Benoit Vicedo
We relate two formalisms recently proposed for describing classical integrable field theories. The first is based on the action of four-dimensional holomorphic Chern-Simons theory introduced and studied by Costello, Witten and Yamazaki. The second makes use of classical generalised Gaudin models associated with untwisted affine Kac-Moody algebras.
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We construct the actions of a very broad family of 2d integrable $σ$-models. Our starting point is a universal 2d action obtained in [arXiv:2008.01829] using the framework of Costello and Yamazaki based on 4d Chern-Simons theory. This 2d action depends on a pair of 2d fields $h$ and $\mathcal{L}$, with $\mathcal{L}$ depending rationally on an auxiliary complex parameter, which are tied together by a constraint. When the latter can be solved for $\mathcal{L}$ in terms of $h$ this produces a 2d integrable field theory for the 2d field $h$ whose Lax connection is given by $\mathcal{L}(h)$. We con…
In this short review we compare constructions of 2d integrable models by means of two gauge field theories. The first one is the 4d Chern-Simons (4d-CS) theory proposed by Costello and Yamazaki. The second one is the 2d generalization of the Hitchin integrable systems constructed by means the Affine Higgs bundles (AHB). We illustrate this approach by considering 1+1 field versions of elliptic integrable systems including the Calogero-Moser field theory, the Landau-Lifshitz model and the field theory generalization of the elliptic Gaudin model.
We show that the off-shell N=3 action of N=4 super Yang-Mills can be written as a holomorphic Chern-Simons action whose Dolbeault operator is constructed from a complex-real (CR) structure of harmonic space. We also show that the local space-time operators can be written as a Penrose transform on the coset SU(3)/(U(1) \times U(1)). We observe a strong similarity to ambitwistor space constructions.
The affine Gaudin model, associated with an untwisted affine Kac-Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological-holomorphic Chern-Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.
We consider finite deformations of the Hull--Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer--Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira--Spencer and holomorphic Chern--Simons theory. The supersymmetric locus of this action is described by an $L_3$ algebra.
Chern-Simons Theories and Integrability
本文系统研究陈-西蒙斯理论与可积场论在不同维度间的对应关系，推广了四维及扭子空间上的已有结果，并新发现五维全纯2-陈-西蒙斯理论与三维可积场论的对应。
Hamiltonian Analysis of Doubled 4d Chern-Simons
该研究通过哈密顿分析揭示了双四维切变斯莫尔理论的泊松代数结构，并推广了康德-肯特-奥利夫构造到可积分模型领域。
We derive the framing anomaly of four-dimensional holomorphic-topological Chern-Simons theory formulated on the product of a topological surface and the complex plane. We show that the presence of this anomaly allows one to couple four-dimensional Chern-Simons theory to holomorphic field theories with Kac-Moody symmetry, where the Kac-Moody level $k$ is critical $k=-h^{\vee}$. Applying this result to a holomorphic sigma model into a complex coadjoint orbit, we derive that four-dimensional Chern-Simons theory admits holomorphic monodromy defects.
On the Holomorphic Gauge Quantization of the Chern-Simons Theory and Laughlin Wave Functions
考虑了具有非紧规范对称性群的Chern-Simons-Matter拉格朗日量。该理论在具有复杂规范固定条件的全纯规范中量子化。讨论了规范场和物质场伴随着复共轭场的模型。有人认为，这样的理论代表了一个足够的框架，量子霍尔态的描述。
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。
Abelian Chern-Simons Vortices and Holomorphic Burgers' Hierarchy
本文讨论了2+1维Abel Chern-Simons规范场理论及其与全纯Burgers族的关系。证明了在不可压无旋流体力学中复势与复规范场之间的关系具有解析Cole-Hopf变换的意义，用全纯薛定谔方程组线性化了Burgers方程组.在Chern-Simons理论中，平面涡旋的运动表现为规范场的极奇点，对应于规范场的零点运动。通过对复Galilean群的boost变换，构造了一组丰富的精确解，并利用广义Kampe de Feriet和Hermite多项式描述了平面涡和涡格的可积动力学.结果被施加到全纯减少的Ishimori模型和相应的层次，描述动态的磁涡旋和相应的晶格在复杂的Calogero-Moser模型。从Moyal时空非对易性的艾里函数的两个涡旋动力学的修正。
Cech, Dolbeault and de Rham cohomologies in Chern-Simons and BF theories
拓扑Chern-Simons（CS）和BF理论及其全纯类似物的de Rham和Dolbeault上同调讨论。我们表明，切赫上同调提供了另一个有用的描述上述拓扑和全纯场论。特别是，所有隐藏的（非局部）对称性的非阿贝尔CS和BF理论可以最清楚地看到在切赫方法。我们考虑多维Manin-Ward可积系统，并描述它们与全纯BF理论的联系。简要讨论了这类一般可积系统的Dressing对称性。
In this note we consider the symplectic reduction of a four-dimensional holomorphic Chern-Simons theory recently introduced in arXiv:1908.02289 for describing integrable field theories. We work out explicitly the case of the lambda deformed Principal Chiral Model (PCM) and show that the symplectic reduction works as a localization mechanism. The reduced Chern-Simons theory restricts to the set of poles of the twist function underlying the theory, where the known classical integrability of the lambda deformed PCM can be reconstructed from the phase space data associated to this set of points in…
We derive, within the Hamiltonian formalism, the classical exchange algebra of a lambda deformed string sigma model in a symmetric space directly from a 4d holomorphic Chern-Simons theory. The explicit forms of the extended Lax connection and R-matrix entering the Maillet bracket of the lambda model are explained from a symmetry principle. This approach, based on a gauge theory, may provide a mechanism for taming the non-ultralocality that afflicts most of the integrable string theories propagating in coset spaces.
In the approach recently proposed by K. Costello and M. Yamazaki, which is based on a four-dimensional variant of Chern-Simons theory, we derive a simple and unifying two-dimensional form for the action of many integrable $σ$-models which are known to admit descriptions as affine Gaudin models. This includes both the Yang-Baxter deformation and the $λ$-deformation of the principal chiral model. We also give an interpretation of Poisson-Lie $T$-duality in this setting and derive the action of the $\mathsf{E}$-model.
We construct an action for holomorphic Chern-Simons theory that couples the gauge field to off-shell gravitational backgrounds, comprising the complex structure and the (3,0)-form of the target space. Gauge invariance of the off-shell action is achieved by enlarging the field space to include an appropriate system of Lagrange multipliers, ghost and ghost-for-ghost fields. Both the BRST transformations and the BV action are compactly and neatly written in terms of superfields which include fields, backgrounds and their antifields. We show that the anti-holomorphic target space derivative can be…
We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.
Complex Counterpart of Chern-Simons-Witten Theory and Holomorphic Linking
在本文中，我们开始探索陈-西蒙-威滕理论的复杂对应物。我们定义了复杂的模拟的高斯环绕数嵌入卡-丘三倍使用正式的路径积分，导致一个严格的数学表达式的复杂曲线。我们给出了一个分析和几何的解释，我们的全纯链接以下平行的真实情况。我们特别表明，绿色内核，出现在明确的积分高斯环绕数被取代的Bochner-Martinelli内核。我们还发现正则表达式的全纯链接使用Grothendieck-Serre对偶局部上同调，后者承认一个推广的任意领域。
We study the approaches to two-dimensional integrable field theories via a six-dimensional(6D) holomorphic Chern-Simons theory defined on twistor space. Under symmetry reduction, it reduces to a four-dimensional Chern-Simons theory, while under solving along fibres it leads to four-dimensional(4D) integrable theory, the anti-self-dual Yang-Mills or its generalizations. From both four-dimensional theories, various two-dimensional integrable field theories can be obtained. In this work, we try to investigate several two-dimensional integrable deformations in this framework. We find that the $λ$-…
On a Deformed Holomorphic Chern-Simons Theory
本文研究了在Calabi-Yau三重展开上的全纯Chern-Simons理论变形，通过引入变形参数h定义新的瞬子解，并探讨了其在特定方向上的异常自由性质。
The physical states of D=5 holomorphic Chern-Simons theory correspond to on-shell D=10 open superstring states in the cohomology of $q_+$, where $q_+$ is one of the 16 spacetime supersymmetry generators. Scattering amplitudes of these states can be computed either using the usual Ramond-Neveu-Schwarz (RNS) superstring prescription with N=1 worldsheet supersymmetry, or using a topological $\hat c$=5 string theory with twisted N=2 worldsheet supersymmetry. It will be argued that the relation between D=5 holomophic Chern-Simons and the RNS superstring is identical to the relation between the the …
Regular-Zero Drinfeld-Sokolov Charges in Doubled Four-Dimensional Chern-Simons Theory
本文研究四维加倍陈-西蒙斯理论有限缺陷约化后的经典局域守恒荷，利用Evans相容不变张量构造局域圆周积分，证明其泊松可交换，在特定假设下得到经典局域阿贝尔泊松子代数的条件存在性。
Multi-pole extension for elliptic models of interacting integrable tops
本文综述并详细描述了${\ m gl}_{NM}$ Gaudin模型，引入其通过满足结合杨-巴克斯特方程的R矩阵的推广，并给出其到Schlesinger系统的自然扩展。
For Chern-Simons-matter theories in three dimensions, gauge invariance may require the Chern-Simons level k to be half-integral, in which case parity is violated. As noted by Pasquetti for abelian theories with N=2 supersymmetry, the partition function on the ellipsoid also admits a suitable holomorphic factorization precisely when the value of k is properly quantized. Using known formulas for the partition function, we investigate analytic aspects of this factorization for non-abelian gauge groups and general matter representations. We verify that factorization occurs in accord with the parit…
We consider Chern-Simons theory with complex gauge group and present a complete non-perturbative evaluation of the path integral (the partition function and certain expectation values of Wilson loops) on Seifert fibred 3-Manifolds. We use the method of Abelianisation. In certain cases the path integral can be seen to factorize neatly into holomorphic and anti-holomorphic parts. We obtain closed formulae of this factorization for the expectation values of torus knots.
The 3d mixed BF Lagrangian 1-form: a variational formulation of Hitchin's integrable system
本文提出了一种三维混合BF作用量的变分公式，用于描述希奇金可积系统的Lax方程和相关级数。
Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions
本文作为“Holomorphic Floer Theory”项目首篇，探讨了指数积分及wall-crossing结构，将Morse-Novikov理论推广至全纯情形，证明了其解析性与微扰展开的resurgent性质，并提出了无限维指数积分的猜想方法。
Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory
本文证明四维Chern-Simons理论与反自对偶Yang-Mills方程的对称性约化在经典层面等价，通过扭量空间全纯Chern-Simons理论统一，并给出二维可积系统及Bogomolny方程的约化实例。
The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technica…
5d 2-Chern-Simons theory and 3d integrable field theories
本文将4维半全纯Chern-Simons理论推广到5维，通过高阶联络$(A,B)$和边界缺陷条件构造3维可积场论的高阶Lax联络，并给出Ward手征模型的推广。
Chern-Simons theory and BCS superconductivity
研究了具有时间Wilson线的Chern-Simons理论的全纯么正联系与约化BCS哈密顿量的Richardson精确解之间的关系。作为Chern-Simons理论的一个极限情形，我们导出了BCS模型的运动积分及其本征值和本征向量。
Hyperbolic Monopoles, (Semi-)Holomorphic Chern-Simons Theories, and Generalized Chiral Potts Models
本文通过（半）全纯场论建立双曲磁单极子与广义手征波茨模型的对应，推广至SU(n)群，提出4d CS构造技术，并揭示其源于10维全纯CS理论的维数约化，统一为同一物理的两种体现。
These lecture notes concern the semi-holomorphic 4d Chern-Simons theory and its applications to classical integrable field theories in 2d and in particular integrable sigma-models. After introducing the main properties of the Chern-Simons theory in 3d, we will define its 4d analogue and explain how it is naturally related to the Lax formalism of integrable 2d theories. Moreover, we will explain how varying the boundary conditions imposed on this 4d theory allows to recover various occurences of integrable sigma-models through this construction, in particular illustrating this on two simple exa…
HyperKähler prequantization of the Hitchin system and Chern-Simons theory with complex gauge group
Hitchin证明了降维自对偶Yang-Mills方程的模空间${\mathcal M}$具有超Kähler结构。在本文中，我们首先明确显示hyperKähler结构，其中的细节是失踪希钦的文件。我们证明了${\mathcal M}$允许三个前量子线丛，对应于三种辛形式。我们使用Quillen的行列式线丛的建设，并表明，这些prequantum线丛的Quillen曲率是成比例的每一个上述辛形式。前量子线丛相对于它们各自的复结构是全纯的。在模空间光滑的情况下，我们证明了这些预量子线丛是如何从具有复规范群的Chern-Simons规范理论的上循环线丛导出的。
We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira-Spencer gravity. We explain a new anomaly cancellation mechanism at all loops in perturbation theory for open-closed topological B-model. At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type I version of Kodaira-Spencer theory in complex dimensions 3 and 5. In complex dimension 5, we show that it can only be coupled consistently at the quantum level to holomorphic Chern-Simons theory with gauge group SO(32). This is analogous to the Green-Schw…
We construct three dimensional N=1 Chern-Simons theories living on M2 branes probing Spin(7) cones. We consider Spin(7) manifolds obtained as quotients of Calabi-Yau four-folds by an anti-holomorphic involution, following a construction by Joyce. The corresponding Chern-Simons theories can be obtained from N=2 theories by an orientifolding procedure. These theories are holographically dual to M theory solutions AdS_4 \times H, where the weak G_2 manifold H is the base of the Spin(7) cone.
We use the 3d-3d correspondence together with the DGG construction of theories $T_n[M]$ labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integral as well as the 3d index, which is k=0. The Chern-Simons partition functions correspond to partition functions of $T_n[M]$ on squashed lens spaces L(k,1). At any k, they admit a holomorphic-antiholomorphic factorization, corresponding to the decomposition of L(k,1) into two so…
R-matrices from Feynman Diagrams in 5d Chern-Simons Theory and Twisted M-theory
本文研究描述扭曲M理论的5维非对易拓扑全纯陈-西蒙斯理论中的R-矩阵类似物，通过费曼图计算关联函数与余积，揭示其与Miura算子、矩阵扩展$W_{\text{∞}}$-代数等的关联。
Chern-Simons Gauge Theory As A String Theory
某些二维拓扑场论可以被解释为弦论背景，在这种背景下，通常的鬼与物质的退耦并不成立。像普通的弦模型一样，这些模型有时也可以被赋予时空解释。例如，三维陈-西蒙斯规范理论可以作为弦论出现。在这种情况下，世界片模型涉及拓扑西格玛模型。瞬子对西格玛模型的贡献引起了时空陈-西蒙斯理论中的威尔逊线插入。陈-西蒙斯理论的某种全纯类似物也可以作为弦论出现。
It has recently pointed out that a four-dimensional analog of Chern-Simons theory provides an elegant framework for understanding integrable models with spectral parameters. The goal of this short note is to better understand the relation of this theory to the more standard three-dimensional Chern-Simons theory. We point out that two Chern-Simons theories, in four dimensions and three dimensions, are related by a novel T-duality in field theory. We then discuss this T-duality in string theory. Our T-duality prescription applies to a more general class of topological quantum field theories, pro…
On the Structure of Wave Functions in Complex Chern-Simons Theory
本文研究复杂陈-西蒙斯理论中波函数结构，揭示holomorphic块的隐藏整性结构，并通过图八结和三扭转结的例子验证了波函数的显式结果。
Chern-Simons and Twisted Supersymmetry in Higher Dimensions
我们描述了三维、五维、七维和九维的特殊超对称规范理论，它们在二、四、六和八重上的紧化产生了全纯丛的模空间上的超对称量子力学和/或更高维中的瞬子方程的类似物的解.这些理论可能发生在包裹特殊完整流形的D-膜的世界体积上。我们还讨论了与物质有关的理论。
A target space string field theory formulation for open and closed B-model is provided by giving a Batalin-Vilkovisky quantization of the holomorphic Chern-Simons theory with off-shell gravity background. The target space expression for the coefficients of the holomorphic anomaly equation for open strings are obtained. Furthermore, open/closed string duality is proved from a judicious integration over the open string fields. In particular, by restriction to the case of independence on continuous open moduli, the shift formulas of [7] are reproduced and shown therefore to encode the data of a c…
In the previous paper arXiv:1711.07122, we show that a holomorphic zero-mode wave function in abelian Chern-Simons theory on the torus can be considered as a quantum version of a modular form of weight 2. Motivated by this result, in this paper we consider an action of a Hecke operator on such a wave function from a gauge theoretic perspective. This leads us to obtain some physical views on the Hecke operators in number theory.
The partition function of complex Chern-Simons theory on a 3-manifold with torus boundary reduces to a finite dimensional state-integral which is a holomorphic function of a complexified Planck's constant $τ$ in the complex cut plane and an entire function of a complex parameter $u$. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a uniq…
Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativit…
Intersecting Surface Operators in 6d Holomorphic Field Theories
通过研究六维全纯场论中的相交曲面算子，计算关联函数得到类似有理R矩阵准经典展开的领头项算子，并证明其满足Yang-Baxter型关系，同时导出曲面算子手征代数的余积。
Quantization of topological-holomorphic field theories: local aspects
本文在模型流形R^m×C^n上严格构造拓扑-全纯混合场论的微扰单圈量子化，证明当m≥1时单圈反常障碍消失，并由此得到纯N=2四维超对称Yang-Mills理论扭曲在单圈水平上存在精确有限量子化。
On the renormalization and quantization of topological-holomorphic field theories
本文研究拓扑-全纯场论的重整化和量子化问题，证明在模型流形R^{d'}×C^d上此类混合理论的紫外有限性，并展示两种消失结果：当d'=1时，奇环量子化障碍消失；当d'>1时，所有障碍消失，允许定义量子可观测量的因子化代数结构。
Bosonic quantum field theories with holomorphic action functionals are realized by two types of constructions involving supersymmetric quantum field theories, compactified on an interval in one type and compactified on a disk and deformed in the other. We establish the equivalence between the two types of constructions by reducing the disk to the interval and the interval to a point. As examples, we discuss constructions of zero-dimensional gauged sigma model, gauged quantum mechanics, gauged symplectic bosons in two dimensions, and Chern-Simons theory and its higher-dimensional variants.
We derive formulas for the classical Chern-Simons invariant of irreducible $SU(n)$-flat connections on negatively curved locally symmetric three-manifolds. We determine the condition for which the theory remains consistent (with basic physical principles). We show that a connection between holomorphic values of Selberg-type functions at point zero, associated with R-torsion of the flat bundle, and twisted Dirac operators acting on negatively curved manifolds, can be interpreted by means of the Chern-Simons invariant. On the basis of Labastida-Marino-Ooguri-Vafa conjecture we analyze a represen…
Renormalisation and the Batalin-Vilkovisky formalism
本文给出了紧致流形上某些量子场论的重整化方法。例子包括杨-米尔斯理论（仅在4维），陈-西蒙斯理论和全纯陈-西蒙斯理论。该方法是在Batalin-Vilkovisky形式主义的框架内。陈-西蒙斯理论在某种程度上是重正化的，它尊重所有的对称性（直到同伦）。这就产生了光滑流形的一个不变量：一个关于流形的上同调的代数结构，张量为一个李代数，它是自然的李无穷大结构的一个“高圈”的丰富。
We derive the six-dimensional (1,0) effective action arising from F-theory on an elliptically fibered Calabi-Yau threefold with multiple sections. The considered theories admit both non-Abelian and Abelian gauge symmetries. Our derivation employs the M-theory to F-theory duality in five-dimensions after circle reduction. Five-dimensional gauge and gravitational Chern-Simons terms are shown to arise at one-loop by integrating out massive Coulomb branch and Kaluza-Klein modes. In the presence of a non-holomorphic zero section, we find an improved systematic for performing the F-theory limit by u…
Ambitwistor Yang-Mills Theory Revisited
本文基于同伦代数视角，在CR ambitwistor空间上构造了最大超对称Yang-Mills理论的扭曲全纯Chern-Simons作用，通过拟同构证明了两者半经典等价并证实了文献猜想。
We show that conformal Chern-Simons gravity in three dimensions has various holographic descriptions. They depend on the boundary conditions on the conformal equivalence class and the Weyl factor, even when the former is restricted to asymptotic Anti-deSitter behavior. For constant or fixed Weyl factor our results agree with a suitable scaling limit of topologically massive gravity results. For varying Weyl factor we find an enhancement of the asymptotic symmetry group, the details of which depend on certain choices. We focus on a particular example where an affine u(1) algebra related to holo…
We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator 'A-hat' is a quantization of the knot complement's classical A-polynomial A(l,m). The construction proceeds by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in TQFT to put them back together. We advocate in particular that, properly interpreted, "gluing = symplectic reduction." We also arrive at a n…
Teichmüller TQFT is a unitary 3d topological theory whose Hilbert spaces are spanned by Liouville conformal blocks. It is related but not identical to PSL(2,R) Chern-Simons theory. To physicists, it is known in particular in the context of 3d-3d correspondence and also in the holographic description of Virasoro conformal blocks. We propose that this theory can be defined by an analytically-continued Chern-Simons path-integral with an unusual integration cycle. On hyperbolic three-manifolds, this cycle is singled out by the requirement of invertible vielbein. Mathematically, our proposal transl…
Integrable Deformations from Twistor Space
本研究基于6维全纯陈-西蒙斯理论，首次完整构建形变sigma模型的可积理论菱形结构，得到全新4维可积场论与多参数2维可积模型，验证了两种可积场论起源途径的统一性。
R-matrices and Miura operators in 5d Chern-Simons theory
本研究在5维非交换gl(1)陈-西蒙斯理论中，通过计算拓扑线缺陷与全纯面缺陷相交的期望值，从第一性原理推导出W-代数与Y-代数的Miura算符，并证明该满足Yang-Baxter方程的期望值与Miura算符及其乘积精确匹配。
Chern-Simons States at Genus One
本文对三维环面$\times$线上的$SU(2)$ Chern-Simons理论的薛定谔图像量子化进行了严格的分析，并引入了Wilson线.定义为环面上光滑$su(2)$联络的规范协变全纯泛函的量子态，用满足附加条件的次数$2k$ θ函数表示.通过将半稳定$su(2)$ -联络空间分裂成9个子流形，并分析了状态在4个余维$1$层上的行为，得到了这些条件.我们构建Knizhnik-Zamolodchikov-Bernard连接允许比较不同复杂结构的环面和威尔逊线的不同位置的状态。通过让两个威尔逊线走到一起，我们证明了一个递归关系的状态空间的尺寸，连同（未经证明）的情况下，自旋$\s>{_1\over^2}$水平的状态意味着Verlinde维数公式。
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