Christopher P. Herzog

A bstract: We argue that interacting conformal line defects in free quantum field theories can exist, provided that inversion symmetry is broken. Important for our demonstration is the existence of a special cross ratio for bulk-defect-defect three point functions that is invariant under the conformal group but picks up a sign under inversion. We examine the particular case of a free scalar field in detail, and provide a toy model example where this bulk field interacts via a Yukawa term with fermions on the line. We expect nontrivial line defects may also exist for free Maxwell theory in four dimensions and free bulk fermions.

A bstract: Aspects of parity-preserving, three-dimensional conformal field theories (CFTs) with a global U(1) symmetry in the presence of a background magnetic field are investigated. A local effective action is constructed to four-derivative order, based on an assumption that the magnetic field drives the theory into a gapped phase. This action is evaluated in a variety of backgrounds, and is used to obtain one- and two-point functions of the conserved current and stress-energy tensor. Dispersive arguments are developed and shown to impose powerful constraints on the Wilson coefficients of the effective action, leading to universal predictions for the CFT response at large magnetic field and the scaling dimensions of background monopole operators. These general results are further examined through explicit calculations in the free complex scalar, free Dirac fermion, and a holographic Einstein-Hilbert-Maxwell model.

We solve a system of massless fermions constrained to two space-time dimensions interacting via a d space-time dimensional Maxwell field. Through dimensional reduction to the defect and bosonization, the system maps to a massless scalar interacting with a nonlocal Maxwell field through a Fϕ-coupling. The d = 2 dimensional case is the usual Schwinger model where the photon gets a mass. More generally, in 2 < d < 4 dimensions, the degrees of freedom map to a scalar which undergoes a renormalization group flow; in the ultraviolet, the scalar is free, while in the infrared it has scaling dimension (4 – d)/2. The infrared is similar to the Wilson-Fisher fixed point, and the physically relevant case d = 4 becomes infrared trivial in the limit of infinite ultraviolet cut-off, consistent with earlier work on the triviality of conformal surface defects in Maxwell theory.

We present a collection of numerical bootstrap computations for 3d CFTs with a U(1) global symmetry. We test the accuracy of our method and fix conventions through a computation of bounds on the OPE coefficients for low-lying operators in the free fermion, free scalar, and generalised free vector field theories. We then compute new OPE bounds for scalar operators in the Gross-Neveu-Yukawa model, O(2) model, and large N limit of the O(N) model. Additionally, we present a number of exclusion plots for such 3d CFTs. In particular, we look at the space of even and odd parity scalar operators in the low-lying spectrum that are compatible with crossing symmetry. As well as recovering the known theories, there are some kinks that indicate new unknown theories.

A bstract: We consider a higher derivative scalar field theory in the presence of a boundary and a classically marginal interaction. We first investigate the free limit where the scalar obeys the square of the Klein-Gordon equation. In precisely d = 6 dimensions, modules generated by d − 2 and d − 4 dimensional primaries merge to form a staggered module. We compute the conformal block associated with this module and show that it is a generalized eigenvector of the Casimir operator. Next we include the effect of a classically marginal interaction that involves four scalar fields and two derivatives. The theory has an infrared fixed point in d = 6 − ϵ dimensions. We compute boundary operator anomalous dimensions and boundary OPE coefficients at leading order in the ϵ expansion for the allowed conformal boundary conditions.

The coupling constants of fixed points in the ϵ expansion at one loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We refer to fixed points that saturate this bound as extremal fixed points. The theories which contain such fixed points are those which undergo a saddle-node bifurcation, entailing the presence of a marginal operator. Among bifundamental theories, a few examples of infinite families of such theories are known. A necessary condition for extremality is that the sizes of the factors of the symmetry group of a given theory satisfy a specific Diophantine equation, given in terms of what we call the extremality polynomial. In this work we study such Diophantine equations and employ a combination of rigorous and probabilistic estimates to argue that these infinite families constitute rare exceptions. The Pell equation, Falting’s theorem, Siegel’s theorem, and elliptic curves figure prominently in our analysis. In the cases we study here, more generic classes of multi-fundamental theories saturate the Rychkov-Stergiou bound only in sporadic cases or in limits where they degenerate into simpler known examples.

We use the numerical conformal bootstrap to study boundary quantum electrodynamics, the theory of a four dimensional photon in a half space coupled to charged conformal matter on the boundary. This system is believed to be a boundary conformal field theory with an exactly marginal coupling corresponding to the strength of the interaction between the photon and the matter degrees of freedom. In part one of this project, we present three results. We show how the Maxwell equations put severe constraints on boundary three-point functions involving two currents and a symmetric traceless tensor. We use semi-definite programming to show that any three dimensional conformal field theory with a global U(1) symmetry must have a spin two gap less than about 1.05. Finally, combining a numerical bound on an OPE coefficient and some Ward identities involving the current and the displacement operator, we bound the displacement operator two-point function above. This upper bound also constrains a boundary contribution to the anomaly in the trace of the stress tensor for these types of theories.

In boundary conformal field theories, global symmetries can be broken by boundary conditions, generating a homogeneous conformal manifold. We investigate these geometries, showing they have a coset structure, and give fully worked out examples in the case of free fields of spin zero and one-half. These results give a simple illustration of the salient features of conformal manifolds while generalizing to interacting and more intricate setups. Our work was inspired by Drukker et al. [ Phys. Rev. Lett. 129, 201603 (2022)].

We use the numerical conformal bootstrap to study boundary quantum electrodynamics, the theory of a four dimensional photon in a half space coupled to charged conformal matter on the boundary. This system is believed to be a boundary conformal field theory with an exactly marginal coupling corresponding to the strength of the interaction between the photon and the matter degrees of freedom. In part one of this project, we present three results. We show how the Maxwell equations put severe constraints on boundary three-point functions involving two currents and a symmetric traceless tensor. We use semi-definite programming to show that any three dimensional conformal field theory with a global U(1) symmetry must have a spin two gap less than about 1.05. Finally, combining a numerical bound on an OPE coefficient and some Ward identities involving the current and the displacement operator, we bound the displacement operator two-point function above. This upper bound also constrains a boundary contribution to the anomaly in the trace of the stress tensor for these types of theories.

We consider free higher derivative theories of scalars and Dirac fermions in the presence of a boundary in general dimension. We establish a method for finding consistent conformal boundary conditions in these theories by removing certain boundary primaries from the spectrum. A rich set of renormalization group flows between various conformal boundary conditions is revealed, triggered by deformations quadratic in the boundary primaries. We compute the free energy of these theories on a hemisphere, and show that the boundary a-theorem is generally violated along boundary flows as a consequence of bulk non-unitarity. We further characterize the boundary theory by computing the two-point function of the displacement operator.