First introduced in 1970, the large-N limit in gauge theories has become a fundamental approach in theoretical physics. Originally developed to understand the strong nuclear force—one of the four fundamental forces of nature—large-N gauge theories have since become critical for addressing challenges in quantum gravity as well.

In this study, we examined the thermodynamic properties of finite-temperature large-N gauge theories. A unique feature of these theories is that they exhibit confinement and deconfinement phase transitions, which vary with temperatures. At high temperatures, the deconfinement phase emerges, describing conditions where particles such as gluons can exist independently instead of being confined.

We made two significant discoveries regarding these confinement/deconfinement transitions. First, we observed that the effective potentials for Polyakov loops follow a special temperature scaling law. Acting as an order parameter, the Polyakov loop indicates phase changes, adopting distinct values in the confinement and deconfinement phases. This scaling law significantly constrains the possible terms in Polyakov loop effective potentials, simplifying the prediction of the temperature dependence of many physical quantities that were previously difficult to analyze.

Second, we found that introducing imaginary angular velocities to these effective potentials leads to transitions to novel Z_m–type deconfinement phases. In these phases, the eigen values of the Polyakov loops form symmetric Z_m configurations, suggesting that even when quarks are deconfined, they are governed by an underlying symmetry at the large-N limit. This Z_m-type deconfinement phase is observed across several models, including Yang-Mills theories and the bosonic BFSS matrix model. This suggests that it is a fundamental feature rather than a peculiarity of a single model. These findings are further supported by Monte-Carlo simulations. In addition, the physical quantities in the Z_m phase obey the temperature scaling law of the effective potential.

Additionally, the holographic principle suggests that finite-temperature large-N gauge theories are equivalent to black holes. This implies that, as per our results, physical quantities in the presence of a black hole would follow the same temperature scaling law. Moreover, if a black hole were to acquire an imaginary angular velocity, it could transition into new states corresponding to the Z_m phases. Overall, these discoveries offer new insights into gauge theories at finite temperatures, with broader implications for better understanding the strong nuclear force and solving various problems of quantum gravity.

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