Area law and OPE blocks in CFT

从三维共形场论的 Carroll 极限得到的二维对称代数的无限塔

Infinite towers of 2d symmetry algebras from Carrollian limit of 3d CFT

本文研究三维共形场论 OPE 块的 Carroll 极限，证明其分解为负整数维数标记的 $\mathfrak{sl}(2,\mathbb{C})$ 模式塔，并由此推导出共形软光子和软引力子定理以及 celestial CFT 的 $S$ 与 $w_{1+\infty}$ 代数。

N=2 SUSY CFT和仿射Laumon空间中手征场的共形块

Conformal blocks of Chiral fields in N=2 SUSY CFT and Affine Laumon Spaces

考虑了N=2超共形块函数的计算问题。我们证明了N=2超共形代数在仿射sl（2）代数上的Kazama-Suzuki陪集实现提供了N=2与仿射sl（2）共形块之间的关系。我们证明了N=2手征场的sl（2）共形块的构造是基于普通的最高权表示.我们利用AGT型对应将四维N=2 SU（2）规范理论的四点sl（2）共形块与Nekrasov瞬子配分函数联系起来，其中包含了一个表面算符.由于以前的关系提出的Alday和立川需要一些特殊的修改的共形块函数，我们重新审视这个问题，并找到直接对应的四点共形块。因此，我们找到了仿射sl（2）四点共形块的一个显式表示，从而得到了N=2手征四点共形块的一个显式组合表示。

Virasoro OPE 块、因果钻石与高维CFT

Virasoro OPE Blocks, Causal Diamonds, and Higher-Dimensional CFT

本文研究了高维CFT中Virasoro恒等式OPE块的构造，利用因果钻石积分推导四点关联函数的单应力张量交换贡献。

N=4 Superconformal Bootstrap of the K3 CFT

We study two-dimensional (4,4) superconformal field theories of central charge c=6, corresponding to nonlinear sigma models on K3 surfaces, using the superconformal bootstrap. This is made possible through a surprising relation between the BPS N=4 superconformal blocks with c=6 and bosonic Virasoro conformal blocks with c=28, and an exact result on the moduli dependence of a certain integrated BPS 4-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We ob…

Worldline approach to semi-classical conformal blocks

We extend recent results on semi-classical conformal blocks in 2d CFT and their relation to 3D gravity via the AdS/CFT correspondence. We consider four-point functions with two heavy and two light external operators, along with the exchange of a light operator. By explicit computation, we establish precise agreement between these CFT objects and a simple picture of particle worldlines joined by cubic vertices propagating in asymptotically AdS$_3$ geometries (conical defects or BTZ black holes). We provide a simple argument that explains this agreement.

一维CFT混合关联子自举中的Polyakov块

Polyakov blocks for the 1D CFT mixed correlator bootstrap

本文引入一维CFT混合关联子自举的交叉对称Polyakov块展开，导出求和规则，并通过对角化基改变，首次给出由$\phi,\phi^2$系统饱和的优化界。

自由 CFT 中块展开的一个公式及其在强耦合 ${\cal N}=4$ SYM 中的应用

A formula for the block expansion in free CFTs and applications to ${\cal N}=4$ SYM at strong coupling

本文提出自由 CFT 中共形（超）块展开的显式公式，并用于证明强耦合 ${\cal N}=4$ SYM 多粒子关联函数的树级 Witten 图表示。

Holographic variables for CFT$_2$ conformal blocks with heavy operators

We consider large-$c$ $n$-point Virasoro blocks with $n-k$ background heavy operators and $k$ perturbative heavy operators. Conformal dimensions of heavy operators scale linearly with large $c$, while splitting into background/perturbative operators assumes an additional perturbative expansion. Such conformal blocks can be calculated within the monodromy method that basically reduces to solving auxiliary Fuchsian second-order equation and finding monodromy of solutions. We show that there exist particular variables that we call holographic, use of which drastically simplifies the whole analysi…

Logarithmic behaviour of connected correlation function in CFT

We study $(m)$-type connected correlation functions of OPE blocks with respect to one spatial region in two dimensional conformal field theory. We find logarithmic divergence for these correlation functions. We justify the logarithmic behaviour from three different approaches: massless free scalar theory, Selberg integral and conformal block. Cutoff independent coefficients are obtained from analytic continuation of conformal blocks. A UV/IR relation has been found in connected correlation functions. We could derive a formal ``first law of thermodynamics'' for a subsystem using deformed reduce…

从CFT Bootstrap看长距离AdS物理的普适性

Universality of Long-Distance AdS Physics from the CFT Bootstrap

我们首先阐明最近证明的集群分解原则，在AdS_{d+1}从CFT_d引导在d > 2。CFT参数还计算了AdS中遥远物体之间的主要相互作用，我们确认了CFT自举和AdS引力在半经典极限下的普遍一致性。我们继续研究推广到二维CFTs，这需要知识的Virasoro共形块在光锥OPE限制。我们计算这些块在一个半经典的，大的中心荷近似，并用它们来证明一个适当修改的定理。特别是，从2D的引导，我们证明了存在的大自旋运营商与固定的“异常尺寸”的赤字角的存在下，在AdS_3的指示。当我们接近BTZ黑洞的阈值时，解释为CFT标度维数，大自旋算符的扭曲谱变得密集。由于Virasoro身份块的交换，BTZ阈值以上的初级状态模拟光算子的热背景。我们导出了BTZ准简正模，并利用自举方程证明了扭曲谱是稠密的。通过对Virasoro共形块进行更精确的计算，可以得到对热性的修正。

Vacuum block thermalization in semi-classical 2d CFT

The universal nature of black hole collapse in asymptotically $AdS_3$ gravitational theories suggests that its holographic dual process, thermalization, should similarly be fixed by the universal features of 2d CFT with large central charge $c$. It is known that non-equilibrium states with scaling dimensions of order $c$ can be sorted into states that eventually thermalize and those that fail to do so. By proving an equivalence between bounded Virasoro coadjoint orbits and certain (in)stability intervals of Hill's equation it is shown that a state that fails to thermalize can be promoted to a …

Liouville CFT、Matrix Models与受约束的WZW

Liouville CFT, Matrix Models and constrained WZW

本文研究了带退化与不规则算子的Liouville CFT共形块的matrix model实现，揭示了其与受共形约束的$SL(2,\mathbb{C})$ WZW模型的联系，并给出了满足不规则KZ方程的广义预解式实现。

Exploring the BTZ bulk with boundary conformal blocks

We point out a simple relation between the bulk field at an arbitrary radial position and the boundary OPE, by placing some old work by Ferrara, Gatto, Grillo and Parisi in the AdS/CFT context. This gives us, in principle, a prescription for extracting the classical bulk field from the boundary conformal block, and also clarifies why the latter is computed by a geodesic Witten diagram. We apply this prescription to the BTZ black hole - viewed as a pure state created by the insertion of a heavy operator in the boundary CFT_2 - and use it to relate a classical field in the bulk to a heavy-light …

Higher-point conformal blocks and entanglement entropy in heavy states

We consider conformal blocks of two heavy operators and an arbitrary number of light operators in a (1+1)-d CFT with large central charge. Using the monodromy method, these higher-point conformal blocks are shown to factorize into products of 4-point conformal blocks in the heavy-light limit for a class of OPE channels. This result is reproduced by considering suitable worldline configurations in the bulk conical defect geometry. We apply the CFT results to calculate the entanglement entropy of an arbitrary number of disjoint intervals for heavy states. The corresponding holographic entangleme…

A conformal block Farey tail

We investigate the constraints of crossing symmetry on CFT correlation functions. Four point conformal blocks are naturally viewed as functions on the upper-half plane, on which crossing symmetry acts by PSL(2,Z) modular transformations. This allows us to construct a unique, crossing symmetric function out of a given conformal block by averaging over PSL(2,Z). In some two dimensional CFTs the correlation functions are precisely equal to the modular average of the contributions of a finite number of light states. For example, in the two dimensional Ising and tri-critical Ising model CFTs, the c…

CFT correlators, ${\cal W}$-algebras and Generalized Catalan Numbers

In two spacetime dimensions the Virasoro heavy-heavy-light-light (HHLL) vacuum block in a certain limit is governed by the Catalan numbers. The equation for their generating function can be generalized to a differential equation which the logarithm of the block satisfies. We show that a similar story holds for the HHLL ${\cal W}_N$ vacuum blocks, where a suitable generalization of the Catalan numbers plays the main role. Moreover, the ${\cal W}_N$ blocks have the same form as the stress tensor sector of HHLL near lightcone conformal correlators in $2(N-1)$ spacetime dimensions. In the latter c…

Seed Conformal Blocks in 4D CFT

We compute in closed analytical form the minimal set of "seed" conformal blocks associated to the exchange of generic mixed symmetry spinor/tensor operators in an arbitrary representation (l,\bar l) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0,|l-\bar l|) and one (|l-\bar l|,0) spinors or tensors. We directly solve the set of Casimir equations, that can elegantly be written in a compact form for any (l,\bar l), by using an educated ansatz and reducing the problem to an algebraic linear system. Various…

CFT和N=2 SYM理论中的递归关系

Recursion relations in CFT and N=2 SYM theory

基于Al.Zamolodchikov关于四点共形块的递归关系的典型例子，并利用最近提出的Alday-Gaiotto-Tachikawa（AGT）猜想，推导出${\cal N}=2$ SYM与$f=0,1,2,3,4$（反）基本波或伴随超多重波的广义预势的递归关系。在所有的情况下，大的期望值限制推导明确。建立了环面上一般1点共形块与球面上特殊4点共形块之间的精确关系。鉴于AGT猜想，这转化为伴随的配分函数和4个基本超多重态之间的关系。

The Probe of Curvature in the Lorentzian AdS$_2$/CFT$_1$ Correspondence

We establish the Lorentzian AdS$_2$/CFT$_1$ correspondence from a reconstruction of all bulk points through the kinematic-space approach. The OPE block is exactly a bulk local operator. We formulate the correspondence between the bulk propagator in the non-interacting scalar field theory and the conformal block in CFT$_1$. When we consider the stress tensor, the variation probes the variation of AdS$_2$ metric. The reparameterization provides the asymptotic boundary of the bulk spacetime as in the derivation of the Schwarzian theory from two-dimensional dilaton gravity theory. Finally, we find…

Classical limit of irregular blocks and Mathieu functions

The Nekrasov-Shatashvili limit of the N=2 SU(2) pure gauge (Omega-deformed) super Yang-Mills theory encodes the information about the spectrum of the Mathieu operator. On the other hand, the Mathieu equation emerges entirely within the frame of two-dimensional conformal field theory (2d CFT) as the classical limit of the null vector decoupling equation for some degenerate irregular block. Therefore, it seems to be possible to investigate the spectrum of the Mathieu operator employing the techniques of 2d CFT. To exploit this strategy, a full correspondence between the Mathieu equation and its …

Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in CFT$_2$

We construct an infinite class of eigenmodes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval $N$ in 2d CFT and study some of its interesting properties, which includes its action on OPE blocks as well as its bulk duals. Our analysis suggests that these eigenmodes, like the OPE blocks have a natural description on the so called kinematic space of CFT$_2$ and in particular realize the Virasoro algebra of the theory on this kinematic space. Taken together, our results hints at the possibility of an effective description of the CFT$_2$ in the kinematic space langua…

AdS/CFT散射矩阵的通用模块

Universal blocks of the AdS/CFT Scattering Matrix

我们确定某些块的S-矩阵描述的散射的束缚态的AdS 5 × S5超弦，允许在通用的R-矩阵的杨双表示。对于这些情况下，我们使用的公式Drinfeld的第二次实现的Yangian在任意的束缚态表示，以获得相应的R-矩阵的显式表达式。然后，我们表明，这些表达式与先前获得的S-矩阵块完全匹配。

The light asymptotic limit of conformal blocks in Toda field theory

We compute the light asymptotic limit of $A_{n-1}$ Toda conformal blocks by using the AGT correspondence. We show that for certain class of CFT blocks the corresponding Nekrasov partition functions in this limit are simplified drastically being represented as a sum of a restricted class of Young diagrams. In the particular case of $A_{2}$ Toda we also compute the corresponding conformal blocks using conventional CFT techniques finding a perfect agreement with the results obtained from the Nekrasov partition functions.

满足AGT需求的CFT练习

CFT exercises for the needs of AGT

要检验W_N-对称控制的共形块与U（N）Nekrasov函数之间的AGT关系，需要知道W-代数后代的Shapovalov矩阵和各种三重分解子.我们收集最简单的表达式，这种类型的N=3和两个最低的后裔水平，连同详细的推导，现在可以计算机化，并用于更一般的研究，在更高的水平上的共形块和AGT关系。

Large-$c$ conformal $(n \leq 6)$-point blocks with superlight weights and holographic Steiner trees

In this note we study CFT$_2$ Virasoro conformal blocks with heavy operators in the large-$c$ limit in the context of AdS$_3$/CFT$_2$ correspondence. We compute the lengths of the holographic Steiner trees dual to the $5$-point and $6$-point conformal blocks using the superlight approximation when one or more dimensions are much less than the others. These results are generalized for $N$-point holographic Steiner trees dual to $(N+1)$-point conformal blocks with superlight weights.

A Stereoscopic Look into the Bulk

We present the foundation for a holographic dictionary with depth perception. The dictionary consists of natural CFT operators whose duals are simple, diffeomorphism-invariant bulk operators. The CFT operators of interest are the "OPE blocks," contributions to the OPE from a single conformal family. In holographic theories, we show that the OPE blocks are dual at leading order in 1/N to integrals of effective bulk fields along geodesics or homogeneous minimal surfaces in anti-de Sitter space. One widely studied example of an OPE block is the modular Hamiltonian, which is dual to the fluctuatio…

blocks_3d: Software for general 3d conformal blocks

We introduce the software blocks_3d for computing four-point conformal blocks of operators with arbitrary Lorentz representations in 3d CFTs. It uses Zamolodchikov-like recursion relations to numerically compute derivatives of blocks around a crossing-symmetric configuration. It is implemented as a heavily optimized, multithreaded, C++ application. We give performance benchmarks for correlators containing scalars, fermions, and stress tensors. As an example application, we recompute bootstrap bounds on four-point functions of fermions and study whether a previously observed sharp jump can be e…

The Goldstone Equivalence Theorem and AdS/CFT

The Goldstone equivalence theorem allows one to relate scattering amplitudes of massive gauge fields to those of scalar fields in the limit of large scattering energies. We generalize this theorem under the framework of the AdS/CFT correspondence. First, we obtain an expression of the equivalence theorem in terms of correlation functions of creation and annihilation operators by using an AdS wave function approach to the AdS/CFT dictionary. It is shown that the divergence of the non-conserved conformal current dual to the bulk gauge field is approximately primary when computing correlators for…

来自平直空间和缺陷CFT的热共形部分波

Thermal conformal partial waves from flat-space and defect CFT

利用影子形式建立平直、热和缺陷背景下的共形部分波之间的对应关系，并通过特定算子配置从平直四点块和缺陷两点块系统推导热单点块。

Minkowski Conformal Blocks and the Regge Limit for SYK-like Models

We discuss scattering in a CFT via the conformal partial-wave analysis and the Regge limit. The focus of this paper is on understanding an OPE with Minkowski conformal blocks. Starting with a t-channel OPE, it leads to an expansion for an s-channel scattering amplitude in terms of t-channel exchanges. By contrasting with Euclidean conformal blocks we see a precise relationship between conformal blocks in the two limits without preforming an explicit analytic continuation. We discuss a generic feature for a CFT correlation function having singular $F^{(M)}(u,v)\sim {u}^{-δ}\,$, $δ>0$, in the li…

伽利略共形场论$_2$的影子形式

The Shadow Formalism of Galilean CFT$_2$

本文为二维伽利略共形场论建立了影子形式，推导了OPE块、共形块和共形分波，并揭示了共形块的新分支点结构、新的反演公式以及与BMS自由标量模型和零缺陷的联系。

AdS/CFT Correspondence与拓扑场论

AdS/CFT Correspondence And Topological Field Theory

在四维$M$上的${\cal N}=4$超杨-米尔斯理论中，可以指定值为$H^2(M,\Z_N)$的离散磁通量。这个通量编码在AdS/CFT对应的五维拓扑场论与陈-西蒙斯行动。一个类似的拓扑场论在七维支配的空间“共形块”的六维$(0,2)$共形场论。

The Mellin Formalism for Boundary CFT$_d$

We extend the Mellin representation of conformal field theory (CFT) to allow for conformal boundaries and interfaces. We consider the simplest holographic setup dual to an interface CFT - a brane filling an $AdS_{d}$ subspace of $AdS_{d+1}$ - and perform a systematic study of Witten diagrams in this setup. As a byproduct of our analysis, we show that geodesic Witten diagrams in this geometry reproduce interface CFT$_d$ conformal blocks, generalizing the analogous statement for CFTs with no defects.

Crossing antisymmetric Polyakov blocks + Dispersion relation

Many CFT problems, e.g. ones with global symmetries, have correlation functions with a crossing antisymmetric sector. We show that such a crossing antisymmetric function can be expanded in terms of manifestly crossing antisymmetric objects, which we call the '+ type Polyakov blocks'. These blocks are built from AdS$_{d+1}$ Witten diagrams. In 1d they encode the '+ type' analytic functionals which act on crossing antisymmetric functions. In general d we establish this Witten diagram basis from a crossing antisymmetric dispersion relation in Mellin space. Analogous to the crossing symmetric case…

Renyi Entropy for Local Quenches in 2D CFTs from Numerical Conformal Blocks

We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of $\log t$ term. Our analysis covers the entire parameter regions with respect to the replica number $n$ and the conformal dimension $h_O$ of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov's recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge $c$…

Reparametrization modes in 2d CFT and the effective theory of stress tensor exchanges

We study the origin of the recently proposed effective theory of stress tensor exchanges based on reparametrization modes, that has been used to efficiently compute Virasoro identity blocks at large central charge. We first provide a derivation of the nonlinear Alekseev--Shatashvili action governing these reparametrization modes, and argue that it should be interpreted as the generating functional of stress tensor correlations on manifolds related to the plane by conformal transformations. In addition, we demonstrate that the rules previously prescribed with the reparametrization formalism for…

On 2d CFTs that interpolate between minimal models

We investigate exactly solvable two-dimensional conformal field theories that exist at generic values of the central charge, and that interpolate between A-series or D-series minimal models. When the central charge becomes rational, correlation functions of these CFTs may tend to correlation functions of minimal models, or diverge, or have finite limits which can be logarithmic. These results are based on analytic relations between four-point structure constants and residues of conformal blocks.

Virasoro Conformal Blocks and Thermality from Classical Background Fields

We show that in 2d CFTs at large central charge, the coupling of the stress tensor to heavy operators can be re-absorbed by placing the CFT in a non-trivial background metric. This leads to a more precise computation of the Virasoro conformal blocks between heavy and light operators, which are shown to be equivalent to global conformal blocks evaluated in the new background. We also generalize to the case where the operators carry U(1) charges. The refined Virasoro blocks can be used as the seed for a new Virasoro block recursion relation expanded in the heavy-light limit. We comment on the im…

Conformal Blocks in Three Dimensions

We derive expressions for conformal blocks involving operators with arbitrary spins in 3-dimensional CFTs. We use previous results on the action of the OPE in the embedding space to derive the conformal blocks. The blocks are given as explicit power series in terms of the conformal cross ratios.

Dimensional reduction of higher-point conformal blocks

Recently, with the help of Parisi-Sourlas supersymmetry an intriguing relation was found expressing the four-point scalar conformal block of a (d-2)-dimensional CFT in terms of a five-term linear combination of blocks of a d-dimensional CFT, with constant coefficients. We extend this dimensional reduction relation to all higher-point scalar conformal blocks of arbitrary topology restricted to scalar exchanges. We show that the constant coefficients appearing in the finite term higher-point dimensional reduction obey an interesting factorization property allowing them to be determined in terms …

From Conformal Blocks to Path Integrals in the Vaidya Geometry

Correlators in conformal field theory are naturally organized as a sum over conformal blocks. In holographic theories, this sum must reorganize into a path integral over bulk fields and geometries. We explore how these two sums are related in the case of a point particle moving in the background of a 3d collapsing black hole. The conformal block expansion is recast as a sum over paths of the first-quantized particle moving in the bulk geometry. Off-shell worldlines of the particle correspond to subdominant contributions in the Euclidean conformal block expansion, but these same operators must …

Comments on Determinant Formulas for General CFTs

We point out that the determinant formula for a parabolic Verma module plays a key role in the study of (super)conformal field theories and in particular their (super)conformal blocks. The determinant formula is known from the old work of Jantzen for bosonic conformal algebras, and we present a conjecture for superconformal algebras. The application of the formula includes derivation of the unitary bound and recursion relations for conformal blocks.

Geodesic description of Heavy-Light Virasoro blocks

We continue to investigate the dual description of the Virasoro conformal blocks arising in the framework of the classical limit of the AdS$_3$/CFT$_2$ correspondence. To give such an interpretation in previous studies, certain restrictions were necessary. Our goal here is to consider more general situation available through the worldline approximation to the dual AdS gravity. Namely, we are interested in computing the spherical conformal blocks without the previously imposed restrictions on the conformal dimensions of the internal channels. The duality is realised as an equality of the so-cal…

Dimensional Reduction for Conformal Blocks

We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO(d+1,1) up into multiplets of SO(d,1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d-1 dimensions. In particular, we obtain a formula for 3d conformal blocks as an infinite sum over 2F1 hypergeometric functions with closed-form coefficients.

针对高度不同的缩放维度的保形块

Conformal blocks for highly disparate scaling dimensions

我们表明，共形块简化大大当有两个外部运营商的缩放尺寸之间的差异很大。特别地，时空维度只出现在我们通过递归关系确定的整体常数中。连接的共形引导程序和AdS / CFT对应进行了讨论。

Holographic two-point functions for Janus interfaces in the $D1/D5$ CFT

This paper investigates scalar perturbations in the top-down supersymmetric Janus solutions dual to conformal interfaces in the $D1/D5$ CFT, finding analytic closed-form solutions. We obtain an explicit representation of the bulk-to-bulk propagator and extract the two-point correlation function of the dual operator with itself, whose form is not fixed by symmetry alone. We give an expression involving the sum of conformal blocks associated with the bulk-defect operator product expansion and briefly discuss finite-temperature extensions. To our knowledge, this is the first two-point function co…

环面阴影形式与精确的全局共形块

Torus shadow formalism and exact global conformal blocks

该研究利用阴影形式推导二维共形场论环面的全局共形块，证明项链道n点环面块可由超几何型的项链函数表达，为环面共形块计算提供了精确形式。

Conformal Blocks Beyond the Semi-Classical Limit

Black hole microstates and their approximate thermodynamic properties can be studied using heavy-light correlation functions in AdS/CFT. Universal features of these correlators can be extracted from the Virasoro conformal blocks in CFT2, which encapsulate quantum gravitational effects in AdS3. At infinite central charge c, the Virasoro vacuum block provides an avatar of the black hole information paradox in the form of periodic Euclidean-time singularities that must be resolved at finite c. We compute Virasoro blocks in the heavy-light, large c limit, extending our previous results by determin…

Perturbative classical conformal blocks as Steiner trees on the hyperbolic disk

We consider the Steiner tree problem in hyperbolic geometry in the context of the AdS/CFT duality between large-$c$ conformal blocks on the boundary and particle motions in the bulk. The Steiner trees are weighted graphs on the Poincare disk with a number of endpoints and trivalent vertices connected to each other by edges in such a way that an overall length is minimum. We specify a particular class of Steiner trees that we call holographic. Their characteristic property is that a holographic Steiner tree with $N$ endpoints can be inscribed into an $N$-gon with $N-1$ ideal vertices. The holog…

On the Kerr-AdS/CFT correspondence

We review the relation between four-dimensional global conformal blocks and field propagation in ${\rm AdS_5}$. Following the standard argument that marginal perturbations should backreact in the geometry, we turn to the study of scalar fields in the generic Kerr-${\rm AdS_5}$ geometry. On one hand, the result for scattering coefficients can be obtained exactly using the isomonodromy technique, giving exact expressions in terms of $c=1$ chiral conformal blocks. On the other hand, one can use the analogy between the scalar field equations to the Level 2 null field Ward identity in two dimension…

Entanglement Entropy, OTOC and Bootstrap in 2D CFTs from Regge and Light Cone Limits of Multi-point Conformal Block

We explore the structures of light cone and Regge limit singularities of $n$-point Virasoro conformal blocks in $c>1$ two-dimensional conformal field theories with no chiral primaries, using fusion matrix approach. These CFTs include not only holographic CFTs dual to classical gravity, but also their full quantum corrections, since this approach allows us to explore full $1/c$ corrections. As the important applications, we study time dependence of Renyi entropy after a local quench and out-of-time ordered correlator (OTOC) at late time. We first show that, the $n$-th ($n>2$) Renyi entropy afte…

Crossing-symmetric Twist Field Correlators and Entanglement Negativity in Minimal CFTs

We study conformal twist field four-point functions on a $\mathbb Z_N$ orbifold. We examine in detail the case $N=3$ and analyze theories obtained by replicated $N$-times a minimal model with central charge $c<1$. A fastly convergent expansion of the twist field correlation function in terms of sphere conformal blocks with central charge $Nc$ is obtained by exploiting covering map techniques. We discuss extensive applications of the formalism to the entanglement of two disjoint intervals in CFT, in particular we propose a conformal block expansion for the partially transposed reduced density m…

Aspects of CFTs on Real Projective Space

We present an analytic study of conformal field theories on the real projective space $\mathbb{RP}^d$, focusing on the two-point functions of scalar operators. Due to the partially broken conformal symmetry, these are non-trivial functions of a conformal cross ratio and are constrained to obey a crossing equation. After reviewing basic facts about the structure of correlators on $\mathbb{RP}^d$, we study a simple holographic setup which captures the essential features of boundary correlators on $\mathbb{RP}^d$. The analysis is based on calculations of Witten diagrams on the quotient space $AdS…

Fermion-Scalar Conformal Blocks

We compute the conformal blocks associated with scalar-scalar-fermion-fermion 4-point functions in 3D CFTs. Together with the known scalar conformal blocks, our result completes the task of determining the so-called `seed blocks' in three dimensions. Conformal blocks associated with 4-point functions of operators with arbitrary spins can now be determined from these seed blocks by using known differential operators.