J. Ramos Cabezas
We explicitly demonstrate the relation between the 2-point t-channel torus block in the large-c regime and the geodesic length of a specific geodesic diagram stretched in the thermal AdS3 spacetime
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We study four types of one-point torus blocks arising in the large central charge regime. According to different limits of conformal dimensions we distinguish between the global block, the light block, the heavy-light block, and the linearized classical block. We show that they are not independent and connected to each other by various links. We find that the global, light, and heavy-light blocks correspond to three different contractions of the Virasoro algebra. Also, we formulate the c-recursive representation of the one-point torus blocks which is relevant in the semiclassical approximation…
The Semiclassical Limit for $SU(2)$ and $SO(3)$ Gauge Theory on the Torus
证明了环面上的$SU(2)$和$SO(3)$量子规范理论，关于杨-米尔斯测度$dμ_T(ø) =N_T^{-1}e^{-S_{YM}(ø)/T}[{\cal D}ø]$的完整性期望值收敛于关于丛上平坦联络的模空间上的辛体积测度$μ_0$的积分，作为$T\downarrow 0$.这些模空间和辛结构的明确描述。
Semiclassical Virasoro Blocks from AdS$_3$ Gravity
我们提出了一个统一的框架，全息计算Virasoro共形块在大中心荷。特别是，我们提供了正确再现所有从共形场论计算中明确已知的半经典Virasoro块的体结构。结果围绕使用测地威滕图，最近在arXiv：1508.00501，在本地AdS $_3$几何产生的重运营商的反作用进行评估。我们还提供了一种替代计算的重轻半经典块-其中两个外部运营商成为参数重-作为一个特定的散射过程，涉及更高的自旋规范场在AdS $_3$，这种方法突出了Virasoro块的手性性质。这些技术可以系统地扩展到计算这些块的校正和不同的半经典制度之间的插值。
We propose the holographic interpretation of the 1-point conformal block on a torus in the semiclassical regime. To this end we consider the linearized version of the block and find its coefficients by means of the perturbation procedure around natural seed configuration corresponding to the zero-point block. From the AdS/CFT perspective the linearized block is given by the geodesic length of the tadpole graph embedded into the thermal AdS plus the holomorphic part of the thermal AdS action.
Recent work has demonstrated that black hole thermodynamics and information loss/restoration in AdS$_3$/CFT$_2$ can be derived almost entirely from the behavior of the Virasoro conformal blocks at large central charge, with relatively little dependence on the precise details of the CFT spectrum or OPE coefficients. Here, we elaborate on the non-perturbative behavior of Virasoro blocks by classifying all `saddles' that can contribute for arbitrary values of external and internal operator dimensions in the semiclassical large central charge limit. The leading saddles, which determine the naive s…
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。
We study CFT2 conformal blocks on a torus and their holographic realization. The classical conformal blocks arising in the regime where conformal dimensions grow linearly with the large central charge are shown to be holographically dual to the geodesic networks stretched in the thermal AdS bulk space. We discuss the n-point conformal blocks and their duals, the 2-point case is elaborated in full detail. We develop various techniques to calculate both quantum and classical conformal block functions. In particular, we show that exponentiated global torus blocks reproduce classical torus blocks …
Global torus blocks in the necklace channel
研究在项链通道中全局环块的性质，发现其在特定条件下满足Casimir方程。
After deriving the classical Ward identity for the variation of the action under a change of the modulus of the torus we map the problem of the sphere with four sources to the torus. We extend the method previously developed for computing the classical conformal blocks for the sphere topology to the tours topology. We give the explicit results for the classical blocks up to the third order in the nome included and compare them with the classical limit of the quantum conformal blocks. The extension to higher orders is straightforward.
Recursive representation of the torus 1-point conformal block
导出了环面上1点共形块的递推关系，并利用它证明了R. Poghossian作为一个例子的递归方法的效率的1点刘维关联函数的模不变性进行了数值分析。
We give a holographic description of global conformal blocks in two dimensional conformal field theory on the sphere and on the torus. We show that the conformal blocks for one-point functions on the torus can be written as Witten diagrams in thermal AdS. This is accomplished by deriving a general conformal Casimir equation for global conformal blocks, and showing that Witten diagrams obey the same equation. We study the semi-classical limit of n-point conformal blocks, and show that these equal the action of a network of bulk world-lines obeying appropriate geodesic equations. We give an alte…
Representations of the Mapping Class Group of the Two Punctured Torus on the Space of \hat {sl}(2,C) Spin1/2-Spin1/2 Kac-Moody Blocks
由Wakimoto和Bernard及Felder的$\hat {sl}(2,C)$卡茨-穆迪代数的自由场表示导出了环面上的$\hat {sl}(2,C)$ Spin 1/2-Spin 1/2 Kac-Moody块的积分表示，并由此导出了两个穿孔环面的映射类群的无限类表示.
We consider the Wilson line networks of the Chern-Simons $3d$ gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus $2d$ CFT. After general discussion that summarizes and further extends results known in the literature we explicitly obtain the one-point torus block and two-point torus blocks through particular matrix elements of toroidal Wilson network operators in irreducible finite-dimensional representations of $sl(2,\mathbb{R})$ algebra. The resulting expressions are given in two alternative forms using diffe…
Semiclassical Analysis of the 3d/3d Relation
我们提供了定量的证据，我们以前的猜想，说明了一个等价的配分函数的3D N=2的规范理论的对偶墙上，SL（2，R）陈-西蒙斯理论的映射环面，为一类的例子与一次穿刺环面。特别地，我们证明了3d N=2配分函数的极限再现了映射环面的双曲体积和Chern-Simons不变量。这是通过分析量子Teichmuller理论的希尔伯特空间中映射类群的元素的迹的经典极限来证明的。我们还表明，subleading校正的配分函数再现Reidemeister扭转。
We consider the conformal block decomposition in arbitrary exchange channels of a two-dimensional conformal field theory on a torus. The channels are described by diagrams built of a closed loop with external legs (a necklace sub-diagram) and trivalent vertices forming trivalent trees attached to the necklace. Then, the $n$-point torus conformal block in any channel can be obtained by acting with a number of OPE operators on the $k$-point torus block in the necklace channel at $k=1,...,n$. Focusing on the necklace channel, we go to the large-$c$ regime, where the Virasoro algebra truncates to …
We chart the breakdown of semiclassical gravity by analyzing the Virasoro conformal blocks to high numerical precision, focusing on the heavy-light limit corresponding to a light probe propagating in a BTZ black hole background. In the Lorentzian regime, we find empirically that the initial exponential time-dependence of the blocks transitions to a universal $t^{-\frac{3}{2}}$ power-law decay. For the vacuum block the transition occurs at $t \approx \frac{πc}{6 h_L}$, confirming analytic predictions. In the Euclidean regime, due to Stokes phenomena the naive semiclassical approximation fails c…
We study large-c SCFT2 on a torus specializing to one-point superblocks in the N=1 Neveu-Schwarz sector. Considering different contractions of the Neveu-Schwarz superalgebra related to the large central charge limit we explicitly calculate three superblocks, osp(1|2) global, light, and heavy-light superblocks, and show that they are related to each other. We formulate the osp(1|2) superCasimir eigenvalue equations and identify their particular solutions as the global superblocks. It is shown that the resulting differential equations are the Heun equations. We study exponentiated global superbl…
Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
本文利用环面上DOZZ量子Liouville理论的半经典极限与N=2*（Ω变形）U（2）超杨-米尔斯理论的Nekrasov-Shatashvili极限之间的对应关系，提出了Lame方程的附加参数的新公式。这个量对于解决单穿孔环面的均匀化问题特别重要。环面和球面的辅助参数的计算是一个长期存在的问题，但如果成功地推导出所谓的经典刘维尔作用量的表达式，这个问题就可以解决。计算后者的方法已经提出了一段时间前由Zamolodchikov兄弟。研究量子刘维尔理论的四点函数在球面上的半经典极限，他们导出了黎曼球面的经典作用量。在目前的工作Zamolodchikov的想法是利用的情况下，刘维尔场论的环面。研究发现，Lame辅助参数由单穿孔环面上的经典Liouville作用决定，或者更具体地说，由鞍点中间经典权上的环面经典块决定.其次，作为上述对应的一个推论，得到了环面辅助参数与N=2* 规范理论中使瞬子“自由能”极值化的所谓“临界”杨氏图的所有重新标度的柱长之和有关。最后，它指出，由于已知的关系的总和超过“临界”列长度可以表示在一个轮廓积分，其中的被积函数是建立在某些特殊的功能。
Monodromy methods for torus conformal blocks and entanglement entropy at large central charge
本文将单值性方法推广至环面并用于副本配分函数，计算大中心荷二维 CFT 的有限尺寸有限温度纠缠熵，证明真空交换主导并与 RT 全息预测一致。
A note on Zamolodchikov's recursion relation for the torus conformal block and its light limit
研究刘维尔和\(A_2\)托达理论中环面单点共形块的类扎莫洛德奇科夫递归关系，推导其光渐近极限下的关系，为\(A_2\)托达理论提供新计算方法，结果与文献一致。
Non-conformal limit of AGT relation from the 1-point torus conformal block
给定一个4d N=2的超对称规范理论，我们可以构造塞伯格-威滕预势，它涉及到瞬子的求和。瞬子模空间上的积分需要正则化。对于UV-有限理论，AGT猜想特别倾向于Nekrasov的正则化方法。这意味着Nekrasov配分函数等于二维W_{N_c}手征代数理论中的共形块。对于$N_c=2$和一个伴随多重态，它与环面1点Virasoro共形块一致。在这种情况下，我们检验了共形维数与伴随多重态质量之间的AGT关系，并研究了共形块的极限，它对应于4d理论的大质量极限，即：渐近自由的4d N=2超对称Yang-Mills理论虽然技术上更复杂，但极限与基本多重态的情况相同，这为AGT猜想提供了一个非平凡的检验。
The $W_n$ Light One-Point Torus Conformal Block
研究$A_{n-1}$Toda场论中单点环面共形块的轻量渐近极限，通过AGT对应关系转化为四维${\cal N}=2^{\ast}$超对称杨-米尔斯理论的Instanton分区函数计算，推导出轻量单点环面$W_n$共形块的显式表示。
Expansions for semiclassical conformal blocks
本文利用BPZ退耦方程辅助参数与等单值tau函数对数导数的联系，提出不同分支点处半经典共形块展开的关系，并通过黑洞微扰理论中的合流Heun方程特征值问题及数值比较准正则谱提供支持。
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 study the multi-boundary entanglement structure of the state associated with the torus link complement $S^3 \backslash T_{p,q}$ in the set-up of three-dimensional SU(2)$_k$ Chern-Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of $k \to \infty$. We present a detailed analysis for several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large $k$ limiting value of the Rényi entropy of torus links of type $T_…
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。
Shadow formalism for supersymmetric conformal blocks
该研究将Shadow形式体系的Shadow算符构造推广到超对称共形场论，验证了其给出大c极限下平面四点、环面一点超对称共形块的已知结果，还求出项链道环面两点超对称共形块并经Casimir方程验证。
Recursion relations for the sphere $4$-point and torus $1$-point ${\cal W}_3$ conformal blocks, generalizing Alexei Zamolodchikov's famous relation for the Virasoro conformal blocks are proposed. One of these relations is valid for any 4-point conformal block with two arbitrary and two special primaries with charge parameters proportional to the highest weight of the fundamental irrep of $SU(3)$. The other relation is designed for the torus conformal block with a special (in above mentioned sense) primary field insertion. AGT relation maps the sphere conformal block and the torus block to the …
One-loop corrections to near extremal Kerr thermodynamics from semiclassical Virasoro blocks
提出利用 Virasoro 半经典块精确计算黑洞熵单圈量子修正的方法，证明四维近极端 Kerr 黑洞中长寿命模式对自旋 $s=1,2$ 粒子贡献 $(s-1/2)\log T_{\text{Hawking}}$ 熵修正，且主导贡献仅源于视界附近区域具有普适性。
Two dimensional field theories invariant under the Bondi-Metzner-Sachs (BMS) group are conjectured to be dual to asymptotically flat spacetimes in three dimensions. In this paper, we continue our investigations of the modular properties of these field theories. In particular, we focus on the BMS torus one-point function. We use two different methods to arrive at expressions for asymptotic structure constants for general states in the theory utilising modular properties of the torus one-point function. We then concentrate on the BMS highest weight representation, and derive a host of new result…
Global conformal blocks via Shadow formalism
本文利用 shadow formalism 研究球面与环面上的 $\mathfrak{sl}_2$ 和 $\mathfrak{sl}_3$ 全局共形块，证明其能复现已知表达式，并推广为 $\mathfrak{sl}_3$ 全局块构建了简单的积分表示。
Spin-k/2-spin-k/2 SU(2) two-point functions on the torus
讨论了SU（2）Wess-Zumino-Witten模型中整数水平k上准素算子环面上的一类两点函数.特别地，我们构造了一个显式表达式的当前块的spin-k/2-spin-k/2环面两点函数的所有k。我们首先检查所提出的当前块的因式分解限制，并测试其单值属性。然后，我们证明了当前块解决相应的Knizhnik-Zamolodchikov类微分方程使用Mathur，Mukhi和Sen的方法。
Virasoro vacuum block at next-to-leading order in the heavy-light limit
本文研究了真空Virasoro块的半经典极限，它描述了球上对角四点关联函数。在中心荷c较大时，取幂后，它取决于两个固定的比值h_H/c和h_L/c，其中h_{H, L}是四点函数算子的共形维数。半经典块可以展开的幂的光比h_L/c和领先的非平凡（线性）的顺序是已知的封闭形式作为一个函数的h_H/c。最近，这一贡献已被匹配到AdS_3重力计算，其中重运营商建立了一个经典的几何对应的BTZ黑洞，而轻运营商描述的测地线在此背景下。在这里，我们第一次计算次领先的二次校正O（（h_L/c）^{2}），再次以一般重算子比h_H/c的封闭形式。结果是前导阶的高度非平凡扩展，并且可能与进一步细化的AdS_{3} / CFT_{2}测试相关。两个区间的雷尼熵的应用也提出了。
Holographic thermal correlators and quasinormal modes from semiclassical Virasoro blocks
本文利用半经典 Virasoro 块及 AGT 对应，精确求解渐近 AdS5 黑洞微扰，给出全息热两点函数与 QNM 量子化条件并加以数值验证。
More on Torus Wormholes in 3d Gravity
本文研究三维引力中环面虫洞对AdS3/CFT2系综对偶的贡献，分析不同插入和Z2商情形下的晚期非衰减行为，并讨论RT对称性对随机矩阵统计类型及虫洞的影响。
This thesis is divided into two parts, where in the first part we investigate the computation of Virasoro 1-point blocks on the torus in the framework of Zamolodchikov's recursion relation. It is widely accepted that this recursion relation contains unphysical poles in the central charge $c$. At each order we conjecture how the pole free expressions depend on the internal and external conformal dimensions and central charge, and propose how to compute it numerically. In this thesis, we have calculated the pole free expression up to order 4. In the second part we introduce a conformal field the…
Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string
本文研究环面上的超刘维尔理论，纠正了超 Virasoro 块并构造模穿越核，进而分析超 Virasoro 极小弦，验证其振幅与拓扑递归一致，并识别出特殊 Ramond 插入。
Modular Exercises for Four-Point Blocks -- I
本文证明有理顶点算子代数的球面四点手征块是模形式，并展示交叉对称性即模对称性，从而利用模形式技术约束手征块和融合系数，并揭示球面-环面对应关系。
Coulomb integrals and conformal blocks in the AdS3-WZNW model
利用球上的库仑气体方法研究了AdS 3-WZNW模型中的谱流保持四点关联函数。我们提出了共形块的多重积分实现，并显式计算涉及算子的振幅，其自旋之和的量化值，即，需要整数个第一种屏蔽装料。结果给出了作为一个总和的独立配置的筛选轮廓产生一个单值不变的扩大的权力的世界表模量。然后，我们研究的因式分解的限制，并表明，领导项的总和可以确定，在半经典的限制，与产品的谱流保存三点功能。这些项可以重写为J. Teschner从H3+-WZNW模型中可归一化态的算子乘积展开的假设得到的积分表达式的m基版本。最后，我们确定一组特定的四点函数的因式分解成两个三点函数的产品之间的等价性，无论是保持或违反谱流数守恒。基于此分析，我们认为，振幅的表达式作为一个整体的自旋的中间运营商持有超出半经典制度，从而证实，频谱流守恒的AdS 3-WZNW模型中的相关因子的解析延拓的关联函数的H3+-WZNW模型。
Modular transformations of tau functions and conformal blocks on the torus
本文研究单孔环面上 tau 函数的模变换，导出连接常数，并首次给出 c=1 Virasoro 模核的闭式公式，证明这些量是特征簇上典范变换的生成函数，且与 N=2* 规范理论相关。
Painlevé/CFT correspondence on a torus
本文在环面上建立了等单值tau函数与共形块之间的对应关系，基于作者在ICMP 2021的报告。
We consider Virasoro conformal blocks in the large central charge limit. There are different regimes depending on the behavior of the conformal dimensions. The most simple regime is reduced to the global sl(2, C) conformal blocks while the most complicated one is known as the classical conformal blocks. Recently, Fitzpatrick, Kaplan, and Walters showed that the two regimes are related through the intermediate stage of the so-called heavy-light semiclassical limit. We study this idea in the particular case of the 5-point conformal block. To find the 5-point global block we use the projector tec…
We analyse Virasoro conformal blocks in the regime of heavy intermediate exchange $(h_p \rightarrow \infty)$. For the 1-point block on the torus and the 4-point block on the sphere, we show that each order in the large-$h_p$ expansion can be written in closed form as polynomials in the Eisenstein series. The appearance of this structure is explained using the fusion kernel and, more markedly, by invoking the modular anomaly equations via the 2d/4d correspondence. We observe that the existence of these constraints allows us to develop a faster algorithm to recursively construct the blocks in th…
Modular bootstrap in Liouville field theory
环面上一般1点共形块的模矩阵表示为球面上4点块的融合矩阵。证明了具有DOZZ结构常数的Liouville场论中复曲面1-点函数的模不变性。
Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
本文考虑球面上的刘维尔场论。我们明确地导出了一个四点关联函数与一个简并场$V_{-\frac{mb}{2}}$的微分方程。我们还介绍和研究了一类四点共形块，它可以精确计算，并表示为有限维积分的椭圆θ-函数的任意中间尺寸。我们还研究了这些共形块的自举方程，并推导出相应的四点关联函数的积分表示。本文给出了环面上原力场的一点关联函数与球面上特殊的四点关联函数之间的关系。
Gauge theory geography: charting a path between semiclassical islands
研究了带 't Hooft 扭曲的 $SU(2)$ Yang-Mills 理论在飞米宇宙和变形极限下的半经典基态，证明它们具有相同的对称性和 $θ$ 依赖真空结构，并通过计算一圈势分析了其量子稳定性。
The Torus One-Point Diagram in Two-Dimensional String Cosmology
本文数值计算二维弦宇宙学背景中的环面单点弦图，通过 Liouville CFT 单点函数的 Virasoro 共形块分解与模空间积分，得到关于出射闭弦能量的简洁结果，并阐明其对宇宙波函数的贡献。
A $T_2 \times R^2$ roadmap to Confinement in SU(2) Yang-Mills theory
本文在带扭曲边界条件的 $T_2\times R^2$ 几何中研究 SU(2) Yang-Mills 禁闭，通过格点 Monte Carlo 验证分数瞬子气体的半经典图像并确定非稀薄转变尺度。
Torus one-point functions in critical loop models
研究通过数值 Bootstrap 方法计算环面临界环模型中的一点函数，发现其可表示为不同中央电荷下球面四点函数，且球面交叉对称性导致环面模自洽性。
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