QFT as a set of ODEs

TL;DR

This work constructs a non-perturbative continuum framework for 2D QFTs in AdS$_2$, characterizing all bulk–boundary correlators through a data set $\{\Delta_i, C_{ijk}, b^{\hat{\mathcal{O}}}_j\}$ and deriving universal first-order ODEs that describe their variation under a bulk relevant deformation. The authors derive three coupled flow equations (dimension flow, BOE-flow, and OPE-flow) using regulated integrals of conformal blocks and then stabilize the formalism by introducing local blocks to ensure convergence across AdS$_2$. They provide explicit expressions for integrated blocks, analyze pole structures, and demonstrate the framework in free theory tests, validating the approach and revealing subtleties around marginal/boundary-marginal cases and level repulsion. The paper further discusses physical implications, such as how marginal/boundary flows generate square-root behaviors at critical couplings, the role of bulk operator renormalization, flat-space limits, and potential large-$N$ dispersive reformulations. Overall, the flow-equation program offers a promising route to follow RG trajectories from solvable AdS QFTs toward strongly coupled phases and toward flat-space scattering data, with potential numerical implementations and broad applicability to BCFTs and holographic contexts.

Abstract

Correlation functions of local operators in Quantum Field Theory (QFT) on hyperbolic space can be fully characterized by the set of QFT data $\lbrace Δ_i,C_{ijk},b^{\hat{\mathcal{O}}}_j\rbrace$. These are the scaling dimensions of boundary operators $Δ_i$, the boundary Operator Product Expansion (OPE) coefficients $C_{ijk}$ and the Boundary Operator Expansion (BOE) coefficients $b^{\hat{\mathcal{O}}}_j$ that characterize how each bulk operator $\hat{\mathcal{O}}$ can be expanded in terms of boundary operators $\mathcal{O}_j$.For simplicity, we focus on two dimensional QFTs and derive a universal set of first order Ordinary Differential Equations (ODEs) that encode the variation of the QFT data under an infinitesimal change of a bulk relevant coupling. In principle, our ODEs can be used to follow a Renormalization Group (RG) flow starting from a solvable QFT into a strongly coupled phase and to the flat space limit.

Figures (30)

• Figure 1: Pictorial derivation of the flow equations for QFT data.
• Figure 2: In the radial quantization picture, the Poincaré half plane is foliated with semicircles. In configurations like the one on the left, corresponding to $\rho>0$, we must first use the BOE of $\hat{\mathcal{O}}$ and then the OPE with the two boundary operators. In cases like the picture on the right, with $\rho<0$, we must first use the OPE of $\mathcal{O}_i$ and $\mathcal{O}_j$ and then the BOE of $\hat{\mathcal{O}}$.
• Figure 3: Sign of $\upsilon$ as a function of $(\tau_1,z_1)$. The dashed line denotes the geodesic connecting the bulk point $(\tau_2,z_2)$ to the boundary point $(\tau_3,0)$.
• Figure 4: In this configuration, we can expand $\hat{\mathcal{O}}_1$ into $\mathcal{O}_l$ and $\hat{\mathcal{O}}_2$ into $\mathcal{O}_j$ via the BOE \ref{['eq:BOE']}, with $\mathcal{O}_l$ inside the red semi-circle and $\mathcal{O}_j$ outside the blue semi-circle. Thus, we get the cyclic ordering corresponding to $C_{jli}$.
• Figure 5: Conformal block decomposition of $\braket{\hat{\mathcal{O}}(\tau,z)\mathcal{O}_i(\tau_1)\mathcal{O}_j(\tau_2)\mathcal{O}_k(\tau_3)}$, where $\hat{\mathcal{O}}(\tau,z)$ is expanded into $\mathcal{O}_l$ and the boundary four-point function admits three possible OPE channels.