S -duality and modular transformation as a non-perturbative deformation of the ordinary pq -duality

摘要

A recent claim that the S -duality between 4 d SUSY gauge theories, which is AGT related to the modular transformations of 2 d conformal blocks, is no more than an ordinary Fourier transform at the perturbative level, is further traced down to the commutation relation $ \left[ {\mathop{P,}\limits^{\vee}\mathop{Q}\limits^{\vee }} \right]=-i\hbar $ between the check-operator monodromies of the exponential resolvent operator in the underlying Dotsenko-Fateev matrix models and β -ensembles. To this end, we treat the conformal blocks as eigenfunctions of the monodromy check operators, what is especially simple in the case of one-point toric block. The kernel of the modular transformation is then defined as the intertwiner of the two monodromies, and can be obtained straightforwardly, even when the eigenfunction interpretation of the blocks themselves is technically tedious. In this way, we provide an elementary derivation of the old expression for the modular kernel for the one-point toric conformal block.