Reiji Yoshioka

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 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.

q-Virasoro/W algebra at root of unity and parafermions

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

q-vertex operator from 5D Nekrasov function

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.

Baxter’sT-Q equation, SU( N )/SU(2) N − 3 correspondence and Ω-deformed Seiberg-Witten prepotential

We study Baxter’s T-Q equation of XXX spin-chain models under the semiclassical limit where an intriguing SU(N)/SU(2)N−3 correspondence is found. That is, two kinds of 4D

Developments of theory of effective prepotential from extended Seiberg–Witten system and matrix models

\mathcal{N} = 2

q-Virasoro algebra at root of unity limit and 2d-4d connection

superconformal field theories having the above different gauge groups are encoded simultaneously in one Baxter’s T-Q equation which captures their spectral curves. For example, while one is SU(Nc) with Nf = 2Nc flavors the other turns out to be

Matrix orientifolding and models with four or eight supercharges

{\text{SU}}{(2)^{{N_c} - 3}}

Cubic constraints for the resolvents of the ABJM matrix model and its cousins

with Nc hyper-multiplets (Nc > 3). It is seen that the corresponding Seiberg-Witten differential supports our proposal.

On non-stationary Lamé equation from WZW model and spin-1/2 XYZ chain

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

Orientifolded matrices and supersymmetries that give rise to spacetime directional asymmetry of effective interactions

We propose 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, 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 4/r. This proceeding is based on arXiv:1308.2068