Lattice Field Theory (LFT) is a rigorous means for both defining and working with quantum field theory beyond perturbation theory, and is an underpinning technology for high precision calculations in Quantum Chromodynamics, the most challenging component of the Standard Model. QCD is at once highly non-linear and scale-dependent, becoming strongly-coupled at the low energy scales relevant for hadrons. LQCD calculations are critical for the interpretation of experimental results ranging from the energy frontier at the LHC, through intensity frontier experiments such as those at KEK and BESIII, the hadron physics programme at Jefferson Lab, down to the low-energy precision determination of the magnetic moment of the muon at Fermilab, and the extreme conditions explored at relativistic ion colliders at RHIC, LHC, and those planned for FAIR and NICA. The advent of realistic LQCD calculations has transformed our ability to test the SM at the 1% level, via the determination of quark masses, the strong coupling constant, and the hadron weak decay and mixing rates needed to constrain Beyond the Standard Model (BSM) scenarios. This is also supported by a significant sub-community developing high-order loop calculations for the precision matching of lattice to experiental results. Detailed determination of the spectrum and structure of hadrons allow us to connect QCD to experimental results and phenomenological models over energy scales ranging from the LHC to nuclear physics. Studies of hot and dense QCD are likewise critical to understanding heavy-ion collisions and the early universe.
The same LFT methodology is also applied to test the viability of BSM theories, where strongly coupled gauge dynamics is thought to play a crucial role in theories of electroweak symmetry breaking in which the Higgs is not fundamental. LFT also offers new approaches to both defining and understanding strongly-interacting quantum theory in other contexts: the study of lattice Green functions calculated in fixed gauge as a means of understanding IR physics; the study of supersymmetric field theory and and the holographic duality conjecture; the world-sheet description of superstrings; discrete models of quantum gravity; and lower-dimensional theories of strongly-interacting electrons relevant for condensed matter systems such as graphene. In parallel, algorithms developed for LFT are increasingly making impact in other fields, such as Bayesian Machine Learning. All of these activities are represented within the UKLFT community.