QALF hyperkähler metrics
Lead Research Organisation:
University College London
Department Name: Mathematics
Abstract
Hyperkähler manifolds are geometric spaces that carry an extremely rich geometric structure. This rich structure makes hyperkähler manifolds particularly beautiful and constrained examples of larger classes of geometric spaces: for example, all hyperkähler manifolds are Ricci-flat and therefore are closely related to solutions to Einstein's equations of General Relativity.
From a completely different perspective, it was understood since the 1980's that hyperkähler manifolds arise naturally as the spaces of vacua (or "equilibrium states") of many gauge theories in theoretical physics, that is, physical theories that generalize Maxwell's equations of electro-magnetism. In a fruitful interaction between mathematics and physics, the geometric properties of the hyperkähler manifolds can be used to derived properties of the corresponding physical theory, while physics predicts the existence of hyperkähler manifolds with distinguished properties in the first place.
A challenging obstacle to our full understanding of hyperkähler manifolds and their behaviour in families is the fact that hyperkähler manifolds can "collapse", that is, they can converge to a limit space of lower dimension. From the physics perspective, collapse of the hyperkähler spaces of vacua arise in certain limits of the corresponding physical theory where coupling constants converge to zero or infinity.
In recent years substantial progress has been made in the study of collapsed degenerations of hyperkähler manifolds in the lowest possible dimension 4. Non-compact hyperkähler manifolds with prescribed asymptotic geometry have played a key role in these recent advances: 4-dimensional hyperkähler manifolds with interesting asymptotic geometry have been constructed since the 1980's using an array of diverse techniques, but only recently they have been completely classified. In this project we aim to construct and classify higher dimensional hyperkähler manifolds with a distinguished asymptotic geometry that we call QALF, solving completely the existence and uniqueness problem for this class of spaces.
Applications of this study are numerous. Within hyperkähler geometry, we aim to use the new examples as building blocks to produce more complicated examples of higher dimensional hyperkähler manifolds and to study their behaviour in families. Beyond pure mathematics, the project has direct applications to theoretical physics, where QALF hyperkähler manifolds arise as spaces of vacua of 3-dimensional quantum gauge theories. While the definition of the quantum theory itself in rigorous mathematical language is currently out of reach, in this project we define rigorously the hyperkähler spaces of vacua of the theory, from which properties of the physical theory can then be derived.
From a completely different perspective, it was understood since the 1980's that hyperkähler manifolds arise naturally as the spaces of vacua (or "equilibrium states") of many gauge theories in theoretical physics, that is, physical theories that generalize Maxwell's equations of electro-magnetism. In a fruitful interaction between mathematics and physics, the geometric properties of the hyperkähler manifolds can be used to derived properties of the corresponding physical theory, while physics predicts the existence of hyperkähler manifolds with distinguished properties in the first place.
A challenging obstacle to our full understanding of hyperkähler manifolds and their behaviour in families is the fact that hyperkähler manifolds can "collapse", that is, they can converge to a limit space of lower dimension. From the physics perspective, collapse of the hyperkähler spaces of vacua arise in certain limits of the corresponding physical theory where coupling constants converge to zero or infinity.
In recent years substantial progress has been made in the study of collapsed degenerations of hyperkähler manifolds in the lowest possible dimension 4. Non-compact hyperkähler manifolds with prescribed asymptotic geometry have played a key role in these recent advances: 4-dimensional hyperkähler manifolds with interesting asymptotic geometry have been constructed since the 1980's using an array of diverse techniques, but only recently they have been completely classified. In this project we aim to construct and classify higher dimensional hyperkähler manifolds with a distinguished asymptotic geometry that we call QALF, solving completely the existence and uniqueness problem for this class of spaces.
Applications of this study are numerous. Within hyperkähler geometry, we aim to use the new examples as building blocks to produce more complicated examples of higher dimensional hyperkähler manifolds and to study their behaviour in families. Beyond pure mathematics, the project has direct applications to theoretical physics, where QALF hyperkähler manifolds arise as spaces of vacua of 3-dimensional quantum gauge theories. While the definition of the quantum theory itself in rigorous mathematical language is currently out of reach, in this project we define rigorously the hyperkähler spaces of vacua of the theory, from which properties of the physical theory can then be derived.
Organisations
People |
ORCID iD |
| Lorenzo Foscolo (Principal Investigator) |
Publications
Bielawski R
(2023)
Hypertoric varieties, $W$-Hilbert schemes, and Coulomb branches
Foscolo L
(2023)
Calorons and Constituent Monopoles
in Communications in Mathematical Physics
Foscolo Lorenzo
(2022)
Calorons and constituent monopoles
in arXiv e-prints
Franchetti Guido
(2023)
The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space
in arXiv e-prints
| Description | The project aimed to study a class of higher dimensional spaces, the QALF hyperkähler manifolds of the title, naturally arising in theoretical physics and geometry. These spaces are expected to have beautiful properties and to have a role in different areas of mathematics and theoretical physics, including geometric representation theory, algebraic and differential geometry, and gauge theory. During the current reporting period, the team working on the project has focused on two main directions: 1) Realising the sought after spaces as moduli spaces of classical gauge theory. A paper by PI L. Foscolo and postdoc C. Ross has been completed and submitted for publication in a very good journal in the research area. The paper provides a new construction of certain gauge theoretic objects known as calorons (or periodic instantons) in terms of simpler and better understood building blocks (monopoles). A follow up paper by the same authors, where these results are generalised to instantons on arbitrary ALF spaces and are used to derive informations about the geometry of the moduli spaces of instantons in this setting, is already in advanced stages of preparation. 2) A new uniform geometric description of Coulomb branches of 3-dimensional quantum gauge theories with N=4 supersymmetries, recently defined within mathematics by Bravermann-Finkelberg-Nakajima, in terms of Hilbert schemes of well-known spaces known as hypertoric varieties. A paper by the PI L. Foscolo and R. Bielawski (Hannover) summarising the results of this part of the project is in final stages of preparation. |
| Exploitation Route | As already explained, there are already two further publications to arise from work on this project. The results developed in the project have been presented at the workshop ALX hyperkähler metrics held at the American Institute of Mathematics in July 2022 and at MSRI/SLMath during the Analytic and Geometric Aspects of Gauge Theory program in Fall 2022. They have attracted interest of various experts, such as S. Cherkis (Arizona), J. Lotay (Oxford), M. Stern (Duke), focusing in two different areas: hyperkähler metrics arising in gauge theory, and gauge theory on higher dimensional manifolds with special holonomy. We expect the results we have established so far will have concrete and fruitful applications within these two areas of research at the interface of geometry and theoretical physics. |
| Sectors | Other |