2d–4d connection between q-Virasoro/W block at root of unity limit and instanton partition function on ALE space

Abstract

We propose and demonstrate a limiting procedure in which, starting from the q-lifted version (or K-theoretic five dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five dimensional instanton partition function onto that on the ALE space R 4 /Z r . This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q → −1, t → −1 limit, from the defining relation of the q-Virasoro algebra and the q-deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N = 1 superconformal algebra. In the r-th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q-analysis * e-mail: itoyama@sci.osaka-cu.ac.jp † e-mail: toota@sci.osaka-cu.ac.jp ‡ e-mail yoshioka@sci.osaka-cu.ac.jp literatures, q-deformed su(n) block is readily generated both at generic q, t and the r-th root of unity limit. In the 4d side, we derive the proper normalization function for general (n, r) that accomplishes the automatic projection through the limit.2

q-Virasoro Modular Double and 3d Partition Functions

“…The q-Virasoro = W q,t (A 1 ) algebra admits several interesting limits in which it reduces to other known algebras, the most famous one being the conformal limit discussed around (2.21). Other interesting limits are: the Hall-Littlewood limit q → 0 with t fixed [109] and recently discussed in [110] in the context of the 5d AGT correspondence; the root of unity limit [111] recently discussed in the context of the 4d AGT correspondence in [112,113]; the special values β = 1, 3/2, 2 in which case connections with Kac-Moody, topological and W 1+∞ algebras respectively were discussed in [109]; the Frenkel-Reshetikhin limit t → 1 with q fixed (classical q-Virasoro algebra) or q → 1 with t fixed [114], in which case the algebra becomes commutative but inherits a natural Poisson algebra structure isomorphic to the Poisson algebra obtained from the difference Drinfeld-Sokolov reduction of SL 2 [115,116]. It would be very interesting to understand all these limits from the viewpoint of the q-Virasoro modular double and 3d gauge theories on compact spaces, but the general discussion is beyond the aim of this work.…”

Elliptic algebra, Frenkel–Kac construction and root of unity limit

“…We have obtained the dictionary between the parameters of the deformed Virasoro algebra and those of U q,p ( sl (2)). We can take the same root of unity limit of the parameters as was done in [13,14].…”

“…One of the generalizations is to consider the 4d SU(N) gauge theory on R 4 /Z r [3,4]. For works in this direction, see for example [5,6,7,8,4,9,10,11,12,13,14,15]. The corresponding CFT is described by a coset sl(N) r ⊕ sl(N) κ sl(N) r+κ , (1.1) posessing the r-th "para-W N symmetry" [6,16].…”

Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra, and 5D AGT correspondence

“…Incidentally, Macdonald functions are reduced to Uglov functions [U] at the root of unity limit of parameters q and t. AGT conjectures at this limit have been also studied. For example, the super Virasoro algebra are generated from the deformed Virasoro algebra at the limit q, t → −1, and it correspond to theories on the ALE space R 4 /Z 2 [IOY1,IOY2]. Moreover, certain conformal algebras A(r, k) are introduced in [BBFLT] and applied to AGT correspondence.…”