This is the second part of the paper 0709.3806v2. Here we show that three-point correlation function with one semi-degenerate field in Toda field theory as well as four-point correlation function with one completely degenerate and one semi-degenerate field can be represented by the finite dimensional integrals.
We compute the light asymptotic limit of $A_{n-1}$ Toda conformal blocks by using the AGT correspondence. We show that for certain class of CFT blocks the corresponding Nekrasov partition functions in this limit are simplified drastically being represented as a sum of a restricted class of Young diagrams. In the particular case of $A_{2}$ Toda we also compute the corresponding conformal blocks using conventional CFT techniques finding a perfect agreement with the results obtained from the Nekrasov partition functions.
This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the $\mathcal{W}$-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra ${\mathfrak s}{\mathfrak l}_3$ on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of f…
Construction of Affine and Conformal Affine Toda Solitons by Hirota's Method
在这次演讲中，我们报告了一些结果的构造孤子解的仿射和共形仿射户田模型使用广田的方法。我们得到新的类孤子连接到Cartan矩阵特征值的退化，以及一些特殊功能的递归计划在这里开发。我们得到了所有这些孤子的普遍质量公式。详细讨论了$SU(6)$和$Sp(3)$的示例。（在第七届J.A. Swieca暑期学校，部分：粒子和场，Campos do Jordão -巴西-1993年1月）
研究了具有极大对称边界的sl（n）共形Toda理论.由于W（n>2）代数的二阶自同构的存在，有两种类型的极大对称边界条件。在这两种情况中的一种情况下，我们发现存在所有可能维数0 =< d =< n-1的D-膜，它们对应于W（n）代数的部分退化表示。我们进行经典和共形的自举分析这样的D-膜，并通过使用半经典光渐近极限这两种方法。特别是，我们确定了批量一点函数。我们观察到显着严重的分歧，在环配分函数，并归因于它们的起源的存在，无限的多重性的融合表示的W（n>2）代数。我们还利用约束泛函形式的微积分讨论了边界作用量的存在性问题，并利用体单点函数的小超空间极限导出了sl（3）Toda经典力学的B“acklund变换的母函数.
We construct correlators in the $W_4$ Toda 2d conformal field theory for a particular class of representations and demonstrate a relation to a $W_2$ (Virasoro) theory with different central charge. The relevance of the classical limits of the constructed 3-point functions and braiding matrices to problems in 4d conformal theories is discussed.
We construct a set of non-rational conformal field theories that consist of deformations of Toda field theory for sl(n). Besides conformal invariance, the theories still enjoy a remnant infinite-dimensional affine symmetry. The case n=3 is used to illustrate this phenomenon, together with further deformations that yield enhanced Kac-Moody symmetry algebras. For generic n we compute N-point correlation functions on the Riemann sphere and show that these can be expressed in terms of sl(n) Toda field theory correlation functions.
Current studies of WN Toda field theory focus on correlation functions such that the WN highest-weight representations in the fusion channels are multiplicity-free. In this work, we study W3 Toda 4-point functions with multiplicity in the fusion channel. The conformal blocks of these 4-point functions involve matrix elements of a fully-degenerate primary field with a highest-weight in the adjoint representation of sl3, and a semi-degenerate primary field with a highest-weight in the fundamental representation of sl3. We show that, when the fusion rules are obeyed, the matrix elements of the fu…
In a recent study we considered W3 Toda 4-point functions that involve matrix elements of a primary field with the highest-weight in the adjoint representation of sl3. We generalize this result by considering a semi-degenerate primary field, which has one null vector at level two. We obtain a sixth-order Fuchsian differential equation for the conformal blocks. We discuss the presence of multiplicities, the matrix elements and the fusion rules.
We describe a compactification of the six-dimensional (2,0) theory on a four-sphere which gives rise to a two-dimensional Toda theory at long distances. This construction realizes chiral Toda fields as edge modes trapped near the poles of the sphere. We relate our setup to compactifications of the (2,0) theory on the five and six-sphere. In this way, we explain a connection between half-BPS operators of the (2,0) theory and two-dimensional W-algebras, and derive an equality between their conformal anomalies. As we explain, all such relationships between the six-dimensional (2,0) theory and Tod…
We construct the free field representation of irregular vertex operators of arbitrary rank which generates simultaneous eigenstates of positive modes of Virasoro and W symmetry generators. The irregular vertex operators turn out to be the exponentials of combinations of derivatives of Liouville or Toda fields, creating irregular coherent states. We compute examples of correlation functions of these operators and study their operator algebra.
We consider the two-dimensional $\mathfrak{sl}_n$ quantum Toda field theory with an imaginary background charge. This conformal field theory has a higher spin symmetry ($W_n$ algebra), a central charge $c \leq n-1$ and a continuous spectrum. Using the conformal bootstrap, we compute structure constants involving two arbitrary scalar fields and a semi-degenerate field of Wyllard type. The solution obtained is not the analytic continuation of the usual Toda three-point function. Non-scalar primary fields and their three-point functions are also discussed. Non-scalar primary fields are classified…
