Conformal Graphs as Twisted Partition Functions.

We show that a class of L-loop conformal ladder graphs are intimately related to twisted partition functions of free massive complex scalars in d=2L+1 dimensions. The graphs arise as four-point functions in certain two- and four-dimensional conformal fishnet models. The twisted thermal two-point function of the scalars becomes a generator of conformal ladder graphs for all loops. We argue that this correspondence is seeded by a system of two decoupled harmonic oscillators twisted by an imaginary chemical potential. We find a number of algebraic and differential relations among the conformal graphs that mirror the underlying free dynamics.

Dynamics of Finite-Temperature Conformal Field Theories from Operator Product Expansion Inversion Formulas.

We apply the operator product expansion inversion formula to thermal two-point functions of bosonic and fermionic conformal field theories in general odd dimensions. This allows us to analyze in detail the operator spectrum of these theories. We find that nontrivial thermal conformal field theories arise when the thermal mass satisfies an algebraic transcendental equation that ensures the absence of an infinite set of operators from the spectrum. The solutions of these gap equations for general odd dimensions are in general complex numbers and follow a particular pattern. We argue that this pattern unveils the large-N vacuum structure of the corresponding theories at zero temperature.

Generalized Wilson-Fisher Critical Points from the Conformal Operator Product Expansion.

We study possible smooth deformations of the generalized free conformal field theory in arbitrary dimensions by exploiting the singularity structure of the conformal blocks dictated by the null states. We derive in this way, at the first nontrivial order in the ε expansion, the anomalous dimensions of an infinite class of scalar local operators, without using the equations of motion. In the cases where other computational methods apply, the results agree.

Gauge fields, membranes, and subdeterminant vector models.

We present a class of classically marginal N-vector models in d=4 and d=3 whose scalar potentials can be written as subdeterminants of symmetric matrices. The d=3 case can be thought of as a generalization of the scalar sector of the Bagger-Lambert-Gustavsson model. Using the Hubbard-Stratonovich transformation we calculate their effective potentials which exhibit intriguing large-N scaling behaviors. We comment on the possible relevance of our models to strings, membranes, and also to a class of novel spin systems that are based on ternary commutation relations.

Conformally coupled scalars, instantons, and vacuum instability in 4D anti-de Sitter space.

We show that a scalar field conformally coupled to AdS gravity in four dimensions with a quartic self-interaction can be embedded into M theory. The holographic effective potential is exactly calculated, allowing us to study nonperturbatively the stability of AdS4 in the presence of the conformally coupled scalar. It is shown that there exists a one-parameter family of conformal scalar boundary conditions for which the boundary theory has an unstable vacuum. In this case, the bulk theory has instanton solutions that mediate the decay of the AdS4 space. These results match nicely with the vacuum structure and the existence of instantons in an effective three-dimensional boundary model.