Exact results in Supersymmetric Gauge Theories
Lead Research Organisation:
King's College London
Department Name: Mathematics
Abstract
Supersymmetric gauge theories are important since they share many features with physical gauge theories (e.g. QCD), but nevertheless allow for certain observables to be computed in a non-perturbative way due to underlying integrability.
Integrability was first developed in the context of classical analytical mechanics, but has since been generalized to various quantum systems. The most prominent example are quantum spin chains, whose spectrum can be obtained by the various forms of the Bethe ansatz. However, this approach also applies to certain two-dimensional, non-trivial CFTs, whose multiparticle S-matrices factor into products of two-particle S-matrices.
The main focus of the project will be N=4 supersymmetric Yang-Mills Theory. This theory is peculiar, because its high degree of symmetry allows for a derivation of many exact results using methods of integrability, whilst still being interesting from a physical point of view as it bears some resemblance to Quantum Chromodynamics, modelling quarks and gluons. Interestingly, it has been shown that certain operators in this theory are directly related to spin-chains, whose spectra can be obtained by standard Bethe-ansatz techniques. For example, the spectrum of the SU(N) Heisenberg spin chain can be obtained by a nested form of the algebraic Bethe ansatz, reducing the problem of finding all eigenfunctions and associated eigenstates of the underlying Hamiltonian to solving a set of algebraic equations for a set of complex numbers, called Bethe roots.
Recently, the Quantum Spectral Curve (QSC) formalism has been introduced, which was originally derived from the analytic Y-system arising in the context of the thermodynamic Bethe ansatz. However, QSC is assumed to hold in more general cases, reproducing many exact results found previously. Additionally, many new results can be obtained from QSC, which are not accessible by other techniques. The involved Q-system can be used to solve the above mentioned SU(N) spin chains in a different way, giving rise to the Bethe equations by making certain analyticity assumptions, rather than by constructing explicit monodromy matrices satisfying the Yang-Baxter equation.
We will investigate the QSC formalism further, as well as other methods used to derive exact results for N=4 SYM and related theories.
Integrability was first developed in the context of classical analytical mechanics, but has since been generalized to various quantum systems. The most prominent example are quantum spin chains, whose spectrum can be obtained by the various forms of the Bethe ansatz. However, this approach also applies to certain two-dimensional, non-trivial CFTs, whose multiparticle S-matrices factor into products of two-particle S-matrices.
The main focus of the project will be N=4 supersymmetric Yang-Mills Theory. This theory is peculiar, because its high degree of symmetry allows for a derivation of many exact results using methods of integrability, whilst still being interesting from a physical point of view as it bears some resemblance to Quantum Chromodynamics, modelling quarks and gluons. Interestingly, it has been shown that certain operators in this theory are directly related to spin-chains, whose spectra can be obtained by standard Bethe-ansatz techniques. For example, the spectrum of the SU(N) Heisenberg spin chain can be obtained by a nested form of the algebraic Bethe ansatz, reducing the problem of finding all eigenfunctions and associated eigenstates of the underlying Hamiltonian to solving a set of algebraic equations for a set of complex numbers, called Bethe roots.
Recently, the Quantum Spectral Curve (QSC) formalism has been introduced, which was originally derived from the analytic Y-system arising in the context of the thermodynamic Bethe ansatz. However, QSC is assumed to hold in more general cases, reproducing many exact results found previously. Additionally, many new results can be obtained from QSC, which are not accessible by other techniques. The involved Q-system can be used to solve the above mentioned SU(N) spin chains in a different way, giving rise to the Bethe equations by making certain analyticity assumptions, rather than by constructing explicit monodromy matrices satisfying the Yang-Baxter equation.
We will investigate the QSC formalism further, as well as other methods used to derive exact results for N=4 SYM and related theories.
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509498/1 | 01/10/2016 | 30/09/2021 | |||
| 1818591 | Studentship | EP/N509498/1 | 01/10/2016 | 31/03/2020 | David Grabner |
| Description | Through the work carried out under this award we were able to better understand how integrability appears in the context of (non) supersymmetric gauge theories. In general it is very hard to carry out calculations in gauge theories, however, QCD is the gauge theory describing the strong force, so that gaining a better understanding of the theories in general will allow us to gain a better understanding of QCD in particular. In our work we focus mainly on N=4 SYM and the bi-scalar fishnet theory. They serve as highly non-trivial toy models for the physical theories describing nature. Through our recent results we hope to contribute significantly to the full solution of these theories. |
| Exploitation Route | The findings of our project can be generalised both in N=4 SYM and the bi-scalar fishnet theory. While a solution to the spectral problem is available in both theories, it remains to find the structure constants for general three-point correlators. Some early results are already available, but we hope that these can be extended through our contribution to obtain the full conformal data, at least in the planar limit. |
| Sectors | Other |