves in a given tropical variety, serving as a tropical analog to classical Gromov-Witten invariants. These invariants help in understanding the geometry of moduli spaces and their compactifications, linking them to enumerative geometry through various structures like Schubert calculus and mirror symmetry.
Tropical Lagrangian Sections: Tropical Lagrangian sections are a crucial concept in tropical geometry, representing Lagrangian submanifolds in a tropical context. They relate to the intersection theory and mirror symmetry in algebraic geometry, providing insights into the duality between toric varieties and their corresponding tropical varieties. Understanding tropical Lagrangian sections is key to grasping how geometry transforms under tropicalization and how these transformations reflect deeper mathematical relationships.
Tropical landau-ginzburg models: Tropical landau-ginzburg models are mathematical constructs that combine tropical geometry with concepts from mirror symmetry and algebraic geometry. They typically involve a polynomial function defined over a tropical semiring, which captures the combinatorial nature of algebraic varieties and their duals. This approach offers a way to study mirror symmetry in a tropical context, highlighting relationships between different geometric structures.
Tropical manifolds: Tropical manifolds are a generalization of classical manifolds in algebraic geometry, constructed using tropical mathematics. They provide a framework for understanding geometric structures through the lens of piecewise-linear functions, enabling the study of complex algebraic varieties in a simpler, combinatorial way. This concept is particularly relevant in exploring enumerative geometry and mirror symmetry, as it allows for counting solutions to geometric problems and establishing connections between seemingly different mathematical structures.
Tropical mirror symmetry: Tropical mirror symmetry refers to a conjectural relationship between two different types of geometric objects: the classical mirrors and their tropical counterparts. This concept suggests that certain invariants of a classical algebraic variety can be computed in a tropical setting, highlighting deep connections between algebraic geometry and combinatorial geometry. The interplay between these two realms is crucial for understanding phenomena like duality in algebraic varieties, leading to insights in both tropical Schubert calculus and the Deligne-Mumford compactification.
Tropical Morse Theory: Tropical Morse Theory is a framework that studies the topology and combinatorial properties of tropical spaces using ideas from classical Morse theory. It focuses on understanding how critical points of tropical functions relate to the geometric and combinatorial structures of tropical varieties, offering insights into their shape and behavior. This theory provides tools to analyze the relationships between tropical geometry and algebraic geometry, especially in the context of mirror symmetry.
Tropical Superpotentials: Tropical superpotentials are functions that play a crucial role in tropical geometry, serving as a bridge between algebraic geometry and combinatorial optimization. They generalize classical potentials and help in understanding the behavior of tropical varieties, particularly in relation to mirror symmetry. These superpotentials encode information about the structure of the underlying geometric objects and can be used to derive various important results in both tropical and algebraic settings.
Tropical syz fibrations: Tropical syz fibrations are a concept in tropical geometry that generalizes the notion of a fibration by utilizing tropical techniques to study algebraic varieties. They provide a framework for understanding how the structure of these varieties can be simplified through tropicalization, allowing researchers to explore properties like mirror symmetry in a more accessible way. This concept plays a significant role in connecting the geometrical properties of algebraic varieties with their combinatorial counterparts, particularly in the context of mirror symmetry.
Tropical toric varieties: Tropical toric varieties are geometric objects that arise from combining tropical geometry with the theory of toric varieties, which are constructed from combinatorial data of fans and their corresponding polytopes. They provide a way to study algebraic varieties through a piecewise-linear lens, allowing for a new perspective on properties like mirror symmetry and compactifications by using combinatorial structures associated with the underlying algebraic geometry.
Tropical welschinger invariants: Tropical Welschinger invariants are mathematical objects that arise in tropical geometry, representing counts of certain types of curves in algebraic geometry. These invariants are particularly useful in understanding the enumerative geometry of real and complex algebraic varieties, connecting classical geometric problems to their tropical counterparts. They play a critical role in the study of mirror symmetry, allowing for the comparison between different geometric objects through tropical techniques.