The Mathematics of String Theory and Gauge Theory
Lead Research Organisation:
City, University of London
Department Name: Sch of Engineering and Mathematical Sci
Abstract
This grant will cover the costs of organizing a workshop to be held between the 3rd and the 5th of May 2012 at City University London and King's College London. The overarching theme of the meeting will be the recent advances in mathematics that have a close connection to string theory and quantum field theories. We aim to bring together the leading international researchers in the field to present their latest results, analyze the state-of-the-art of the field, and inspire collaborations to identify and tackle the outstanding problems in this research area.
The interface between mathematics on the one hand and string theory and quantum field theories on the other, has often given rise to novel advances in mathematics, such as mirror symmetry, Seiberg-Witten invariants, or the study of integrable systems. Quantum field theories can have local, or gauge, symmetries, which provide them with a particularly rich mathematical structure. String theory has provided novel ways of tackling long-standing problems in this area, for example through the study of dualities between gauge theories or the gauge/string correspondence.
However, many questions about gauge theories remain unanswered. There is a shared recognition amongst many mathematicians and physicists that new insights into these questions are likely to lead to profound new developments in both disciplines in the future. Understanding the non-perturbative properties of gauge theories has been identified as a key problem of modern mathematics - indeed the solution of the four-dimensional Yang-Mills theory is one of the Clay Institute's Millennium Mathematics prizes.
Recently, significant new advances have occurred which have allowed mathematicians and physicists to address some of the long-standing gaps in our understanding of quantum gauge theories. To name just three: Firstly, localisation techniques have been used to compute the partition functions of supersymmetric gauge theories on S^4 or S^3, allowing for not only a much better handle on these gauge theories, but also allowing for the calculation of related mathematical invairants such as the Khovanov-Rozansky homology. Secondly, the AdS/CFT correspondence has recently led to both a study of sophisticated geometrical structures using advanced techniques such as equivariant localisation on symplectic manifolds and an investigation of new integrable systems and their Yangian representations. Thirdly, the investigation of supersymmetric vacua and operators of gauge theories has, in the last year or two, been given a new impetus as a result of our improved understanding of stable objects in the derived category of coherent sheafs on the one hand and a way to implement a type of deformation quantisation of the integrable structure underlyig the vacua of supersymmetric gauge theories.
The proposed meeting will focus on four areas in which such novel developments have recently taken place:
- Localisation, geometrical invariants and supersymmetric gauge theories.
- The AdS/CFT correspondence.
- Supersymmetric vacua and operators of gauge theories.
- New integrable systems and strongly coupled gauge theories.
We have chosen these four topics not only because there have been considerable advances in the recent year or two but also because we think there is considerable scope for the ideas from one area to benefit from interactions with another, and may stimulate further significant breakthroughs.
The interface between mathematics on the one hand and string theory and quantum field theories on the other, has often given rise to novel advances in mathematics, such as mirror symmetry, Seiberg-Witten invariants, or the study of integrable systems. Quantum field theories can have local, or gauge, symmetries, which provide them with a particularly rich mathematical structure. String theory has provided novel ways of tackling long-standing problems in this area, for example through the study of dualities between gauge theories or the gauge/string correspondence.
However, many questions about gauge theories remain unanswered. There is a shared recognition amongst many mathematicians and physicists that new insights into these questions are likely to lead to profound new developments in both disciplines in the future. Understanding the non-perturbative properties of gauge theories has been identified as a key problem of modern mathematics - indeed the solution of the four-dimensional Yang-Mills theory is one of the Clay Institute's Millennium Mathematics prizes.
Recently, significant new advances have occurred which have allowed mathematicians and physicists to address some of the long-standing gaps in our understanding of quantum gauge theories. To name just three: Firstly, localisation techniques have been used to compute the partition functions of supersymmetric gauge theories on S^4 or S^3, allowing for not only a much better handle on these gauge theories, but also allowing for the calculation of related mathematical invairants such as the Khovanov-Rozansky homology. Secondly, the AdS/CFT correspondence has recently led to both a study of sophisticated geometrical structures using advanced techniques such as equivariant localisation on symplectic manifolds and an investigation of new integrable systems and their Yangian representations. Thirdly, the investigation of supersymmetric vacua and operators of gauge theories has, in the last year or two, been given a new impetus as a result of our improved understanding of stable objects in the derived category of coherent sheafs on the one hand and a way to implement a type of deformation quantisation of the integrable structure underlyig the vacua of supersymmetric gauge theories.
The proposed meeting will focus on four areas in which such novel developments have recently taken place:
- Localisation, geometrical invariants and supersymmetric gauge theories.
- The AdS/CFT correspondence.
- Supersymmetric vacua and operators of gauge theories.
- New integrable systems and strongly coupled gauge theories.
We have chosen these four topics not only because there have been considerable advances in the recent year or two but also because we think there is considerable scope for the ideas from one area to benefit from interactions with another, and may stimulate further significant breakthroughs.
Planned Impact
The proposed workshop will bring together the leading international researchers in the field to present and discuss the
state-of-the-art of the field. Interactions and discussions amongst such a critical mass of world experts investigating
mathematical aspects of gauge and string theories is likely to generate a high impact on several areas of mathematics
including (i) string theory and its relation to gauge and other quantum field theories, (ii) the mathematics of mirror
symmetry, topological invariants and the Langlands programme (iii) integrability (iv) geometry and representation theory.
The workshop will allow ample time for formal and informal discussions amongst participants across these diverse fields
with the aim of identifying key problems and establishing new collaborations amongst the attendees.
We aim to make the workshop a high-visibility event, in order to increase its impact on theoretical and mathematical
physics research given above and so as to promote the high standing of UK research in this area. To acheive these
objectives we will: (i) publicise the workshop beforehand through the established channels in the field (see impact plan)
so as to reach the widest possible audience (ii) a number of prominent researchers in the field have already agreed to
participate in the workshop (see Workplan). (iii) Participants will be asked to acknowledge the workshop and EPSRC
funding for it in any research papers that they may write which have benefited from the workshop (iv) build and promote a
website for the conference; this will contain all the standard workshop information, as well as the transparencies of the
talks given at the workshop so as to reach the widest possible audience in the post-conference phase.
Finally, we will not be charging a conference fee and will encourage early-stage researchers to attend the meeting with
the aim of promoting informal discussions with the world's leading experts in the field. Such exposure is often a vital part
of training of young researchers and can significantly build their academic capacity.
state-of-the-art of the field. Interactions and discussions amongst such a critical mass of world experts investigating
mathematical aspects of gauge and string theories is likely to generate a high impact on several areas of mathematics
including (i) string theory and its relation to gauge and other quantum field theories, (ii) the mathematics of mirror
symmetry, topological invariants and the Langlands programme (iii) integrability (iv) geometry and representation theory.
The workshop will allow ample time for formal and informal discussions amongst participants across these diverse fields
with the aim of identifying key problems and establishing new collaborations amongst the attendees.
We aim to make the workshop a high-visibility event, in order to increase its impact on theoretical and mathematical
physics research given above and so as to promote the high standing of UK research in this area. To acheive these
objectives we will: (i) publicise the workshop beforehand through the established channels in the field (see impact plan)
so as to reach the widest possible audience (ii) a number of prominent researchers in the field have already agreed to
participate in the workshop (see Workplan). (iii) Participants will be asked to acknowledge the workshop and EPSRC
funding for it in any research papers that they may write which have benefited from the workshop (iv) build and promote a
website for the conference; this will contain all the standard workshop information, as well as the transparencies of the
talks given at the workshop so as to reach the widest possible audience in the post-conference phase.
Finally, we will not be charging a conference fee and will encourage early-stage researchers to attend the meeting with
the aim of promoting informal discussions with the world's leading experts in the field. Such exposure is often a vital part
of training of young researchers and can significantly build their academic capacity.
Organisations
| Description | This was a workshop grant. The workshop was very successful and allowed for an excellent exchange of ideas between leading researchers in the field. |
| Exploitation Route | by exchanging ideas to further develop mathematical and theoretical physics |
| Sectors | Other |
| URL | http://kings-city.wikidot.com/ |
| Description | The workshop explored cutting edge research in Mathematical and Theoretical Physics. Researchers presented their findings and exchange ideas. |
| First Year Of Impact | 2012 |
| Sector | Other |
| Description | Workshop Website |
| Form Of Engagement Activity | A magazine, newsletter or online publication |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The workshop website http://kings-city.wikidot.com/ was set-up in order to provide information on and provide a record of the workshop. Participants could find practical information about the organisation of the workshop. The website contains a record of the talks given at the workshop. Since the talks were at the cutting edge of the subjects, the information is of use to researchers in the field by providing a complementary description of the research presented by the researchers. |
| Year(s) Of Engagement Activity | 2012 |