R-Twisting and 4d/2d Correspondences (1006.3435v2)

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.

• The paper reveals that quantum monodromy from R-twisting in 4d N=2 theories encodes the BPS spectrum and exhibits finite-order behavior under common denominator R-charges.
• It employs cluster algebras and quiver mutations to systematically classify 4d SCFTs, mapping wall-crossing phenomena to precise algebraic exchange relations.
• The study establishes correspondences with 2d integrable models and RCFT characters, highlighting the deep interplay between 4d BPS states and 2d conformal theories.

Review of "R-Twisting and 4d/2d Correspondences"

This paper investigates the deep connections between four-dimensional (4d) ${\cal N}=2$ supersymmetric field theories and two-dimensional (2d) conformal field theories (CFTs), primarily focusing on their BPS spectra and associated structures. The authors consider these connections via the twisting of R-symmetries, providing insights into the 4d/2d correspondences and the potential for classification of such theories through quiver diagrams and cluster algebras.

A core part of the paper involves analyzing the BPS monodromy, which is constructed from the BPS states in the four-dimensional theories. The main results are as follows:

1. Quantum Monodromy and R-Twisting: The quantum monodromy operator $M$ reflects the states generated by the twisting of R-charge in 4d ${\cal N}=2$ theories. This operator emerges from the quantum dilogarithms associated with the solitonic spectrum and is expected to have finite order whenever the R-charges have a common denominator.
2. Connection with Cluster Algebras: For a class of theories where the BPS spectrum can be encoded using quivers, the combinatorial structure of wall-crossing is elegantly captured by the cluster algebra framework. This formalism allows representing the BPS jumping phenomenon using the algebra's exchange relations.
3. Classification via Quiver Mutations: The paper extends the understanding of quiver diagrams, proposing that BPS spectra associated with these quivers provide a systematic method to classify 4d ${\cal N}=2$ SCFTs. The mutation operations on these quivers parallel the wall-crossing phenomena encountered in the studies of the BPS spectrum.
4. Correspondences with 2d Theories: The paper reveals correspondences between 4d theories and 2d integrable models. Specifically, the authors show that the monodromy/operator traces in the four-dimensional theories correspond to the characters of finite-type RCFTs in two dimensions, exemplified by the ${\cal N}=2$ Landau-Ginzburg models.
5. Role of TBA and 2d CFTs in 4d Models: The paper hypothesizes that certain thermal Bethe Ansatz (TBA) systems for 2d CFTs that describe integrable deformation of CFTs have analogs in 4d ${\cal N}=2$ theories. This reflects the rich interplay between two-dimensional integrable systems and higher-dimensional supersymmetric theories.
6. Solitonic Solutions and Fixed Points: Furthermore, the path-integral interpretation tying to the quantization of these theories suggests a deep relation to the fixed points under the R-symmetry actions, which in some scenarios realize the Verlinde algebra structure linked to the corresponding RCFT characters.

Overall, this exploration provides a robust framework to connect 4d ${\cal N}=2$ theories with quantum monodromy and 2d integrable systems, leveraging quiver mutations and cluster algebras. These insights paves the way for potential classification frameworks and deepens the understanding of the underlying algebraic and geometrical structures in quantum field theories. Future directions will likely involve a refined understanding of these correspondences, especially in elucidating the physical implications of these algebraic structures on the dynamics of the theories. Additionally, it will be valuable to further explore the reduction to three dimensions and investigate related connections with known 2d theories for a comprehensive classification theory across dimensions.