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3d Chern--Simons matter theories from generalized Argyres--Douglas theories

Takahiro Nishinaka, Yutaka Yoshida

TL;DR

This work develops a precise dictionary between 4d Argyres–Douglas theories and 3d N=2 Chern–Simons matter theories obtained from U(1)_r twisted S^1 reductions, using the IR Schur index as a guiding tool. For specific coprime pairs (M,N)—notably (A2,A_{N-1}) with gcd(N,3)=1 and (A3,A_{N-1}) with gcd(N,4)=1—the authors identify explicit CS gauges, matter content, mixed CS levels, and monopole superpotentials that drive the IR to N=4 fixed points, with half-indices matching the parent 4d Schur indices. They further uncover a novel Nahm-sum expression for the vacuum character of the (3,8) W_3 minimal model and derive a CS-matter description for the corresponding twisted reduction (A2,A4). A conjectural general framework for (A_{M-1},A_{N-1}) at coprime M,N is proposed via a quantum monodromy ansatz, linking Schur indices to W_M minimal-model vacuum characters. The results illuminate IR dualities among UV UV-complete CS theories and provide a structured path toward understanding the 4d–2d VOA connections through 3d boundary data and localization techniques.

Abstract

We study 3d $\mathcal{N}=2$ Chern--Simons matter theories describing the R-twisted $S^1$-reduction of Argyres--Douglas theories of $(A_{M-1},A_{N-1})$ type with $\text{gcd}(M,N)=1$, via a recently-proposed 4d/3d correspondence. In particular, for the $(A_2,A_{N-1})$ and $(A_3,A_{N-1})$ theories, we identify a series of Chern--Simons matter theories with monopole superpotentials consistent with an $\mathcal{N}=4$ supersymmetry enhancement in the infrared. As a by-product, we also find a novel Nahm sum formula for the vacuum character of $(3,8)$ $W_3$ minimal model, from which we find another Chern--Simons matter theory describing the R-twisted $S^1$-reduction of the $(A_2,A_4)$ theory.

3d Chern--Simons matter theories from generalized Argyres--Douglas theories

TL;DR

This work develops a precise dictionary between 4d Argyres–Douglas theories and 3d N=2 Chern–Simons matter theories obtained from U(1)_r twisted S^1 reductions, using the IR Schur index as a guiding tool. For specific coprime pairs (M,N)—notably (A2,A_{N-1}) with gcd(N,3)=1 and (A3,A_{N-1}) with gcd(N,4)=1—the authors identify explicit CS gauges, matter content, mixed CS levels, and monopole superpotentials that drive the IR to N=4 fixed points, with half-indices matching the parent 4d Schur indices. They further uncover a novel Nahm-sum expression for the vacuum character of the (3,8) W_3 minimal model and derive a CS-matter description for the corresponding twisted reduction (A2,A4). A conjectural general framework for (A_{M-1},A_{N-1}) at coprime M,N is proposed via a quantum monodromy ansatz, linking Schur indices to W_M minimal-model vacuum characters. The results illuminate IR dualities among UV UV-complete CS theories and provide a structured path toward understanding the 4d–2d VOA connections through 3d boundary data and localization techniques.

Abstract

We study 3d Chern--Simons matter theories describing the R-twisted -reduction of Argyres--Douglas theories of type with , via a recently-proposed 4d/3d correspondence. In particular, for the and theories, we identify a series of Chern--Simons matter theories with monopole superpotentials consistent with an supersymmetry enhancement in the infrared. As a by-product, we also find a novel Nahm sum formula for the vacuum character of minimal model, from which we find another Chern--Simons matter theory describing the R-twisted -reduction of the theory.

Paper Structure

This paper contains 26 sections, 153 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. General framework and strategy
  3. Half index from IR formula for 4d Schur index
  4. Ellipsoid partition function from IR formula
  5. General formula for 3d CS matter theory from IR formulas
  6. Example: review of $(A_1,A_{2N})$ case
  7. $(A_1,A_2)$ theory
  8. $(A_1,A_{2N})$ theories
  9. Quantum monodromy of $(A_{M-1},A_{N-1})$
  10. $(A_2,A_{N-1})$ theory
  11. Schur index for $(A_2, A_{N-1})$
  12. Half index and CS matter theory
  13. Monopole superpotential
  14. Residual global symmetry
  15. Example: $(A_2,A_1)$
  16. ...and 11 more sections

Figures (8)

  • Figure 1: The BPS quiver for the $(A_{M-1},A_{N-1})$ theory for odd $M$. When $M$ is even, the orientation of the arrows attached to the nodes in the bottom row are reversed. The diagram has $(M-1)$ rows and $(N-1)$ columns, and therefore $(M-1)(N-1)$ gauge nodes. The charge shown inside each circle stands for the primitive charge associated with the corresponding node.
  • Figure 2: The BPS quiver for the $(A_2,A_3)$ theory. The charge shown inside each circle stands for the primitive charge associated with the corresponding node.
  • Figure 3: The BPS quiver after one operation of $\mu_{+-}$
  • Figure 4: The BPS quiver after the second operation of $\mu_{+-}$
  • Figure 5: The BPS quiver for the $(A_2,A_{N-1})$ theory. The charge shown inside each circle stands for the primitive charge associated with the corresponding node.
  • ...and 3 more figures