arXiv subjects

Christopher P. Herzog

Publications and source records attributed to Christopher P. Herzog.

At least 19 recordsLinked to original sources

arXiv2026

Magnetised Bounds for Conformal Field Theories

Aspects of parity-preserving, charge-conjugation-invariant, 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.

hep-th↗

arXiv2026

Scattering off Chamblin-Reall Branes

We study the linearized scattering of dilaton-graviton waves from a thin brane in three-dimensional spacetime. Holographically, the setup models scattering from an interface in a family of strongly coupled theories related to dimensional reductions of higher-dimensional $AdS_{d+2}$ gravity. Unlike the pure $AdS_3$ case, for $d>1$ the physical bulk mode allows incident radiation to be redistributed into reflected, transmitted, and evanescent components. For the $d=2$ background, we obtain a controlled solution in which the interface acts like a rough translucent window, producing diffuse angular scattering and absorption into surface modes. From the dual perspective, the scattering process is suggestive of dissipative flow toward the infrared. For $d=4$, the same analysis reveals a sensitivity to the infrared boundary condition, suggesting that the singular zero-temperature geometry must be regulated in order to have a well-defined scattering process. The structure of the equations nevertheless suggests that a regulated $d=4$ problem may exhibit the same qualitative redistribution of incident flux.

hep-th↗

arXiv2026

How to get an interacting conformal line defect for free theories

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.

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arXiv2025

An Atlas for 3d Conformal Field Theories with a U(1) Global Symmetry

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.

hep-th↗

arXiv2024

A Nonlocal Schwinger 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 $F ϕ$-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.

hep-th↗

arXiv2024

An Interacting, Higher Derivative, Boundary Conformal Field Theory

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.

hep-th↗

arXiv2024

Fermions in Boundary Conformal Field Theory : Crossing Symmetry and $ε$-Expansion

We use the equations of motion in combination with crossing symmetry to constrain the properties of interacting fermionic boundary conformal field theories. This combination is an efficient way of determining operator product expansion coefficients and anomalous dimensions at the first few orders of the $ε$ expansion. Two necessary ingredients for this procedure are knowledge of the boundary and bulk spinor conformal blocks. The bulk spinor conformal blocks are derived here for the first time. We then consider a number of examples. For $ϕ$ a scalar field and $ψ$ a fermionic field, we study the effects of a $ϕ\bar ψψ$ coupling in $4- ε$ dimensions, a $ϕ^2 \bar ψψ$ coupling in $3 -ε$ dimensions, and a $(\bar ψψ)^2$ coupling in $2+ε$ dimensions. We are able to compute some new anomalous dimensions for operators in these theories. Finally, we relate the anomalous dimension of a surface operator to the behavior of the charge density near the surface.

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arXiv2024

Extremal fixed points and Diophantine equations

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.

hep-th↗

arXiv2023

Bootstrapping Boundary QED Part I

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.

hep-th↗

arXiv2023

Boundaries in Free Higher Derivative Conformal Field 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.

hep-th↗

arXiv2023

Weyl Anomalies of Four Dimensional Conformal Boundaries and Defects

Motivated by questions about quantum information and classification of quantum field theories, we consider Conformal Field Theories (CFTs) in spacetime dimension $d\geq 5$ with a conformally-invariant spatial boundary (BCFTs) or $4$-dimensional conformal defect (DCFTs). We determine the boundary or defect contribution to the Weyl anomaly using the standard algorithm, which includes imposing Wess-Zumino consistency and fixing finite counterterms. These boundary/defect contributions are built from the intrinsic and extrinsic curvatures, as well as the pullback of the ambient CFT's Weyl tensor. For a co-dimension one boundary or defect (i.e. $d=5$), we reproduce the $9$ parity-even terms found by Astaneh and Solodukhin, and we discover $3$ parity-odd terms. For larger co-dimension, we find $23$ parity-even terms and $6$ parity-odd terms. The coefficient of each term defines a "central charge" that characterizes the BCFT or DCFT. We show how several of the parity-even central charges enter physical observables, namely the displacement operator two-point function, the stress-tensor one-point function, and the universal part of the entanglement entropy. We compute several parity-even central charges in tractable examples: monodromy and conical defects of free, massless scalars and Dirac fermions in $d=6$; probe branes in Anti-de Sitter (AdS) space dual to defects in CFTs with $d \geq 6$; and Takayanagi's AdS/BCFT with $d=5$. We demonstrate that several of our examples obey the boundary/defect $a$-theorem, as expected.

hep-th↗

arXiv2023

The Tilting Space of Boundary Conformal Field 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 generalising to interacting and more intricate setups. Our work was inspired by [2203.17157]

hep-th↗

arXiv2022

A Sum Rule for Boundary Contributions to the Trace Anomaly

In the context of boundary conformal field theory, we derive a sum rule that relates two and three point functions of the displacement operator. For four dimensional conformal field theory with a three dimensional boundary, this sum rule in turn relates the two boundary contributions to the anomaly in the trace of the stress tensor. We check our sum rule for a variety of free theories and also for a weakly interacting theory, where a free scalar in the bulk couples marginally to a generalized free field on the boundary.

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arXiv2022

Conformal Surface Defects in Maxwell Theory are Trivial

We consider a free Maxwell field in four dimensions in the presence of a codimension two defect. Reflection positive, codimension two defects which preserve conformal symmetry in this context are very limited. We show only generalized free fields can appear in the defect operator product expansion of the bulk Maxwell field; in particular correlation functions of these defect operators can be evaluated via Wick's Theorem.

hep-th↗

arXiv2021

Hydrodynamics of quantum spin liquids

Quantum spin liquids are topological states of matter that arise in frustrated quantum magnets at low temperatures. At low energies, such states exhibit emergent gauge fields and fractionalized quasiparticles and can also possess enhanced global symmetries compared to their parent microscopic Hamiltonians. We study the consequences of this emergent gauge and symmetry structure for the hydrodynamics of quantum spin liquids. Specifically, we analyze two cases, the $U(1)$ spin liquid with a Fermi surface and the $SU(4)$-symmetric "algebraic" spin liquid. We show that the emergent degrees of freedom in the spin liquid phase lead to a variety of additional hydrodynamic modes compared to the high-temperature paramagnetic phase. We identify a hydrodynamic regime for the internal $U(1)$ gauge field common to both states, characterized by slow diffusion of the internal transverse photon.

cond-mat.str-el↗

arXiv2021

Anomalies from correlation functions in defect conformal field theory

In previous work, we showed that an anomaly in the one point function of marginal operators is related by the Wess-Zumino condition to the Euler density anomaly on a two dimensional defect or boundary. Here we analyze in detail the two point functions of marginal operators with the stress tensor and with the displacement operator in three dimensions. We show how to get the boundary anomaly from these bulk two point functions and find perfect agreement with our anomaly effective action. For a higher dimensional conformal field theory with a four dimensional defect, we describe for the first time the anomaly effective action that relates the Euler density term to the one point function anomaly, generalizing our result for two dimensional defects.

hep-th↗

arXiv2020

Two Point Functions in Defect CFTs

This paper is designed to be a practical tool for constructing and investigating two-point correlation functions in defect conformal field theory, directly in physical space, between any two bulk primaries or between a bulk primary and a defect primary, with arbitrary spin. Although geometrically elegant and ultimately a more powerful approach, the embedding space formalism gets rather cumbersome when dealing with mixed symmetry tensors, especially in the projection to physical space. The results in this paper provide an alternative method for studying two-point correlation functions for a generic $d$-dimensional conformal field theory with a flat $p$-dimensional defect and $d-p=q$ co-dimensions. We tabulate some examples of correlation functions involving a conserved current, an energy momentum tensor and a Maxwell field strength, while analysing the constraints arising from conservation and the equations of motion. A method for obtaining bulk-to-defect correlators is also explained. Some explicit examples are considered: free scalar theory on $\mathbb{R}^p \times ({\mathbb R}^q / {\mathbb Z}_2)$ and a free four dimensional Maxwell theory on a wedge.

hep-th↗

arXiv2020

Interface Conformal Anomalies

We consider two $d \geq 2$ conformal field theories (CFTs) glued together along a codimension one conformal interface. The conformal anomaly of such a system contains both bulk and interface contributions. In a curved-space setup, we compute the heat kernel coefficients and interface central charges in free theories. The results are consistent with the known boundary CFT data via the folding trick. In $d=4$, two interface invariants generally allowed as anomalies turn out to have vanishing interface charges. These missing invariants are constructed from components with odd parity with respect to flipping the orientation of the defect. We conjecture that all invariants constructed from components with odd parity may have vanishing coefficient for symmetric interfaces, even in the case of interacting interface CFT.

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