How to remove the boundary in CFT - an operator algebraic procedure
Roberto Longo, Karl-Henning Rehren
探讨了有无类时边界的二维共形量子场论之间的关系。
The relation between two-dimensional conformal quantum field theories with and without a timelike boundary is explored.
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Algebraic Structures in Euclidean and Minkowskian Two-Dimensional Conformal Field Theory
我们回顾如何模块范畴，交换和非交换Frobenius代数出现在合理的共形场论。对于欧几里德CFT，我们使用的方法的基础上缝合的表面，并在闵可夫斯基的情况下，我们描述CFT的净算子代数。
Algebraic properties of CFT coset construction and Schramm-Loewner evolution
Schramm-Loewner演化是晶格模型中界面在临界点处的标度极限。这些模型的临界行为可以用共形场论的极小模型来描述。某些CFT关联函数是关于SLE的鞅。本文将李群$G$上带附加布朗运动的Schramm-Loewner演化推广到因子空间$G/A$上.然后，我们研究了临界行为的SLE描述与共形场论陪集模型之间的联系。为了保持一致性，这种构造应该为某些组的选择给出极小模型。
One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the $p$-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the b…
Conformal Field Theory and Algebraic Structure of Gauge Theory
我们考虑与杨-米尔斯理论和二维共形场论有关的各种同伦代数。我们的主要研究对象是Yang-Mills $L_{\infty}$和$C_{\infty}$代数及其与CFT中Lian-Zuckerman型代数结构的关系。我们还考虑了几个与规范理论有关的代数的例子，涉及一阶公式和物质场的规范理论。
Recurrence Relations for Finite-Temperature Correlators via AdS$_{2}$/CFT$_{1}$
本文旨在提出一种新的代数方法来处理共形场论中的动量空间关联函数。作为例子，我们提出了一种新的Lie代数方法来计算具有恒定背景电场的CFT $_{1}$与AdS $_{2}$黑洞对偶的带电标量算符的频率空间两点函数.我们的方法是基于实时处方的AdS/CFT对应，Euclideanization的AdS $_{2}$黑洞和投影酉表示的李代数$\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R})$。我们得到了新的递归关系的欧几里得CFT $_{1}$两点函数，这是完全可解的，并完全确定的频率和电荷依赖的两点函数。通过Wick旋转回到Lorentz签名，得到了延迟和超前的CFT $_{1}$两点函数，与已知结果一致.
Algebraic Conformal Field Theories II
本文在代数量子场论的框架下研究了陪集共形域论中的一些数学问题。我们考虑了A型对角陪集上的不动点分解问题，并给出了如何将可约表示分解为不可约表示。我们证明了相应的陪集CFT在Turaev意义下产生了一个酉张量模范畴，因此可以用来构造3-流形不变量。我们还表明，Kac-Wakimoto假设（KWH）和Kac-Wakimoto猜想（KWC）是等价的，在一般条件下，可以检查的例子，这一结果似乎是很难证明纯粹的代表性考虑。还提供了示例。
Generalized free fields and the AdS-CFT correspondence
受AdS-CFT对应中的结构问题的启发，重新考虑了广义自由场理论。构造了广义自由场的应力-能量张量作为Wightman场的极限。虽然这个极限是奇异的，但它满足了庞加莱生成元的守恒局部密度的要求。一个明确的“全息”公式有关的克莱因-戈登场的AdS的广义自由场闵可夫斯基时空提供，并与“代数”概念的全息。奇异应力-能量张量和正则AdS应力-能量张量之间的一个简单的关系。
In this article, we provide a detailed construction and analysis of the mathematical conformal field theory of the free fermion, defined in the sense of Graeme Segal. We verify directly that the operators assigned to disks with two disks removed correspond to vertex operators, and use this to deduce analytic properties of the vertex operators. One of the main tools used in the construction is the Cauchy transform for Riemann surfaces, for which we establish several properties analogous to those of the classical Cauchy transform in the complex plane.
An algebraic quantum field theory (AQFT) may be expressed as a functor from a category of spacetimes to a category of algebras of observables. However, a generic category $\mathsf{C}$ whose objects admit interpretation as spacetimes is not necessarily viable as the domain of an AQFT functor; often, additional constraints on the morphisms of $\mathsf{C}$ must be imposed. We introduce disjointness relations, a generalisation of the orthogonality relations of Benini, Schenkel and Woike (arXiv:1709.08657). In any category $\mathsf{C}$ equipped with a disjointness relation, we identify a subcategor…
A Simple Derivation of Finite-Temperature CFT Correlators from the BTZ Black Hole
本文给出了有限温度下与BTZ黑洞对偶的二维共形场论动量空间两点函数的一个简单的李代数方法。利用AdS/CFT对应的实时描述和李代数$\mathfrak{so}(2,2) \cong \mathfrak{sl}(2,\mathbb{R})_{L} \oplus \mathfrak{sl}(2,\mathbb{R})_{R}$的梯形方程，证明了动量空间中有限温度两点函数关于左、右动量满足线性递推关系.这些递推关系式是精确可解的，并完全决定了有限温度共形场论推迟和超前两点函数的动量依赖性。
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk Ad…
Algebraic approach to spacetime bulk reconstruction
本文受AdS/CFT全息量子纠错理论及扭结变换猜想启发，用体与边界余圈导数交织刻画互补恢复，建立Klein-Gordon场的子区域对偶性，揭示扭结变换的代数本质并得到冯·诺依曼代数类型的副产品结论。
We review the algebraic and analytic aspects of the conformal field theory (CFT) and its relation to the stochastic Loewner evolution (SLE) in an example of the Ising model. We obtain the scaling limit of the correlation functions of Ising free fermions on an arbitrary simply connected two-dimensional domain $D$. Then, we study the analytic and algebraic aspects of the fermionic CFT on $D$, using the Fock space formalism of fields, and the Clifford vertex operator algebra (VOA). These constructions lead to the conformal field theory of the Fock space fields and the fermionic Fock space of stat…
Quantum Operations in Algebraic QFT
本文研究手征共形量子场论的冯·诺依曼代数表述，综述共形子网研究进展，基于已有工作构建共形网的量子伽罗瓦理论，提出作为广义规范对称性的量子操作可描述同中心荷的所有共形子网，并涉及高维AQFT情况。
Entanglement Entropy in CFT and Modular Nuclearity
本文研究了共形场论中纠缠熵的有限性，证明了真空态的规范纠缠熵在广泛共形网类中有限，并探讨了模$p$-核性条件对局部量子场论中互信息有限性的影响。
Observables in Schrödinger CFTs: How Aliens Built the Pyramids
本文通过双重态-算符对应，研究了零质量算符在Schrödinger CFT中的代数结构，揭示了由“外星算符”构成的交错“金字塔表示”，并推广了异常对称性守恒律。
CFTs on Riemann Surfaces of genus $g\geq 1$
$N$ -共形域论中全纯场的点函数可以用代数几何的方法计算。建立了亏格为$g\geq 1$的超椭圆Riemann曲面上Virasoro场的2点函数的显式公式。Virasoro $N$ -点函数更高的$N$归纳，我们表明，他们有一个很好的图形表示。讨论了三点函数在$(2,5)$极小模型中的应用.
We propose a procedure to derive quantum spectral curves of AdS/CFT type by requiring that a specially designed analytic continuation around the branch point results in an automorphism of the underlying algebraic structure. In this way we derive four new curves. Two are based on SU(2|2) symmetry, and we show that one of them, under the assumption of square root branch points, describes Hubbard model. Two more are based on SU(2|2) x SU(2|2). In the special subcase of zero central charge, they both reduce to the unique nontrivial curve which furthermore has analytic properties compatible with PS…
Structural and Dynamical Aspects of the AdS/CFT Correspondence: a Rigorous Approach
我们详细研究了由Maldacena和Witten证明的AdS/CFT对应（的一个版本），在由渐近反德西特（AAdS）时空给出的引力背景中的量子场论与在后者的共形无穷大中的共形协变量子场论之间（在彭罗斯的意义上），方面，这样：（a）是独立于（对）具体模型在量子场论，和（b）易于重铸在一个数学上严格的模具。我们采用的出发点定理证明的局部量子物理学（也称为代数量子场论）在反德西特（AdS）时空，称为代数全息或厄勒伦对偶。目前工作的主要内容是将Wehren的结果推广到一个相当一般的d维AAdS时空（d>3），仔细研究这种推广的性质与AdS时空相比是如何被削弱和/或修改的，并探索非平凡的引力效应如何在共形无穷大的量子理论中表现出来。语言雇用的量子理论有关我们的推广厄勒伦对偶遵循函式制定的局部量子物理由于Brunetti，Fredenaldi和Verch，扩大后，索默，以纳入边界条件。
We provide a detailed construction of the quantum theory of the massless scalar field on 2-dimensional, globally-hyperbolic (in particular, Lorentzian) manifolds using the framework of perturbative algebraic quantum field theory. From this we subalgebras of observables isomorphic to the Heisenberg and Virasoro algebras on the Einstein cylinder. We also show how the conformal version of general covariance, as first introduced by Pinamonti as an extension of the construction due to Brunetti, Fredenhagen and Verch, may be applied to the concept of natural Lagrangians in order to obtain a simple c…
We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed $gl(1|1)$ spin-chain and its continuum limit - the $c=-2$ symplectic fermions theory - and rely on two technical companion papers, "Continuum limit and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over i…
Quantization and Algebraic Index
本文综述BV量子化与代数指标理论的关联研究，通过拓扑量子力学BV量子化证明经典代数指标定理，推广至二维手征CFT提出椭圆手征指标类比，并验证其与椭圆曲线Gromov-Witten不变量生成函数的镜像等价性。
We explore the role of sandwiched Renyi relative entropy in AdS/CFT and in finite-dimensional models of holographic quantum error correction. In particular, in the context of operator algebra quantum error correction, we discuss a suitable generalization of sandwiched Renyi relative entropy over finite-dimensional von Neumann algebras. It is then shown that the equality of bulk and boundary sandwiched relative Renyi entropies is equivalent to algebraic encoding of bulk and boundary states, the Ryu-Takayanagi formula, the equality of bulk and boundary relative entropy, and subregion duality. Th…
Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
本文提出Z_N扩展的chiral和bulk共形场论中的融合规则及其对应的拓扑有序，通过构建非平凡的子代数结构揭示拓扑双影的对应关系，并统一了对称性和拉格朗日子代数。
Ryu-Takayanagi Formula for Multi-Boundary Black Holes from 2D Large-$c$ CFT Ensemble
该研究通过2D大c CFT集合推导出多边界黑洞的Ryu-Takayanagi公式，揭示了纠缠熵与CFT数据的统计结构之间的联系。
BRST, Generalized Maurer-Cartan Equations and CFT
本文研究了二维扰动共形场论中的BRST荷。主要目标是写的运营商方程表示守恒定律的BRST收费在微扰理论的纯代数运算相应的运营商代数，这是通过OPE定义。构造了相应的方程，并研究了它们的对称性，直到形式耦合常数的二阶。看来，得到的方程可以解释为广义Maurer-Cartan的。我们详细研究了两个具体的例子：玻色非线性西格玛模型和微扰一阶理论。特别是，我们表明，爱因斯坦方程，这是共形不变性的条件，这两个扰动理论，扩大到二阶，可以改写在这样的广义Maurer-Cartan形式。
Local fields in boundary conformal QFT
从代数的角度分析了Minkowski时空的半空间x>0上的共形场论（“边界CFT”），特别阐明了局部代数的代数结构和局部场的双定域电荷结构.场的内容和允许的边界条件的特点是在一个非局部手征场代数。
We give a derivation of quantum spectral curve (QSC) - a finite set of Riemann-Hilbert equations for exact spectrum of planar N=4 SYM theory proposed in our recent paper Phys.Rev.Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system -- a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compat…
Virasoro constraint is the operator algebra version of one-loop equation for a Hermitian one-matrix model, and it plays an important role in solving the model. We construct the realization of the Virasoro constraint from the Conformal Field Theory (CFT) method. From multi-loop equations of the one-matrix model, we get a more general constraint. It can be expressed in terms of the operator algebras, which is the Virasoro subalgebra with extra parameters. In this sense, we named as generalized Virasoro constraint. We enlarge this algebra with central extension, this is a new kind of algebra, and…
A UV-Finite Refinement of the Ryu-Takayanagi Relation in AdS$_3$/CFT$_2$
利用相对熵替代紫外发散的冯·诺依曼纠缠熵，在AdS$_3$/CFT$_2$中建立了Ryu-Takayanagi关系，通过将手征相对熵精确结果推广到菱形区域，得到边界相对熵的Schwarzian泛函形式，并证明在线性阶相对熵等于RT测地线长度变化除以$4G_N$。
As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular vectors of a certain (e.g. Virasoro or super-Virasoro) underlying algebra. In this paper we reformulate this problem in the language of conformal field theory. Such a reformulation has several advantages: it leads to the identification of quantum curves more efficiently, it proves in full generality that they indeed have the structure of singular vectors, it e…
Topological and conformal field theory as Frobenius algebras
二维共形域论（CFT）可以通过其相关函数来定义。它们必须满足某些一致性条件，这些条件是由沿着圆或间隔切割世界片而产生的。一个（有理）CFT的构造可以分为两个步骤，其中一个是复解析的，另一个是纯代数的。我们实现的代数部分的建设与三维拓扑域论的帮助下，并表明，任何对称的特殊Frobenius代数在适当的编织幺半范畴产生的解决方案。二维拓扑域论提供了一类特殊的例子，其相关的幺半范畴是向量空间范畴。
Multi-Point Affine Bialgebras for AdS/CFT Integrability
本文严格分析AdS/CFT可积性中仿射双代数的奇点结构，提出三点和四点仿射代数，证明其构成一致拟三角李双代数。
SLE martingales in coset conformal field theory
Scharmm-Loewner演化（SLE）和共形场论（CFT）是研究二维模型临界行为的常用工具，但它们使用的对象不同。虽然SLE与格点模型有着天然的联系，适合于严格的证明，但它缺乏共形场论的计算和预测能力。为了提供一种同时使用SLE和CFT的方法，我们考虑CFT关联函数，它是关于SLE的鞅。建立了陪集空间上Schramm-Loewner演化的参数与陪集共形场论的代数数据之间的联系。然后，我们检查我们的方法与行为的parafermionic和极小模型的一致性。陪集模型与非临界大规模场论，我们讨论SLE的影响。
The Nichols algebra of screenings
在二维共形域论模型中，两个相关的构造与屏蔽算子相关联。一个是根据给定CFT模型中的屏蔽物种所张成的编织向量空间X和顶点算子空间Y构造的局部系统，另一个是Nichols代数B（X）及其Yetter-Drinfeld模的范畴，我们提出作为代数对应物，在Kazhdan-Lusztig对偶的“编织”版本中，在对数CFT模型中实现的顶点算子代数的表示范畴。
By studying the space of geodesics in $ADS_3/CFT_2$ and quantizing the geodesic motion, we relate scattering data to boundary entanglement of the CFT vacuum. The basic idea is to use a family of plane waves parametrized by coordinates of the space of geodesics i.e. kinematic space. This idea enables a simple calculation of the Berry curvature living on kinematic space. As a result we recover the Crofton form with a coefficient depending on the scattering energy. In arriving at these results the space of horocycles is used. We show that this new space used in concert with kinematic space incorp…
Non-unitary Wightman CFTs and non-unitary vertex algebras
本文建立了带拟初级生成族的 Möbius 顶点代数与单位圆上不必幺正的 Möbius 协变 Wightman CFT 之间最一般的范畴等价，并证明非幺正 Reeh–Schlieder 定理。
We use the formalism of strange correlators to construct a critical classical lattice model in two dimensions with the \emph{Haagerup fusion category} $\mathcal{H}_3$ as input data. We present compelling numerical evidence in the form of finite entanglement scaling to support a Haagerup conformal field theory (CFT) with central charge $c=2$. Generalized twisted CFT spectra are numerically obtained through exact diagonalization of the transfer matrix and the conformal towers are separated in the spectra through their identification with the topological sectors. It is further argued that our mod…
In this paper we study the covering vertex model of the one-dimensional Hubbard Hamiltonian constructed by Shastry in the realm of algebraic geometry. We show that the Lax operator sits in a genus one curve which is not isomorphic but only isogenous to the curve suitable for the AdS/CFT context. We provide an uniformization of the Lax operator in terms of ratios of theta functions allowing us to establish relativistic like properties such as crossing and unitarity. We show that the respective $\mathrm{R}$-matrix weights lie on an Abelian surface being birational to the product of two elliptic …
Generalizing quantum dimensions: Symmetry-based classification of local pseudo-Hermitian systems and the corresponding domain walls
本文基于对称性拓扑场论研究CFT的分类，提出伪自伴系统的量子维度推广，用于描述无质量及有质量的流行为，以及领域壁问题与拓扑量子场论之间的关系。
Classifying fusion rules of anyons or SymTFTs: A general algebraic formula for domain wall problems and quantum phase transitions
本文提出一种公式，用于描述在拓扑有序相或拓扑量子场论中任意子的变换规律，通过有隙或对称保持的域壁。基于复数线性交换融合环的环同态，结合Verlinde公式，提供任意子分类及CFT与TQFT之间的无质量重整化群流和测量诱导量子相变的描述。
Tensor complex renormalization with generalized symmetry and topological bootstrap
本研究提出了一种基于广义对称性的 TC 重整化算法，通过拓扑数据重构完整 FP 张量，揭示了 CFT 中拓扑与动力学的深层联系。
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