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R-Twisting and 4d/2d Correspondences

Sergio Cecotti, Andrew Neitzke, Cumrun Vafa

TL;DR

The paper builds a deep bridge between 4d N=2 SCFT data and 2d RCFTs by encoding the 4d BPS spectrum into a q-deformed Kontsevich-Soibelman monodromy. It shows that when R-charges are rational the monodromy has finite order and its traces reproduce RCFT characters, with connections to Y-systems and the Thermodynamic Bethe Ansatz, while the Verlinde algebra emerges from line operators in fixed hyperkahler loci. A unifying framework using ADE quivers, hypersurface CY singularities, and 4d/2d worldsheet correspondences enables a classification of 4d theories with up to three BPS generators and links to 2d classifications. The work also provides both M5-brane/topological-string and purely 4d derivations, highlights the quantum Frobenius property at roots of unity, and outlines a program to classify 4d N=2 theories via 2d quiver data, with many explicit examples and q-series identities arising from wall-crossing.

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.

R-Twisting and 4d/2d Correspondences

TL;DR

The paper builds a deep bridge between 4d N=2 SCFT data and 2d RCFTs by encoding the 4d BPS spectrum into a q-deformed Kontsevich-Soibelman monodromy. It shows that when R-charges are rational the monodromy has finite order and its traces reproduce RCFT characters, with connections to Y-systems and the Thermodynamic Bethe Ansatz, while the Verlinde algebra emerges from line operators in fixed hyperkahler loci. A unifying framework using ADE quivers, hypersurface CY singularities, and 4d/2d worldsheet correspondences enables a classification of 4d theories with up to three BPS generators and links to 2d classifications. The work also provides both M5-brane/topological-string and purely 4d derivations, highlights the quantum Frobenius property at roots of unity, and outlines a program to classify 4d N=2 theories via 2d quiver data, with many explicit examples and q-series identities arising from wall-crossing.

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.

Paper Structure

This paper contains 99 sections, 476 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Open A-branes and the physical interpretation of topological amplitudes
  3. The open topological A model
  4. Embedding in the physical superstring
  5. Integrating out the BPS states
  6. The case of one M5-brane on $\Sigma \times S^1$
  7. BPS states and R-twisting
  8. Our setup
  9. Topological A model
  10. SCFT
  11. Deforming away from the SCFT point
  12. The half-monodromy $Y$, and fractional monodromy $K$
  13. The four-dimensional perspective
  14. A purely four-dimensional approach
  15. Hilbert space and operators
  16. ...and 84 more sections

Figures (4)

  • Figure 1: The $A_m\,\square\,A_n$ quiver.
  • Figure 2: The $A_m\,\boxtimes\, A_n$ quiver. Mutation at the nodes $\otimes$ gives back $A_m\,\square\,A_n$.
  • Figure 3: The allowed quivers with three nodes. (The two $A_3$'s are equivalent).
  • Figure 4: The quiver $A_{N-1}\, \square\, \widehat{A}_1$