arXiv subjects

Jiaxin Qiao

Publications and source records attributed to Jiaxin Qiao.

At least 19 recordsLinked to original sources

arXiv2026

Bulk OPE Coefficients of the $E$-Series Virasoro Minimal Models

We present bulk-primary operator-product coefficients for every Virasoro minimal model with an $E_6$, $E_7$, or $E_8$ modular invariant. Dividing by a universal product of chiral generalized-minimal-model OPE coefficients reduces the conformal bootstrap problem to a finite algebraic problem governed by root-of-unity quantum-group $6j$ symbols. The exceptional factors of the OPE coefficients can be chosen independent of the second Kac indices $s$, and their dependence on the central charge reduces to finitely many reference cases and explicit phases. A finite computer-assisted calculation yields candidate algebraic $s=1$ seed data and high-precision numerical evidence for existence and uniqueness modulo primary-field signs. We prove that exact existence and uniqueness for these seeds imply the corresponding result for all Kac labels within the genus-zero bulk locality and crossing equations; the finite seed step remains uncertified. The three families require $18$, $32$, and $107$ representative reduced OPE coefficients for $E_6$, $E_7$, and $E_8$, respectively. We provide reconstruction conventions and coefficient tables for both unitary and nonunitary models. Our calculation assumes only Virasoro symmetry, bulk locality (permutation symmetry) and OPE associativity (crossing equations).

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arXiv2026

QFT as a set of ODEs

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.

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arXiv2026

Protected operators in non-local defect CFTs from AdS

For a local quantum field theory in anti-de Sitter space with conformal boundary conditions but without dynamical gravity, the boundary theory is generically a non-local conformal field theory. Such theories can support conformal defects, but the standard local-CFT arguments based on a boundary stress tensor and conserved currents do not apply. We argue that, under general assumptions, displacement and tilt operators nevertheless exist and have protected quantum numbers. The mechanism is a Goldstone-type phenomenon in AdS: defect-induced symmetry breaking on the boundary is spontaneous from the viewpoint of the local bulk theory, whose Ward identities enforce the corresponding protected defect operators. We illustrate the mechanism in weakly coupled defect RG flows, long-range Landau--Ginzburg models, 4D Maxwell theory, and Yang--Mills theory in AdS.

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arXiv2026

Locality constraints in AdS$_2$ without parity

We study bulk locality constraints in quantum field theories in AdS$_2$. The known derivation of locality sum rules in AdS$_{d+1}$ does not apply for $d=1$ due to the different singularity structure of the conformal blocks and the inequivalence of operator orderings on the boundary. Assuming unitarity and a mild growth condition, we establish power-law bounds for correlators, derive dispersion relations and an expansion in terms of ``even'' and ``odd'' local blocks that converges in the entire AdS$_2$. These yield two novel families of symmetric and antisymmetric locality sum rules. We test these sum rules explicitly in the free scalar field theory.

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arXiv2026

Yang-Mills Flux Tube in AdS

We initiate the study of flux tubes in confining gauge theories placed in a rigid AdS background, which serves as an infrared regulator. Varying the AdS radius from large to small allows us to interpolate between the flat space confining string, and a weakly coupled string-like object which is held together by the AdS gravitational potential. At any radius, the string preserves a subgroup of AdS isometries equivalent to the one-dimensional conformal group and hence, from the boundary point of view, can be thought of as a conformal defect. The defect hosts a protected operator, called displacement, which nonlinearly realizes the broken AdS isometries. At small radius the displacement corresponds to the gauge field strength inserted at the boundary, while at large radius it is mapped to the Goldstone mode living on the string worldsheet. This relates gauge field and worldsheet degrees of freedom. We propose a hypothesis according to which the large and small radius perturbative expansions can be smoothly matched with each other. As a test, we calculate the leading order corrections to the scaling dimensions and OPE coefficients of a set of defect operators at weak coupling in planar 3D Yang-Mills.

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arXiv2025

Quantum Groups as Global Symmetries II. Coulomb Gas Construction

We study a conformal field theory that arises in the infinite-volume limit of a spin chain with $U_q(sl_2)$ global symmetry. Most operators in the theory are defect-ending operators which allows $U_q(sl_2)$ symmetry transformations to act on them in a consistent way. We use Coulomb gas techniques to construct correlation functions and compute all OPE coefficients of the model, as well as to prove that the properties imposed by the quantum group symmetry are indeed satisfied by the correlation functions. In particular, we treat the non-chiral operators present in the theory. Free boson realization elucidates the origin of the defects attached to the operators. We also comment on the role of quantum group in generalized minimal models.

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arXiv2025

Symmetry-deformed toric codes and the quantum dimer model

Motivated by the recent introduction of a $U(1)$-symmetric toric code model, we investigate symmetry-based deformations of topological order by systematically deconstructing the Gauss-law-enforcing star terms of the toric code (TC) Hamiltonian. This "term-dropping" protocol introduces global symmetries that go beyond the alternative framework of "ungauging" topological order in symmetry-deformed models and gives rise to models such as the $U(1)$TC or $XY$TC. These models inherit (emergent) subsystem symmetries (from the original 1-form symmetry of the TC) that can give rise to (subextensive) ground-state degeneracies, which can still be organized by the eigenvalues of Wilson loop operators. However, we demonstrate that these models do not support topological or fracton order (as has been conjectured in the literature) due to the loss of (emergent) gauge symmetry. An extreme deformation of the TC is the quantum dimer model (QDM), which we discuss along the family of symmetry-deformed models from the perspective of subsystem symmetries, sublattice modulation, and quantum order-by-disorder mechanisms resulting in rich phase diagrams. For the QDM, this allows us to identify an emergent SO(2) symmetry for what appears to be a gapless ground state (by numerical standards) that is unstable to the formation of a plaquette valence bond solid upon sublattice modulation.

cond-mat.str-el↗

arXiv2025

Quantum Groups as Global Symmetries

We study quantum field theories which have quantum groups as global internal symmetries. We show that in such theories operators are generically non-local, and should be thought as living at the ends of topological lines. We describe the general constraints of the quantum group symmetry, given by Ward identities, that correlation functions of the theory should satisfy. We also show that generators of the symmetry can be represented by topological lines with some novel properties. We then discuss a particular example of $U_q(sl_2)$ symmetric CFT, which we solve using the bootstrap techniques and relying on the symmetry. We finally show strong evidence that for a special value of $q$ a subsector of this theory reproduces the fermionic formulation of the Ising model. This suggests that a quantum group can act on local operators as well, however, it generically transforms them into non-local ones.

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arXiv2025

Universality of the microcanonical entropy at large spin

We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with $c > 1$. Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin-$J$ operators must grow like $\exp\left( π\sqrt{\frac{2}{3}(c-1) J} \right)/\sqrt{2J}$ in any twist interval at or above $(c-1)/12$, with a known twist-dependent prefactor. This proves that the large $J$ spectrum becomes dense even without averaging over spins. For twists below $(c-1)/12$ we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin-$J$ operators must go to zero as $J$ becomes large.

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arXiv2025

A universal inequality on the unitary 2D CFT partition function

We prove the conjecture proposed by Hartman, Keller and Stoica [HKS14]: the grand-canonical free energy of a unitary 2D CFT with a sparse spectrum below the scaling dimension $\frac{c}{12}+ε$ and below the twist $\frac{c}{12}$ is universal in the large $c$ limit for all $β_Lβ_R \neq 4π^2$. The technique of the proof allows us to derive a one-parameter (with parameter $α\in(0,1]$) family of universal inequalities on the unitary 2D CFT partition function with general central charge $c\geqslant 0$, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the inequality on the $(β_L,β_R)$ plane. The infinite iteration of this equation gives the boundary of maximal-validity domain, which depends on the parameter $α$ in the inequality. In the $c \to \infty$ limit, with the additional assumption of a sparse spectrum below the scaling dimension $\frac{c}{12} + ε$ and the twist $\frac{αc}{12}$ (with $α\in (0,1]$ fixed), our inequality shows that the grand-canonical free energy exhibits a universal large $c$ behavior in the maximal-validity domain. This domain, however, does not cover the entire $(β_L, β_R)$ plane, except in the case of $α= 1$. For $α= 1$, this proves the conjecture proposed by [HKS14], and for $α< 1$, it quantifies how sparseness in twist affects the regime of universality. Furthermore, this implies a precise lower bound on the temperature of near-extremal BTZ black holes, above which we can trust the black hole thermodynamics.

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arXiv2023

A radial variable for de Sitter two-point functions

We introduce a "radial" two-point invariant for quantum field theory in de Sitter (dS) analogous to the radial coordinate used in conformal field theory. We show that the two-point function of a free massive scalar in the Bunch-Davies vacuum has an exponentially convergent series expansion in this variable with positive coefficients only. Assuming a convergent Källén-Lehmann decomposition, this result is then generalized to the two-point function of any scalar operator non-perturbatively. A corollary of this result is that, starting from two-point functions on the sphere, an analytic continuation to an extended complex domain is admissible. dS two-point configurations live inside or on the boundary of this domain, and all the paths traced by Wick rotations between dS and the sphere or between dS and Euclidean Anti-de Sitter are also contained within this domain.

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arXiv2023

Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-like Formula for Near-extremal Black Holes

We show that for a unitary modular invariant 2D CFT with central charge $c>1$ and having a nonzero twist gap in the spectrum of Virasoro primaries, for sufficiently large spin $J$, there always exist spin-$J$ operators with twist falling in the interval $(\frac{c-1}{12}-\varepsilon,\frac{c-1}{12}+\varepsilon)$ with $\varepsilon=O(J^{-1/2}\log J)$. We establish that the number of Virasoro primary operators in such a window has a Cardy-like i.e. $\exp\left(2π\sqrt{\frac{(c-1)J}{6}}\right)$ growth. We make further conjectures on potential generalization to CFTs with conserved currents. A similar result is proven for a family of holographic CFTs with the twist gap growing linearly in $c$ and a uniform boundedness condition, in the regime $J\gg c^3\gg1$. From the perspective of near-extremal rotating BTZ black holes (without electric charge), our result is valid when the Hawking temperature is much lower than the "gap temperature".

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arXiv2023

Twist accumulation in conformal field theory. A rigorous approach to the lightcone bootstrap

We prove that in any unitary CFT, a twist gap in the spectrum of operator product expansion (OPE) of identical scalar primary operators (i.e. $ϕ\times ϕ$) implies the existence of a family of primary operators $\mathcal{O}_{τ, \ell}$ with spins $\ell \rightarrow \infty$ and twists $τ\rightarrow 2 Δ_ϕ$ in the same OPE spectrum. A similar twist-accumulation result is proven for any two-dimensional Virasoro-invariant, modular-invariant, unitary CFT with a normalizable vacuum and central charge $c > 1$, where we show that a twist gap in the spectrum of Virasoro primaries implies the existence of a family of Virasoro primaries $\mathcal{O}_{h, \bar{h}}$ with $h \rightarrow \infty$ and $\bar{h} \rightarrow \frac{c - 1}{24}$ (the same is true with $h$ and $\bar{h}$ interchanged). We summarize the similarity of the two problems and propose a general formulation of the lightcone bootstrap.

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arXiv2022

On the Wick Rotation of the Four-point Function in Conformal Field Theories

Conformal field theories (CFTs) in Euclidean signature satisfy well-accepted rules, such as conformal invariance and the convergent Euclidean operator product expansion (OPE). Nowadays, it is common to assume that CFT correlators exist and have various properties in the Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. In this thesis, we clarify that at the level of four-point functions, the Euclidean CFT axioms imply the standard quantum field theory axioms such as Osterwalder-Schrader axioms (in Euclidean) and Wightman axioms (in Lorentzian).

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arXiv2022

Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions

We analyze the convergence properties of operator product expansions (OPE) for Lorentzian CFT four-point functions of scalar operators. We give a complete classification of Lorentzian four-point configurations. All configurations in each class have the same OPE convergence properties in s-, t- and u-channels. We give tables including the information of OPE convergence for all classes. Our work justifies that in a subset of the configuration space, Lorentzian CFT four-point functions are genuine analytic functions. Our results are valid for unitary CFTs in $d\geq2$. Our work also provides some Lorentzian regions where one can do bootstrap analysis in the sense of functions.

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arXiv2021

Distributions in CFT II. Minkowski Space

CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios $ρ, \barρ$. We prove a key fact that $|ρ|, |\barρ| < 1$ inside the forward tube, and set bounds on how fast $|ρ|, |\barρ|$ may tend to 1 when approaching the Minkowski space. We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).

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arXiv2020

Distributions in CFT I. Cross-Ratio Space

We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations involving the boundary of the region of convergence, e.g. for deriving the dispersion relations. We establish our results by elementary methods, relying only on crossing symmetry and the standard convergence properties of the conformal block expansion. This is the first in a series of papers on distributional properties of correlation functions in conformal field theory.

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arXiv2017

A tauberian theorem for the conformal bootstrap

For expansions in one-dimensional conformal blocks, we provide a rigorous link between the asymptotics of the spectral density of exchanged primaries and the leading singularity in the crossed channel. Our result has a direct application to systems of SL(2,R)-invariant correlators (also known as 1d CFTs). It also puts on solid ground a part of the lightcone bootstrap analysis of the spectrum of operators of high spin and bounded twist in CFTs in d>2. In addition, a similar argument controls the spectral density asymptotics in large N gauge theories.

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