Curve counting at conifold singularities
Lead Research Organisation:
University of Sheffield
Abstract
How can we understand the geometry of a very complicated space? An efficient method is to probe it by counting how many subspaces of a certain type we can possibly fit inside it. For example, we can geometrically tell apart a rugby ball (an ellipsoid) from a miniature cooling tower (a one-sheeted hyperboloid) by asking “how many straight lines can they possibly contain?”: this number is zero for the oval ball, but infinite for the model tower.
As with all things in Maths, when it comes to ramping up the level of difficulty of the question, sky’s the limit. We could for example ask, using the same logic, how to distinguish the quintic (2875 straight lines) and sextic (528 straight lines) Calabi-Yau manifolds, which appear in certain models of quantum gravity containing the minimal number of copies of the electromagnetic field. In general, the information amassed by counting curves of all possible shapes and forms gives a sort of “biometric passport” of a geometric space: on top of their intrinsic value, the curve-counts can be effectively employed to distinguish shapes for which the same enumerative problem gives different answers.
The quest for an effective solution to enumerative questions received impressive momentum from a revolutionary discovery made in the early ‘90s by both physicists and geometers: solving the geometrical problem of counting curves in a given ambient space is tantamount to understanding a class of energy-minimising field configurations of an associated physical model defined on that same space. This surprising connection between Mathematics and Physics is important for theoretical physicists, as it puts on a rigorous footing the calculation of quantum mechanical observables for certain effective theories of matter, radiation, and gravity in four space-time dimensions. At the same time, it has been of invaluable importance for geometers: the intuition from quantum field theory has led to a series of remarkable (if conjectural) predictions about the original enumerative problem, often implying its complete solution.
The most difficult setup in this subject is when the geometry of the space becomes singular by developing “spikes” somewhere, like the tip of a cone. On the physics side, this means that the interactions in the corresponding model become violently coupled, with light particles drastically affecting the long-distance behaviour of the system. Quantum field theory considerations here prescribe a mysterious universal law obeyed by curve-counting invariants near the “spikes” of the geometry (in jargon, its conifold singularities): this is a major conjectural statement where mathematicians have yet to rigorously make sense of the physics heuristics.
This proposal delineates two pathways to the mathematical proof of this universal conjectural behaviour – known as the Conifold Gap Conjecture – for non-trivial families of Calabi-Yau manifolds. These methods sidestep the intricacies posed by the singular geometry by recasting the original algebro-geometric problem as a question in either
i) the asymptotic analysis of a specific class of measures on flat space, describing the statistical mechanics of a certain gas of 2-dimensional particles, or
ii) the combinatorics of certain sums over graphs, which appear in the study of Landau-Ginzburg models at higher genus.
As an application, we will completely solve the all-genus Gromov-Witten theory of an important family of Calabi-Yau varieties.
As with all things in Maths, when it comes to ramping up the level of difficulty of the question, sky’s the limit. We could for example ask, using the same logic, how to distinguish the quintic (2875 straight lines) and sextic (528 straight lines) Calabi-Yau manifolds, which appear in certain models of quantum gravity containing the minimal number of copies of the electromagnetic field. In general, the information amassed by counting curves of all possible shapes and forms gives a sort of “biometric passport” of a geometric space: on top of their intrinsic value, the curve-counts can be effectively employed to distinguish shapes for which the same enumerative problem gives different answers.
The quest for an effective solution to enumerative questions received impressive momentum from a revolutionary discovery made in the early ‘90s by both physicists and geometers: solving the geometrical problem of counting curves in a given ambient space is tantamount to understanding a class of energy-minimising field configurations of an associated physical model defined on that same space. This surprising connection between Mathematics and Physics is important for theoretical physicists, as it puts on a rigorous footing the calculation of quantum mechanical observables for certain effective theories of matter, radiation, and gravity in four space-time dimensions. At the same time, it has been of invaluable importance for geometers: the intuition from quantum field theory has led to a series of remarkable (if conjectural) predictions about the original enumerative problem, often implying its complete solution.
The most difficult setup in this subject is when the geometry of the space becomes singular by developing “spikes” somewhere, like the tip of a cone. On the physics side, this means that the interactions in the corresponding model become violently coupled, with light particles drastically affecting the long-distance behaviour of the system. Quantum field theory considerations here prescribe a mysterious universal law obeyed by curve-counting invariants near the “spikes” of the geometry (in jargon, its conifold singularities): this is a major conjectural statement where mathematicians have yet to rigorously make sense of the physics heuristics.
This proposal delineates two pathways to the mathematical proof of this universal conjectural behaviour – known as the Conifold Gap Conjecture – for non-trivial families of Calabi-Yau manifolds. These methods sidestep the intricacies posed by the singular geometry by recasting the original algebro-geometric problem as a question in either
i) the asymptotic analysis of a specific class of measures on flat space, describing the statistical mechanics of a certain gas of 2-dimensional particles, or
ii) the combinatorics of certain sums over graphs, which appear in the study of Landau-Ginzburg models at higher genus.
As an application, we will completely solve the all-genus Gromov-Witten theory of an important family of Calabi-Yau varieties.
Organisations
People |
ORCID iD |
| Andrea Brini (Principal Investigator) |