In theories with topological sectors, such as lattice QCD and four-dimensional SU(N) gauge theories with periodic boundary conditions, conventional update algorithms suffer from topological freezing due to large action barriers separating distinct sectors. With appropriately constructed bias potentials, Metadynamics and related enhanced sampling techniques can mitigate this problem and significantly reduce the integrated autocorrelation times of the topological charge and associated observables. We test strategies to accelerate the buildup of bias potentials and the possibility of extrapolating potentials from small to large volumes. We also investigate the effectiveness of orthogonal algorithmic improvements, such as longer HMC trajectories and HMC variants, which may benefit conventional simulations as well.

In this kick-off presentation for the "Recent developments in QCD" session at Baryons 2025 I will tie together the recent progress made on the extraction of parton distribution functions (PDFs) in lattice QCD and the long standing efforts in solving the inverse problem in the form of spectral function reconstruction.

We investigate the symmetry structure of the $3+1$ D staggered fermion Hamiltonian and its implications for anomalies. Since the spin and flavor degrees of freedom of Dirac fermions are distributed over the lattice, in addition to the standard on-site mass term, the staggered fermion system also admits one-, two-, and three-link bilinear terms within a unit cube as local, charge conserving mass terms with different spin and flavor dependence. We identify the spin flavor structures of all those bilinear mass terms and determine the symmetries preserved by each of them. Among them, one of the one-link mass terms preserves a larger residual symmetry associated with conserved charges that generate the Onsager algebra. Motivated by this structure, we consider a kink profile of the one-link mass and analyze the resulting domain-wall system. In the low-energy limit, the $3+1$ D bulk becomes gapped, while two-flavor massless Dirac fermions appear as localized modes on the $2+1$ D domain wall. We show that the bulk conserved charges act on the wall as generators of a flavor $\mathrm{SU}(2)$ symmetry, and that no symmetric mass gap is allowed for the boundary theory when this $\mathrm{SU}(2)$ symmetry and space reflection symmetry are both imposed. This realizes the parity anomaly of the boundary theory and shows that the boundary flavor symmetry and anomaly descend from the ultraviolet staggered-fermion Hamiltonian rather than emerging only in the infrared.

The energy-momentum tensor (EMT) is the conserved current corresponding to space-time translation symmetry. Its applications are remarkably diverse, ranging from the thermodynamics to the calculation of transport coefficients. While the EMT is well-defined in the continuum up to a total derivative, with its coefficients fixed by Ward identities, its extension to lattice QCD is not straightforward. The primary challenge arises from the breaking of continuous space-time symmetries by the discrete lattice regulator. Although the EMT can be constructed on the lattice in a way that yields the correct continuum limit, the operators are not uniquely defined. In this proceeding, we construct the EMT for both pure-gauge theory and full QCD, discussing its renormalization in the specific context of determining the coefficients required for shear viscosity. In this context, we present a comparative analysis of the trace anomaly, number density, pressure, energy density and enthalpy density with imaginary chemical potential for multiple $\beta$ values at approximately the same temperature, aimed for the continuum limit.

Some algorithms for the numerically exact treatment of fermion determinants are summarised. This is not supposed to be a review, rather a concise handbook. The audience is expected to have a basic understanding of how to put fermions on a computer. We primarily discuss different ways to work with the fermion matrix in the "sausage" (Green's function) formulation for quantum Monte Carlo (QMC). We emphasise the need for varied approaches in different space-time volume regimes. In particular, for small spatial volumes we describe a numerically stable method based on dense matrix operations. It is designed specifically to deal with very low temperature regimes. On the other hand, for (relatively) large volumes we describe a highly efficient and scalable sparse matrix approach.

Within Yang-Mills-Proca theory with external sources in the form of three static quarks, regular, finite energy solutions are obtained. It is shown that color electric/magnetic fields have two components: the first part is a gradient/curl component, respectively, and the second part is a nonlinear component. It is shown that the color electric field has a Y-like spatial distribution provided by three static quarks. Such a Y-like behavior arises because the gradient component of the electric field is present. The nonlinear component of the electric field is a curl one, and it appears because the vector potential sourced by a solenoidal current is present. The color magnetic field is purely curl one, since its nonzero color components do not contain a nonlinear component; this results in the fact that its force lines lie on the surface of a torus. It is shown that the results obtained are in satisfactory agreement with the results obtained in lattice calculations in quantum chromodynamics. To discuss such an agreement, we have shown that the Yang-Mills-Proca equation can be obtained from the Lagrangian describing a gluon condensate varying in space. Also, we compare the energy profile obtained by us with that obtained in lattice calculations with a static potential.