8.4 Tropical Hurwitz numbers
9 min read•august 20, 2024
offer a fresh perspective on counting branched covers of surfaces. They translate classical geometric problems into the language of , providing a combinatorial approach to enumerative geometry.
This tropical viewpoint bridges classical and modern mathematics, connecting to mirror and string theory. It simplifies calculations while revealing deep structural insights, making it a powerful tool in algebraic geometry and related fields.
Definition of tropical Hurwitz numbers
- Tropical Hurwitz numbers are a tropical geometric analog of classical Hurwitz numbers, which count branched covers of the projective line with specified ramification profiles
- They are defined in terms of weighted counts of certain tropical curves, which are piecewise linear graphs embedded in the plane
- The definition involves fixing the degree of the covering map and the ramification profiles at the branch points
Connection to classical Hurwitz numbers
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- Classical Hurwitz numbers count branched covers of the Riemann sphere by compact Riemann surfaces with prescribed ramification profiles over a fixed set of points
- Tropical Hurwitz numbers can be seen as a combinatorial analog of classical Hurwitz numbers, where the role of Riemann surfaces is played by tropical curves
- The correspondence between classical and tropical Hurwitz numbers can be made precise through a process called , which relates algebraic curves to tropical curves
Relationship with tropical curves
- Tropical Hurwitz numbers are defined in terms of counts of tropical curves satisfying certain conditions
- A tropical curve is a metric graph embedded in the plane, where each edge has a positive real length and is equipped with a integer weight
- The weights on the edges encode information about the ramification profiles of the corresponding branched cover
- The tropical curves counted by Hurwitz numbers have a fixed number of unbounded edges (legs) corresponding to the branch points, and the weights of these legs determine the ramification profiles
Role of branch points
- Branch points play a crucial role in the definition of Hurwitz numbers, both in the classical and tropical settings
- In the classical case, branch points are the points on the Riemann sphere where the branched cover fails to be a local isomorphism, and the ramification profile at a branch point describes the behavior of the map near that point
- In the tropical case, branch points correspond to the unbounded edges (legs) of the tropical curve, and the weights of these legs encode the ramification profiles
- The number and location of branch points, as well as their ramification profiles, are fixed in the definition of Hurwitz numbers, and the count is taken over all possible covers with these prescribed data
Computation of tropical Hurwitz numbers
- Tropical Hurwitz numbers can be computed using combinatorial methods, by enumerating tropical curves with the specified properties
- The computation involves counting weighted graphs satisfying certain conditions, such as having a fixed number of legs with prescribed weights and satisfying a balancing condition at each vertex
- The balancing condition ensures that the graphs correspond to valid tropical curves and is related to the Riemann-Hurwitz formula in the classical setting
Combinatorial approach
- The combinatorial approach to computing tropical Hurwitz numbers involves enumerating all possible graphs satisfying the given conditions and adding up their weights
- The weights of the graphs are determined by the edge lengths and the weights of the edges, which encode the ramification data
- The enumeration can be done systematically using techniques from combinatorics, such as generating functions and bijective methods
- The combinatorial approach is particularly well-suited to studying the structure and properties of Hurwitz numbers, such as their recursion relations and asymptotic behavior
Graphical interpretation
- Tropical Hurwitz numbers have a natural graphical interpretation in terms of weighted graphs embedded in the plane
- Each tropical curve counted by a Hurwitz number corresponds to a graph with a certain number of legs and vertex degrees determined by the ramification data
- The graphical interpretation provides a convenient way to visualize and manipulate the tropical curves, and to study their combinatorial properties
- Many questions about Hurwitz numbers can be translated into questions about the corresponding graphs, such as counting the number of graphs with certain properties or studying their connectivity and genus
Examples and calculations
- Example: The simplest nontrivial tropical Hurwitz number counts covers of degree 2 of the projective line with two simple branch points. This number is equal to 1, corresponding to a unique tropical curve with two legs of weight 1 connected by an edge of length 1/2
- Example: For covers of degree 3 with two simple branch points, the tropical Hurwitz number is 3, corresponding to three distinct tropical curves: two curves with two legs of weight 1 and one leg of weight 2, and one curve with three legs of weight 1
- More complex examples involve higher degree covers and more complicated ramification profiles, leading to more intricate tropical curves and combinatorial structures
- Efficient algorithms and software tools have been developed to compute tropical Hurwitz numbers for a wide range of input data, allowing for the exploration of their properties and applications
Applications and significance
- Tropical Hurwitz numbers have found applications in various areas of mathematics and mathematical physics, providing new insights and connections between different fields
- They serve as a bridge between the classical theory of Hurwitz numbers and the modern developments in tropical geometry and enumerative geometry
- The study of tropical Hurwitz numbers has led to new results and conjectures in related areas, such as mirror symmetry and topological string theory
Enumerative geometry
- Hurwitz numbers are a fundamental object of study in enumerative geometry, which deals with counting geometric objects satisfying certain conditions
- Tropical Hurwitz numbers provide a new perspective on enumerative problems, by translating them into the language of tropical geometry and combinatorics
- Many classical enumerative problems, such as counting curves with prescribed singularities or intersecting subvarieties, have tropical analogs that can be studied using Hurwitz numbers
- The tropical approach often leads to new results and insights, such as recursive formulas, generating functions, and connections to integrable systems
Mirror symmetry
- Mirror symmetry is a profound duality between complex geometry and symplectic geometry, with far-reaching implications in string theory and enumerative geometry
- Hurwitz numbers have been shown to play a key role in the mirror symmetry correspondence, providing a link between the enumeration of curves and the computation of
- Tropical Hurwitz numbers have been used to provide a tropical analog of mirror symmetry, relating the enumeration of tropical curves to the study of tropical Gromov-Witten invariants and Landau-Ginzburg models
- The tropical approach has led to new proofs and generalizations of classical mirror symmetry results, as well as to the discovery of new mirror pairs and correspondences
Topological string theory
- Hurwitz numbers have a natural interpretation in terms of topological string theory, which studies the quantum mechanics of strings propagating on curved backgrounds
- In this context, Hurwitz numbers appear as certain amplitudes or partition functions that encode the degeneracies of string states
- Tropical Hurwitz numbers provide a new way to compute and study these amplitudes, by relating them to the enumeration of tropical curves
- The tropical approach has led to new results and conjectures in topological string theory, such as the correspondence between Hurwitz numbers and Gromov-Witten invariants, and the relation to integrable hierarchies and matrix models
Generalizations and variations
- The theory of tropical Hurwitz numbers has been generalized and extended in various directions, leading to new concepts and results
- These generalizations often involve modifying the definition of Hurwitz numbers by changing the underlying geometric objects, the ramification conditions, or the weights assigned to the covers
- Some of the most important generalizations include higher genus Hurwitz numbers, labeled Hurwitz numbers, and weighted Hurwitz numbers
Higher genus surfaces
- While the classical definition of Hurwitz numbers involves covers of the projective line (genus 0), the theory can be generalized to covers of higher genus surfaces
- Higher genus Hurwitz numbers count covers of a fixed Riemann surface of genus g by surfaces of a specified genus, with prescribed ramification profiles
- The tropical analog of higher genus Hurwitz numbers involves counts of tropical curves on higher genus tropical surfaces, such as tropical abelian varieties or tropical Jacobians
- Higher genus Hurwitz numbers have important applications in the study of moduli spaces of curves and in the computation of intersection numbers on these spaces
Labeled vs unlabeled points
- The classical definition of Hurwitz numbers involves covers with unlabeled branch points, meaning that the order of the branch points is not taken into account
- A variation of the theory considers labeled Hurwitz numbers, where the branch points are labeled and the covers are counted with respect to this labeling
- Labeled Hurwitz numbers have a slightly different combinatorial structure than unlabeled ones, and are related to the representation theory of the symmetric group
- The tropical analog of labeled Hurwitz numbers involves counts of tropical curves with labeled legs, and has applications in the study of double Hurwitz numbers and the ELSV formula
Weighted Hurwitz numbers
- Another variation of the theory involves assigning weights to the Hurwitz numbers, which can depend on various parameters such as the degree of the cover, the genus of the source and target surfaces, or the ramification profiles
- Weighted Hurwitz numbers arise naturally in the study of the moduli space of curves and its tautological intersection theory, and have connections to integrable systems and matrix models
- The tropical analog of weighted Hurwitz numbers involves counts of tropical curves with weights assigned to the vertices, edges, or legs of the curve
- Weighted Hurwitz numbers have been used to provide new proofs and generalizations of classical results, such as the Ekedahl-Lando-Shapiro-Vainshtein formula for the intersection numbers of tautological classes on the moduli space of curves
Open problems and current research
- Despite the significant progress made in the study of tropical Hurwitz numbers, there are still many open problems and active areas of research in the field
- Some of the most important questions concern the geometric and algebraic meaning of Hurwitz numbers, their relation to other enumerative invariants, and their role in the study of moduli spaces and integrable systems
- Current research in the field is focused on exploring these connections and on developing new tools and techniques for computing and studying Hurwitz numbers
Connections to integrable systems
- Hurwitz numbers have deep connections to integrable systems, such as the Kadomtsev-Petviashvili (KP) hierarchy and its dispersionless limit
- These connections have been explored in the classical setting, where Hurwitz numbers are related to certain tau functions and solutions of integrable hierarchies
- In the tropical setting, there are still many open questions about the relation between tropical Hurwitz numbers and integrable systems, such as the existence of tropical tau functions and the meaning of the dispersionless limit
- Current research is focused on understanding these connections and on using them to derive new properties and recursion relations for tropical Hurwitz numbers
Geometric meaning of generating functions
- The generating functions of Hurwitz numbers, which encode the information about all Hurwitz numbers of a given type, have a rich geometric and algebraic structure
- In the classical setting, these generating functions are related to the intersection theory of tautological classes on the moduli space of curves, and to the geometry of the Deligne-Mumford compactification
- In the tropical setting, the geometric meaning of the generating functions is still not fully understood, and there are many open questions about their relation to tropical moduli spaces and intersection theory
- Current research is focused on exploring these connections and on using them to derive new results and insights about the structure and properties of Hurwitz numbers
Hurwitz numbers and moduli spaces
- Hurwitz numbers are closely related to the geometry and topology of moduli spaces of curves, such as the Deligne-Mumford compactification and its tautological ring
- In the classical setting, Hurwitz numbers appear as intersection numbers of tautological classes, and provide a way to compute and study these numbers
- In the tropical setting, there is still much to be understood about the relation between tropical Hurwitz numbers and tropical moduli spaces, such as the moduli space of tropical curves or the tropical Hodge bundle
- Current research is focused on exploring these connections and on using them to derive new results and insights about the geometry and combinatorics of moduli spaces
- Some open problems in this area include the computation of tropical intersection numbers, the study of the cohomology and Chow rings of tropical moduli spaces, and the relation to the tautological ring and its tropical analog.
Key Terms to Review (18)
Boris A. Shapiro: Boris A. Shapiro is a mathematician known for his contributions to tropical geometry, particularly in relation to Hurwitz numbers. His work focuses on understanding the combinatorial aspects of these numbers and how they relate to various mathematical structures, including moduli spaces and algebraic curves. Shapiro's insights have advanced the field by connecting classical ideas to modern concepts in tropical geometry.
Counting Covers: Counting covers refers to the method of enumerating certain types of algebraic curves, specifically in the context of Hurwitz numbers, which count the number of ways to cover a target space with a given domain space using holomorphic maps. This concept is essential for understanding the combinatorial aspects of algebraic geometry and plays a critical role in calculating tropical Hurwitz numbers, where the geometry is studied through tropical methods.
Decorated trees: Decorated trees are combinatorial structures used in tropical geometry to represent certain algebraic and geometric objects. These trees have vertices that can be decorated with additional data, such as markings or weights, which help in counting and categorizing different configurations. They play a key role in calculating tropical Hurwitz numbers, connecting the combinatorial aspects of trees with algebraic curves and their mappings.
Degeneration techniques: Degeneration techniques are methods used to study and understand geometric objects by analyzing their limiting behaviors as they 'degenerate' into simpler or singular forms. This approach helps mathematicians gain insights into the properties of these objects by examining how they change under certain conditions, particularly in the context of algebraic geometry and tropical geometry, where understanding their structure is crucial for calculating invariants like Hurwitz numbers.
Gabriel F. T. Farkas: Gabriel F. T. Farkas is a mathematician known for his contributions to the study of tropical geometry and the combinatorial aspects of Hurwitz numbers. His work often explores the relationship between algebraic curves and their tropical counterparts, particularly in the context of counting branched covers, which are essential in understanding tropical Hurwitz numbers and their applications in enumerative geometry.
Genus g tropical Hurwitz numbers: Genus g tropical Hurwitz numbers are a count of the different ways to map a Riemann surface of genus g to the projective line, considering the combinatorial structure induced by tropical geometry. These numbers provide insight into the relationship between algebraic geometry and combinatorial topology, linking complex curves and their mappings to simpler, piecewise-linear objects in tropical geometry.
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.
Nodal curve contributions: Nodal curve contributions refer to the specific contributions of curves with nodes when calculating Tropical Hurwitz numbers, which count the number of ways to connect a given number of points on a curve while considering various geometric and combinatorial factors. These contributions take into account how nodes affect the topology of the curves, as nodes introduce singularities that alter the usual degree of the mapping, thereby impacting the counting process and the overall result.
Rationality: Rationality in mathematics often refers to a property of a number, specifically whether it can be expressed as the quotient of two integers. In the context of Tropical Geometry, rationality takes on a broader meaning, involving the classification of geometric objects based on their properties and their relation to rational points. This concept is important for understanding how these objects behave under various operations and transformations, particularly when analyzing tropical Hurwitz numbers.
Real Algebraic Geometry: Real algebraic geometry studies the properties of solutions to polynomial equations with real coefficients. This field explores how these solutions can be understood geometrically, particularly focusing on the real points of algebraic varieties and their interactions with topology and combinatorial structures. It connects to concepts such as genus, amoebas, and Hurwitz numbers, which provide deeper insights into how algebraic structures behave under tropicalization and other transformations.
Stable Maps: Stable maps are a type of morphism from a pointed, possibly nodal, curve to a target space that maintains certain geometric properties, making them important in understanding moduli spaces. They are particularly crucial for ensuring that the moduli spaces of curves are compact and well-behaved, providing a foundation for further explorations in algebraic geometry and enumerative geometry.
Symmetry: Symmetry refers to a balanced and proportionate similarity in the arrangement of parts on opposite sides of a dividing line or around a central point. In the context of Tropical Hurwitz numbers, symmetry plays a crucial role in understanding the properties and behaviors of these numbers, particularly in relation to their combinatorial aspects and geometric interpretations.
Tropical addition: Tropical addition is a fundamental operation in tropical mathematics, defined as the minimum of two elements, typically represented as $x \oplus y = \min(x, y)$. This operation serves as the backbone for tropical geometry, connecting to various concepts such as tropical multiplication and providing a distinct algebraic structure that differs from classical arithmetic.
Tropical Curves: Tropical curves are piecewise-linear structures that serve as a tropical analog to classical algebraic curves. These curves arise from the study of tropical geometry and are constructed by considering the valuation of polynomials over the tropical semiring, providing a framework for understanding properties such as intersections and moduli.
Tropical Hurwitz numbers: Tropical Hurwitz numbers are combinatorial objects that count the number of ways to 'decoratively' cover a given algebraic curve with specified data such as branch points and ramification types, using the framework of tropical geometry. These numbers provide a tropical analogue to classical Hurwitz numbers, which count the different ways to factor polynomials over the complex numbers, connecting algebraic geometry and combinatorial topology.
Tropical maps: Tropical maps are mathematical constructs that extend classical algebraic geometry into the realm of tropical geometry by utilizing the tropical semiring, where addition is replaced by taking minimums and multiplication is replaced by addition. This transformation allows for a combinatorial and piecewise linear approach to study geometric objects, leading to insights in areas such as enumerative geometry and the computation of tropical Hurwitz numbers.
Tropical Multiplication: Tropical multiplication is a mathematical operation in tropical geometry where the standard multiplication of numbers is replaced by taking the minimum of their values, thus transforming multiplication into an addition operation in this new framework. This concept connects deeply with tropical addition, allowing for the exploration of various algebraic structures and their properties.
Tropicalization: Tropicalization is the process of translating algebraic varieties and their properties into a piecewise-linear setting using tropical geometry. This allows for the study of complex geometric structures through combinatorial means, enabling a more accessible approach to problems involving algebraic curves and surfaces.