Difference between revisions of "Slorg"
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= | bt: I realized that I was not being psceire in saying that multi-trace operators dominate the thermodynamics at high temperature. Let us consider the free energy F(T) in the high temperature limit, of a U(N) gauge theory on the sphere (say N=4 SYM). F(T) goes like N^2 T^{d+1}, where d is the number of spatial dimensions. There is also a dependence on the t Hooft coupling, and we know it is a function of order 1, and let's ignore it for now. e^{F(T)/T} goes like d(E) e^{-E/T}, where d(E) is the number of states of a typical energy E, the latter scales like N^2 T^{d+1}, the same way as the free energy, by saddle point approximation. So the energy of a typical state scales with N like N^2. In the weakly coupled theory, at least, this means that we have an operator made out of the product of N^2 fields, and for operators of such high dimension there are nontrivial relations between single and multi-trace operators ( despite that N is large, normally we ignore the relation between single and multiple traces, but this breaks down when the dimension of the operator is high enough). Indeed, this phenomenon is the origin of the Hagedorn like phase transition in the free large N gauge theory on the sphere. | ||
==See also== | ==See also== | ||