Abstract
A recent claim that the S-duality between 4d SUSY gauge theories, which is AGT related to the modular transformations of 2d conformal blocks, is no more than an ordinary Fourier transform at the perturbative level, is further traced down to the commutation relation [P,Q]=-i\hbar between the check-operator monodromies of the exponential resolvent operator in the underlying Dotsenko-Fateev matrix models and beta-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.
Highlights
Sums [19,20,21,22], expressed via Nekrasov functions [23, 24], with the conformal blocks of 2d conformal theories [25, 26]
A recent claim that the S-duality between 4d SUSY gauge theories, which is AGT related to the modular transformations of 2d conformal blocks, is no more than an ordinary Fourier transform at the perturbative level, is further traced down to the commutation relation [P, Q] = −i between the check-operator monodromies of the exponential resolvent operator in the underlying Dotsenko-Fateev matrix models and β-ensembles
We treat the conformal blocks as eigenfunctions of the monodromy check operators, what is especially simple in the case of one-point toric block
Summary
An archetypical example of duality is provided by the switch between coordinate and momentum operators. Consider the two operators A = eiPand B = eiQ, with the Their eigenfunctions are related by the Fourier transform in the eigenvalue space: AZa(Q) = eiaZa(Q) BZa (Q) = eia Za (Q). The only difference of the modular S-duality from the above ordinary pq-duality is that the relevant operators Aand Bin the case of non-perturbative conformal blocks commute in a little less trivial way, but perturbatively they satisfy exactly (2.1) This explains the perturbative result of [60, 61] for the properly normalized conformal blocks and straightforwardly provides its non-perturbative generalization, which is in accordance with [27, 28]
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