HET Journal club: David McGady
Speaker: David McGady
Title: Hagedorn poles, modularity, and Casimir energies in field theories
Abstract: Planar limits of gauge theories and field theory limits of string theories both have numbers of states, c(E), that grow exponentially with energy, c(E) ~ Exp[\beta_H E]. This so-called Hagedorn growth is thought to be generic to tractable limits of the strong interaction. In this talk, we extend the notion of spectral zeta functions to physical systems with c(E) ~ Exp[\beta_H E]. Specifically, we focus on quantum field theory partition functions that are modular forms with poles.
Any modular partition function with a pole at finite temperature has Hagedorn growth. Its spectral zeta function is defined by its Mellin transform. In this talk, we define Mellin transforms of modular forms with poles. These define qualitatively *new* zeta functions, which in turn define imply universal sum-rules that equate the total number of excited states to (minus) the number of ground states: \sum_E c(E) = -c(0). These robust and general sum-rules may suggest insights into Casimir energies, central charges, and L-functions in number theory.