arXiv subjects

Edoardo Lauria

Publications and source records attributed to Edoardo Lauria.

At least 19 recordsLinked to original sources

arXiv2026

Long-range minimal models

We study a class of nonlocal conformal field theories in two dimensions which are obtained as deformations of the Virasoro minimal models. The construction proceeds by coupling a relevant primary operator $ϕ_{r,s}$ of the $m$-th minimal model to a generalized free field, in such a way that the interaction term has scaling dimension $2-δ$. Flowing to the infrared, we reach a new class of CFTs that we call long-range minimal models. In the case $r=s=2$, the resulting line of fixed points, parametrized by $δ$, can be studied using two perturbative expansions with different regimes of validity, one near the mean-field theory end, and one close to the long-range to short-range crossover. This is due to a straightforward generalization of an infrared duality which was proposed for the long-range Ising model ($m = 3$) in 2017. We find that the large-$m$ limit is problematic in both perturbative regimes, hence nonperturbative methods will be required in the intermediate range for all values of $m$. For the models based on $ϕ_{1,2}$, the situation is rather different. In this case, only one perturbative expansion is known but it is well behaved at large $m$. We confirm this with a computation of infinitely many anomalous dimensions at two loops. Their large-$m$ limits are obtained from both numerical extrapolations and a method we develop which carries out conformal perturbation theory using Mellin amplitudes. For minimal models, these can be accessed from the Coulomb gas representations of the correlators. This method reveals analytic expressions for some integrals in conformal perturbation theory which were previously only known numerically.

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arXiv2025

A strong-weak duality for the 1d long-range Ising model

We investigate the one-dimensional Ising model with long-range interactions decaying as $1/r^{1+s}$. In the critical regime, for $1/2 \leq s \leq 1$, this system realizes a family of nontrivial one-dimensional conformal field theories (CFTs), whose data vary continuously with $s$. For $s>1$ the model has instead no phase transition at finite temperature, as in the short-range case. In the standard field-theoretic description, involving a generalized free field with quartic interactions, the critical model is weakly coupled near $s=1/2$ but strongly coupled in the vicinity of the short-range crossover at $s=1$. We introduce a dual formulation that becomes weakly coupled as $s \to 1$. Precisely at $s=1$, the dual description becomes an exactly solvable conformal boundary condition of the two-dimensional free scalar. We present a detailed study of the dual model and demonstrate its effectiveness by computing perturbatively the CFT data near $s=1$, up to next-to-next-to-leading order in $1-s$, by two independent approaches: (i) standard renormalization of our dual field-theoretic description and (ii) the analytic conformal bootstrap. The two methods yield complete agreement.

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arXiv2025

Analytic and numerical bootstrap for the long-range Ising model

We combine perturbation theory with analytic and numerical bootstrap techniques to study the critical point of the long-range Ising (LRI) model in two and three dimensions. This model interpolates between short-range Ising (SRI) and mean-field behaviour. We use the Lorentzian inversion formula to compute infinitely many three-loop corrections in the two-dimensional LRI near the mean-field end. We further exploit the exact OPE relations that follow from bulk locality of the LRI to compute infinitely many two-loop corrections near the mean-field end, as well as some one-loop corrections near SRI. By including such exact OPE relations in the crossing equations for LRI we set up a very constrained bootstrap problem, which we solve numerically using SDPB. We find a family of sharp kinks for two- and three-dimensional theories which compare favourably to perturbative predictions, as well as some Monte Carlo simulations for the two-dimensional LRI.

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arXiv2025

1d Ising model with $1/r^{1.99}$ interaction

We study the 1d Ising model with long-range interactions decaying as $1/r^{1+s}$. The critical model corresponds to a family of 1d conformal field theories (CFTs) whose data depends nontrivially on $s$ in the range $1/2\leq s\leq 1$. The model is known to be described by a generalized free field with quartic interaction, which is weakly coupled near $s=1/2$ but strongly coupled near the short-range crossover at $s=1$. We propose a dual description which becomes weakly coupled at $s=1$. At $s=1$, our model becomes an exactly solvable conformal boundary condition for the 2d free scalar. We perform a number of consistency checks of our proposal and calculate the perturbative CFT data around $s=1$ analytically using both 1) our proposed field theory and 2) the analytic conformal bootstrap. Our results show complete agreement between the two methods.

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arXiv2025

A relativistic continuous matrix product state study of field theories with defects

We propose a method to compute expectation values in 1+1-dimensional massive Quantum Field Theories (QFTs) with line defects using Relativistic Continuous Matrix Product State (RCMPS). Exploiting Euclidean invariance, we use a quantization scheme where (imaginary) time runs perpendicularly to the defect. With this choice, correlation functions of local operators in the presence of the defect can be computed as expectation values of extended operators in the no-defect vacuum, which can be approximated by a homogeneous RCMPS. We demonstrate the effectiveness of this machinery by computing correlation functions of local bulk and defect operators in $ϕ^4$ theory with a magnetic line defect, in perturbative, strong coupling, critical, and symmetry-broken regimes.

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arXiv2024

A bootstrap study of minimal model deformations

For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the $T\bar T$ deformation in two dimensions.

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arXiv2024

Perturbative RG flows in AdS: an étude

We discuss general properties of perturbative RG flows in AdS with a focus on the treatment of boundary conditions and infrared divergences. In contrast with flat-space boundary QFT, general covariance in AdS implies the absence of independent boundary flows. We illustrate how boundary correlation functions remain conformally covariant even if the bulk QFT has a scale. We apply our general discussion to the RG flow between consecutive unitary diagonal minimal models which is triggered by the $ϕ_{(1,3)}$ operator. For these theories we conjecture a flow diagram whose form is significantly simpler than that in flat-space boundary QFT. In several stand-alone appendices we discuss two-dimensional BCFTs in general and the minimal model BCFTs in particular. These include both an extensive review as well as the computation of several new BCFT correlation functions.

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arXiv2024

Conformal boundary conditions for a 4d scalar field

We construct unitary, stable, and interacting conformal boundary conditions for a free massless scalar in four dimensions by coupling it to edge modes living on a boundary. The boundary theories we consider are bosonic and fermionic QED$_3$ with $N_f$ flavors and a Chern-Simons term at level $k$, in the large-$N_f$ limit with fixed $k/N_f$. We find that interacting boundary conditions only exist when $k\neq 0$. To obtain this result we compute the $β$ functions of the classically marginal couplings at the first non-vanishing order in the large-$N_f$ expansion, and to all orders in $k/N_f$ and in the couplings. To check vacuum stability we also compute the large-$N_f$ effective potential. We compare our results with the the known conformal bootstrap bounds.

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arXiv2023

Vacuum stability, fixed points, and phases of QED$_3$ at large $N_f$

We consider three-dimensional Quantum Electrodynamics in the presence of a Chern-Simons term at level $k$ and $N_f$ flavors, in the limit of large $N_f$ and $k$ with $k/N_f$ fixed. We consider either bosonic or fermionic matter fields, with and without quartic terms at criticality: the resulting theories are critical and tricritical bosonic QED$_3$, Gross-Neveu and fermionic QED$_3$. For all such theories we compute the effective potentials and the $β$ functions of classically marginal couplings, at the leading order in the large $N_f$ limit and to all orders in $k/N_f$ and in the couplings. We determine the RG fixed points and discuss the quantum stability of the corresponding vacua. While critical bosonic and fermionic QED$_3$ are always stable CFTs, we find that tricritical bosonic and Gross-Neveu QED$_3$ exist as stable CFTs only for specific values of $k/N_f$. Finally, we discuss the phase diagrams of these theories as a function of their relevant deformations.

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arXiv2022

Bootstrapping line defects with $O(2)$ global symmetry

We use the numerical bootstrap to study conformal line defects with $O(2)$ global symmetry. Our results are very general and capture in particular conformal line defects originating from bulk CFTs with a continuous global symmetry, which can either be preserved or partially broken by the presence of the defect. We begin with an agnostic approach and perform a systematic bootstrap study of correlation functions between two canonical operators on the defect: the displacement and the tilt. We then focus on two interesting theories: a monodromy line defect and a localized magnetic field line defect. To this end, we combine the numerical bootstrap with the $\varepsilon$-expansion, where we complement existing results in the literature with additional calculations. For the monodromy defect our numerical results are consistent with expectations, with known analytic solutions sitting inside our numerical bounds. For the localized magnetic field line defect our plots show a series of intriguing cusps which we explore.

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arXiv2022

Bootstrapping boundary-localized interactions II: Minimal models at the boundary

We provide evidence for the existence of non-trivial unitary conformal boundary conditions for a three-dimensional free scalar field, which can be obtained via a coupling to the m'th unitary diagonal minimal model. For large m we can demonstrate the existence of the fixed point perturbatively, and for smaller values we use the numerical conformal bootstrap to obtain a sharp kink that smoothly matches onto the perturbative predictions. The wider numerical analysis also yields universal bounds for the spectrum of any other boundary condition for the free scalar field. A second kink in these bounds hints at a second class of non-standard boundary conditions, as yet unidentified.

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arXiv2021

3d Large $N$ Vector Models at the Boundary

We consider a 4d scalar field coupled to large $N$ free or critical $O(N)$ vector models, either bosonic or fermionic, on a 3d boundary. We compute the $β$ function of the classically marginal bulk/boundary interaction at the first non-trivial order in the large $N$ expansion and exactly in the coupling. Starting with the free (critical) vector model at weak coupling, we find a fixed point at infinite coupling in which the boundary theory is the critical (free) vector model and the bulk decouples. We show that a strong/weak duality relates one description of the renormalization group flow to another one in which the free and the critical vector models are exchanged. We then consider the theory with an additional Maxwell field in the bulk, which also gives decoupling limits with gauged vector models on the boundary.

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arXiv2020

Line and surface defects for the free scalar field

For a single free scalar field in $d \geq 2$ dimensions, almost all the unitary conformal defects must be `trivial' in the sense that they cannot hold interesting dynamics. The only possible exceptions are monodromy defects in $d \geq 4$ and co-dimension three defects in $d \geq 5$. As an intermediate result we show that the $n$-point correlation functions of a conformal theory with a generalized free spectrum must be those of the generalized free theory.

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arXiv2020

Bootstrapping boundary-localized interactions

We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a `shadow pair' of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a `kink' in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

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arXiv2020

${\cal N}=2$ Supergravity in $D=4,5,6$ Dimensions

An overview of matter-coupled ${\cal N}=2$ supergravity theories with 8 real supercharges, in 4,5 and 6 dimensions is given. The construction of the theories by superconformal methods is explained from basic principles. Special geometry is obtained and characterized. The relation between the theories in those dimensions is discussed. This leads to the concepts of very special geometry and quaternionic-Kähler manifolds, whose structures are explained.

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arXiv2019

3d Abelian Gauge Theories at the Boundary

A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a $U(1)$ symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFT) parametrized by the gauge coupling $τ$ in the upper-half plane and by the choice of the CFT in the decoupling limit $τ\to \infty$. Upon performing an $SL(2,\mathbb{Z})$ transformation in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's $SL(2, \mathbb{Z})$ action [1]. In particular the cusps on the real $τ$ axis correspond to the 3d gauging of the original CFT. We study general properties of this BCFT. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. We then consider the case in which the 3d CFT is one Dirac fermion. Thanks to 3d dualities this BCFT is mapped to itself by a bulk $S$ transformation, and it also admits a decoupling limit which gives the $O(2)$ model on the boundary. We compute scaling dimensions of boundary operators and the hemisphere free-energy up to two loops. Using an $S$-duality improved ansatz, we extrapolate the perturbative results and find good approximations to the observables of the $O(2)$ model. We also consider examples with other theories on the boundary, such as large-$N_f$ Dirac fermions --for which the extrapolation to strong coupling can be done exactly order-by-order in $1/N_f$-- and a free complex scalar.

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arXiv2018

Radial coordinates for defect CFTs

We study the two-point function of local operators in the presence of a defect in a generic conformal field theory. We define two pairs of cross ratios, which are convenient in the analysis of the OPE in the bulk and defect channel respectively. The new coordinates have a simple geometric interpretation, which can be exploited to efficiently compute conformal blocks in a power expansion. We illustrate this fact in the case of scalar external operators. We also elucidate the convergence properties of the bulk and defect OPE decompositions of the two-point function. In particular, we remark that the expansion of the two-point function in powers of the new cross ratios converges everywhere, a property not shared by the cross ratios customarily used in defect CFT. We comment on the crucial relevance of this fact for the numerical bootstrap.

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arXiv2018

Spinning operators and defects in conformal field theory

We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.

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