We calculate the anomalous dimensions of operators with large global charge J in certain strongly coupled conformal field theories in three dimensions, such as the O(2) model and the supersymmetric fixed point with a single chiral superfield and a W = Φ 3 superpotential. Working in a 1/J expansion, we find that the large-J sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge J is always a scalar operator whose dimension ∆ J satisfies the sum rule 16 ∆ J+1 = 0.04067 , up to corrections that vanish at large J. The spectrum of low-lying excited states is also calculable explcitly: For example, the second-lowest primary operator has spin two and dimension ∆ J + √ 3. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order J + 1 2 . The propagation speeds of the Goldstone waves and heavy fermions are 1 √ 2 and ± 1 2 times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large J.

We advocate a framework for constructing perturbative closed string compactifications which do not have large-radius limits. The idea is to augment the class of vacua which can be described as fibrations by enlarging the monodromy group around the singular fibers to include perturbative stringy duality symmetries. As a controlled laboratory for testing this program, we study in detail six-dimensional (1,0) supersymmetric vacua arising from two-torus fibrations over a two-dimensional base. We also construct some examples of two-torus fibrations over four-dimensional bases, and comment on the extension to other fibrations.August, 2002 The setupThe idea is as follows: spacetime is a product of 10 − n Minkowski dimensions with an internal space X n . The internal space is a fiber product of a k-dimensional fiber space G over a n − k-dimensional base B. Fiberwise, G solves the string equations of motion 1 Some families of solutions of kind (III) will also contain solutions of kind (I), in the same way that ordinary Calabi-Yau manifolds often have orbifold limits at special points in moduli space.

We prove that every unitary two-dimensional conformal field theory (with no extended chiral algebra, and with c,c > 1) contains a primary operator with dimension ∆ 1 that satisfies 0 < ∆ 1 < c+c 12 + 0.473695. Translated into gravitational language using the AdS 3 /CFT 2 dictionary, our result proves rigorously that the lightest massive excitation in any theory of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than 1/(4G N ) + o( √ −Λ). In the flat-space approximation, this limiting mass is twice that of the lightest BTZ black hole. The derivation applies at finite central charge for the boundary CFT, and does not rely on an asymptotic expansion at large central charge. Neither does our proof rely on any special property of the CFT such as supersymmetry or holomorphic factorization, nor on any bulk interpretation in terms of string theory or semiclassical gravity. Our only assumptions are unitarity and modular invariance of the dual CFT. Our proof demonstrates for the first time that there exists a universal center-of-mass energy beyond which a theory of "pure" quantum gravity can never consistently be extended.