Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs

TL;DR

This work builds a 4d–2d bridge by reducing 4d $ ext{N}=2$ SCFTs on a twisted circle to 3d EFTs, and then analyzing high-temperature (Cardy) limits of the index on higher sheets to extract 3d TQFT data and boundary VOAs. Using Di Pietro–Komargodski-type EFT techniques and Maruyoshi–Song Lagrangians, it systematizes the 3d Chern–Simons couplings and monopole superpotentials that encode the 4d/2d SCFT–VOA correspondence, revealing a Galois-orbit structure among TQFTs tied to $M(2,2n+3)$ minimal models. The second sheet yields non-unitary Lee–Yang-type TQFTs with boundary $M(2,5)$ VOAs, while higher sheets realize Galois-conjugate MTCs such as Fibonacci and their conjugates, including spin-TQFTs, with boundary VOAs computed via half-indices. Extending to $(A_1,A_4)$, the authors connect to the $ ext{T}_2$ theory and uncover richer dualities and non-abelian TQFTs, highlighting a versatile 4d–3d–2d framework for understanding the SCFT/VOA correspondence through interpolating 3d EFTs and their modular data.

Abstract

The high-temperature limit of the superconformal index, especially on higher sheets, often captures useful universal information about a theory. In 4d $\mathcal{N}=2$ superconformal field theories with fractional r-charges, there exists a special notion of high-temperature limit on higher sheets that captures data of three-dimensional topological quantum field theories arising from r-twisted circle reduction. These TQFTs are closely tied with the VOA of the 4d SCFT. We study such high-temperature limits. More specifically, we apply Di~Pietro-Komargodski type supersymmetric effective field theory techniques to r-twisted circle reductions of $(A_1,A_{2n})$ Argyres-Douglas theories, leveraging their Maruyoshi-Song Lagrangian with manifest $\mathcal{N}=1$ supersymmetry. The result on the second sheet is the Gang-Kim-Stubbs family of 3d $\mathcal{N}=2$ SUSY enhancing rank-$0$ theories with monopole superpotentials, whose boundary supports the Virasoro minimal model VOAs $M(2,2n+3)$. Upon topological twist, they give non-unitary TQFTs controlled by the $M(2,2n+3)$ modular tensor category (MTC). The high-temperature limit on other sheets yields their unitary or non-unitary Galois conjugates. This opens up the prospect of a broader four-supercharge perspective on the celebrated correspondence between 4d $\mathcal{N}=2$ SCFTs and 2d VOAs via interpolating 3d EFTs. Several byproducts follow, including a systematic approach to 3d SUSY enhancement from 4d SUSY enhancement, and a 3d QFT handle on Galois orbits of various MTCs associated with 4d $\mathcal{N}=2$ SCFTs.

Figures (10)

• Figure 1: The odd function $\overline{B}_1(x)$ versus $x$. It is periodic with period $1$, and discontinuous only across $\mathbb{Z}$, where it attains the average value of its left and right limits.
• Figure 2: Decomposition of the moduli-space of holonomies as in Ardehali:2021irq, for the integral \ref{['eq:I_4 for n=1']}. The domain $-1/2<x<0$ is neglected due to the $\mathbb Z_2$ Weyl redundancy. The constant $\epsilon$ is taken suitably small, for instance $\epsilon=0.01$
• Figure 3: The plot of $12Q^{\gamma=1}_h(x)$ versus $x$. The minima at $x=\pm2/10$ are exactly zero.
• Figure 4: The plot of $L^{\gamma=1}_h(x)$ versus $x$ for $(A_1,A_2)$. The minima at $x=\pm2/10$ are exactly zero.
• Figure 5: The plot of $12Q^{\gamma=2}_h(x)$ versus $x$. The flat direction in the middle signals a gauge-invariant monopole.