  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.
