N=4 Superconformal Bootstrap of the K3 CFT

TL;DR

This work develops a c=6, (4,4) K3 CFT bootstrap by exploiting an exact link between N=4 blocks and Virasoro blocks at c=28 and a moduli-dependent integrated four-point function A_{ijk ilde l}. The authors derive moduli-dependent bounds on the non-BPS spectrum, observe a continuum signaling at ADE singularities, and connect the bounds to the A1 cigar CFT, the free T^4/Z_2 orbifold, and large-volume geometric data. They also obtain analytic and numerical bounds on the critical dimension where OPE coefficients can diverge, as well as a large-volume bound on the first nonzero eigenvalue of the scalar Laplacian on K3, tying CFT data to geometric properties. Overall, the results provide a coherent CFT-based picture of spectrum structure across the K3 moduli space and offer universal bounds applicable to higher-dimensional CFTs.

Abstract

We study two-dimensional (4,4) superconformal field theories of central charge c=6, corresponding to nonlinear sigma models on K3 surfaces, using the superconformal bootstrap. This is made possible through a surprising relation between the BPS N=4 superconformal blocks with c=6 and bosonic Virasoro conformal blocks with c=28, and an exact result on the moduli dependence of a certain integrated BPS 4-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We observe directly from the CFT perspective the signature of a continuous spectrum above a gap at the singular moduli, and find numerically an upper bound on this gap that is saturated by the $A_1$ N=4 cigar CFT. We also derive an analytic upper bound on the first nonzero eigenvalue of the scalar Laplacian on K3 in the large volume regime, that depends on the K3 moduli data. As two byproducts, we find an exact equivalence between a class of BPS N=2 superconformal blocks and Virasoro conformal blocks in two dimensions, and an upper bound on the four-point functions of operators of sufficiently low scaling dimension in three and four dimensional CFTs.

Figures (9)

• Figure 1: The chiral-anti-chiral $c=6$ NS $\mathcal{N}=4$ superconformal block with external BPS primaries $\mathcal{O}^\pm$ and intermediate non-BPS primary of weight $h$.
• Figure 2: The chiral-anti-chiral $c={3(k+2)\over k}$ NS $\mathcal{N}=2$ superconformal block with external chiral/anti-chiral primaries ${\phi}^\pm$ of weight ${|q|\over 2}$ and $U(1)_R$ charge $\pm q = \pm\left( {\ell+2\over k}\right)$, and intermediate $U(1)_R$ neutral non-BPS primary of weight $h$.
• Figure 3: The eye-shaped region bounded by the dashed line is the range of $q(z)$ under one branch of the $q$-map \ref{['qmap']}. The regions $D_1$, $D_2$ and $D_3$ each contains two fundamental domains of the $S_3$ crossing symmetry group. See Appendix \ref{['Sec:qmap']}.
• Figure 4: The dots indicate the upper bound $\widehat{\Delta}_{gap}$ on the gap versus $f(1/2)$, the four-point function evaluated at the crossing symmetric point, at derivative orders ranging from 8 to 20. The solid line plots the extrapolation to infinite order using a quadratic fit. The minimal $f(1/2)$ and maximal gap are simultaneously saturated by an untwisted sector correlator at the free orbifold point. The shaded region represents the gap in the OPE of twist fields at a fixed point of $T^4/\mathbb{Z}_2$ with a rectangular $T^4$, where the minimal $f(1/2)$ and maximal gap are achieved by a square $T^4$ at radii $R_i=1$ ($1/\sqrt{2}$ times the self-dual radius).
• Figure 5: The integration region ${\cal D} = D' \setminus E$. The left is in the $q$-plane, and the right in the $\tau$-plane. The entire region enclosed by the solid line is $D'$. The region between the solid and dashed lines is $D' \setminus D_1$, and the shaded region is its image $E$ under $z \to 1-z$ for the right half and $z \to 1/z$ for the left half. The entire unshaded region inside solid line is the integration region $\cal D$. See Appendix \ref{['Sec:Aijkl1']}.