Chaiho Rim, Hong Zhang
Virasoro irregular conformal block with arbitrary rank is obtained for the classical limit or equivalently Nekrasov-Shatashvili limit using the beta-deformed irregular matrix model (Penner-type matrix model for the irregular conformal block). The same result is derived using the generalized Mathieu equation which is equivalent to the loop equation of the irregular matrix model.
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We present a new systematic way to evaluate the classical limit of the Virasoro irregular conformal block for arbitrary rank n based on the irregular partition function. In addition, we prove that the classical irregular conformal block has the exponential form as suggested by A. Zamolodchikov and Al. Zamolodchikov for the regular case. We provide an explicit calculation for the rank 2 case in detail.
Virasoro irregular conformal block is presented as the expectation value of Jack-polynomials of the beta-deformed Penner-type matrix model and is compared with the inner product of Gaiotto states with arbitrary rank. It is confirmed that there are non-trivial modifications of the Gaiotto states due to the normalization of the states. The relation between the two is explicitly checked for rank 2 irregular conformal block.
Notes on crossing transformations of Virasoro conformal blocks
本文从第一性原理出发，利用 Hilbert 空间结构自举推导 Virasoro 共形块的交叉核，并给出教学式背景说明、应用解释及新结果证明。
WKB-asymptotics for multipoint Virasoro conformal blocks and applications
本文通过WKB方法研究多点Virasoro共形块的渐近行为，并探讨其在扩展Zamolodchikov递归关系和最小弦理论幅值计算中的应用。
Virasoro OPE and Conformal Blocks from the Inverse Shapovalov Form
本文通过最近确定的身份解析方法推导出维拉索罗OPE和球面上四点共形块的表达式，其结果不依赖于维拉索罗奇点向量的具体形式，仅依赖其已知的共形权重，并探讨了大中央电荷极限的影响。
Irregular conformal block is motivated by the Argyres-Douglas type of N=2 super conformal gauge theory. We investigate the classical/NS limit of the irregular conformal block using spectral curve on a Riemann surface with irregular punctures, which is equivalent to the loop equation of irregular matrix model. The spectral curve is reduced to the second order (Virasoro symmetry, $SU(2)$ for the gauge theory) and third order ($W_3$ symmetry, $SU(3)$) differential equations of a polynomial with finite degree. The Virasoro and W symmetry generate flow equations in the spectral curve and determine …
The theory of elasticity (a.k.a. Riva-Cardy model) has been regarded as an example of scale invariant but not conformal field theories. We argue that in $d=2$ dimensions, the theory has hidden global conformal symmetry of $SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$ without its Virasoro extension. More precisely, we can embed all the correlation functions of the displacement vector into a global conformal field theory with four-derivative action in terms of two scalar potential variables, which necessarily violates the reflection positivity. The energy-momentum tensor for the potential variables…
Exponentiation of higher-point and higher-genus Virasoro conformal blocks in the semiclassical limit
本文证明了在c→∞且h/c有限的半经典极限下，任意通道的高点多点函数和高亏格背景的Virasoro共形块均指数化，通过将振荡子方法推广到三内线顶点情形，并提供了计算2d CFT中全局共形块的新构造方法。
Spinor representations of the Virasoro and super-Virasoro algebras for conformal spin to be equal 1/k
本文研究了Virasoro代数和超维拉索罗代数的旋量表示为1/k的可能性，其中k为整数.
A note on half-integer irregular representations of Virasoro algebra
本文提出两种等价方法构造半整数阶Virasoro代数不规则表示，揭示其结构并提供计算Argyres-Douglas理论相关态的工具。
The Virasoro Completeness Relation and Inverse Shapovalov Form
本文利用奇异向量算子及其共形维数构造了Virasoro Verma模Shapovalov形式的显式逆，并由此得到单位分解，为通过缝合程序计算Virasoro共形块提供了工具。
Promoting finite to infinite symmetries: the 3+1-dimensional analogue of the Virasoro algebra and higher-spin fields
在这封信中明确提供了有限伪酉对称的无限放大。u（2，2）=so（4，2）+u（1）的特殊情况构成了3+1维共形对称的无限维推广，除了具有所有共形自旋的物质场之外。这些代数为更高维度的可积场模型提供了一个新的舞台;例如，高SO（4，2）-自旋场的反德西特和共形规范理论。一个非交换的空间几何解释的建议也概述。
Higher conformal variations and the Virasoro vertex operator algebra
我们开发了一个微积分的变化，在一定的空间上的共形映射的泛函。这样的空间Ω是由所有在包含黎曼球面的固定紧环集的域上共形的映射组成的，并且是“保义的”。变分法的基础是使用向量场描述这种映射的无穷小变化。我们表明，衍生物的所有订单就这样的共形映射，共轭后，由一个适当的功能，产生的Virasoro顶点算子代数的结构。我们的建设分三步进行。我们首先在Ω上放置一个自然拓扑，定义光滑路径和对所有阶的微分运算（“共形导数”）。然后，我们研究某些二阶变分方程及其解决方案。我们最后表明，这样的解决方案产生的陈述的维拉索罗代数的共形衍生物，从以下的Virasoro顶点算子代数的建设。证明了多重共变共形导数的级数展开式等价于多重顶点算子的乘积。本文推广了作者最近的一项工作，把一阶共形导数与共形场论的应力-能量张量联系起来。
While the argument by Zamolodchikov and Polchinski suggests global conformal invariance implies Virasoro invariance in two-dimensional unitary conformal field theories with discrete dilatation spectrum, it is not the case in more general situations without these assumptions. We indeed show that almost all the globally conformal invariant differential equations in two dimensions are neither Virasoro invariant nor Weyl invariant. The only exceptions are the higher spin conservation laws, conformal Killing tensor equations and the Laplace equation of a conformal scalar.
Hypotrochoids in conformal restriction systems and Virasoro descendants
共形约束系统是一个交换的结合的酉代数，它在黎曼球面的连通开集上具有单叶共形映射的广群的表示，并且在子代数上具有一族线性泛函，满足一组性质，包括共形不变性和一类约束。这体现了共形循环系综CLE中期望值的一些期望性质。在共形约束系统的背景下，我们研究了与次摆线简单曲线（包括椭圆）相关的某些代数元素。这些有CLE的解释是“重正化随机变量”，只有当至少有一个环的次摆线形状是非零的。每条曲线都有一个中心w，一个尺度ε和一个旋转角θ，我们将重正化随机变量分析为u=εe^{iθ}和w的函数。我们发现它在u和u* 的正幂上有一个展开式，并且纯u（u*）的幂的系数在w（w*）中是全纯的.我们确定这些系数（“次摆线领域”）与某些Virasoro后裔的身份字段在共形场论，从而表明，他们形成的顶点算子代数结构的一部分。这在很大程度上推广了作者（在CLE中）以及作者与他的合作者V. Riva和J. Cardy（在SLE 8/3和其他限制措施中）的工作，其中椭圆的情况，在u^2阶，导致了CFT的应力-能量张量。推导中使用的Virasoro顶点算子代数结构的共形导数的作者最近建立的一个重要的方式。结果表明，特别是准确的评价CLE期望的产品的hypotrochoid领域，以及它们之间的非平凡的关系，通过顶点算子代数，并进一步阐明CLE和CFT之间的关系。
Conformal Mechanics and the Virasoro Algebra
证明了任何单变量标度不变力学不仅具有0+1共形对称性，而且具有全维拉索罗代数的对称性.我们讨论的adS/CFT对应的影响。
Super Virasoro Ghost Algebra using the Conformal Ghosts
利用Clifford代数的共形鬼和反交换无穷维Gamma矩阵构造了共形维数为3/2的超共形鬼电流发生器。超级维拉索罗代数的幽灵在两个国家。和R扇区。在这两种情况下的反常项推导出使用雅可比恒等式。
We derive recursive representations in the internal weights of N-point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.
We construct confluent conformal blocks of the second kind of the Virasoro algebra. We also construct the Stokes transformations which map such blocks in one Stokes sector to another. In the BPZ limit, we verify explicitly that the constructed blocks and the associated Stokes transformations reduce to solutions of the confluent BPZ equation and its Stokes matrices, respectively. Both the confluent conformal blocks and the Stokes transformations are constructed by taking suitable confluent limits of the crossing transformations of the four-point Virasoro conformal blocks.
Non-stationary difference equation for q-Virasoro conformal blocks
本文研究q,t-Virasoro块的非静态差分方程，探讨退化表示下的conformal块性质及与Macdonald多项式的关系。
We present recent developments of irregular conformal conformal states. Irregular vertex operators and their adjoint in a new formalism are used to define the irregular conformal states and their inner product instead of using the colliding limit procedure. Free field formalism can be augmented by screening operators which provide more degrees of freedom. The inner product is conveniently given as the partition function of an irregular matrix model. (Deformed) spectral curve is the loop equation of the matrix model at Nekrasov-Shatashivili limit. We present the details of analytic structure of…
A pseudo-conformal representation of the Virasoro algebra
推广了准素域的概念，我们找到了维拉索罗代数的一个新的表示，我们称之为伪共形表示。在特殊情况下，这种表示可简化为普通的或对数的共形场论。然而，在其他情况下，格林函数不同于普通或对数共形场论的格林函数。该表示由两个矩阵参数化。我们对这两个矩阵进行了分类，并计算了一个简单例子中的一些矩阵。
Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide three closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge $c$. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. Another is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large $c$. In both …
We present the irregular matrix model which has contains $\mathcal{W}_3$ and Virasoro symmetry. The irregular matrix model is obtained using the colliding limit of the Toda field theories and produces the inner product between irregular modules of $\mathcal{W}_3$ symmetry. We evaluate the partition function using the flow equation which is the realization of the Virasoro and $\mathcal{W}$-symmetry.
Factorized Combinations of Virasoro Characters
我们调查的最小Virasoro模型，这是表示为几个基本块的产品的字符的线性组合。我们的分析是基于考虑准经典极限的字符的渐近行为。特别地，我们引入了次级有效中心荷的概念。我们发现所有可能的情况下，分解发生的基础上的高斯-雅可比或沃森身份。利用这些结果，我们建立不同的字符之间的各种类型的身份。特别是，我们提出了几个身份推广罗杰斯-拉马努金身份。应用准粒子表示，模不变配分函数，超共形理论和边界的共形模型进行了简要讨论。
Flat Connections from Irregular Conformal Blocks
本文通过代数方法推导修正BPZ方程，构造非紧Lefschetz环积分表示，并由此在不规则共形块丛上导出新型平坦联络。
A four-dimensional conformal construction of Virasoro-Shapiro amplitudes
本文提出一种四维共形构造，通过共形场论原理构建了与弦理论相关的Virasoro-Shapiro幅，并为幅bootstrap方法提供了新的研究方向。
We compute 5-point classical conformal blocks with two heavy, two light, and one superlight operator using the monodromy approach up to third order in the superlight expansion. By virtue of the AdS/CFT correspondence we show the equivalence of the resulting expressions to those obtained in the bulk computation for the corresponding geodesic configuration.
Irregular conformal block and its matrix model
不规则共形块是研究Argyres-Douglas理论的一种新工具，它的不规则向量表示为一组正Virasoro生成元的同时本征态。利用厄米矩阵模型的β系综的配分函数是寻找非规则共形块的一种方法。到目前为止，该方法仅限于偶次不规则奇异性的情况。本文提出了一种新的奇次矩阵模型，并计算了它的配分函数。该模型与前一模型的不同之处在于其势函数中增加了矩阵平方根因子。
Solutions of the Master Virasoro Equations as Conformal Points of Perturbed WZNW Model
证明了酉仿射代数上的仿射Virasoro构造的主方程自然地出现在非酉水平$k$ WZNW模型的融合代数中.对应于主方程的解的算子适合于围绕给定的共形非酉WZNW模型进行单参数可重整化扰动。在大的$|k|$极限下，找到了重整化群β函数的红外不动点，红外共形点的个数与主方程的解的个数$\frac{1}{2}N(N+1)$一样多，其中$N$是主方程的解的个数.
Conformal Models of Thirring Type and the Affine Virasoro Construction
研究了一类具有四个费米子相互作用项的1+1维模型。在每一阶的微扰展开，模型是紫外有限和洛伦兹非不变的。我们表明，某些特权值的耦合常数，洛伦兹对称性恢复，事实上，该模型原来是共形不变的。这种现象既是量子力学的，又是非微扰的。
Conformal blocks in Virasoro and W theories: duality and the Calogero-Sutherland model
研究了具有Virasoro对称性和W-扩展对称性的共形场论的共形块的性质。当共形块只包含二阶简并场时，共形块服从二阶微分方程，它们可以被解释为具有非平凡辫状性质的三角Calogero-Sutherland哈密顿量的基态波函数.两类二阶退化场之间存在一个广义对偶性质。通过研究这种对偶性，我们发现Calogero-Sutherland哈密顿量的激发态可以用两个分区来表征，或者在WA_{k-1}理论的情况下可以用k个分区来表征。通过将所考虑的共形场论推广到一个u（1）场，我们发现可以将推广的共形场论的Hilbert态与Calogero-Sutherland哈密顿量的激发非多项式本征态对应起来。当Calogero-Sutherland运动积分的作用量在希尔伯特空间上平移时，它们与Alba，Fateev，Litvinov和Tarnopolsky最近在刘维理论中发现的AGT猜想中的运动积分相同。在玻色子化时，这些运动积分可以表示为两个或一般k个玻色子Calogero-Sutherland哈密顿量的和，该玻色子Calogero-Sutherland哈密顿量由具有三角形结构的相互作用项耦合。对于特定的耦合常数，共形块可以用具有配对性质的Jack多项式表示，并给出了特定分数量子霍尔态的电子波函数
By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the $\mathfrak{bms}_{3}$ algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite dimensional lifts of the so-called $\mathfrak{B}_{k}$, $\mathfrak{C}_{k}$ and $\mathfrak{D}_{k}$ algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Kač-Moody algebras using modified Su…
We study Virasoro minimal-model 4-point conformal blocks on the sphere and 0-point conformal blocks on the torus (the Virasoro characters), as solutions of Zamolodchikov-type recursion relations. In particular, we study the singularities due to resonances of the dimensions of conformal fields in minimal-model representations, that appear in the intermediate steps of solving the recursion relations, but cancel in the final results.
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…
We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conform…
Ghost Systems Revisited: Modified Virasoro Generators and Logarithmic Conformal Field Theories
研究了将高自旋鬼系统推广到对数共形场论的可能性。特别是，我们对c=-26很感兴趣，它的行为与已知的c=-2的情况非常不同。能量动量张量不能再由广义辛费米子场的导数的组合来建立。此外，代数扩展理论只有在考虑非平凡黎曼曲面时才是相容的。这导致LCFT具有一些意想不到的性质。例如，Virasoro模L_0是对角的，对于某些变形参数，甚至整个全局共形群都是非对数的。
We derive new constraints on the spectrum of two-dimensional conformal field theories with central charge $c>1.$ Employing the pillow representation of the four point correlator of identical scalars with dimension $Δ_{\mathcal{O}}$ and positivity of the coefficients of its expansion in the elliptic nome we place central charge dependent bounds on the dimension of the first excited Virasoro primary the scalar couples to, in the form $Δ_1<f(c,Δ_{\mathcal{O}}).$ We give an analytic expression for $f(c,Δ_{\mathcal{O}})$ and write down transcendental equations that significantly improve the analyti…
We discuss the resummation procedure of the superpotential in 4d $\mathcal{N}=2$ SYM theory without matter from the point of view of 2d Liouville conformal field theory, utilizing the AGT correspondence. We identify contributions of different descendants of the intermediate state, defining a conformal block in CFT, to the superpotential and answer the question, which descendants are responsible for the appearance of the branch cuts in the superpotential.
Following Nag-Sullivan, we study the representation of the group ${\rm Diff}^+(S^1)$ of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides a triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott-Virasoso cocylce class, and leads to a surprising identity for the Takhtajan-Teo energy functional on ${\rm Diff}^+(S^1)$.
Three-Dimensional Gravity with Conformal Scalar and Asymptotic Virasoro Algebra
Strominger用渐近维拉索罗代数导出了BTZ黑洞的贝肯斯坦-霍金熵。我们将Strominger方法应用于Martinez和Zanelli（MZ）发现的一个黑洞解。这是具有共形标量场的三维引力的解。解不是$AdS_3$，但它是渐近的$AdS_3$ ;因此，它具有渐近的维拉索罗代数。我们计算了该理论的中心荷，并将Cardy公式与Bekenstein-Hawking熵进行了比较。事实证明，函数形式确实一致，但总的数值系数不一致。这是因为这种方法给出了数值系数的“最大可能熵”。
Virasoro conformal blocks are expected to exponentiate in the limit of large central charge $c$ and large operator dimensions $h_i$, with the ratios $h_i/c$ held fixed. We prove this by employing the oscillator formulation of the Virasoro algebra and its representations. The techniques developed are then used to provide new derivations of some standard results on conformal blocks.
The conformal bootstrap is applied to percolation and dilute self-avoiding polymers, two theories with Virasoro central charge $c=0$ in two dimensions. In both cases we propose a spectrum of operators motivated by Virasoro symmetry which is devoid of a stress energy tensor as an approximate means of enforcing $c=0$. Percolation is treated in $2\leq D \leq 6$ dimensions, and the self-avoiding walk in $2 \leq D \leq 4$.
Conformal nets from minimal W-algebras
本文证明了所有幺正极小W-代数的强分级局部性，从而产生不可约分级局部共形网，并基于此提出了统一构造以获得包含N=3,4超Virasoro网在内的新共形网，同时证明了完全有理猜想对N=0至4超Virasoro代数成立。
We study certain six dimensional theories arising on $(p,q)$ brane webs living on $\mathbb{R}\times S^1$. These brane webs are dual to toric elliptically fibered Calabi-Yau threefolds. The compactification of the space on which the brane web lives leads to a deformation of the partition functions equivalent to the elliptic deformation of the Ding-Iohara algebra. We compute the elliptic version Dotsenko-Fateev integrals and show that they reproduce the instanton counting of the six dimensional theory.
Spinon Bases, Yangian Symmetry and Fermionic Representations of Virasoro Characters in Conformal Field Theory
研究了用自旋为1/2的仿射初级场$ϕ^α$的模描述$SU(2)$、能级$k=1$、Wess-Zumino-Witten共形场论.这些被证明是满足广义“典型的交换关系”，我们用它来构建一个基础的希尔伯特空间的杨的$Y(sl_2)$的表示。利用这种描述，我们明确地导出了Virasoro特征的所谓“费米子表示”，这是Kedem等人首先提出的。\cite{kedem}。我们指出，类似的结果是预期的合理的共形场论广泛的一类。
Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs' classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, enta…
We revisit the critical two-dimensional Ashkin-Teller model, i.e. the $\mathbb{Z}_2$ orbifold of the compactified free boson CFT at $c=1$. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at $c=1$. We show that blocks with a degenerate channel field should be computed by…
On the space of generic conformal blocks the modular transformation of the underlying surface is realized as a linear integral transformation. We show that the analytic properties of conformal block implied by Zamolodchikov's formula are shared by the kernel of the modular transformation and illustrate this by explicit computation in the case of the one-point toric conformal block.
Motivated by sine-square deformation (SSD) for quantum critical systems in 1+1-dimension, we discuss a Mobius quantization approach to the two-dimensional conformal field theory (CFT), which bridges the conventional radial quantization and the dipolar quantization recently proposed by Ishibashi and Tada. We then find that the continuous Virasoro algebra of the dipolar quantization can be interpreted as a continuum limit of the Virasoro algebra for scaled generators in the SSD limit of the Mobius quantization approach.
To a given algebraic curve we assign an infinite family of quantum curves (Schrödinger equations), which are in one-to-one correspondence with, and have the structure of, Virasoro singular vectors. For a spectral curve of a matrix model we build such quantum curves out of an appropriate representation of the Virasoro algebra, encoded in the structure of the $α/β$-deformed matrix integral and its loop equation. We generalize this construction to a large class of algebraic curves by means of a refined topological recursion. We also specialize this construction to various specific matrix models w…
Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
径向sine-Gordon/Painlevé III方程的τ函数的短程展开式由一个收敛级数给出，该级数包含不规则的$c=1$共形块，并且对于单值数据具有一定的周期性.长距离的不规则膨胀表现出类似的周期性相对于不同的一对坐标的monodromy流形。这一观察是用来推测一个确切的表达式的连接常数提供相对归一化的两个系列。直到一个初等前因子，它由两组坐标之间的正则变换的生成函数给出。
Conformal transformations and the SLE partition function martingale
我们提出了一个实现在共形场论（CFT）的局部有限共形变换固定一个点。当不动点是原点或无穷远点时，我们给出明确的构造。这两种情况都涉及到维拉索罗代数的Borel子代数的幂。我们用它来建立相干态表示，并推导出一个密切的模拟威克定理的维拉索罗代数。这允许计算共形配分函数在非平凡的几何形状，从上半平面去除外壳获得。然后将其应用于随机Loewner演化（SLE）。我们给出了严格的推导方程，以前获得的作者，连接随机Loewner方程的表示理论的维拉索罗代数。给出了一个新的证明，证明了该构造枚举所有多项式SLE鞅。当从上半平面移除的壳之一是SLE壳时，我们证明了配分函数是概率学家已知的著名的局部鞅，从而揭开了它的CFT起源。
Unification of the General Non-Linear Sigma Model and the Virasoro Master Equation
Virasoro主方程在WZW模型的算子代数（仿射李代数）中描述了一个被称为仿射-Virasoro结构的共形场论的大集合，而一般非线性西格玛模型的爱因斯坦方程描述了另一个共形场论的大集合。这次谈话总结了最近的工作，统一这两套共形场论，连同一个假定的大类新的共形场论。其基本思想是考虑在一般西格玛模型的背景下的$L_{ij} \partial x^i \partial x^j$形式的自旋二算子。这些算符满足维拉索罗代数的要求导致了一组被称为统一爱因斯坦-维拉索罗主方程的方程，其中自旋为2的时空场$L_{ij}$耦合到西格玛模型的通常时空场。给出了这个统一系统的单圈形式，并讨论了它的一些代数和几何性质。
Conformal Transformation, Near Horizon Symmetry, Virasoro Algebra and Entropy
在广义相对论（GR）中存在某些与定态解共形相关的黑洞解，但这些时空的视界熵与定态解一样也是视界面积的四分之一并不明显.在这里，我研究这个话题的背景下，维拉索罗代数和卡迪公式。利用共形时空中存在共形Killing向量，视界由其范数的消失决定的事实，得到了保持近视界结构不变的微分同胚.计算了该区域的Noether荷和其中对应于这些矢量的一个括号。最后，他们评估的Sultana-Dyer（SD）黑洞，这是共形的史瓦西度规。证明了该括号与通常的具有中心扩张的维拉索罗代数完全相同.通过对零模本征值和中心荷的识别，利用Cardy公式得到了SD视界的熵。有趣的是，这又是地平线面积的四分之一。在这种情况下，唯一的区别是，与静止的面积相比，面积被保形因子修改。这一分析直接证明了先前的假设。
We revisit the construction of multi-centered solutions in three-dimensional anti-de Sitter gravity in the light of the recently discovered connection between particle worldlines and classical Virasoro conformal blocks. We focus on multi-centered solutions which represent the backreaction of point masses moving on helical geodesics in global AdS$_3$, and argue that their construction reduces to a problem in Liouville theory on the disk with Zamolodchikov-Zamolodchikov boundary condition. In order to construct the solution one needs to solve a certain monodromy problem which we argue is solved …
Nonlocal Conformal Field Theory
本文利用分数维拉宿代数探索非局域共形场论中的算子乘积展开，发现非局域量子动力学仅以态依赖中心荷实现该代数，并证明非高斯不动点符合此标准。
We study the relation of irregular conformal blocks with the Painlevé III$_3$ equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painlevé III$_3$. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both $c=1$ and $c\to\infty$ conformal blocks. We extend this analysis to the domain of strong-coupling regim…
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…
We perform a detailed study of a class of irregular correlators in Liouville Conformal Field Theory, of the related Virasoro conformal blocks with irregular singularities and of their connection formulae. Upon considering their semi-classical limit, we provide explicit expressions of the connection matrices for the Heun function and a class of its confluences. Their calculation is reduced to concrete combinatorial formulae from conformal block expansions.
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