A large N phase transition in the continuum two dimensional SU(N) X SU(N) principal chiral model
R. Narayanan, H. Neuberger, E. Vicari
用数值方法证明了二维连续大N主手征模型在关联长度量级的临界距离处存在光滑两点函数的相变。
It is established by numerical means that the continuum large N principal chiral model in two dimensions has a phase transition in a smoothed two point function at a critical distance of the order of the correlation length.
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A non perturbative approach of the principal chiral model between two and four dimensions
利用非微扰Wilson类重整化群方程研究了二维和四维主手征模型。因此，我们能够在整个尺寸范围内跟踪有效耦合常数的演变，而无需求助于任何类型的小参数展开。这使我们能够确定其三维临界物理，并解决长期存在的差异，不同的微扰方法，其特点的主要手征模型所属的类模型。
In this paper, which is the second one in a series of two papers, we shall present two more solutions, non-minimal ones, for the Z_{3} parafermionic chiral algebra with Delta_{psi}=Delta_{psi^{+}}=8/3, psi(z), psi^{+}(z) being the principal parafermionic fields.
The Wilson Renormalization Group Approach of the Principal Chiral Model around Two Dimensions
在精确重整化群方程的局域势近似下研究了二维主手征Ginzburg-Landau-Wilson模型。该模型与经典阻挫自旋系统的长距离物理有关，它与相应的非线性西格玛模型具有相同普适类的不动点。这就揭示了受抑自旋系统的不同微扰方法之间长期存在的差异。
We define a two-parameter family of integrable deformations of the principal chiral model on an arbitrary compact group. The Yang-Baxter sigma-model and the principal chiral model with a Wess-Zumino term both correspond to limits in which one of the two parameters vanishes.
Lattice chiral fermions from continuum defects
我们考虑余维2的缺陷是否能产生新的格手征费米子。
Master Ward Identity for Nonlocal Symmetries in D=2 Principal Chiral Models
本文用路径积分方法导出了二维主手征模型中与无穷多个非定域守恒律相联系的反常主沃德恒等式
Motivated by our previous study of the Twisted Eguchi-Kawai model for non minimal twists, we re-examined the behaviour of the reduced version of the two dimensional principal chiral model. We show that this single matrix model reproduces the same features as the standard lattice model. In particular, scaling towards the continuum limit, the correct value of the internal energy, the magnetic susceptibility and the mass gap. Given our capacity to reach larger values of $N$, we use the reduced model to study the nature and properties of its large $N$ phase transition existing at intermediate coup…
$SU(2)$ principal chiral model with tensor renormalization group on a cubic lattice
本文用tTRG和ATRG方法研究三维SU(2)主手征场模型的连续相变与热力学量，结果与蒙特卡洛及重整化群标度一致，为张量网络研究四维有限密度QCD奠定基础。
Reflection Factors for the Principal Chiral Model
本文考虑半直线上SU（2）主手征模型（$k=1$水平）的尺度不变边界条件.通过对红外极限共形场论的研究，并与Kondo问题进行比较，我们提出了一组允许边界条件和相应的反射因子。
Hidden Symmetries of the Principal Chiral Model and a Nonstandard Loop Algebra
我们研究的精确结构的循环代数的'敷料'对称性的主要手征模型，并讨论了一个新的无限集的阿贝尔对称性的场方程保持一个辛形式的空间的解决方案。
We study the sigma model with $SU(N)\times SU(N)$ symmetry in 1+1 dimensions. The two- and four-particle form factors of the Noether current operators are found, by combining the integrable-bootstrap method with the large-$N$ expansion.
Form factors for principal chiral field model with Wess-Zumino-Novikov-Witten term
构造了水平1上WZNW项的$SU(2)$主手征场模型描述的无质量模型的能量动量张量迹的形状因子。我们解释了如何将这种构造推广到一类可积的无质量模型，包括从三临界流到临界流的伊辛模型。
A Concentration of Measure Phenomenon in the Principal Chiral Model
本文利用测度集中现象研究主奇点模型在大N极限下的行为，发现其配分函数等价于自由质量型理论。
Using the Poisson current algebra of the supersymmetric principal chiral model, we develop the algebraic canonical structure of the model by evaluating the fundamental Poisson bracket of the Lax matrices that fits into the rs matrix formalism of non-ultralocal integrable models. The fundamental Poisson bracket has been used to compute the Poisson bracket algebra of the monodromy matrix that gives the conserved quantities in involution.
Lax pair and Darboux transformation of noncommutative U(N) principal chiral model
利用单参数平坦联络族给出了U（N）主手征模型的Lax形式的非对易推广。进一步利用Lax形式导出了一组参数非对易贝克隆变换和一组无穷守恒量。从拉克斯对出发，我们导出了该模型的达布变换的非对易形式。
Zero-Curvature Formalism of Supersymmetric Principal Chiral Model
研究了超主手征模型超场的单参数变换族，得到了超主手征模型的不同零曲率表示。将参数变换与超Riccati方程联系起来，导出了一组无限的局部和非局部守恒律。一个Lax表示的模型，从而产生一个超空间monodromy运营商。
Boundary scattering in the principal chiral model
非正式地介绍了我们最近关于带边界的主手征模型的工作。我们发现经典可积边界条件和量子边界S-矩阵都可以用对称空间G/H来分类。的连接被解释为存在一个扭曲的Yangian代数的非本地收费。
Collective field approach to gauged principal chiral field at large N
考虑了主手征场与无动力学项的规范场相互作用的格点模型。该模型可以看作是有限温度下格点规范理论的强耦合极限。在大N极限下导出了集体场变量的完整方程组，并研究了模型的相结构。
From Principal Chiral Model to Self-dual Gravity
证明了SU $(N)$主手征模型的作用量导致了$N \to {\infty}$对Husain天体方程作用量的极限。本文考虑了Hilbert空间中的主手征模型$L^2(\Re^1)$，并证明了在这种情况下手征方程等价于Husain天体方程的Moyal变形。通过$L^2(\Re^1)$中的李代数表示，给出了寻找后一个方程的解的新方法。
Conserved Charges in the Principal Chiral Model on a Supergroup
研究了目标空间为紧李群的1+1维经典主手征模型.给出了李超代数的一个不变张量，给出了如何构造局部守恒荷。我们计算了这些电流的超泊松括号，并认为它们是非线性生成的。我们展示了如何在对合中导出无穷多个局部电荷。我们证明，这些费用泊松交换与非本地收费的模型。
We explain the physical role of non-perturbative saddle points of path integrals in theories without instantons, using the example of the asymptotically free two-dimensional principal chiral model (PCM). Standard topological arguments based on homotopy considerations suggest no role for non-perturbative saddles in such theories. However, resurgence theory, which unifies perturbative and non-perturbative physics, predicts the existence of several types of non-perturbative saddles associated with features of the large-order structure of perturbation theory. These points are illustrated in the PC…
The SO(N) principal chiral field on a half-line
研究了SO（N）主手征模型在半直线上的可积性，发现混合Dirichlet/Neumann边界条件（以及纯Dirichlet或Neumann边界条件）导致无穷多个守恒荷经典地对合.我们使用的异常计数方法表明，至少有一个非平凡的例子生存的量化，比较我们的结果与建议的反射矩阵，并在此基础上，提出了一些初步的意见预期的边界束缚态。
Principal Chiral Model on Superspheres
研究了奇维超球上主手征模型（PCM）的谱随曲率半径R的变化。对于S^{3}上的体积填充膜|2}，我们计算了作为R的函数的精确边界谱。扩展到更高的维超球进行了讨论，但没有进行详细的。我们的结果为超球PCM和OSP（2S+2）之间的强弱耦合对偶性提供了非常有说服力的证据|2S）Gross-Neveu模型最近由Candu和Saleur在arXiv中验证：0801.0444。
Conserved Quantities in Noncommutative Principal Chiral Model with Wess-Zumino Term
构造了U（N）主手征模型的Wess-Zumino非对易扩张，并利用Brezin { \it et.al} \cite{BIZZ}的迭代过程得到了该模型的定域和非定域守恒量.我们还提出了等价的描述为Lax形式主义的模型。我们展开的领域微扰和零阶和一阶运动方程，零曲率条件，迭代方法，Lax形式主义，本地和非本地守恒量。
Hidden Symmetries of the Principal Chiral Model Unveiled
通过将二维U（N）主手征模型与一个简单的线性系统联系起来，我们得到了解的自由场参数化。自由场数据上的明显对称变换给出了模型的对称性。这样，所有已知的“隐藏对称性”和贝克隆变换，以及一系列新的对称性就产生了。
If the space of minima of the effective potential of a weakly coupled 2d quantum field theory is not connected, then a mass gap will be nonpertubatively generated. As examples, we consider two sigma models compactified on a small circle with twisted boundary conditions. In the compactified SO(3) model the vacuum manifold consists of two points and the mass gap is nonperturbative. In the case of the compactified SU(2) principal chiral model the vacuum manifold is a single circle and the mass gap is perturbative.
A continuum limit of the chiral Jacobian in lattice gauge theory
本文研究了Hasenfratz，Laliena和Niedermayer以及Lüscher在格点规范理论中提出的指数定理和手征雅可比矩阵对手征雅可比矩阵和反常矩阵的连续形式的影响。我们在不考虑微扰展开的情况下，取了格点雅可比因子的一个连续极限，并仅利用格点Dirac算子的一般性质恢复了连续理论的结果.这个过程是基于一组定义良好的规则，从而提供了一种替代的方法，在连续介质理论的手征雅可比矩阵和相关的异常的传统分析。通过使用显式形式的格Dirac算子介绍了纽伯格，它满足Ginsparg-Wilson关系，我们说明了我们的计算在一些细节。本文还从这一观点对有限截断的指数定理作了简要的评述。
Non-commutative principal chiral models
扭曲的Eguchi-Kawai约化手征模型被证明形式上等价于U（1）非对易父理论。非对易理论描述了膜系统中非对易带电快子场的真空动力学。为了与连续非对易理论联系起来，简化模型需要一个双标度大N极限。我们展示了一个可能的限制程序，我们建议调查数字。我们的数值计算结果表明，与概述的程序基本一致
The exact mass-gaps of the principal chiral models
对于与所有经典李代数相关的主要手征西格玛模型，我们发现了质量间隙（物理粒子质量与$Λ$参数之比）的精确表达式。该计算是基于自由能的比较，在存在源耦合到保守的电荷的理论计算的两种方式：通过热力学贝特Anglands从精确的散射矩阵和直接在微扰理论。计算提供了一个非平凡的测试的形式的确切的散射矩阵。
A large class of integrable deformations of the Principal Chiral Model, known as the Yang-Baxter deformations, are governed by skew-symmetric R-matrices solving the (modified) classical Yang-Baxter equation. We carry out a systematic investigation of these deformations in the presence of the Wess-Zumino term for simple Lie groups, working in a framework that treats both inhomogeneous and homogeneous deformations on the same footing. After analysing the cohomological conditions under which such a deformation is admissible, we consider an action for the general Yang-Baxter deformation of the Pri…
Large-N Principal Chiral Model in Arbitrary External Fields
本文给出二维大 N $SU(N)$ 主手征模型在任意化学势和耦合强度下的真空显式解，并揭示其弱耦合微扰一致性与强耦合下的空间维度涌现。
Typically, the exact ground state energy of integrable models at finite volume can be computed using two main methods: the thermodynamic Bethe ansatz approach and the lattice discretization technique. For quantum sigma models (with non-ultra local Poisson structures) the bridge between these two approaches has only been done through numerical methods. We briefly review these two techniques on the example of the SU(2) principal chiral field model and derive a single integral equation based on the Faddeev-Reshetikhin discretization of the model. We show that this integral equation is equivalent …
Quantum Mechanically Induced Wess-Zumino Term in the Principal Chiral Model
在二维主手征模型中，Wess-Zumino项可以通过量子力学方法引入，使得耦合常数$λ^2 = 8π/|k|$达到临界值的模型在量子水平上转变为Wess-Zumino-Novikov-Witten模型. Wess-Zumino项来自隐藏在原始模型的位形空间中的球体上可能的不等价量子化。结果表明，在不等价量子化中，球上的Dirac位势在整个位形空间中是Wess-Zumino项.
We analyze, and prove, the associativity of the new Z_{3} parafermionic chiral algebra which has been announced some time ago, with principal parafermionic fields having the conformal dimension Δ_ψ=8/3. In doing so we have developed a new method for analyzing the associativity of a given chiral algebra of parafermionic type, the method which might be of a more general significance than a particular conformal field theory studied in detail in this paper. Still, even in the context of our particular chiral algebra, of Z_{3} parafermions with Δ_ψ=8/3, the new method allowed us to give a proof of …
Boundary scattering, symmetric spaces and the principal chiral model on the half-line
本文研究了半直线上主手征模型的可积边界条件和边界杨-巴克斯特方程的有理解。在每一个案例中，我们都发现了一种联系，（I型，黎曼，整体）对称空间G/H：存在一类边界场被限制在H的陪集内的可积BC，这些BC被G/H × G/H参数化，在所有由G/H参数化的经典G的定义表示中存在BYBE的有理解;利用这些，我们提出了边界S-矩阵的主手征模型，参数化G/H × G/H，这对应于我们的边界条件。
Current Algebra and Integrability of Principal Chiral Model on the World-sheet with General Metric
研究了二维世界片上具有一般度规的主手征模型的经典流代数。我们开发的哈密顿形式主义，并确定电流之间的泊松括号的形式。然后我们确定了Lax联络的Poisson括号，并证明了这个Poisson括号不依赖于世界片度量。我们还研究了这个模型的Nambu-Gotto形式。我们证明了Lax连接的存在并确定了它们的Poisson括号。
An elliptic integrable deformation of the Principal Chiral Model
本研究提出SL_R(N)群上主手征模型的双参数椭圆可积σ模型，推广Cherednik的N=2构造，给出其Lax联络与R-矩阵，阐释其源自环面等变半全纯4维Chern-Simons理论，为同类模型构建开辟路径。
We study $η$-deformations of principal chiral model (PCM) from the viewpoint of a 4D Chern-Simons (CS) theory. The $η$-deformed PCM has originally been derived from the 4D CS theory by Delduc, Lacroix, Magro and Vicedo [arXiv:1909.13824]. The derivation is based on a twist function in the rational description. On the other hand, we start with a twist function in the trigonometric description and discuss possible boundary conditions. We show that a certain boundary condition reproduces the usual $η$-deformed PCM and another one leads to a new kind of Yang-Baxter deformation.
2D Principal Chiral Field at Large N as a Possible Solvable 2D String Theory
给出了无限大秩群流形上二维主手征场的精确显式解。明确地找到了任意大小的外部诺特场的基态的能量。在小场下，我们发现在质量隙处基态能量存在一个反对数奇异性，这表明在$N=\infty$处理论谱中包含扩展的物体而不是点状粒子.
Principal chiral model scattering and the alternating quantum spin chain
考虑了具有${q_{+}\over 2}$，${q_{-} \over2}$自旋的临界交替量子自旋链.使用Bethe animals技术，我们找到明确的表达式的$S$ -矩阵的模型。我们表明，在$q_{\pm} \rightarrow \infty$的限制，我们的结果与获得的主要手征模型水平一，LL（RR）LR散射。我们还研究了该模型的束缚态的散射，并恢复了XXZ（sine-Gordon）模型的结果。
Local conserved charges in principal chiral models
研究了1+1维主手征模型中的局域守恒荷.对于基础群的每个全对称不变张量，都有一个经典守恒的局部荷。这些本地收费被证明是在对合与非本地的Yangian收费。然后研究了局部荷的Poisson括号代数。对于每一个经典代数，一个无限集的本地收费与自旋等于指数模的Coxeter数的构造，它表明，这些交换与彼此。简要评论的证据，并影响，生存的这些费用在量子理论。
We present a finite set of equations for twisted PCF model. At the special twist in the root of unity we demonstrate that the vacuum energy is exactly zero at any size L. Also in SU(2) case we numerically calculate the energy of the single particle state with zero rapidity, as a function of L.
Operator Products in the SU($\infty$) Principal Chiral Model
本文研究SU($\infty$)主手征模型的算符乘积，回顾局域算符形状因子与精确两点函数，并讨论非真空乘积，旨在从公理化形式体系重建拉格朗日场论。
We obtain exact matrix elements of physical operators of the (1+1)-dimensional nonlinear sigma model of an SU(N)-valued bare field, in the 't Hooft limit N goes to infinity. Specifically, all the form factors of the Noether current and the stress-energy-momentum tensor are found with an integrable bootstrap method. These form factors are used to find vacuum expectation values of products of these operators.
Perturbative expansions in many physical systems yield 'only' asymptotic series which are not even Borel resummable. Interestingly, the corresponding ambiguities point to nonperturbative physics. We numerically verify this renormalon mechanism for the first time in two-dimensional sigma models, that, like four-dimensional gauge theories, are asymptotically free and generate a strong scale through dimensional transmutation. We perturbatively expand the energy through a numerical version of stochastic quantization. In contrast to the first energy coefficients, the high order coefficients are ind…
Quantum equivalence of the Freedman-Townsend model and the principal chiral $σ$-model
本文通过Batalin-Vilkovisky量子化方法处理Freedman-Townsend模型，并利用路径积分论证证明其与主手性$\sigma$-模型的量子等价性。
Principal Chiral Field at Large N
给出了无限大秩群流形上二维主手征场的精确显式解。明确地找到了任意大小的外部诺特场的基态的能量。精确的Gell-Mann - Low函数在大的场值下表现出与微扰计算一致的渐近自由行为。计算了重整化荷中的微扰展开系数。它们以阶乘的方式生长，其顺序表明存在renormalons。在小场下，我们发现在质量隙处基态能量存在一个反对数奇异性，这表明在$N=\infty$处理论谱中包含扩展的物体而不是点状粒子.
A multi-parameter integrable deformation of the principal chiral model is presented. The Yang-Baxter and bi-Yang-Baxter sigma-models, the principal chiral model plus a Wess-Zumino term and the TsT transformation of the principal chiral model are all recovered when the appropriate deformation parameters vanish. When the Lie group is SU(2), we show that this four-parameter integrable deformation of the SU(2) principal chiral model corresponds to the Lukyanov model.
It is commonly asserted that there is a \hat G\times G centreless Kac-Moody extension of the manifest G\times G global symmetry of the two-dimensional principal chiral model (PCM) for the group manifold G. Here, we show that the symmetry is in fact larger, namely \hat G\times \hat G, the full centreless Kac-Moody extension of the entire manifest G\times G global symmetry. Extending previous results in the literature, we also obtain an explicit realisation of the Virasoro-like symmetry of the PCM, generated by K_n=L_{n+1} - L_{n-1} for both positive and negative n. We show that these generators…
SO(3) vortices as a mechanism for generating a mass gap in the 2d SU(2) principal chiral model
我们提出了一种机制，可以在任意小的温度下在SU（2）手征自旋模型中产生质量间隙。利用扭曲边界条件引入的外Z（2）通量的性质，给出了质量间隙非零的充分条件。这个条件又转化为一个有效的对偶伊辛模型，外部磁场由SO（3）涡旋产生。我们证明了在有效伊辛模型中有一个非零的磁场就足以使SU（2）系统有一个质量隙。我们还表明，某些涡旋相关不等式，如果满足，将意味着一个非零的有效磁场。最后，我们给出了所需的相关不等式的一些可验证性参数和Monte Carlo证据。
A heterotic integrable deformation of the principal chiral model
受杂化σ模型玻色部分及相关哈密顿表述启发，研究提出二维主手征模型的杂化规范场形变新模型，通过构造Lax对和满足经典Yang-Baxter方程的近标准形式R矩阵证明其经典可积性。
Recently, Kazakov, Gromov and Vieira applied the discrete Hirota dynamics to study the finite size spectra of integrable two dimensional quantum field theories. The method has been tested from large values of the size L down to moderate values using the SU(2) x SU(2) principal chiral model as a theoretical laboratory. We continue the numerical analysis of the proposed non-linear integral equations showing that the deep ultraviolet region L -> 0 is numerically accessible. To this aim, we introduce a relaxed iterative algorithm for the numerical computation of the low-lying part of the spectrum …
Boundary scattering in the SU(N) principal chiral model on the half-line with conjugating boundary conditions
本文研究了半直线上SU（N）主手征模型在一组特殊边界条件下的动力学行为。在以前的工作中，这些BC已经被证明对应于边界散射矩阵（K-矩阵），它们是表示共轭的，并且其矩阵结构对应于对称空间SU（N）/SO（N）或SU（N）/Sp（N）之一。从整体粒子谱和粒子在矢量表示中的K-矩阵出发，我们构造了粒子在高秩表示中散射离开边界的K-矩阵。然后，我们进行了分析的物理带极结构，并提供了一套完整的边界科尔曼-图恩机制，这些极点不对应于粒子耦合到边界。我们发现，该模型没有非平凡的边界状态。
Conserved Charges and Supersymmetry in Principal Chiral Models
我们报告调查的本地（和非本地）收费玻色子和超对称的主手征模型在1+1维。在玻色PCM中，对于基础群的每个对称不变张量，都存在经典守恒的局部荷。这些都与非本地Yangian收费通勤。局域电荷之间的代数关系则更为微妙。我们给出了一个通用的公式，无限套相互交换的本地收费与自旋等于指数的基础古典代数模其Coxeter数。这些结果中的许多扩展到超对称PCM，但与当地保守的费用与反对称不变量的李代数。我们简单地评论了玻色子和超相变材料中局域电荷的量子守恒。
We discuss noncollinear magnetic phenomena whose local order parameter is characterized by more than one spin vector. By focusing on the simple cases of 2D triangular and 3D pyrochlore lattices, we demonstrate that their low-energy theories can be described by a one-parametric class of sigma models continuously interpolating between the classical Heisenberg model and the principal chiral model ${\rm Tr} \,\left( \partial_a U \partial_a U^\dagger\right)$ for all $U\in \text{SU}(2)$. The target space can be viewed as a U$(1)$ fibration over the $CP(1)$ space. The 3D version of our model is furth…
We initiate a systematic, non-perturbative study of the large-$N$ expansion in the two-dimensional $\text{SU}(N)\times \text{SU}(N)$ Principal Chiral Model (PCM). Starting with the known infinite-$N$ solution for the ground state at fixed chemical potential \cite{Fateev:1994dp,Fateev:1994ai}, we devise an iterative procedure to solve the Bethe ansatz equations order by order in $1/N$. The first few orders, which we explicitly compute, reveal a systematic enhancement pattern at strong coupling calling for the near-threshold resummation of the large-$N$ expansion. The resulting double-scaling li…
In this letter we explore different representations of the SU(2) principal chiral model on the lattice. We couple chemical potentials to two of the conserved charges to induce finite density. This leads to a complex action such that the conventional field representation cannot be used for a Monte Carlo simulation. Using the recently developed Abelian color flux approach we derive a new worldline representation where the partition sum has only real and positive weights, such that a Monte Carlo simulation is possible. In a second step we transform the model to new dual variables in the Kramers-W…
Continuum and Lattice Overlap for Chiral Fermions on the Torus
应用重叠公式计算了二维环面上扭曲边界条件下的手征行列式。我们首先评估连续谱重叠，它是收敛的和良好的定义，并产生正确的弦论结果的实部和虚部的有效作用。然后，我们表明，晶格版本的重叠给连续重叠的结果在极限时，晶格间距趋于零，并在该极限的subleading项是无关紧要的。
Integrable Generalized Principal Chiral Models
研究了群流形上具有特殊形式度量的二维非线性西格玛模型。我们解决这个特殊的一类西格玛模型的可积性问题。给出了群流形上度量的两个代数条件。这些条件的每一个解都定义了一个可积模型。虽然代数系统是超定的一般，我们给出两个例子的解决方案。我们找到了这些模型的Lax场，并计算了它们的泊松括号。我们还得到了一般模型的一阶重整化群（RG）方程。我们解决RG方程的例子，我们有，并表明，他们是可积的沿RG流。
The properties of (N X N)-matrix-valued-field theories, in the limit N goes to infinity, are harder to obtain than those for isovector-valued field theories. This is because we know less about the sum of planar diagrams than the sum of bubble/linear diagrams. Combining the 1/N-expansion with the axioms for form factors, exact form factors can be found for the integrable field theory of an SU(N)-valued field in 1+1 dimensions. These form factors can be used to find the vacuum expectation value of the product of two field operators. We briefly mention how the results can be applied to 2+1 dimens…
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