Central charges of para-Liouville and Toda theories from M5-branes
Central charges of para-Liouville and Toda theories from M5-branes
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来自 M5 膜的准刘维尔理论和户田理论的中心指控
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发表时间：
2011
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通讯作者：
Yuji Tachikawa
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作者：
T. Nishioka;Yuji Tachikawa
我们提出了N个m5膜，在R/Zm上加上形变参数ǫ1，2，实现了具有ŜU(M)N对称性和第m阶准WN对称性的二维理论。这包括m=1的标准WN对称性和m=N=2的超Viraroro对称性。我们通过从6d理论的反常多项式计算2d理论的中心电荷，对这一建议进行了小的检验。∗离开东京工业大学简介：N个M5膜，用奈克拉索夫的形变参数ǫ1，2加在R上，现在被认为实现了具有WN对称性的二维理论[1，2]。在[4]中的观察之后，在[3]中对这一陈述进行了一次检查。也就是说，文[5]中确定的N个m5-膜的反常多项式在R上的等变积分给出了二维理论的反常多项式，由此可以再现具有WN对称性的Toda理论的中心电荷。对于6dN=(2，0)型G=A，D，E型理论也可以进行同样的分析，它的反常多项式也是已知的[6，7]，它正确地再现了G型Toda理论的中心电荷。在最近的一篇论文[8]中，提出了R/Z2上两个变形ǫ1，2的M5膜产生一个具有ŜU(2)2对称性和超Virasoro对称性的系统。作为推广，我们提出了R/Zm上的N个m5膜实现了一个具有自由玻色子、ŜU(M)N和第m个准WN对称性的二维系统。这里Zm在(z，w)→C∈R上充当(z，w)7≃(ez，ew)。我们通过计算6d反常多项式的中心电荷，给出了一个小的支持证据。我们还将推测，如果改用G型的6dN=(2，0)理论，会发生什么。下式中，G表示An、Dn或En中的一个；Rg、Hg和Dg分别是G的秩数、(对偶)Coxeter数和维度。它们满足Dg=Rg(Hg+1)。仿W对称和仿Toda理论：实现Thew(Ĝ)对称[10，11，12，13，14，15]的一个方法是考虑陪集的手征代数
We propose that N M5-branes, put on R/Zm with deformation parameters ǫ1,2, realize two-dimensional theory with ŜU(m)N symmetry and m-th para-WN symmetry. This includes the standard WN symmetry for m = 1 and super-Viraroro symmetry for m = N = 2. We provide a small check of this proposal by calculating the central charge of the 2d theory from the anomaly polynomial of the 6d theory. ∗on leave from IPMU, the University of Tokyo Introduction: N M5-branes, put on R with Nekrasov’s deformation parameters ǫ1,2, are now believed to realize two-dimensional theory with WN symmetry [1, 2]. One check of this statement was given in [3] following the observation made in [4]. Namely, the equivariant integral on R of the anomaly polynomial of N M5-branes determined in [5] gives the anomaly polynomial of the 2d theory, from which the central charge of the Toda theory with WN symmetry can be reproduced. The same analysis can be performed for 6d N = (2, 0) theory of type G = A,D,E whose anomaly polynomial is also known [6, 7]; and it correctly reproduces the central charge of the Toda theory of type G. In a recent paper [8], it was proposed that two M5-branes on R/Z2 with deformations ǫ1,2 give rise to a system with the ŜU(2)2 symmetry and the super-Virasoro symmetry. As a generalization, we propose that N M5-branes on R/Zm realize a 2d system with a free boson, ŜU(m)N , and the m-th para-WN symmetry . Here Zm acts as (z, w) 7→ (ez, ew) on (z, w) ∈ C ≃ R. We give a small piece of supporting evidence by calculating the central charge from the 6d anomaly polynomial. We will also speculate what happens if the 6d N = (2, 0) theory of type G is used instead. In the following G stands for one of An, Dn or En; rG, hG and dG are the rank, the (dual) Coxeter number and the dimension of G, respectively. They satisfy dG = rG(hG + 1). Para-W symmetry and para-Toda theory: One way to realize theW (Ĝ) symmetry [10, 11, 12, 13, 14, 15] is to consider the chiral algebra of the coset