(2,2) Superconformal Bootstrap in Two Dimensions

TL;DR

This work develops a two-dimensional ${ m N}=(2,2)$ superconformal bootstrap to bound non-BPS spectra across moduli by exploiting a bridge between BPS ${ m N}=2$ blocks and bosonic Virasoro blocks. By formulating crossing equations for CA and CC OPE channels and enforcing positivity via semidefinite programming, it yields numerical bounds on CC/CA gaps that depend on chiral ring data. The analysis identifies exact saturations by orbifold and LG/cy points and reveals moduli-dependent structures, including kinks corresponding to large-volume Calabi–Yau limits and conifold behavior described by Liouville theory. These results provide moduli-aware constraints on the massive spectrum of ${ m N}=(2,2)$ theories and offer a framework for exploring Calabi–Yau moduli spaces through bootstrap data.

Abstract

We find a simple relation between two-dimensional BPS N=2 superconformal blocks and bosonic Virasoro conformal blocks, which allows us to analyze the crossing equations for BPS 4-point functions in unitary (2,2) superconformal theories numerically with semidefinite programming. We constrain gaps in the non-BPS spectrum through the operator product expansion of BPS operators, in ways that depend on the moduli of exactly marginal deformations through chiral ring coefficients. In some cases, our bounds on the spectral gaps are observed to be saturated by free theories, by N=2 Liouville theory, and by certain Landau-Ginzburg models.

Figures (14)

• Figure 1: Possible unitary multiplets allowed by the $\mathcal{N}=2$ superconformal symmetry in the $\phi_1\phi_1$ OPE between two chiral primaries of R-charge $q=1$. The solid/dashed lines are the degenerate multiplets satisfying $g_r =0$. In particular, the $r=\frac{1}{2}$ degenerate primary is the chiral primary of R-charge $2q=2$, while all the non-BPS primaries carry R-charge $2q-1=1$. The internal chiral primary is shown in dashed lines for $3\le c<6$ because it violates the unitarity bound $|2q|\le c/3$ and is not present in unitary CFTs. The gray shaded region corresponds to the continuum of non-degenerate multiplets. Note there is necessarily a gap in the weight above the chiral primary if $3\le c < 6$.
• Figure 2: The limits that relate ${\cal N}=2$ super-Virasoro blocks with BPS external primaries and non-BPS as well as BPS internal representations.
• Figure 3: Relation between $\mathcal{N}=2$ super-Virasoro blocks with external BPS primaries and bosonic Virasoro blocks.
• Figure 4: Left: Upper bounds on the gap in the CA channel $\Delta^{CA}_{gap}$, with the minimal assumption \ref{['minimalCCgap']} on the gap in the CC channel $\Delta^{CC}_{gap}$, as a function of the central charge $c$, at derivative orders $4, \, 8, \, 12, \, 16$ (from green to red). Right: The same plot zoomed into $3\leq c \leq 4$.
• Figure 5: Left: Upper bounds on the gap in the CA channel $\Delta^{CA}_{gap}$ from the numerical bootstrap in the case of $c=3$ and external charge $q=\frac{1}{3}$. The bounds depend on the chiral ring coefficient $\lambda$ and the gap in the CC channel $\Delta^{CC}_{gap}$ we put into the crossing equation. The blue, yellow, and green curves are the bootstrap bounds with $\Delta^{CC}_{gap} = {2 \over 3}$, ${2 \over 3} < \Delta^{CC}_{gap} \leq {4 \over 3}$, and $\Delta^{CC}_{gap} > {4 \over 3}$, respectively. Right: The three-dimensional visualization of the same plot. The peak for $\frac{2}{3}<\Delta^{CC}_{gap} \leq \frac{3}{4}$ is saturated by the point $(R, b) = (\sqrt{4\over3}, 0)$ on the moduli space of the $T^2/\mathbb{Z}_3$ theory, at which the OPE coefficient for the $(a,a)$ primary $\overline\phi'_{-\frac{1}{3}}$ vanishes and hence $\Delta^{CC}_{gap}$ increases to $4\over3$.