3d $\mathcal{N}=2$ minimal SCFTs from Wrapped M5-branes

TL;DR

The paper constructs 3d ${\cal N}=2$ SCFTs $T_N[M]$ from wrapped M5-branes on closed hyperbolic 3-manifolds and connects their central charges to an $SL(2)$ Chern-Simons invariant via a rigorously defined 3d/3d correspondence using resurgence. It develops a state-integral formalism for complex CS theory, enabling Borel-resummed perturbative invariants to compute the squashed-sphere free energy and hence ${c_T}$; explicit calculations for Weeks, Thurston, and related Dehn fillings reveal small central charges that scale with hyperbolic volume. The bootstrap analysis of these theories points to a distinctive third kink at ${\Delta_\Phi}\approx0.86$ with ${c_T}\approx0.93$, but the studied manifolds Weeks/Thurston do not realize this kink, suggesting minimal SCFTs arise from more general wrapped-M5 configurations or defects. Overall, the work links hyperbolic geometry to concrete 3d SCFT data, offering a geometry-driven path to classifying simple ${\cal N}=2$ CFTs and guiding future spectrum refinements and topological invariant developments.

Abstract

We study CFT data of 3-dimensional superconformal field theories (SCFTs) arising from wrapped two M5-branes on closed hyperbolic 3-manifolds. Via so-called 3d/3d correspondence, central charges of these SCFTs are related to a $SL(2)$ Chern-Simons (CS) invariant on the 3-manifolds. We give a rigorous definition of the invariant in terms of resurgence theory and a state-integral model for the complex CS theory. We numerically evaluate the central charges for several closed 3-manifolds with small hyperbolic volume. The computation suggests that the wrapped M5-brane systems give infinitely many discrete SCFTs with small central charges. We also analyze these `minimal' SCFTs in the eye of 3d $\mathcal{N}=2$ superconformal bootstrap.

Figures (7)

• Figure 1: Space of closed 3-manifolds $M$ with $\textrm{vol}(M) <\textrm{vol}{(}(S^3\backslash {\bf 5^2_1})_{(5,-1)}{)}\simeq 2.02988$. ${\rm vol(M)}$ denotes a topological invariant of 3-manifold $M$ called hyperbolic volume, the volume measured in the unique hyperbolic metric $(R_{\mu\nu}=-2g_{\mu\nu})$. For each non-zero hyperbolic volume plotted in the graph, there are only finitely many (mostly unique) ${\rm CH3}$s. The spectrum is discrete and infinite and has a (non-zero) lower bound 0.9427 which is saturated by the Weeks manifold ($(S^3\backslash {\bf 5^2_1})_{(5,-1)(5,-2)}$) 2007arXiv0705.4325G.
• Figure 2: An ideal tetrahedron $\Delta$, tetrahedron with truncated vertices. Hyperbolic structures of $\Delta$ are parameterized by edge parameters $(z:=e^{Z},z':=e^{Z'},z"=e^{Z"})$ satisfying relations $z'=\frac{1}{1-z}$ and $z"=1-z^{-1}$. These parameters assigned to each pair of boundary edges, as shown in the figure. Geometrically, the logarithm parameters ($Z,Z',Z"$) measure complex dihedral angles between two faces meeting at the edges. Imaginary parts of these logarithm parameters take values between 0 and $\pi$.
• Figure 3: White-head link (${\bf {\color{red}5}^{2}_{\color{blue}1}}$), the 1st one among links with 2 components and 5 crossings.
• Figure 4: Lower bound of $c_T$ for 3d ${\cal N}=2$ SCFTs near the third kink point. The bound is obtained with most general assumption about SCFT spectrum consistent with unitarity. There is no interesting feature even at the kink point. For reference we included $c_T$ values computed for the $T[{\it Weeks}]$ and $T[{\it Thurston}]$ wrapped M5-brane SCFTs obtained in the previous section.
• Figure 5: (a) Upper bound for dimension of $\bar{\Phi} \Phi$ with explicit assumption of $\Phi^2$ operator decoupling. The jump is at $(\Delta_\Phi, \Delta_{\bar{\Phi} \Phi}) = (0.8598(2),2.3937(5)).$ (b) Same bound overlapped with $\Phi^2$ included in the spectrum (red dots). The third kink can be identified as sudden jump (black dots) in the bound when spectrum excludes $\Phi^2$. The numerics was obtained with $\Lambda =13$.