Andreas Gustavsson
We compute the partition function of five-dimension Abelian sYM including a graviphoton term on a five-torus. The result agrees with Abelian M5 brane partition function with zero characteristics on an associated six-torus.
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T-duality of string theory can be extended to the Poisson-Lie T-duality when the target space has a generalized isometry group given by a Drinfel'd double. In M-theory, T-duality is understood as a subgroup of U-duality, but the non-Abelian extension of U-duality is still a mystery. In this paper, we study membrane theory on a curved background with a generalized isometry group given by the $\mathcal{E}_n$ algebra. This provides a natural setup to study non-Abelian U-duality because the $\mathcal{E}_n$ algebra has been proposed as a U-duality extension of the Drinfel'd double. We show that the…
Anti-field Formalism and Non-Abelian Duality
在二维西格玛模型中实现非阿贝尔对偶性的行为必然导致额外的可约对称性。Batalin-Vilkovisky形式主义被用来处理这种新的对称性。在非阿贝尔对偶性方面，我们学到了宝贵的经验。我们强调，特别是对应于这种对称性的非阿贝尔对偶的鬼部门的影响。
Non-abelian self-duality from self-interaction
利用自相互作用机制推导出三维非阿贝尔自对偶作用量。
Abelian Duality and Abelian Wilson Loops
考虑一般黎曼紧致四维流形上的纯U（1）量子规范场理论.我们用阿贝尔威尔逊圈插入来计算配分函数。我们发现它的对偶协方差性质，并推导出拓扑选择规则。最后，我们表明，有明显的对偶性，必须假设存在扭曲的拓扑部门除了标准的untwisted之一。
Advent of Non-Abelian Vortices and Monopoles-- further thoughts about duality and confinement
本文综述了非阿贝尔涡旋和磁单极的最新研究进展，着重讨论了它们与禁闭和对偶性的关系。在这种情况下，一个非常新的非阿贝尔涡旋的建设，不动态阿贝尔化是至关重要的。
We start by describing two of the main proposals for duality in Abelian gauge theories, namely $F$(ield strength)-duality approach and the $S$% -duality formalism. We then discuss how $F$-duality and $S$-duality can be applied to the case of linearized gravity. By emphasizing the similarities and differences between these two type of dualities we explore the possibility of combining them in just one duality formalism.
Generalized Duality Symmetry of Non-Abelian Theories
量子Yang-Mills理论描述了对偶（$\tilde g$）和非对偶（$g$）电荷并揭示了广义对偶对称性，它是通过类比QED中的Zwanziger形式发展起来的。
Duality transformation of non-Abelian tensor gauge fields
对于非阿贝尔张量规范场，我们找到了对偶变换的另一种形式，它具有正变换和逆变换重合的性质。这种对偶变换具有所需的性质，即直接变换和逆变换将拉格朗日形式映射到彼此。
On Non-Abelian Duality in Sigma Models
给出了在弦背景上实现非阿贝尔对偶的一种方法。它表明，一个直接的推广熟悉的阿贝尔对偶诱导一个额外的局部对称性的规范不变量理论。非阿贝尔等距群被证明是扩大到一个非半单群。然而，在消除规范场以获得对偶理论时，新代数并不封闭。因此，规范固定程序变得有问题。这里提出的新方法避免了这些问题，并导致对偶理论在适当的对偶意义。
We find new type II backgrounds with non-relativistic symmetries via non-Abelian T-duality. First we consider geometries with Galilean symmetries in type IIA, which have been identified as non-relativistic generalizations of the ABJM background and massive IIA supergravities. We then consider the non-Abelian T-duality transformation on the backgrounds with Lifshitz symmetry constructed by Donos and Gauntlett. Using gauge/gravity duality we study aspects of the field theory dual to these backgrounds.
We construct the duality-symmetric actions for a large class of six-dimensional models describing hierarchies of non-Abelian scalar, vector and tensor fields related to each other by first-order (self-)duality equations that follow from these actions. In particular, this construction provides a Lorentz invariant action for non-Abelian self-dual tensor fields. The class of models includes the bosonic sectors of the 6d (1,0) superconformal models of interacting non-Abelian self-dual tensor, vector, and hypermultiplets.
Non-Abelian Duality and Canonical Transformations
在具有任意群G的主手征模型中，构造了产生非交换环的正则变换。一些评论有关的扩展到更一般的$σ$ -模型，如WZW模型，给出。
In this short article we develop recent proposals to relate Yang-Baxter sigma-models and non-abelian T-duality. We demonstrate explicitly that the holographic space-times associated to both (multi-parameter)-$β$-deformations and non-commutative deformations of ${\cal N}=4$ super Yang-Mills gauge theory including the RR fluxes can be obtained via the machinery of non-abelian T-duality in Type II supergravity.
Dual Non-Abelian Duality and the Drinfeld Double
弦论中非阿贝尔对偶的标准概念被推广到允许“非交换守恒电荷”的$\si$ -模型类。这样的$\si$ -模型可以与每一个李双代数$(\cg ,\cgt)$相关联，并且它们具有等距群当且仅当交换子$[\cgt,\cgt]$不等于$\cgt$ .在背景的扩大类中，非阿贝尔对偶{ \it是}一个在适当意义上的对偶变换。它交换了$\cg$和$\cgt$的角色，并且可以解释为相互对偶理论的相空间的辛同胚。我们给出了任何$(\cg,\cgt)$的非阿贝尔对偶变换的显式公式。阿贝尔模空间$O(d,d;{\bf Z})$的非阿贝尔类似物由对应的德林费尔德二重的所有极大迷向分解组成。
A canonical approach to S-duality in Abelian gauge theory
研究了一般四维流形上U（1）规范理论的电磁对偶性。这种理论的配分函数变换为比重的模形式。然而，在规范的方法，我们表明，S-对偶，T-对偶，是由一个规范的变换导致一个模块不变的配分函数。
Type IIA/IIB string duality for targets with abelian isometries
我们研究T-对偶变换的超对称弦与目标空间几何紧凑的阿贝尔等距。我们认为这些模型的配分函数，我们表明，虽然T-对偶不是一个对称性，由于异常，它涉及IIA型IIB型字符串。这样，我们将相应的结果推广到一般情况下的非平凡的背景与阿贝尔等距和世界片的任何属的环面紧化。
Abelian duality in WZW models
我们分析了单连通群的WZW模型的交换T-对偶。我们证明了对偶理论是原理论的某个轨道，并检验了它是共形不变的。我们确定了对偶理论的谱，并证明了它与原理论的谱是一致的。
Non--Abelian Duality, Parafermions and Supersymmetry
非阿贝尔对偶关系的超对称性进行审查。当等距群对复杂结构的作用是非平凡的时，利用路径有序Wilson线，在对偶后非局部地实现了扩展超对称性。原型的例子被详细考虑，超-Kahler度量与SO（3）等距和超对称WZW模型。对于后者，二元性后产生的超对称性非局域实现中的自然对象是经典的非阿贝尔副费米子。本文还建立了WZW模型及其非阿贝尔模型关于向量子群的典范等价性。
We briefly review the essential points of our recent work in non-Abelian T-duality. In particular, we show how non-abelian T-duals can effectively describe infinitely high spin sectors of a parent theory and how to implement the transformation in the presence of non-vanishing Ramond fields in type-II supergravity.
We examine non-Abelian T-duality on backgrounds preserving N=1 supersymmetry. We show how backgrounds with SU(3)-structure are mapped to backgrounds with SU(2)-structure. We provide the transformation rules for the corresponding pure spinors. We use these techniques to construct new flavoured solutions of type IIA supergravity.
AdS/CFT Duality and Conformality for Non-Abelian Manifold
一个轮廓的保形方法的规范层次，包括使用非阿贝尔orbifolds给统一的模型的左右型。
Generalized Abelian S-duality and coset constructions
电磁对偶和高维类似物作为对称性在广义陪集结构中得到，类似于Rocek和Verlinde描述的二维陪集模型的轴向量对偶。我们还研究了d-流形中p-形式和（d-p-2）-形式之间的对偶性的整体性质。特别是，模对偶异常由四维和二维中的欧拉特征支配。Wilson线算子插入的对偶变换也被考虑。
On Non-Abelian Duality
本文对非阿贝尔对偶作了一般性的研究。我们首先确定一个可能的障碍，共形不变性的对偶理论的非半单群。对任意无反常子群的Wess-Zumino-Witten（WZW）模型构造了精确的非交换对偶，并对陪集共形场论进行了相应的推广.我们刻画了具有手征等距的$σ$ -模型的精确非阿贝尔对偶，并将对偶的标准概念推广到WZW模型的反常子群，从而给出了构造左右不同群的对偶变换的方法.我们还提出了一些新的例子，四维引力瞬子的非阿贝尔对偶。
Non-Abelian Duality Based on Non-Semi-Simple Isometry Groups
分析了建立在非半单等距群上的非阿贝尔对偶变换。首先给出了原始非线性西格玛模型与其非阿贝尔对偶模型等价的条件。等距李代数的不变和非退化的双线性形式的存在是这个等价的关键。构造了具有非半单等距的共形不变西格玛模型的非阿贝尔对偶，证明了其β函数为零.本研究解决了一个明显的障碍，共形不变性的西格玛模型通过非阿贝尔对偶的基础上非半单群。
Exact non-abelian duality
我们研究WZNW模型的非阿贝尔规范。当规范群是半单时，我们能够给出对偶共形场论的精确公式，适用于$k$能级的所有值。然后将所得结果应用于弦论中的非阿贝尔目标空间对偶性，表明如果规范群是半单的，则标准公式在低能极限下是量子力学定义良好的。
We analyse super non-Abelian T-duality for principal chiral models, symmetric space sigma models, and semi-symmetric space sigma models for general Lie supergroups. This includes T-duality along both bosonic and fermionic directions. As an example, we perform the explicit dualisation of the OSp(1|2) principal chiral model, and, whilst the target superspace of this model is a three-dimensional supergravity background, we find that its super non-Abelian T-dual falls outside the class of such backgrounds.
A Vector Non-abelian Chern-Simons Duality
已知阿贝尔Chern-Simons规范理论具有“$S$ -自对偶”作用量，其中它的耦合常数$k$是反演的{ \it，即} $k \leftrightarrow {1 \over k}$。在这里，一个向量非阿贝尔对偶被发现在纯非阿贝尔的陈-西蒙斯作用在经典水平。将对偶Chern-Simons作用量降维为二维，构成了文献中已经给出的对偶Wess-Zumino-Witten作用量。
Non-Abelian Duality for Open Strings
我们研究非阿贝尔对偶变换在开弦的情况下。在度量了目标空间的等距并发展了一般形式之后，我们详细研究了具有SO（N）等距的目标空间的等距，对于SO（2）情形，它约化为已知的阿贝尔T-等距。我们应用的形式主义，以电气和磁荷4D黑洞解，在阿贝尔的情况下，对偶坐标满足狄利克雷条件。
Dilaton transformation under abelian and non-abelian T-duality in the path integral approach
本文给出了一种在T-对偶下用路径积分方法导出非线性子变换的简便方法。在执行积分的规范场的微妙之处进行了仔细分析，使用保利-维拉尔正规化，从而澄清现有的文献中的歧义。形式主义不仅可以应用于阿贝尔的情况下，但这是第一次，非阿贝尔的情况下，以及。此外，通过选择一个特定的规范，我们直接得到了阿贝尔情形下对偶几何的目标空间协变表达式。最后，它表明，衡量非阿贝尔等距的条件是弱于文献中通常发现的。
Remarks on Non-Abelian Duality in N=1 Supersymmetric Gauge Theories
最近Seiberg证明了一种连接超对称QCD型不同理论的对偶对称性。我们通过分析一个平坦的方向的理论，沿着这两个对偶理论去到同一理论的IR提供支持这一猜想。
Quantum corrections of Abelian Duality Transformations
提出了阿贝尔对偶变换的一个修正，保证了$σ$ -模型（不一定保形不变）与其对偶是量子等价的（在微扰论中至少有两个环）.这需要对{ \sl dual} $σ$ -模型进行某种非标准的微扰处理。对一类特殊的块对角纯度量$σ$模型给出了修正对偶变换的显式公式。
Relation between S-duality in N=2 SQCD and Non-Abelian Duality in N=1 SQCD
利用N=1超对称QCD的M理论组态，分析了N=2超对称QCD中N=1理论的强耦合模及其与S对偶变换的关系。结果表明，N=1超对称QCD的两种IIA型描述，一种是强耦合描述，另一种是弱耦合描述，对应于N=1超对称QCD的单一M理论描述。讨论了单态场的存在性。
Parent Action Approach for the Duality between Non-Abelian Self-Dual and Yang-Mills-Chern-Simons Models
一些作者认为，不能使用父作用方法来建立2+1阿贝尔和非阿贝尔自对偶（SD）和Yang-Mills-Chern-Simons（YMCS）模型之间的对偶。我们在这里提出了一个替代（微扰）的观点，并表明，这种等价性可以实现与父行动的方法。
A Problem with Non-Abelian Duality
我们调查的对偶变换在一类背景与非阿贝尔等距，即Bianchi型（齐次）宇宙学在任意尺寸。简单的对偶变换的度量和反对称张量字段，推广那些已知的阿贝尔等距（比安奇我）的情况下，得到使用拉格朗日或哈密顿方法。将这些处方应用到一个特定的共形不变的$\s$模型，我们表明，没有一个子变换导致一个新的共形背景。提出了解决这一问题的一些可能途径。
Defects, Non-abelian T-duality, and the Fourier-Mukai transform of the Ramond-Ramond fields
本文构造了生成无迷向等距群非阿贝尔T-对偶性的拓扑缺陷，证明其由对应空间上的线丛给出，并利用缺陷规范不变通量生成的Ramond-Ramond场Fourier-Mukai变换，提供了一种紧凑的场变换计算方法。
Towards Mirror Symmetry as Duality for Two-Dimensional Abelian Gauge Theories
超共形西格玛模型的Calabi-Yau目标空间被描述为复曲面簇中的完全相交子簇，可以作为某些二维阿贝尔规范理论的低能极限得到。我们制定镜像对称这类Calabi-丘空间作为一个对偶的阿贝尔规范理论，给出明确的映射有关的两个拉格朗日。对偶性与导致低能极限同构理论的不等价理论有关。这个公式表明镜像对称可以用阿贝尔对偶性推导出来。在这种情况下，对偶性的应用由于非平凡动力学的存在和全局对称性的缺乏而变得复杂。我们提出了一种方法来克服这些障碍，导致一个更对称的拉格朗日。然而，这一论点并没有推导出这一猜想。
Non-abelian U-duality originates from the construction of exceptional Drinfel'd algebra (EDA), which extends the constriction of the classical Drinfel'd double. This symmetry is a natural extension of Poisson--Lie T-duality and is believed to be a symmetry of Type II string/M-theory or their low-energy effective theories. In this paper, we consider non-abelian U-dualities of 11- or 10-dimensional backgrounds starting with E${}_{n(n)}$ EDA with $n\leq 6$ with vanishing trombone gauging. The latter guarantees that all dual backgrounds satisfy the standard supergravity equations of motion. In par…
Duality and Canonical Transformations
本文对弦理论和规范理论中某些对偶对称性的正则变换描述作了简要的评述。特别是，我们考虑阿贝尔和非阿贝尔的T-对偶在封闭和开放的弦论以及S-对偶在阿贝尔和非阿贝尔的非超对称规范理论。
For a general class of SO(4) symmetric backgrounds in type II-supergravity, we show that the action of non-Abelian T-duality can be described via consistent truncation to seven dimensional theories with seemingly massive modes. As such, any solution to these theories uplifts to both massive type IIA and IIB supergravities presenting an invertible map between the two. For supersymmetric backgrounds, we show that for spinors transforming under SO(4) the non-Abelian T-duality transformation breaks the original supersymmetry by half. We use these mappings to generate the non-Abelian T-duals of the…
Abelian duality, walls and boundary conditions in diverse dimensions
我们系统地应用对偶墙的形式主义来研究二元、三维和四维自由场论中对偶变换对边界条件和局部与非局部算子的作用。特别地，我们构造了一大类具有环形目标的二维sigma模型的D-膜，并确定了T-对偶群在其上的作用，证明了D-膜上的T-对偶变换是由微分几何形式的Fourier-Mukai变换给出的.
In this letter we generalised the procedure of non-abelian T-duality based on a B-shift and a sequence of formal abelian T-dualities in non-isometric directions to 11-dimensional backgrounds. This consists of a C-shift followed by either a formal U-duality transformation or taking a IIB section. We investigate restrictions and applicability of the procedure and find that it can provide supergravity solutions for the SL(5) exceptional Drinfeld algebra only when a spectator field is present, which is consistent with examples known in the literature.
We study the effect of T-duality on supersymmetry in the context of type II supergravity. For both U(1) Abelian and SU(2) non-Abelian T-duality, we demonstrate that the supersymmetry variations after T-duality are related to the variations before T-duality through the Kosmann spinorial Lie derivative, which vanishes when the Killing spinors are independent of the T-duality directions. As a byproduct of our analysis, we present closed expressions for SU(2) T-duality in a class of spacetimes with diagonal Bianchi IX symmetry and comment on specific examples of T-dual geometries, including a nove…
In this paper we propose a linear quiver with gauge groups of increasing rank as field theory dual to the AdS_5 background constructed by Sfetsos and Thompson through non-Abelian T-duality. The formalism to study 4d N=2 SUSY CFTs developed by Gaiotto and Maldacena is essential for our proposal. We point out an interesting relation between (Hopf) Abelian and non-Abelian T-dual backgrounds that allows to see both backgrounds as different limits of a solution constructed by Maldacena and Nunez. This suggests different completions of the long quiver describing the CFT dual to the non-Abelian T-dua…
We obtain an AdS$_{2}$ solution to Type IIA supergravity with 4 Poincare supersymmetries, via non-Abelian T-duality with respect to a freely acting SL(2,$\mathbf{R}$) isometry group, operating on the AdS$_3\times$S$^3\times$CY$_2$ solution to Type IIB. That is, non-Abelian T-duality on AdS$_3$. The dual background obtained fits in the class of AdS$_2\times$S$^3\times$CY$_2$ solutions to massive Type IIA constructed in [1]. We propose and study a quiver quantum mechanics dual to this solution that we interpret as describing the backreaction of the baryon vertex of a D4-D8 brane intersection.
Recent progress which relates non-abelian T-duality of $\mathcal{N}=1$ SuGra solutions to the powerful techniques of Generalised geometry is reviewed. It is shown that SU(3) structure solutions are mapped to SU(2) structures and the transformation rule of the corresponding pure spinors is presented. This constitutes an important step on the road towards the utility of the duality within holography, showing for example, how smeared sources must transform and so how to add flavour to the T-duals.
A comment on ``On Non-Abelian Duality'' by Enrique Álvarez, Luis Álvarez-Gaumé and Yolanda Lozano
评论的文件给人的印象是，无论是规范版本的西格玛模型产生的原始或双重模型取决于规范固定的选择。但事实证明并非如此。
Abelian and Non-Abelian Dualities in String Bacgrounds
我们提出了一个简短的讨论最近的工作对偶对称性在非平凡弦背景。对偶性是从规范场强度为零的规范非线性σ模型得到的。标准的结果被复制的阿贝尔规范群，而一种新的类型的对偶被确定为非阿贝尔规范群。文中给出了WZW模型和四维黑洞的反算实例。（发表于《从超弦到超引力》，Erice，1992年）
We study the action of non-Abelian T-duality in the context of N=1 geometries with well understood field theory duals. In the conformal case this gives rise to a new solution that contains an AdS_5 X S^2 piece. In the case of non-conformal geometries we obtain a new background in massive IIA supergravity that presents similar behaviour to the cascade of Seiberg dualities. Some physical observables are discussed.
We consider the action of a non-abelian T-duality on backgrounds with an AdS${}_5$ factor in type IIA supergravity, finding a new type IIB background, as well as the non-abelian T-dual of a domain wall solution which has as limits the non-abelian T-dual of $AdS_5\times T^{1,1}$ and the non-abelian T-dual of $AdS_3\times \mathbb{R}^2\times S^2\times S^3$. We explore some consequences of non-abelian T-duality for the dual conformal field theories.
T-duality for open strings with respect to non-abelian isometries
我们度量了一类带边界的西格玛模型的非阿贝尔等距性。强迫规范场的场强为零，使得规范场模型等价于未规范场模型，只要适当地考虑边界条件。对规范场积分得到T-对偶模型。我们观察到，T-对偶交换诺依曼（或混合）边界条件与狄利克雷边界条件。
(Non-)Abelian Kramers-Wannier duality and topological field theory
我们研究对偶性和拓扑场论之间的联系。首先，二维Kramers-Wannier对偶被公式化为一个简单的三维拓扑声明（或多或少的庞加莱对偶），并给出了一个类似的公式用于高维情况。在这种形式下，它们导致具有两种颜色的边界的简单TFT。统计模型存在于这些TFT的边界上，如CS/WZW或AdS/CFT对应关系中。经典模型（Poisson-Lie T-对偶）在2d情形下提出了一个非阿贝尔推广，用量子群代替阿贝尔群.令人惊讶的是，TFT公式不用计算就解决了这个问题：量子群出现在图像中，与经典动机无关。与陈-西蒙斯理论的联系出现在辛层次上，也出现在德林费尔德双像的图像中：从双像计算的3流形中的链接的Reshetikhin-Turaev不变量包括在这些TFT中。所有这些都暗示了更高维度的美好现象。
Supersymmetic Extension of the Non-Abelian Scalar-Tensor Duality
非阿贝尔反对称张量场的Freedman-Townsend模型的对偶场论是群流形G上的非线性西格玛模型。这可以推广到耦合到Yang-Mills场的Freedman-Townsend模型和陪集空间G/H上的非线性西格玛模型之间的对偶性。我们给出了这种对偶的超对称扩张，并发现这种非线性西格玛模型的目标空间是一个复陪集空间GC/HC。
Electric-Magnetic duality and the "Loop Representation" in Abelian Gauge Theories
阿贝尔规范理论是量子化的几何表示，推广了环表示和治疗的电和磁运营商在同一个基础上。通常的规范代数变成一个拓扑代数的非本地运营商，类似于有序无序对偶代数的特胡夫特。这些对偶算子提供了理论的物理相空间的完整描述。
Duality and Global Symmetries
这是对场论中对偶性的一般介绍。以整体对称性的存在与破缺为指导，系统地证明了不同场论之间的对偶性。讨论的系统包括交换和非交换的T-对偶在弦论，交换和nonabelian玻色子化，和对偶的无质量和大量的反对称张量场论在任意数量的维度。有关这些技术的公开问题也进行了讨论。（在第33届卡尔帕奇冬季学校的讲座`二元性：字符串和字段'。
Using the homogeneous Gödel spacetimes we find some new solutions for the field equations of bosonic string effective action up to first order in $α'$ including both dilaton and axion fields. We then discuss in detail the (non-)Abelian T-dualization of Gödel string cosmologies via the Poisson-Lie (PL) T-duality approach. In studying Abelian T-duality of the models we get seven dual models in such a way that they are constructed by one-, two- and three-dimensional Abelian Lie groups acting freely on the target space manifold. The results of our study show that the Abelian T-dual models are, und…
We present a formulation of Double Field Theory with a Drinfeld double as extended spacetime. It makes Poisson-Lie T-duality (including abelian and non-abelian T-duality as special cases) manifest. This extends the scope of possible applications of the theory, which so far captured abelian T-duality only, considerably. The full massless bosonic subsector (NS/NS and R/R) of type II string theories is covered.
Non-Abelian Duality in the String Effective Action
研究了玻色弦论单圈有效作用量在非阿贝尔T-对偶变换下的对称性。证明了原拉格朗日量及其对偶是成比例的。这一结果意味着相应的约化低能有效作用量是等价的，并导致原始和对偶二维非线性西格玛模型的Weyl反常系数之间的函数关系。{}最后，我们将这种形式主义应用于一些简单的例子。
The dynamics of abelian vector and antisymmetric tensor gauge fields can be described in terms of twisted self-duality equations. These first-order equations relate the p-form fields to their dual forms by demanding that their respective field strengths are dual to each other. It is well known that such equations can be integrated to a local action that carries on equal footing the p-forms together with their duals and is manifestly duality invariant. Space-time covariance is no longer manifest but still present with a non-standard realization of space-time diffeomorphisms on the gauge fields.…
Non-Abelian Axial-Vector Duality: a Geometric Description
我们给出了一个几何特征的准轴向量（Kiritsis-Obers）的目标空间对偶的双代数（Klimcik-Severa）的方法的精神。我们表明，通过采取concentients构建的西格玛模型有非阿贝尔手征电流，遵守“非交换守恒律”，并提供了标准的西格玛模型有一个对偶使用轴向量程序。
Non-Abelian Vortices on Cylinder -- Duality between vortices and walls
本文研究了超对称非阿贝尔规范理论中超多重态在基本表象下的圆柱上的涡旋。我们确定了周期性涡旋的模空间，发现随着涡旋模的变化，会出现一对壁状物体。通过引入扭曲边界条件，也可以由圆柱上的单涡得到二维畴壁。我们可以把这些现象理解为II型超弦理论中D-膜组态之间的T-对偶。使用这个T-对偶图片，我们发现一个一对一的对应关系之间的模空间的非阿贝尔涡旋和扭结的D-膜配置域壁。
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