Narrow Results

A matrix model approach to proof of the AGT relation is briefly reviewed. It starts from the substitution of conformal blocks by the Dotsenko–Fateev β-ensemble averages and Nekrasov functions by a double deformation of the exponentiated Seiberg–Witten prepotential in β≠1 and g_s≠0 directions. Establishing the equality of these two quantities is a typical matrix model problem.

We discuss our results on QCD with a number of fundamental fermions ranging from zero to sixteen. These theories exhibit a wide array of fascinating phenomena which have been under close scrutiny, especially in recent years, first and foremost is the approach to conformality. To keep this review focused we have chosen scale generation, or lack thereof, as a guiding theme, however the discussion will be set in the general framework of the analysis of the phases and phase transitions of strong interactions at zero and nonzero temperature.

Bootstrap equations for conformal correlators that mimic the early theory of conformal bootstrap are written down in frames of the AdS/CFT approach. The simplified version of these equations, that may be justified if Schwinger–Keldysh formalism is used in AdS/CFT instead of conventional Feynman–Witten diagrams technique, permits to calculate values of conformal dimensions in the $O(N)$ symmetric model with conformal or composite Habbard–Stratonovich field.

In this paper, we shall review the equivalence between Palatini-$f({ℛ})$ theories and Brans–Dicke (BD) theories at the level of action principles. We shall define the Helmholtz Lagrangian associated to Palatini-$f({ℛ})$ theory and we will define some transformations which will be useful to recover Einstein frame and BD frame. We shall see an explicit example of matter field and we will discuss how the conformal factor affects the physical quantities.

We discuss various aspects of Conformal Field Theories on the Lattice. We mainly investigate the SU(3) gauge theory with N_f degenerate fermions in the fundamental representation, employing the one-plaquette gauge action and the Wilson fermion action.

First we make a brief review of our previous works on the phase structure of lattice gauge theories in terms of the gauge coupling constant and the quark mass. We thereby clarify the reason why we conjecture that the conformal window is 7 ≤ N_f ≤ 16.

Secondly, we introduce a new concept, “conformal theories with IR cutof” and point out that any numerical simulation on a lattice is bounded by an IR cutoff ∧_IR. Then we make predictions that when N_f is within the conformal window, the propagator of a meson G(t) behaves at large t, as G(t) = c exp (−m_Ht)/t^α, that is, a modified Yukawa-type decay form, instead of the usual exponential decay form exp (−m_Ht), in the small quark mass region. This holds on an any lattice for any coupling constant g, as far as g is between 0 and g*, where g* is the IR fixed point. We verify that numerical results really satisfy the predictions for the N_f = 7 case and the N_f = 16 case.

Thirdly, we discuss small number of flavors (N_f = 2 ∼ 6) QCD at finite temperatures. We point out theoretically and verify numerically that the correlation functions at T/T_c > 1 exhibit the characteristics of the conformal function with IR cutoff, an exponential decay with power correction.

Investigating our numerical data by a new method which we call the “local-analysis” of propagators, we observe that the N_f = 7 case and the N_f = 2 at T ∼ 2T_c case are similar to each other, while the N_f = 16 case and the N_f = 2 at T = 10² ∼ 10⁵T_c cases are similar to each other.

Further, we observe our data are consistent with the picture that the N_f = 7 case and the N_f = 2 at T ∼ 2T_c case are close to the meson unparticle model. On the other hand, the N_f = 16 case and the N_f = 2 at T = 10² ∼ 10⁵T_c cases are close to a free state in the Z(3) twisted vacuum. All results are consistent with naive physical intuition and give clues for long standing issues at high temperatures such as why the free energy at high temperatures does not reach the Stefan-Boltzmann ideal gas limit even at T = 100T_c.