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

We demonstrate that the parafermions appear in the r-th root of unity limit of q-Virasoro/W n algebra. The proper value of the central charge of the coset model sl(n)r⊕ sl(n) m−n sl(n) m−n+r is given from the parafermion construction of the block in the limit. *

The five dimensional AGT correspondence implies the connection between the q-deformed Virasoro block and the 5d Nekrasov partition function. In this paper, we determine a q-deformation of the four-point block in the Coulomb gas representation from the 5d Nekrasov function, and obtain an expression of the q-deformed vertex operator. If we use only one kind of the q-vertex operators, one of the insertion points of them must be modified in order to hold the 2d/5d correspondence. *

This is a semi-pedagogical review of a medium size on the exact determination of and the role played by the low energy effective prepotential F in QFT with (broken) extended supersymmetry, which began with the work of Seiberg and Witten in 1994. While paying an attention to an overall view of this subject lasting long over the two decades, we probe several corners marked in the three major stages of the developments, emphasizing uses of the deformation theory on the attendant Riemann surface as well as its close relation to matrix models. Examples picked here in different contexts tell us that the effective prepotential is to be identified as the suitably defined free energy F of a matrix model: F = F .To be submitted to PTEP as an invited review article and based in part on the talk delivered by one of the authors (H.I.) in the workshop held at

A novel functorial relationship in perturbative quantum field theory is pointed out that associates Feynman diagrams (FD) having no external line in one theory Th 1 with singlet operators in another one Th 2 having an additional U (N ) symmetry and is illustrated by the case where Th 1 and Th 2 are respectively the rank r − 1 and the rank r complex tensor model. The values of FD in Th 1 agree with the large N limit of the Gaussian average of those operators in Th 2 . The recursive shift in rank by this FD functor converts numbers into vectors, then into matrices, then into rank 3 tensors and so on. This FD functor can straightforwardly act on the d dimensional tensorial quantum field theory (QFT) counterparts as well. In the case of rank 2-rank 3 correspondence, it can be combined with the geometrical pictures of the dual of the original FD, namely, equilateral triangulations (Grothendieck's dessins d'enfant) to form a triality which may be regarded as a bulk-boundary correspondence.