[Paper Review] R-Twisting and 4d/2d Correspondences

Sergio Cecotti, Andrew Neitzke+1 more|arXiv (Cornell University)|Jun 17, 2010

Black Holes and Theoretical PhysicsPhysics and Astronomy102 references209 citations

TL;DR

This paper establishes a correspondence between 4d N=2 superconformal field theories (SCFTs) and 2d rational conformal field theories (RCFTs) via the q-deformed Kontsevich-Soibelman monodromy operator constructed from BPS dyon spectra. It shows that when R-charges are rational, the monodromy has finite order, and its trace matches the characters of associated 2d RCFTs. The Verlinde algebra emerges from line operator insertions at R-symmetry fixed points on hyperKähler manifolds, and the TBA systems of Zamolodchikov arise naturally from the 4d theory's BPS spectrum.

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.

Motivation & Objective

• To establish a 4d/2d correspondence between N=2 superconformal field theories and 2d rational conformal field theories.
• To show that rational R-charges in 4d N=2 SCFTs imply finite-order monodromy in the q-deformed Kontsevich-Soibelman operator.
• To identify the trace of the monodromy operator with characters of 2d RCFTs associated with TBA systems.
• To realize the Verlinde algebra as the algebra of line operators evaluated at R-symmetry fixed points on hyperKähler manifolds.
• To classify 4d N=2 SCFTs using quiver data and map this to 2d N=2 theory classification, especially for theories with up to three BPS generators.

Proposed method

• Construct the q-deformed Kontsevich-Soibelman monodromy operator from the BPS dyon spectrum of 4d N=2 SCFTs.
• Use the half-monodromy Y(q) and fractional monodromy K(q) to define the quantum monodromy in terms of quantum torus algebras.
• Map the monodromy to Y-systems of Zamolodchikov via cluster algebra mutations and quantum dilogarithm identities.
• Realize the Verlinde algebra as the algebra of line operators evaluated at fixed points of the R-symmetry action on the hyperKähler moduli space.
• Use the quantum Frobenius property when q is a root of unity to constrain monodromy eigenvalue multiplicities and relate them to RCFT characters.
• Apply the canonical trace to the monodromy operator to extract 2d RCFT characters, verified numerically for (An, A1) and (A3, A1) models.

Experimental results

Research questions

• RQ1 Does the q-deformed Kontsevich-Soibelman monodromy operator of a 4d N=2 SCFT have finite order when R-charges are rational?
• RQ2 Can the trace of the monodromy operator be identified with the characters of a 2d RCFT associated with the TBA system of Zamolodchikov?
• RQ3 How does the Verlinde algebra of a 2d RCFT emerge from the structure of line operators in the 4d N=2 theory?
• RQ4 What is the role of quantum cluster algebras and quantum dilogarithm identities in connecting 4d BPS spectra to 2d TBA systems?
• RQ5 How does the classification of 4d N=2 SCFTs via quiver data relate to the classification of 2d N=2 SCFTs?

Key findings

• For 4d N=2 SCFTs with rational R-charges, the q-deformed monodromy operator M(q) has finite order, and M^N = 1 when q is a primitive Nth root of unity.
• The trace of the monodromy operator Tr[K(q)] matches the characters of the 2d RCFT associated with the corresponding TBA system, as verified for (A2, A1), (A3, A1), and (A4, A1) models.
• In the (A2, A1) model, the canonical trace of M(q) reproduces the characters of the (2,5) minimal model, and the eigenvalue multiplicities of M(q) follow a periodic pattern with period 6.
• For the (A3, A1) model, the eigenvalue multiplicities of M(q) are approximately equidistributed as N_k(N) = [N/3] + a_k(N), with |a_k(N)| ≤ 1 and periodicity of 6.
• For the (A4, A1) model, the monodromy satisfies M^7 = 1, and the eigenvalue multiplicities are N_k(N) = [N²/7] + a_k(N) with |a_k(N)| ≤ 1, indicating strong equidistribution.
• The Verlinde algebra of the (An, A1) theory is realized through the evaluation of line operators at R-symmetry fixed points on the hyperKähler moduli space, with the fusion rules encoded in the monodromy action.