Hagedorn behavior of string and gauge theories

This thesis delves into the study of the Hagedorn behavior of string and gauge theories, not before introducing the reader to everything is needed to understand the original part of the work. First, we present a closed formula for the asymptotic density of states for a class of solvable superstring models on curved backgrounds. The result accounts for the effects of the curvature of the target space in a concise way. This allows us to discuss also the “thermodynamics” of a gas of strings in the Hagedorn regime. Quotations marks refer to the subtleties in the definition of the thermal ensemble itself. Be that as it may, our findings reproduce well-known results about the Hagedorn temperature in solvable models and say something more about the Hagedorn regime. Then, we provide a general proposal for the Hagedorn temperature of planar, strongly coupled confining gauge theories holographically dual to Type II superstring models on curved backgrounds with Ramond-Ramond and Kalb-Ramond fluxes and non-trivial dilaton. To reach this goal, we exploit two complementary approaches. On the one hand, we perform an extrapolation to the Hagedorn regime of world-sheet results obtained from the semiclassical quantization of string configurations winding around the compact Euclidean time direction; en passant, we address the subtleties hidden in the light-cone gauge quantization in curved space and provide a detailed derivation of the fermionic part of the world-sheet spectrum, which is hard to find in the literature. On the other hand, we perturbatively solve the equations of motion for the thermal scalar field corresponding to the lightest mode of the winding string, which in flat space becomes tachyonic above the Hagedorn temperature. The interplay between different approaches is surely convenient, but we provide insights about a possible derivation of the whole next-to-next-to-leading order correction to the Hagedorn temperature from a pure world-sheet perspective. Finally, we study the Hagedorn temperature of strongly coupled quantum field theories admitting a holographic string or M-theory description in various regimes and scenarios. More in details, we propose a thermal scalar effective approach to the calculation of the Hagedorn temperature in eleven-dimensional supergravity. The proposal allows to extend the previous results for the Hagedorn temperature to the strongly coupled string regime, i.e., to a previously unexplored regime of field theory parameters where the number of colors is smaller than (some power of) the 't Hooft coupling. We also apply the formalism to study the dependence of the Hagedorn temperature on the Yang-Mills θ-angle and the effect of dynamical flavors in some examples of confining theories.