Surfaces on fourfolds

Project Information

FourSurf

Grant agreement ID: 101087365

DOI

10.3030/101087365

EC signature date 5 April 2023

Start date 1 September 2023

End date 31 August 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 870 000,00

EU contribution

€ 1 870 000,00

Coordinated by

UNIVERSITEIT UTRECHT
Netherlands

Periodic Reporting for period 1 - FourSurf (Surfaces on fourfolds)

Reporting period: 2023-09-01 to 2026-02-28

Summary of the context and overall objectives of the project

The ERC CoG Foursurf is concerned with the subject of enumerative geometry: a discipline within geometry dating back to the Ancient Greeks. A typical problem from enumerative geometry is the determination of the number of circles tangent to three given circles in the plane (Apollonius's problem). It is also a field with many exciting modern aspects due to unexpected connections to theoretical physics (specifically, supersymmetric quantum field theory and string theory).

Traditionally, enumerative geometry was mostly concerned with counting curves. E.g. Apollonius's problem asks for the number of circles tangent to three given circles. Another 19th century example is the statement that a cubic surface, a geometric shape defined by an equation of degree three in three variables, contains precisely 27 lines (when viewed over the complex numbers and in projective space). Building on some recent developments in the field, this ERC CoG is about the next chapter in enumerative geometry: counting surfaces.

Specifically, this project explores the counting of surfaces on so-called Calabi-Yau fourfolds --- a type of space with four complex dimensions and many symmetries. One reason for studying this subject is because it connects to numerous other fields in mathematics such as Hodge theory, combinatorics, modular forms, representation theory as well as theoretical physics (specifically, supersymmetric quantum field theory and string theory). The impact of this project is on creating new connections of this kind. This will reveal hidden relations between previously unrelated problems in mathematics and physics and may help us to solve some of those problems.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

One place where the enumerative geometry on Calabi-Yau fourfolds connects to both combinatorics and theoretical physics is in the Magnificent Four formula. In the early 20th century (around 1915) MacMahon asked in how many ways can you pile a given number of boxes in the corner of a large room (without leaving gaps). He gave a beautiful formula for the answer in terms of an infinite product. He then moved ahead to the analogous problem in four dimensions, where he initially proposed an incorrect infinite product formula. A correct infinite product formula was never found since. In 2017 in physics, Nekrasov discovered the correct formula if one allows to count the (4-dimensional) piles of boxes with appropriate weights! He was led to this solution through the study of a quantum field theory: supersymmetric Yang-Mills theory of (complex) 4-dimensional space. As part of this ERC CoG project, jointly with J. V. Rennemo, we finished a purely geometric proof of this formula and we explain where the weight comes from geometrically.

In a different direction, in physics Nekrasov formulated another beautiful theory: gauge origami. Roughly speaking, it describes supersymmetric quantum field theory on a (4-dimensional) space-time that can break into pieces along lines and points. Imagine two universes meeting in one point in space-time. He describes how certain objects called instantons move in this space-time. Jointly with Arbesfeld-Lim, we are finishing a purely geometric description of Nekrasov's theory which provides us with novel connections between geometry, physics, and representation theory. The viewpoint of this ERC CoG project is to regards gauge origami theory as a special kind of "surface counting" (technically realized by a so-called virtual cycle due to Borisov-Joyce and Oh-Thomas).

These are two sample problems explored in this ERC CoG project. Other problems are related to Hodge theory and modular forms, where new insights have been obtained by members of our group.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The proof of Nekrasov's Magnificent Four formula and the geometric formulation of Nekrasov's gauge origami theory have moved the field beyond the state-of-the-art by creating new connections between geometry and physics. Other members of the group have developed novel insights into the intersection theory on moduli spaces of stable sheaves on complex surfaces, logarithmic tautological rings, and Hodge theory. All of these developments open up new directions.