EPSRC-SFI - Solving Spins and Strings
Lead Research Organisation:
University of Surrey
Department Name: Mathematics
Abstract
In recent years, stemming from the study of a particular class of integrable systems, new remarkable mathematical structures have been discovered. These exotic algebraic constructions extend the standard framework of quantum groups to situations where novel unexpected phenomena are seen to emerge. Integrable systems have the property that their evolution equations can be exactly solved via reduction to an auxiliary linear problem. When these systems are combined with Lie superalgebras - that is, Lie algebras for which there exists a notion of "even" (commuting) and "odd" (anti-commuting) generators - new exciting facts occur. This has been partly established through the work of the applicants. The so called "Hopf" algebra describing multiple (tensorial) products of these algebras, for instance, acquires non-trivial deformations, whose consequences have not yet been fully understood. Furthermore, the systems in question exhibit a symmetry-enhancement which is not manifest from the Hamiltonian formulation. This "secret" symmetry results in novel more complicated quantities being conserved during the time evolution. A complete mathematical formulation of these effects has yet to be developed, and it is believed to be crucial to understand potential implications for branches of mathematics such as algebra, geometry, the topology of knots and link-invariants, and integrable systems.
The aim of this research project is to understand such exotic structures, and use this new understanding to attack challenging problems at the interface between Mathematical Physics and these contiguous areas. One such problem is the so-called "non-ultralocality" of Poisson structures, governing the formulation of integrable systems in their semi-classical approximation. Non-ultralocality makes the algebraic interpretation of the solution to these systems dramatically more obscure, and it is a difficult problem which has challenged mathematicians for years. We believe that the key to significant progress in this direction is a rigorous understanding of the underlying exotic algebras. Any progress in this area will have a major long-term impact on the mathematical community, and on the scientific environment in the UK and internationally.
We plan to attack the problem by constructing a diverse set of "representations" which explicitly realise the action of these exotic algebras; especially important will be what we call the "massless" ones. These are special representations occurring when the parameters satisfy very particular relations, and have recently been found to play a crucial role in the associated spectral analysis. This will be combined with the development of new techniques to treat quantum superalgebras and the so-called Bethe ansatz. From this work, we plan to derive new results on quantum groups and apply them to the problem of non-ultralocality in integrable systems. The intradisciplinary character of the project, combining ideas and techniques from different areas of mathematics, will lead to new results across a broad range of topics, from group theory to geometry (Hamiltonian structures), topology (knot invariants, Grassmannian manifolds) and combinatorics (Bethe equations, Baxter operators and Yangians).
The aim of this research project is to understand such exotic structures, and use this new understanding to attack challenging problems at the interface between Mathematical Physics and these contiguous areas. One such problem is the so-called "non-ultralocality" of Poisson structures, governing the formulation of integrable systems in their semi-classical approximation. Non-ultralocality makes the algebraic interpretation of the solution to these systems dramatically more obscure, and it is a difficult problem which has challenged mathematicians for years. We believe that the key to significant progress in this direction is a rigorous understanding of the underlying exotic algebras. Any progress in this area will have a major long-term impact on the mathematical community, and on the scientific environment in the UK and internationally.
We plan to attack the problem by constructing a diverse set of "representations" which explicitly realise the action of these exotic algebras; especially important will be what we call the "massless" ones. These are special representations occurring when the parameters satisfy very particular relations, and have recently been found to play a crucial role in the associated spectral analysis. This will be combined with the development of new techniques to treat quantum superalgebras and the so-called Bethe ansatz. From this work, we plan to derive new results on quantum groups and apply them to the problem of non-ultralocality in integrable systems. The intradisciplinary character of the project, combining ideas and techniques from different areas of mathematics, will lead to new results across a broad range of topics, from group theory to geometry (Hamiltonian structures), topology (knot invariants, Grassmannian manifolds) and combinatorics (Bethe equations, Baxter operators and Yangians).
Planned Impact
This project stands at the interface between different mathematical disciplines, with the potential of benefiting several distinct communities of researchers.
In the short term (5 to 10 years), this will strongly impact the UK scientific community, currently very active in the area, and the international stage, by generating new trends in Lie superalgebras and quantum groups. A wide response in terms of publications, Masters and PhD theses and scientific exchanges is foreseen. In the longer run (50+ years), quantifiable progress on important questions, such as those concerning non-ultralocality in Poisson structures, will give great prestige to the project, and will significantly affect our mathematical understanding of integrability.
In order to deliver this impact, several activities are foreseen:
- Conferences: we intends to disseminate our research at the main UK and international events.
- Research visits and Collaborations: we have strong links with researchers at various UK institutions (in particular City University) whom we will frequently visit during the project. We will also pursue international collaborations which will greatly contribute to the worldwide impact of the project.
- Seminars: a rich program of seminars and journal clubs will be in place at both respective host institutions. This will provide an informal setting for discussion with potential contributors to the project, and an opportunity for PhD students to be exposed to cutting-edge research. It will also facilitate cross-fertilisation with the other departmental research groups.
- Mentoring and training high-skill researchers: we will i) hire and mentor the PDRAs, ii) advertise the research to potential MSc or PhD applicants, in particular the PhD Open Days. One of the applicants has extensively published with the junior members of his group, and was recently awarded Best Lecturer at the Durham Integrability School - summer 2015.
Societal Impact
The applicants strongly believe in the need of inspiring the new generations towards Mathematics and the STEM subjects.
- They will be ambassadors for local activities aimed at raising awareness on the importance of Mathematics in Education
- One of the applicant organised and fund-raised for the 2014 edition of the Tomorrow's Mathematicians Today (TMT) undergraduate conference. This was an astounding success, with press releases by the IMA and LMS (who co-funded the event) and the University of Surrey Bulletin. Sponsors were IMA, GCHQ, GSK, MedImmune, Springer and Taylor & Francis, emphasising the importance for mathematicians to connect with the world outside academia.
In the short term (5 to 10 years), this will strongly impact the UK scientific community, currently very active in the area, and the international stage, by generating new trends in Lie superalgebras and quantum groups. A wide response in terms of publications, Masters and PhD theses and scientific exchanges is foreseen. In the longer run (50+ years), quantifiable progress on important questions, such as those concerning non-ultralocality in Poisson structures, will give great prestige to the project, and will significantly affect our mathematical understanding of integrability.
In order to deliver this impact, several activities are foreseen:
- Conferences: we intends to disseminate our research at the main UK and international events.
- Research visits and Collaborations: we have strong links with researchers at various UK institutions (in particular City University) whom we will frequently visit during the project. We will also pursue international collaborations which will greatly contribute to the worldwide impact of the project.
- Seminars: a rich program of seminars and journal clubs will be in place at both respective host institutions. This will provide an informal setting for discussion with potential contributors to the project, and an opportunity for PhD students to be exposed to cutting-edge research. It will also facilitate cross-fertilisation with the other departmental research groups.
- Mentoring and training high-skill researchers: we will i) hire and mentor the PDRAs, ii) advertise the research to potential MSc or PhD applicants, in particular the PhD Open Days. One of the applicants has extensively published with the junior members of his group, and was recently awarded Best Lecturer at the Durham Integrability School - summer 2015.
Societal Impact
The applicants strongly believe in the need of inspiring the new generations towards Mathematics and the STEM subjects.
- They will be ambassadors for local activities aimed at raising awareness on the importance of Mathematics in Education
- One of the applicant organised and fund-raised for the 2014 edition of the Tomorrow's Mathematicians Today (TMT) undergraduate conference. This was an astounding success, with press releases by the IMA and LMS (who co-funded the event) and the University of Surrey Bulletin. Sponsors were IMA, GCHQ, GSK, MedImmune, Springer and Taylor & Francis, emphasising the importance for mathematicians to connect with the world outside academia.
People |
ORCID iD |
| Alessandro Torrielli (Principal Investigator) |
Publications
Bielli D
(2023)
A study of integrable form factors in massless relativistic AdS2
in Journal of High Energy Physics
Cavaglià A
(2021)
Quantum Spectral Curve for AdS3/CFT2: a proposal
in Journal of High Energy Physics
De Leeuw M
(2021)
Free fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT
in Journal of High Energy Physics
De Leeuw M
(2020)
Free Fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT
Fontanella A
(2022)
Non-relativistic string monodromies
Fontanella A
(2023)
Light-cone gauge in non-relativistic AdS5×S5 string theory
in Journal of High Energy Physics
Fontanella A
(2022)
Extending the nonrelativistic string AdS coset
in Physical Review D
Fontanella A
(2022)
Extending the non-relativistic string AdS coset
| Description | The grand challenge which has occupied our recent work and which we plan to focus on in the next 3-5 years is the integrable structure of the massless sector of quantum superstrings in Anti de Sitter spaces. This is a significantly novel challenge for the international string theory community working on exact results in holography and AdS/CFT, and it involves going beyond the standard paradigms of integrability in supersymmetric theories. 1. During the past three years my collaborators and I have made significant progress in understanding the integrable massless sector of quantum strings on AdS3 spaces. This has been partly determined by the discovery that the S-matrix in the entire massless sector is of difference form, thanks to a particular change of variables which A. Fontanella (Perimeter Institute) and I have found. We call this form pseudo-relativistic, and it allows an enormous simplification in the calculations and a powerful use of analyticity in one single difference variable. We have exploited the presence of this remnant of relativistic invariance in the gauge-fixed sigma model to conjecture the Thermodynamic Bethe ansatz (TBA) for the full massless sector in a paper with Bogdan Stefanski (City U.) and Olof Ohlsson Sax (Nordita - Stockholm). In this paper we also include an analysis of the special point in the mixed-flux case, which is a hot topic of debate in the literature. 2. By using the pseudo-relativistic invariance, my postdoc Juan Miguel Nieto Garcia (now at Hamburg U.) and I we have obtained formulas for the norm of the quantum Bethe states and their scalar products, and in collaboration with Marius de Leeuw, Chiara Paletta, Ana Retore and Anton Pribytok at Trinity College Dublin we have shown that all the integrable S-matrices relevant to the holographic dualities in lower dimensions satisfy the free-fermion condition. We have used this fact to find a Bogoliubov transformation which recasts the transfer matrix of the AdS_3 and AdS_2 string theories in free-fermion language, and in the case of AdS2 (8 vertex type Hamiltonian) we have found a state which plays the same role of the pseudovacuum in the algebraic Bethe ansatz. The lack of pseudovacuum is a difficult problem to overcome in models without U(1) symmetry: the way we got around this issue was by using the free-fermion condition and by discovering a peculiar particle-hole transformation which allows us to very easily diagonalise the transfer matrix. We have called the special state a ``pseudo-pseudovacuum''. 3. More recently I have exploited the pseudo-relativistic invariance to derive the form factors for the massless relativistic limit of AdS_3 integrability, in a solo paper. To achieve this result I have solved Watson's equations directly for the minimal two-particle form factor, and I have then adapted the off-shell Bethe ansatz approach of Babujian et al to the case of the three-particle quantum states. I have conjectured a multiparticle expression, and it will be of paramount importance to test this proposal in specific examples. I am currently working with a team of collaborators on adopting Lukyanov's method of vertex operators for this problem. 4. In a recent paper, in collaboration with B Stefanski, N Gromov (King's College London) and A Cavaglia' (Torino U.), we have conjectured the quantum spectral curve (QSC) for the massive sector of AdS_3 strings. This work has been made possible by a combination of techniques borrowed from AdS_5, where the QSC was originally developed, and methods adapted specifically for the more complex setup provided by AdS_3. This work is the first application of the technique of the QSC to lower-dimensional AdS/CFT integrability. It opens the way to the possibility of deriving exact non-perturbative predictions for the dual boundary CFT, which is non yet under control in the entirety of its moduli space. It also opens the door to trying to use the pseudo-relativistic invariance to extend this to the massless sector, which is a challenge which we are undertaking at the moment. |
| Exploitation Route | 1. The discovery of pseudo-relativistic invariance in the massless AdS/CFT S-matrix has had an immediate resonance and a number of papers have been adopting our new variables, most notably three recent papers by A. Sfondrini (Padua U.) and S. Frolov (Trinity College Dublin). 2. My solo paper was highly praised by the referees of J. Phys. A as a calculational ``tour de force'' and I expect that we can build on the exact integral formulas which I have constructed. 3. My collaboration with the Dublin group is very promising towards continuing to produce high quality outputs. Our introduction of a pseudo-pseudovacuum is very novel and not fully explored, which leaves a large room for growth in this research direction. 4. Our QSC paper was immediately welcomed by the community with an invitation to give a plenary talk at the major AdS/CFT integrability conference IGST 2022 (talk delivered by A. Cavaglia'), and it has been considered by experts as a game changer in this particular specialism. |
| Sectors | Other |