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Conformal Manifolds in Four Dimensions and Chiral Algebras

Matthew Buican, Takahiro Nishinaka

TL;DR

This work establishes a tight link between the conformal manifold of four-dimensional N=2 SCFTs and the structure of their associated two-dimensional chiral algebras. It proves a general bound that any such theory with an exactly marginal (gauge-like) deformation has a chiral algebra with at least three generators, and it demonstrates a saturating example realized by the A(6) algebra at a special conformal point. The analysis combines 4d representation theory, moduli-space arguments, and the 4d/2d Schur correspondence, showing that conformal gauging enforces nontrivial chiral content, including hidden fermionic generators when a>=c. The results connect conformal manifolds, moduli spaces, and 2d chiral algebras, with explicit constructions from Argyres-Douglas theories and a concrete test via Schur/Macdonald indices, indicating a rich interplay between higher- and lower-dimensional symmetries with potential mathematical structure yet to be uncovered.

Abstract

Any N=2 superconformal field theory (SCFT) in four dimensions has a sector of operators related to a two-dimensional chiral algebra containing a Virasoro sub-algebra. Moreover, there are well-known examples of isolated SCFTs whose chiral algebra is a Virasoro algebra. In this note, we consider the chiral algebras associated with interacting N=2 SCFTs possessing an exactly marginal deformation that can be interpreted as a gauge coupling (i.e., at special points on the resulting conformal manifolds, free gauge fields appear that decouple from isolated SCFT building blocks). At any point on these conformal manifolds, we argue that the associated chiral algebras possess at least three generators. In addition, we show that there are examples of SCFTs realizing such a minimal chiral algebra: they are certain points on the conformal manifold obtained by considering the low-energy limit of type IIB string theory on the three complex-dimensional hypersurface singularity x_1^3+x_2^3+x_3^3+A x_1x_2x_3+w^2=0. The associated chiral algebra is the A(6) theory of Feigin, Feigin, and Tipunin. As byproducts of our work, we argue that (i) a collection of isolated theories can be conformally gauged only if there is a SUSY moduli space associated with the corresponding symmetry current moment maps in each sector, and (ii) N=2 SCFTs with a>=c have hidden fermionic symmetries (in the sense of fermionic chiral algebra generators).

Conformal Manifolds in Four Dimensions and Chiral Algebras

TL;DR

This work establishes a tight link between the conformal manifold of four-dimensional N=2 SCFTs and the structure of their associated two-dimensional chiral algebras. It proves a general bound that any such theory with an exactly marginal (gauge-like) deformation has a chiral algebra with at least three generators, and it demonstrates a saturating example realized by the A(6) algebra at a special conformal point. The analysis combines 4d representation theory, moduli-space arguments, and the 4d/2d Schur correspondence, showing that conformal gauging enforces nontrivial chiral content, including hidden fermionic generators when a>=c. The results connect conformal manifolds, moduli spaces, and 2d chiral algebras, with explicit constructions from Argyres-Douglas theories and a concrete test via Schur/Macdonald indices, indicating a rich interplay between higher- and lower-dimensional symmetries with potential mathematical structure yet to be uncovered.

Abstract

Any N=2 superconformal field theory (SCFT) in four dimensions has a sector of operators related to a two-dimensional chiral algebra containing a Virasoro sub-algebra. Moreover, there are well-known examples of isolated SCFTs whose chiral algebra is a Virasoro algebra. In this note, we consider the chiral algebras associated with interacting N=2 SCFTs possessing an exactly marginal deformation that can be interpreted as a gauge coupling (i.e., at special points on the resulting conformal manifolds, free gauge fields appear that decouple from isolated SCFT building blocks). At any point on these conformal manifolds, we argue that the associated chiral algebras possess at least three generators. In addition, we show that there are examples of SCFTs realizing such a minimal chiral algebra: they are certain points on the conformal manifold obtained by considering the low-energy limit of type IIB string theory on the three complex-dimensional hypersurface singularity x_1^3+x_2^3+x_3^3+A x_1x_2x_3+w^2=0. The associated chiral algebra is the A(6) theory of Feigin, Feigin, and Tipunin. As byproducts of our work, we argue that (i) a collection of isolated theories can be conformally gauged only if there is a SUSY moduli space associated with the corresponding symmetry current moment maps in each sector, and (ii) N=2 SCFTs with a>=c have hidden fermionic symmetries (in the sense of fermionic chiral algebra generators).

Paper Structure

This paper contains 17 sections, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Chiral algebra / Schur sector basics
  3. Gauging
  4. Phenomenology
  5. An argument in favor of $|\chi|\ge3$
  6. The case of $N\ge4$
  7. The case of $N\ge4$
  8. The case of $N=1$
  9. The case of $N=1$
  10. The case of $N\ge2$
  11. The case of $N\ge2$
  12. An example with $|\chi|=3$
  13. An example with $|\chi|=3$
  14. Explicit check of multiplet recombination in four dimensions
  15. Explicit check of multiplet recombination in four dimensions
  16. ...and 2 more sections

Figures (1)

  • Figure 1: A quiver diagram describing the theory $\hat{\mathcal{T}}$ near a weak coupling point on the conformal manifold. The middle circle stands for an $SU(2)$ vector multiplet gauging the diagonal $SU(2)$ flavor symmetry of three $(A_1,A_3)$ theories.