In this thesis, we will explore certain aspects of thermal phases in gauge theories and their implications through holography. While thermal phases and phase transitions in gauge theories have been widely studied, several unresolved issues remain, such as the Yang-Mills mass gap in the confined phase and the confinement-to-deconfinement crossover in QCD. The development of holography offers a new perspective on gauge theories by examining the dual gravitational theory, which also provides insights into gravity via the dual gauge theory description. Inspired by this duality, the partially deconfined phase has been proposed in recent years for gauge theories, with its dual being the small black hole phase in gravity.

We will investigate the microscopic origin of the linear potential with Casimir scaling in the confined phase of gauge theory. We will examine the similarities between Bose-Einstein condensation and confinement in gauge theories, as both experience an enhancement factor in the partition function proportional to the volume of the gauge group. At low temperatures in the confined phase, the Polyakov loop is slowly varying and Haar random, with small corrections in inverse temperature. We demonstrate how this property of the Polyakov loop leads to a linear confining potential at intermediate distances. String breaking at long distances results from small corrections to the Haar randomness of the Polyakov loops. Additionally, we will investigate how the slowly varying Haar random behavior extends to the rectangular Wilson loop by numerically studying the 3D SU(2) pure Yang-Mills theory. We will then study the partially confined phase for strongly coupled pure Yang-Mills theory at large N and provide evidence for the formation of flux tubes in the confined sector of this phase.

Next, we will focus on holographic theories and examine how the bulk geometry is encoded in the matrix degrees of freedom on the gauge theory side. The partially confined phase is crucial in this identification. We propose a method for factorizing the extended Hilbert space of gauge theories by separating the colors and computing the von Neumann entanglement entropy. Using this method, we will study the evaporation of a small black hole and the recovery of the Page curve, highlighting the role of the confined sector of the partially confined phase in this process. Finally, we will discuss how entanglement between matrix degrees of freedom leads to the emergence of spacetime in the bulk gravitational theory and how we can construct the operator algebra of small excitations in the gauge theory that describes an arbitrary region of the bulk geometry.