We present lattice QCD calculations of higher order cumulants of electric charge distributions for small baryon chemical potentials $\mu_B$ by using up to NNNLO Taylor expansions. Ratios of these cumulants are evaluated on the pseudo-critical line, $T_{pc}(\mu_B)$, of the chiral transition and compared to corresponding measurements in heavy ion collision experiments by the STAR and PHENIX Collaborations. We demonstrate that these comparisons give strong constraints on freeze-out parameters. Furthermore, we use strangeness fluctuation observables to compute the ratio $\mu_S/\mu_B$ on the crossover line and compare it to $\mu_S/\mu_B$ at freeze-out stemming from fits to strange baryon yields measured by the STAR Collaboration.

In recent years there has been much progress on the investigation of the QCD phase diagram with lattice QCD simulations. In this review I focus on the developments in the last two years. Especially the addition of external influences or new parameter ranges yield an increasing number of interesting results. I discuss the progress for small, finite densities from both analytical continuation and Complex Langevin simulations, for heavy quark bound states (quarkonium), the dependence on the quark masses (Columbia plot) and the influence of a magnetic field. Many of these conditions are relevant for the understanding of both the QCD transition in the early universe and heavy ion collision experiments which are conducted for example at the LHC and RHIC.

We present a lattice QCD based determination of the chiral phase transition temperature in QCD with two massless (up and down) and one strange quark having its physical mass. We propose and calculate two novel estimators for the chiral transition temperature for several values of the light quark masses, corresponding to Goldstone pion masses in the range of $58~{\rm MeV}\lesssim m_\pi\lesssim 163~{\rm MeV}$. The chiral phase transition temperature is determined by extrapolating to vanishing pion mass using universal scaling relations. After thermodynamic, continuum and chiral extrapolations we find the chiral phase transition temperature $T_c^0=132^{+3}_{-6}$ MeV. We also show some preliminary calculations that use the conventional estimator for the pseudo-critical temperature and compare with the new estimators for $T_c^0$. Furthermore, we show results for the ratio of the chiral order parameter and its susceptibility and argue that this ratio can be used to differentiate between $O(N)$ and $Z_2$ universality classes in a non-parametric manner.

We discuss the discretization of Yang-Mills theories on Dynamical Triangulations in the compact formulation, with gauge fields living on the links of the dual graph associated with the triangulation, and the numerical investigation of the minimally coupled system by Monte Carlo simulations. We provide, in particular, an explicit construction and implementation of the Markov chain moves for 2D Causal Dynamical Triangulations coupled to either $U(1)$ or $SU(2)$ gauge fields; the results of exploratory numerical simulations on a toroidal geometry are also presented for both cases. We study the critical behavior of gravity related observables, determining the associated critical indices, which turn out to be independent of the bare gauge coupling: we obtain in particular $\nu = 0.496(7)$ for the critical index regulating the divergence of the correlation length of the volume profiles. Gauge observables are also investigated, including holonomies (torelons) and, for the $U(1)$ gauge theory, the winding number and the topological susceptibility. An interesting result is that the critical slowing down of the topological charge, which affects various lattice field theories in the continuum limit, seems to be strongly suppressed (i.e., by orders of magnitude) by the presence of a locally variable geometry: that may suggest possible ways for improvement also in other contexts.

Motivated in part by the pseudo-Nambu Goldstone Boson mechanism of electroweak symmetry breaking in Composite Higgs Models, in part by dark matter scenarios with strongly coupled origin, as well as by general theoretical considerations related to the large-N extrapolation, we perform lattice studies of the Yang-Mills theories with $Sp(2N)$ gauge groups. We measure the string tension and the mass spectrum of glueballs, extracted from appropriate 2-point correlation functions of operators organised as irreducible representations of the octahedral symmetry group. We perform the continuum extrapolation and study the magnitude of finite-size effects, showing that they are negligible in our calculation. We present new numerical results for $N=1$, $2$, $3$, $4$, combine them with data previously obtained for $N=2$, and extrapolate towards $N\rightarrow \infty$. We confirm explicitly the expectation that, as already known for $N=1,2$ also for $N=3,4$ a confining potential rising linearly with the distance binds a static quark to its antiquark. We compare our results to the existing literature on other gauge groups, with particular attention devoted to the large-$N$ limit. We find agreement with the known values of the mass of the $0^{++}$, $0^{++*}$ and $2^{++}$ glueballs obtained taking the large-$N$ limit in the $SU(N)$ groups. In addition, we determine for the first time the mass of some heavier glueball states at finite $N$ in $Sp(2N)$ and extrapolate the results towards $N \rightarrow +\infty$ taking the limit in the latter groups. Since the large-$N$ limit of $Sp(2N)$ is the same as in $SU(N)$, our results are relevant also for the study of QCD-like theories.

In the heavy, static quark mass regime of QCD, the Polyakov loop is well known to be an order parameter of the deconfinement phase transition; however, the sensitivity of the Polyakov loop to the deconfinement of light, dynamical quarks is less clear. On the other hand, from the perspective of an effective Lagrangian written in the vicinity of the chiral transition, the Polyakov loop is an energy-like operator and should hence scale as any energy-like operator would. We show here that the Polyakov loop and heavy-quark free energy are sensitive to the chiral transition, i.e. their scaling is consistent with energy-like observables in 3-$d$ $O(N)$ universality classes.