$\mathcal{PT}$ Symmetry and QCD: Finite Temperature and Density

The relevance of $\mathcal{PT}$ symmetry to quantum chromodynamics (QCD), the gauge theory of the strong interactions, is explored in the context of finite temperature and density. Two significant problems in QCD are studied: the sign problem of finite-density QCD, and the problem of confinement. It is proven that the effective action for heavy quarks at finite density is $\mathcal{PT}$-symmetric. For the case of $1+1$ dimensions, the $\mathcal{PT}$-symmetric Hamiltonian, although not Hermitian, has real eigenvalues for a range of values of the chemical potential $\mu$, solving the sign problem for this model. The effective action for heavy quarks is part of a potentially large class of generalized sine-Gordon models which are non-Hermitian but are $\mathcal{PT}$-symmetric. Generalized sine-Gordon models also occur naturally in gauge theories in which magnetic monopoles lead to confinement. We explore gauge theories where monopoles cause confinement at arbitrarily high temperatures. Several different classes of monopole gases exist, with each class leading to different string tension scaling laws. For one class of monopole gas models, the $\mathcal{PT}$-symmetric affine Toda field theory emerges naturally as the effective theory. This in turn leads to sine-law scaling for string tensions, a behavior consistent with lattice simulations.