Abstract
We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs with c < 8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists for c < 25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how “extremal spectra” which maximize the degeneracy at the gap can be determined numerically.
Highlights
Refining the modular bootstrap approach of [16,17,18,19,20,21]
A surprisingly rich set of new constraints on the CFT spectrum will be uncovered from unitarity and the modular invariance of the torus partition function alone
By optimizing the linear functional acting on the modular crossing equation, we have uncovered a surprisingly rich set of constraints on the spectrum
Summary
The vacuum Virasoro character χ0 and the non-degenerate Virasoro character χh are given by q−. The spectrum may contain conserved currents of spin s, along with twist-2 primaries of dimension s + 1 and spin s − 1 We refer to such a partition function as that of the generic type. In optimizing the bounds with increasing derivative order N , we find that at larger values of the central charge c, one must work to higher values of N for the bound ∆(mNo)d to stabilize When such a stabilization is unattainable due to the computational complexity, we will need to numerically extrapolate ∆(mNo)d to N = ∞, by fitting with a polynomial in 1/N (say, of linear or quadratic order)
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