String theory is first and foremost a theory that combines quantum mechanics with the theory of gravity into one unified theoretical framework. However its relevance surpasses far beyond and reaches deep into our understanding of the fundamentals of quantum field theories.
A remarkable connection that string theory induces is that of quantum field theories (QFTs) and mathematical structures, such as geometry.
There are various points of contact how this connection can arise, but all of them in one way or another follow from studying a type of Hitchin system, whose solutions are so-called Higgs bundles. This connection, and both its physics and mathematics implications, is what this ERC Consolidator grant team set out to explore.
This geometrization program of QFTs is however not a mere translation of QFT properties into geometric terms: the connection to string theoretic realizations becomes essential (and oftentimes, the only approach) to studying strongly-coupled QFTs. The standard approach to interacting QFTs is to study them as small perturbations to an essentially free or exactly solvable theory. However, if these interactions become strong, such a perturbative approach is void, and alternative means of describing such theories need to be developed. Tackling this problem of strong coupling is where string theory has established itself as a vital tool: both in terms of realization QFTs from a dimensional reductions, as well as holographic description (using dual gravitational theories).
It is addressing this challenge of characterizing stongly-coupled QFTs, which evolved to be the main motivation and goal of the ERC Consolidator group. It is also here, where the group produced the most impactful results. At the start of the grant the geometrization program was the main focus:
string theory compactified on suitably singular spaces realizes QFTs, including such strongly-coupled theories -- in a multitude of spacetime dimension. Examples are QFTs with scale invariance in 5d and 6d, but also 4d theories that are closely related to realistic QFTs. The geometry of the compactification space determines in this instance the properties of the QFT and can be utilized even if the QFT itself is strongly-coupled. The team led two research directions: results on geometric explorations of strongly-coupled QFTs (in 5d) and so-called generalized symmetries in string theory.
Core to the research of this ERC Consolidator grant was the geometrization program of QFTs, leading to new approaches to strongly coupled theories, with and without scale invariance, as well as new symmetries and their realization in string theory. In turn the insights from QFTs has enabled vital progress in geometry, in particular the notoriously difficult exceptional holonomy spaces. The synergy between physics and mathematics has again led to exciting progress in both fields leading to a ubituity of new research directions for the future.
