R-Twisting and 4d/2d Correspondences

TL;DR

This paper explores how R-charge properties of 4D N=2 conformal field theories are reflected in q-deformed monodromy operators, linking them to 2D conformal characters, Y-systems, and classifying theories via quiver data.

Contribution

It introduces a novel connection between R-charges, monodromy operators, and 2D CFT characters, and proposes a classification scheme for 4D N=2 theories based on quiver data.

Findings

Monodromy operators have finite order for rational R-charges.

Trace of q-deformed monodromy relates to 2D CFT characters.

Classification of 4D N=2 theories via quiver data and 2D theory correspondence.

Abstract

We show how aspects of the R-charge of N=2 CFTs in four dimensions are encoded in the q-deformed Kontsevich-Soibelman monodromy operator, built from their dyon spectra. In particular, the monodromy operator should have finite order if the R-charges are rational. We verify this for a number of examples including those arising from pairs of ADE singularities on a Calabi-Yau threefold (some of which are dual to 6d (2,0) ADE theories suitably fibered over the plane). In these cases we find that our monodromy maps to that of the Y-systems, studied by Zamolodchikov in the context of TBA. Moreover we find that the trace of the (fractional) q-deformed KS monodromy is given by the characters of 2d conformal field theories associated to the corresponding TBA (i.e. integrable deformations of the generalized parafermionic systems). The Verlinde algebra gets realized through evaluation of line operators at the loci of the associated hyperKahler manifold fixed under R-symmetry action. Moreover, we propose how the TBA system arises as part of the N=2 theory in 4 dimensions. Finally, we initiate a classification of N=2 superconformal theories in 4 dimensions based on their quiver data and find that this classification problem is mapped to the classification of N=2 theories in 2 dimensions, and use this to classify all the 4d, N=2 theories with up to 3 generators for BPS states.