Nonrational conformal field theory
J. Teschner
讨论了一类非有理共形场论（CFT）的数学理论的发展问题，其中包含酉CFT。提出了一个模块函子的概念的一个变种，似乎是适合这样的CFT。
We discuss the problem to develop a mathematical theory of a certain class of nonrational conformal field theories (CFT) which contain the unitary CFT. A variant of the concept of a modular functor is proposed that appears to be suitable for such CFT.
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The integrable structure of nonrational conformal field theory
以刘维理论为例，我们展示了如何分离成左，右移动自由度的非理性共形场论可以明确的可积结构。关键的观察是，存在单独的Baxter Q算子的左，右移动的自由度。结合对Q算子的分析性质的研究和Sklyanin的分离变量法，得到了谱的完整刻画。采取连续极限，使我们特别是重新推导的刘维尔反射振幅只使用可积结构。
Chern-Simons Gauge Theory and the AdS(3)/CFT(2) Correspondence
三元流形上纯Chern-Simons理论的体配分函数是对偶边界RCFT的共形块空间中的一个状态，因此在边界模群下非平凡地变换。相反，AdS（3）弦论的体配分函数是对偶CFT在边界上的模不变的配分函数。这是一个难题，因为AdS（3）弦论在远距离上形式上简化为纯陈-西蒙斯理论。我们研究这个难题的背景下，大量的陈-西蒙斯理论。我们发现，在这种情况下解决的难题是由一个手性的“旁观者玻色子”在边界CFT恢复模块不变性的外观。它耦合到共形度规，但不耦合到边界上的规范场。因此，我们找到了一个推广的标准陈-西蒙斯/RCFT对应涉及“非全纯共形块”和非理性边界CFTs。这些推广出现在AdS（3）弦论的长距离极限中，其中旁观者玻色子的角色由理论中的其他自由度扮演。
We review various aspects of two dimensional conformal field theories paying close attention to the algebraic structures that intervene. We provide a compact description regarding the appearance of a chiral algebra as the symmetry algebra related to local conformal symmetry, namely the Virasoro algebra. We then introduce two dimensional conformal field theories with additional symmetries in which extended chiral algebras emerge as a natural generalization of the conformal case.
Permutation Orbifolds in Conformal Field Theories and String Theory
本文综述了近几年来关于二维共形场论中置换轨道折叠的结果，它们在弦论中的应用，以及它们在四维杂化弦模型中的应用。
The problem of constructing gauge-invariant actions for conformal higher-spin fields in curved backgrounds is known to be notoriously difficult. In this paper we present gauge-invariant models for conformal maximal depth fields with spin $s=5/2$ and $s=3$ in four-dimensional Bach-flat backgrounds. We find that certain lower-spin fields must be introduced to ensure gauge invariance when $s>2$, which is analogous to a conjecture made earlier in the literature for conformal higher-spin fields of minimal depth.
Generalized BRST symmetry for arbitrary spin conformal field theory
我们发展了任意自旋s共形场论的有限场相关BRST（FFBRST）变换。我们讨论了这些系统中的FFBRST变换的新功能。为了说明结果，我们考虑自旋1和自旋2的共形场论在两个例子。在形式体系中，我们发现FFBRST变换连接了线性和非线性规范中自旋1和自旋2共形场论的生成泛函。此外，共形场论的FFBRST变换的框架下，也分析了Batalin-Vilkovisky（BV）制定建立的结果。
Comments on Galilean conformal field theories and their geometric realization
我们讨论推广薛定谔代数的非相对论共形代数。这些代数的一个例子是一个共形的，加速扩展的，伽利略代数，它也出现了相对论性共形代数的收缩。在两个维度上，这承认了一个“奇异”的中心延伸，因此助推器不会通勤。我们研究具有这种对称性的非相对论性共形场论的一般性质。我们实现几何对称的度量不变下的异国情调的共形伽利略代数，虽然它的签名既不是洛伦兹也不是欧几里得。我们在此背景下评论全息型计算。
In generic conformal field theories with $W_3$ symmetry, we identify a primary field $σ$ with rational Kac indices, which produces the full $\mathbb{Z}_3$ charged and neutral sectors by the fusion processes $σ\times σ$ and $σ\times σ^*$, respectively. In this sense, this field generalises the $\mathbb{Z}_3$ fundamental spin field of the three-state Potts model. Among the degenerate fields produced by these fusions, we single out a `parafermion' field $ψ$ and an `energy' field $\varepsilon$. In analogy with the Virasoro case, the exact curves for conformal dimensions $(h_σ,h_ψ)$ and $(h_σ,h_\va…
We review conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to BPZ equations and conformal blocks. Examples include Liouville theory, (generalized) minimal models, free bosonic theories, the $H_3^+$ model, and the $SU_2$ and $\widetilde{SL}_2(\mathbb{R})$ WZW models. We also discuss relations betwe…
It is known that there are 48 Virasoro algebras acting on the monster conformal field theory. We call conformal field theories with such a property, which are not necessarily chiral, code conformal field theories. In this paper, we introduce a notion of a framed algebra, which is a finite-dimensional non-associative algebra, and showed that the category of framed algebras and the category of code conformal field theories are equivalent. We have also constructed a new family of integrable conformal field theories using this equivalence. These conformal field theories are expected to be useful f…
Conformal field theories in random domains and stochastic Loewner evolutions
我们回顾了最近发展的传统的代数方法之间的关系共形场论和最近的概率方法的基础上随机Loewner演变。它基于在共形场论中实现随机共形映射。
We study four dimensional quantum gravity formulated as a certain conformal field theory at the ultraviolet fixed point, whose dynamics is described by the combined system of Riegert-Wess-Zumino and Weyl actions. Background free nature comes out as quantum diffeomorphism symmetry by quantizing the conformal factor of the metric field nonperturbatively. In this paper, Minkowski background M^4 is employed in practice. The generator of quantum diffeomorphism that forms conformal algebra is constructed. Using it, we study the composite scalar operator that becomes a good conformal field. We find t…
Instantons on ALE spaces and Super Liouville Conformal Field Theories
我们证明了N=1超刘维尔共形场论的共形块可以用ALE空间O_{P^1}（-2）上的SU（2）Nekrasov配分函数来描述.
Scale and conformal invariance in field theory: a physical counterexample
在本文中，我们举例说明了二维弹性理论如何提供了一个物理例子，说明场论显示尺度而不是共形不变性。
Non-Equilibrium Conformal Field Theories with Impurities
本文提出了一种在共形场论中构造非平衡定态的方法。这些状态维持着两个量子系统之间的能量流，这两个量子系统最初是在不同的温度下制备的，它们的动力学性质由两个可能不同的、通过杂质连接的共形场论表示。这种结构依赖于一个实时配方的共形缺陷动力学的基础上的场散射图片平行化-但又不同-欧几里德公式。我们提出了这个公式的基本特点，并给出了一个代数结构的实时散射映射，我们说明的情况下，SU（2）为基础的共形场论。
Conformal Field Theory on R x S^3 from Quantized Gravity
研究了由量子化的引力场导出的R × S^3上的共形代数。我们研究的模型是一个四维引力的重正化量子理论，它由无痕张量模的Weyl作用和度量场中共形因子的非微扰动力学的诱导Wess-Zumino作用的组合系统描述。证明了辐射^+规范中的剩余规范同态不变性等于共形对称性，并通过朴素共形变换和一定的依赖于场的规范变换的组合给出了保规范固定条件的共形变换，从而形成量子力学上的闭代数.在共形场论的背景下讨论了引力的么正性问题。我们通过求解共形不变性条件构造物理态，并计算它们的标度维数。结果表明，共形对称混合的正度规和负度规模式，因此，负度规模式并不独立出现在所有的规范不变状态。
We reexamine two-dimensional Lorentzian conformal field theory using the formalism previously developed in a study of sine-square deformation of Euclidean conformal field theory. We construct three types of Virasoro algebra. One of them reproduces the result by Lüscher and Mack, while another type exhibits the divergence in the central charge term. The other leads the continuous spectrum and contains no closed time-like curve in the system.
Quantum Langlands duality and conformal field theory
Drinfeld提出了几何Langlands对应及其量子形变的理论。我们改进了这些结构，并提出了它们与代数共形域论的关系。
Conformal nets and local field theory
我们描述了一个坐标自由的概念共形网作为共形域论的数学模型。我们定义共形网之间的缺陷，并介绍组成的缺陷，从而提供了一个概念的共形域论之间的态射。总之，我们的特征的代数结构的收集共形网作为一个对称的monoidal三范畴。这三个范畴中的可对偶对象对应于保形网值三维局部拓扑量子场论。证明了可对偶共形网是具有有限μ-指标的不可约网的有限和。这种分类提供了各种各样的三维局部域论，包括与特殊酉群的环群的中心扩展相关联的局部域论。
Differential equations and conformal field theories
本文讨论了作者最近关于共形域论中手征亏格为零和亏格为一的相关函数的微分方程组的存在性的结果。
This article develops new techniques for understanding the relationship between the three different mathematical formulations of two-dimensional chiral conformal field theory: conformal nets (axiomatizing local observables), vertex operator algebras (axiomatizing fields), and Segal CFTs. It builds upon previous work which introduced a geometric interpolation procedure for constructing conformal nets from VOAs via Segal CFT, simultaneously relating all three frameworks. In this article, we extend this construction to study the relationship between the representation theory of conformal nets and…
This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or quantum field theory is assumed.
Galois groups in rational conformal field theory II. The discriminant
在任何共形场论中，我们将熔合环的多项式关系的判别式表示为模矩阵行的-2次方的乘积。判别式被证明是一个整数，总是，这是一个产品的素数划分的水平。给出了所有WZW共形场论的判别式的详细公式。
Diffeomorphism group and conformal fields
共形场是一类新的比张量场更一般的$Vect(N)$模。构造了相应的群作用.共形域因此被不变地定义。
Vect(N) conformal fields and their exterior derivatives
共形场是最近发现的一类向量场在$N$维上的代数表示。构造了共形域上的不变一阶微分算子（外导数）。
Conformal Field Theory in Conformal Space
我们提出了一个新的框架，拉格朗日描述的共形场论在各种尺寸的基础上的局部版本的d+2维共形空间。其结果包括d=（1，3）的共形引力的真规范理论和与之耦合的任何标准物质，其重要特征是共形引力约束的自动推导，这是分析物质系统所必需的。
Logarithmic Conformal Field Theory Through Nilpotent Conformal Dimensions
通过引入幂零共形权研究了对数共形场论。利用该装置，我们导出了LCFT的变换律、奇异向量和关联函数的结构等性质。我们讨论了一个额外的能量动量张量的出现，这是能量动量张量的对数伙伴。
We give a general definition of the conformal anomaly for theories that are not classically Weyl invariant and show that this definition yields a quantity that is both finite and local. As an example we study the conformal anomaly for a non-minimally coupled massless scalar and show that our definition coincides with results obtained using the heat kernel method.
In the framework of metric-like approach, totally symmetric arbitrary spin bosonic conformal fields propagating in flat space-time are studied. Depending on the values of conformal dimension, spin, and dimension of space-time, we classify all conformal field as long, partial-short, short, and special conformal fields. An ordinary-derivative (second-derivative) Lagrangian formulation for such conformal fields is obtained. The ordinary-derivative Lagrangian formulation is realized by using double-traceless gauge fields, Stueckelberg fields, and auxiliary fields. Gauge-fixed Lagrangian invariant …
An Introduction to Conformal Field Theory
对二维共形场论作了全面的介绍。
Talk presented at the conference on representation theory and harmonic analysis at Saclay, the talk presented the development in conformal field theory since 1968
Arbitrary spin conformal fields in (A)dS
研究了（A）dS空间中偶数维大于等于4的全对称任意自旋共形场.得到了这类场的常导数和规范不变的拉格朗日公式。规范对称性是用辅助场和Stueckelberg场实现的。我们证明了共形场的拉格朗日量可以分解为无质量场、部分无质量场和有质量场的规范不变拉格朗日量之和。我们得到了部分无质量场和有质量场的质量谱，证实了早期文献中关于质量谱的猜想。与平坦空间中的共形场相反，（A）dS空间中的共形场的动力学项相对于进入拉格朗日量的场是对角的。给出了将平坦空间中的共形场映射到（A）dS空间中的共形场的共形变换的显式形式。提出了共形场的简单规范固定拉格朗日量的类Lorentz和类de-Donder规范条件。使用这种规范固定的拉格朗日量，这是不变的整体BRST变换下，我们解释了如何在我们的方法的框架下获得共形场的配分函数。
Correlation functions for Galilean conformal invariant field theory
本文考虑非相对论共形群，计算了伽利略共形不变场的两点函数，证明了$d=2$中伽利略共形不变场的关联函数与$d\neq2$中伽利略共形不变场的相关函数不同。我们在$d\neq2$中的结果与以前在\cite{3}中的结果不同。
On conformal field theories with low number of primary fields
利用Verlinde公式和模矩阵的对称性，给出了一个求所有具有少量主场的共形场论的算法。我们采用的算法上多达八个主要领域。四个新的共形场论被发现，这似乎不是来自当前代数。这支持了有理共形场论比以前所怀疑的要丰富得多的事实。
Logarithmic conformal field theories with continuous weights
研究共形权为实数连续子集的对数共形场论。本文研究了对数场与普通共形场构成的复变子之间的一般关系。作为一个例子，库仑气体模型的分解子明确研究。
Two-dimensional fractional supersymmetric conformal- and logarithmic conformal- field theories and the two point functions
研究了一般的二维分数阶超对称共形场论。研究了该理论的对称性结构。利用由$(L_{-1},L_{0},G_{-1/(F+1)})$和$(\bar{L}_{-1},\bar{L}_{0},\bar{G}_{-1/(F+1)})$生成的闭子代数的生成元，计算了超多重态分量场的两个点函数。然后研究了对数超共形场论，得到了手征和满两点函数。
Characters for Coset Conformal Field Theories
通过求解Kac-Moody分支方程，得到了陪集共形场论特征的显式表达式，并应用于具体实例，确定了准素场共形权的整数移位.我们还给出了一个不能用识别流方法描述的陪集共形场论的例子。
Four-dimensional conformal field theory using quaternions
利用四元数全纯函数作为四元数共形变换的生成元，建立了四维四元数参数化共形场论。本文以二维复参数化共形场论（CCFT）为模型，研究了应力张量、守恒电荷、对称生成元、量子化条件和几种算子乘积展开（OPE）。还讨论了未来的应用。
Conformal Field Theories: From Old to New
在最近的工作的简短回顾中，我们讨论了从旧的共形场论构造新的共形场论的作用量的一般问题。当旧的共形场论在其手征代数中引入新的共形应力张量时，这种结构就成立了，新的共形场论一般是一个新的自旋2规范理论。作为一个例子，我们讨论了新的双自旋规范西格玛模型，它以这种方式从一般共形非线性西格玛模型中产生。
Light-cone formulation of conformal field theory adapted to AdS/CFT correspondence
建立了任意维时空中共形场论的光锥公式。研究了具有任意共形维数和任意自旋的共形基本场和阴影场。找到了共形基本场和阴影场空间上的共形代数生成元用AdS空间中场动力学的光锥规范形式中的自旋算符表示的方法，作为应用光锥形式的一个例子，讨论了有质量任意自旋AdS场和相应的边界CFT场在两点函数水平上的AdS/CFT对应.
A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
本文提出了二维A_{N-1}共形Toda理论中的关联函数与某些共形N = 2 SU（N）四维规范规范理论中的Nekrasov瞬子配分函数之间的关系。我们的建议推广了最近发现的刘维尔理论和SU（2）箭之间的关系。新的特点出现在分析中没有对应的刘维的情况下。
Operator algebra of the SL(2) conformal field theories
计算了SL（2）退化共形场论的算子代数的结构常数。
Curvature formula for the space of 2-d conformal field theories
本文导出了二维共形场论空间上自然黎曼度量的曲率张量公式和边界共形场论空间的曲率张量公式。
Conformal Field Theory with Two Kinds of Bosonic Fields and Two Linear Dilatons
本文考虑一个含有两种玻色自由度的二维共形场论。两个线性介子场使我们能够研究更一般的情况。各种性质的模型，如OPEs，中心荷，共形性质的领域和相关的代数进行了研究。
Conformal totally symmetric arbitrary spin fermionic fields propagating in the flat space-time of even dimension greater than or equal to four are investigated. First-derivative metric-like formulation involving Fang-Fronsdal kinetic operator for such fields is developed. Gauge invariant Lagrangian and the corresponding gauge transformations are obtained. Gauge symmetries of the Lagrangian are realized by using auxiliary fields and the Stueckelberg fields. Realization of conformal algebra symmetries on the space of conformal gauge fermionic fields is obtained. The on-shell degrees of freedom o…
We study integrable deformations of sine-Liouville conformal field theory. Every integrable perturbation of this model is related to the series of quantum integrals of motion (hierarchy). We construct the factorized scattering matrices for different integrable perturbed conformal field theories. The perturbation theory, Bethe ansatz technique, renormalization group and methods of perturbed conformal field theory are applied to show that all integrable deformations of sine-Liouville model possess non-trivial duality properties.
Irrational Conformal Field Theory
这是一个无理共形场论的回顾，其中包括有理共形场论作为一个小子空间。审查的中心议题包括Virasoro主方程，其解决方案和动力学的非理性共形场论。动力学的讨论包括球上的广义Knizhnik-Zamolodchikov方程、环面上相应的类热系统和无理共形场论的一般世界片作用量。
Conformal Invariance in Classical Field Theory
本文利用一阶拉格朗日形式的几何推广，分析了任意主场的共形场论。我们要求，全球共形变换是诺特对称性，我们证明了行动功能可以采取严格不变的关于这些变换。换句话说，对于共形不变的拉格朗日理论，不存在“Chern-Simons”型拉格朗日量。
Conformal primary fields are of central importance in a conformal field theory with d > 2 spacetime dimensions. They can be defined in two ways. A first definition involves commutators between the field and the generators of the conformal group; a second definition characterizes a primary field according to its behavior under a finite conformal transformation. In the existing literature, the proof of the equivalence of the definitions is either omitted or carried out with little details. In this paper we present a clear and concise review of the two definitions and provide a simple and detaile…
A general transformation formula for conformal fields
建立了手征共形场在任意全纯坐标变换下的显式变换公式。作为应用，计算了一般准素场在第四级的变换律。
D-brane conformal field theory
我们概述了共形场论中边界条件的结构。边界条件由圆盘上体场的反射系数的一致集合以及保持共形权重的融合规则的自同构ω的选择来指定。非平凡自同构ω对应于任意共形场论的D-膜组态。
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]
Anti--de Sitter/ boundary conformal field theory correspondence in the non-relativistic limit
边界共形场论是研究半无限时空中的共形场论的理论。在非相对论极限（$x\rightarrowεx, t\rightarrow t, ε\rightarrow 0$）下，边界共形代数转化为边界伽利略共形代数（BGCA）。本文对AdS/BCFT在非相对论极限下的某些性质进行了研究。我们用BGCA生成元来约束伽利略共形不变场的关联函数。对于表面具有边界条件的情况$x=0$（$z=\bar{z}$），我们的结果与BCFT两点函数的非相对论极限一致。我们还介绍了边界伽利略共形场论的全息对偶。
2D conformal field theory and vertex operator algebras
这篇文章将出现在数学物理百科全书（爱思唯尔，2006年）。
Conformal Operators for Partially Massless States
AdS/CFT的对应关系进行了探讨的“部分无质量”领域在AdS空间（它有更少的螺旋状态比一个大规模的领域，但比传统的无质量领域）。在边界共形场论中，这样的场对应于服从某个共形不变微分方程的场，该方程已由伊斯特伍德等人描述。这样的场的第一个后代是负范数的共形场。因此，部分无质量场在德西特空间中可能比在反德西特空间中更有物理意义。
Conformal field theory and graphs
我们回顾了一种特殊的图形推广ADE Dynkin图编码的频谱共形场论所描述的共形嵌入的$\widehat{sl}(n)_k$建设的方法。
Holomorphic Chern-Simons-Witten Theory: from 2D to 4D Conformal Field Theories
众所周知，有理二维共形场论与定义在三维实流形上的Chern-Simons场论是联系在一起的。我们考虑全纯的类似物的Chern-Simons理论定义在三维复流形（六个实维度）和描述4D共形场论与它们连接。所有这些模型都是可积的。我们描述类似的Virasoro和仿射李代数，其中的本地行动的全纯类似的陈-西蒙斯理论领域成为非本地后，推到可积的四维共形场论领域的行动。简要讨论了可积四维共形场论的量子化及其与弦论的关系。
Light-cone gauge formulation of fields in AdS space and conformal field theory in flat space adapted for the study of AdS/CFT correspondence is developed. Arbitrary spin mixed-symmetry fields in AdS space and arbitrary spin mixed-symmetry currents, shadows, and conformal fields in flat space are considered on an equal footing. For the massless and massive fields in AdS and the conformal fields in flat space, simple light-cone gauge actions leading to decoupled equations of motion are found. For the currents and shadows, simple expressions for all 2-point functions are also found. We demonstrat…
Quantum Group Symmetries in Conformal Field Theory
量子群在某些二维场论中扮演着隐藏对称性的角色。我们讨论了它们是如何在共形场论的Wess-Zumino-Witten模型中扮演这个角色的。
It was argued recently that conformal invariance in flat spacetime implies Weyl invariance in a general curved background for unitary theories and possible anomalies in the Weyl variation of scalar operators are identified. We argue that generically unitarity alone is not sufficient for a conformal field theory to be Weyl invariant. Furthermore, we show explicitly that when a unitary conformal field theory couples to gravity in a Weyl invariant way, each primary scalar operator that is either relevant or marginal in the unitary conformal field theory corresponds to a Weyl-covariant operator in…
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