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EPSRC Reference: EP/V047698/1
Title: QALF hyperkähler metrics
Principal Investigator: Foscolo, Dr L
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Department: Mathematics
Organisation: UCL
Scheme: Standard Research - NR1
Starts: 01 July 2021 Ends: 30 June 2023 Value (£): 202,146
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
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Summary on Grant Application Form
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.

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