Periodic Reporting for period 4 - HIGGSBNDL (Higgs bundles: Supersymmetric Gauge Theories and Geometry)
Periodo di rendicontazione: 2021-03-01 al 2022-02-28
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.
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.
Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far (For the final period please include an overview of the results and their exploitation and dissemination)
The first half of the project focused on excpetional holonomy spaces and compactifications of branes (M5-branes specifically) and resulting lower-dimensional QFTs.
In the real of special holonomy the output of the team's research is a construction of new exceptional holonomy spaces using string theory motivated results (both in G2 and Spin (7)) and a general analysis of the Higgs bundles associated to G2-compactifications. This led to a strong interaction with the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, where Team members regularly presented at meetings, and the PI was invited to join the collaboration.
The program of M5-brane compactifications has two strands:
1. Higgs bundles for compactifications to 3d.
2. The second strand resulted in an analysis of M5-branes in so-called F-theory compactifications. The team studied compactifications, where the coupling is spacetime-varying, and pioneered new holographic duals to 2d SCFTs obtained from reductions with space-time varying coupling. This work was presented at Strings 2017 and String-Math 2017.
The second half of the project developed two of the most impactful strands in the project:
I). Geometric classification of 5d Superconformal Field Theories (SCFTs)
II). Generalized global symmetries in String theory.
In a series of papers the team developed a systematic geometric description of 5d SCFTs -- which are intrinsically strongly-coupled QFTs. This includes characterization of their global (0-form) flavor symmetries, BPS states, RG-flows after mass deformations, and characterization of parameter spaces (Higgs and Coulomb branch moduli spaces). This work was presented at String-Math in 2019, and numerous other conferences, workshops and seminars (later on largely virtually). The Oxford team played a leading role in these developments on 5d SCFTs and related 4d and 6d theories.
Generalized global symmetries have seen a huge development in the past years, and the team provided the string theoretic realizations of these symmetries: pioneering the realization of higher-form and higher-group symmetries in M-theory compactifications, and developing new approaches to studying so-called non-invertible symmetries in 4d QFTs. The Oxford group has played a central role in these developments, and presented this at Strings 2021 and String-Math 2022.
The first half of the project focused on excpetional holonomy spaces and compactifications of branes (M5-branes specifically) and resulting lower-dimensional QFTs.
In the real of special holonomy the output of the team's research is a construction of new exceptional holonomy spaces using string theory motivated results (both in G2 and Spin (7)) and a general analysis of the Higgs bundles associated to G2-compactifications. This led to a strong interaction with the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, where Team members regularly presented at meetings, and the PI was invited to join the collaboration.
The program of M5-brane compactifications has two strands:
1. Higgs bundles for compactifications to 3d.
2. The second strand resulted in an analysis of M5-branes in so-called F-theory compactifications. The team studied compactifications, where the coupling is spacetime-varying, and pioneered new holographic duals to 2d SCFTs obtained from reductions with space-time varying coupling. This work was presented at Strings 2017 and String-Math 2017.
The second half of the project developed two of the most impactful strands in the project:
I). Geometric classification of 5d Superconformal Field Theories (SCFTs)
II). Generalized global symmetries in String theory.
In a series of papers the team developed a systematic geometric description of 5d SCFTs -- which are intrinsically strongly-coupled QFTs. This includes characterization of their global (0-form) flavor symmetries, BPS states, RG-flows after mass deformations, and characterization of parameter spaces (Higgs and Coulomb branch moduli spaces). This work was presented at String-Math in 2019, and numerous other conferences, workshops and seminars (later on largely virtually). The Oxford team played a leading role in these developments on 5d SCFTs and related 4d and 6d theories.
Generalized global symmetries have seen a huge development in the past years, and the team provided the string theoretic realizations of these symmetries: pioneering the realization of higher-form and higher-group symmetries in M-theory compactifications, and developing new approaches to studying so-called non-invertible symmetries in 4d QFTs. The Oxford group has played a central role in these developments, and presented this at Strings 2021 and String-Math 2022.
The team produced 67 publications (and several more preprints, under review), which in the four categories have made huge contributions:
1. QFTs and geometry of Higgs bundles
2. M5-brane compactifications and Higgs bundles
3. Geometric Classification of 5d SCFTs
4. Generalized Symmetries in string theory
1. and 2. were building on the state of the art at the start of the project, and amount to achieving the goals set out in the proposal. There are still questions about exceptional holonomy spaces, which are compact and realize chiral matter. However what the results of the team showed is that QFT applications (i.e. non-compact geometries, where gravity is decoupled) equally have intricate geometric and more importantly physical implications. This is what led to the new objectyive 3., where completely new methods were developed to tackle 5d SCFTs.
Likewise, the work on 2. resulted in new constructions of 3d N=1 theories from M5-branes, but the crucial insight that then drove the research direction to developing the new objective 4. was the study of generalized symmetries, which were the missing piece in a complete description of the M5-brane reductions.
That realization led to the natural implication, that such generalized symmetries need to also be incorporated into any string theoretic realization of QFTS. Since 2020 when e.g. the work on generalized symmetries in 5d SCFTs appeared this has become a very active research direction in the field and led to many other generalizations both in the team and in the research community.
Thus in summary: the research has both addressed the initial objectives, but then led to new areas of investigation, that are far beyond the state of the art, and expectations of the initial proposal.
1. QFTs and geometry of Higgs bundles
2. M5-brane compactifications and Higgs bundles
3. Geometric Classification of 5d SCFTs
4. Generalized Symmetries in string theory
1. and 2. were building on the state of the art at the start of the project, and amount to achieving the goals set out in the proposal. There are still questions about exceptional holonomy spaces, which are compact and realize chiral matter. However what the results of the team showed is that QFT applications (i.e. non-compact geometries, where gravity is decoupled) equally have intricate geometric and more importantly physical implications. This is what led to the new objectyive 3., where completely new methods were developed to tackle 5d SCFTs.
Likewise, the work on 2. resulted in new constructions of 3d N=1 theories from M5-branes, but the crucial insight that then drove the research direction to developing the new objective 4. was the study of generalized symmetries, which were the missing piece in a complete description of the M5-brane reductions.
That realization led to the natural implication, that such generalized symmetries need to also be incorporated into any string theoretic realization of QFTS. Since 2020 when e.g. the work on generalized symmetries in 5d SCFTs appeared this has become a very active research direction in the field and led to many other generalizations both in the team and in the research community.
Thus in summary: the research has both addressed the initial objectives, but then led to new areas of investigation, that are far beyond the state of the art, and expectations of the initial proposal.