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Macdonald index from 3d TQFT

Heeyeon Kim, Hongseok Kim, Jaewon Song

TL;DR

The paper develops a unified 3d perspective on the Macdonald index for a broad class of Argyres-Douglas theories by flowing to 3d abelian Chern-Simons matter theories under twisted reductions, where a distinguished axial U(1)_A symmetry governs the Macdonald grading. It proposes a fermionic, IR-driven formula for the Macdonald index by refining the quantum monodromy/operator construction in the 3d setup, and validates this framework across numerous (A1,G) theories via explicit computations and F-maximization checks. The approach connects the Macdonald index to refined VOA characters, Schur-like IR traces, and arc-space Hilbert series, offering multiple cross-checks and strong evidence for the conjectured expressions. The results deliver concrete closed-form fermionic sums for many AD theories (including A-series, D-series, and E-series cases) and illuminate how UV/global symmetry data flow to the IR U(1)_A grading that captures the Macdonald refinement, with broader implications for IR BPS dynamics and 4d/3d correspondences.

Abstract

We propose a new fermionic sum formula for the Macdonald index of a class of Argyres-Douglas theories. The formula arises naturally from a three-dimensional topological field theory obtained via a twisted dimensional reduction of the 4d theory. Such a reduction often gives rise to a 3d ${\mathcal N}=2$ abelian Chern-Simons matter theory, which is expected to flow to an ${\mathcal N}=4$ superconformal fixed point. After performing a topological twist, we obtain a 3d TFT admitting boundary conditions that support the vertex operator algebra associated with the original 4d theory. In this framework, the Macdonald index appears as a half-index of the 3d gauge theory, with the Macdonald grading determined by a distinguished $U(1)_A$ symmetry in the infrared ${\mathcal N}=4$ superconformal algebra. We present a general procedure to identify this $U(1)_A$ symmetry and, whenever possible, show that it reproduces the refined character of the associated vertex operator algebra, thereby recovering the Macdonald index. Our construction also gives a hint towards the IR formula for the Macdonald index in terms of 4d BPS particles on the Coulomb branch.

Macdonald index from 3d TQFT

TL;DR

The paper develops a unified 3d perspective on the Macdonald index for a broad class of Argyres-Douglas theories by flowing to 3d abelian Chern-Simons matter theories under twisted reductions, where a distinguished axial U(1)_A symmetry governs the Macdonald grading. It proposes a fermionic, IR-driven formula for the Macdonald index by refining the quantum monodromy/operator construction in the 3d setup, and validates this framework across numerous (A1,G) theories via explicit computations and F-maximization checks. The approach connects the Macdonald index to refined VOA characters, Schur-like IR traces, and arc-space Hilbert series, offering multiple cross-checks and strong evidence for the conjectured expressions. The results deliver concrete closed-form fermionic sums for many AD theories (including A-series, D-series, and E-series cases) and illuminate how UV/global symmetry data flow to the IR U(1)_A grading that captures the Macdonald refinement, with broader implications for IR BPS dynamics and 4d/3d correspondences.

Abstract

We propose a new fermionic sum formula for the Macdonald index of a class of Argyres-Douglas theories. The formula arises naturally from a three-dimensional topological field theory obtained via a twisted dimensional reduction of the 4d theory. Such a reduction often gives rise to a 3d abelian Chern-Simons matter theory, which is expected to flow to an superconformal fixed point. After performing a topological twist, we obtain a 3d TFT admitting boundary conditions that support the vertex operator algebra associated with the original 4d theory. In this framework, the Macdonald index appears as a half-index of the 3d gauge theory, with the Macdonald grading determined by a distinguished symmetry in the infrared superconformal algebra. We present a general procedure to identify this symmetry and, whenever possible, show that it reproduces the refined character of the associated vertex operator algebra, thereby recovering the Macdonald index. Our construction also gives a hint towards the IR formula for the Macdonald index in terms of 4d BPS particles on the Coulomb branch.

Paper Structure

This paper contains 41 sections, 153 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Macdonald index formula
  3. Definition of the Macdonald index
  4. Computation of Macdonald index
  5. $\mathcal{N}=1$ Lagrangian flowing to $\mathcal{N}=2$ AD theory
  6. Class $\mathcal{S}$ description and TQFT structure
  7. Refined character for the VOA
  8. Hilbert series of the Arc space of a scheme
  9. Conjectural formula for the Macdonald index from IR formula
  10. $(A_1, A_{2n})$ theory
  11. $(A_1, D_{2n+1})$ theory
  12. $(A_1, E_n)$ theory
  13. $(A_1, A_{2n+1})$ theory
  14. $(A_1, D_{2n+2})$ theory
  15. Macdonald index from the twisted compactification
  16. ...and 26 more sections

Figures (12)

  • Figure 1: BPS quiver for the $(A_1, A_4)$ theory in the canonical chamber
  • Figure 2: BPS quiver for the $(A_1, A_{2n})$ theory in the canonical chamber
  • Figure 3: The absolute value of the three-sphere partition function of the 3d theory for $(A_{1},A_{2})$ as a function of mixing parameter $\nu$. The function is minimized at $\nu=0$.
  • Figure 4: The absolute value of three-sphere partition function for the 3d theory of $(A_{1},A_{4})$ as a function of the mixing parameter $\nu$ with $U(1)_{A}=-U(1)_{\Phi_{1}}+U(1)_{\Phi_{3}}$. The function is minimized at $\nu=0$.
  • Figure 5: BPS quiver for the $(A_1, A_{2n+1})$ theory in the canonical chamber
  • ...and 7 more figures