DiRAC Project dp162/PPSP311, “Lattice studies of 3d super-Yang–Mills and holography”, is using the Cambridge icelake cluster to explore conjectured holographic dualities that relate supersymmetric quantum field theories to quantum gravity in a higher number of space-time dimensions. Such holographic dualities are widely employed in theoretical physics, and are beginning to benefit from first-principles lattice field theory studies. While holography is best understood in the limit of a large number of colours (N) with strong coupling, lattice field theory provides a powerful non-perturbative tool to explore the validity and applications of holography away from this limit.

The specific focus of this project is on maximally supersymmetric Yang–Mills (SYM) theory in three space-time dimensions, with gauge group U(N). Building on our prior work arXiv:2010.00026 that confirmed consistency between the low-temperature, large-volume behaviour of this field theory and the `D2′ phase of a homogeneous euclidean D2-brane black hole in the dual IIA supergravity, we are carrying out numerical lattice field theory computations to study the expected phase transition between this homogeneous D2 phase and a localized D0 phase as the volume decreases.

On the field-theory side, this corresponds to a `spatial deconfinement’ transition signalled by the Wilson lines that wrap around the spatial cycles of the lattice. The localized D0 black hole is dual to the small-volume spatially deconfined phase in which the Wilson line has a large magnitude |W|, relative to its maximum value |W|=1. The opposite regime of a large spatial volume with |W| vanishing in the large-N limit corresponds to the homogeneous D2 black hole we previously investigated.

The accompanying figure (from doi:10.22323/1.430.0221) presents preliminary results for the Wilson line magnitude |W|, demonstrating the behaviour described above. Here we work with fixed N=8 and 16^3 lattice volumes, which allows us to change the spatial volume by varying the temporal extent (dimensionless inverse temperature) r_beta. As described above, |W| increases as the spatial volume shrinks to enter the spatially deconfined D0 phase, while the smaller values of |W| for larger volumes in the spatially confined D2 phase would decrease even further for N>8. The four different data sets in the figure consider different values for certain tunable parameters in the SYM lattice action, as described in doi:10.22323/1.430.0221. Through ongoing analyses of the corresponding Wilson line susceptibilities, we will be able to carry out novel tests of holography based on our non-perturbative numerical lattice calculations.

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