We explore the large spin spectrum in two-dimensional conformal field theories with a finite twist gap, using the modular bootstrap in the lightcone limit. By recursively solving the modular crossing equations associated to different P SL(2, Z) elements, we identify the universal contribution to the density of large spin states from the vacuum in the dual channel. Our result takes the form of a sum over P SL(2, Z) elements, 1 A priori, there could be theories without conserved currents, but with an accumulation of operators towards vanishing twist, and therefore have zero twist gap.2 These are the P SL(2, Z) images of the cusp at τ = i∞. 3 More precisely, this is a sum over the coset P SL(2, Z)/Γ ∞ where Γ ∞ is the subgroup generated by T : τ → τ + 1 that stabilizes the cusp at τ = i∞.4 In [1] this argument was credited to Tom Hartman.

Abstract:We derive an explicit bound on the dimension of the lightest charged state in two dimensional conformal field theories with a global abelian symmetry. We find that the bound scales with c and provide examples that parametrically saturate this bound. We also prove that any such theory must contain a state with charge-to-mass ratio above a minimal lower bound. We comment on the implications for charged states in three dimensional theories of gravity.

Site-directed spin labeling in combination with paramagnetic relaxation enhancement (PRE) measurements is one of the most promising techniques for studying unfolded proteins. Since the pioneering work of Gillespie and Shortle (J Mol Biol 1997;268:158), PRE data from unfolded proteins have been interpreted using the theory that was originally developed for rotational spin relaxation. At the same time, it can be readily recognized that the relative motion of the paramagnetic tag attached to the peptide chain and the reporter spin such as 1 H N is best described as a translation. With this notion in mind, we developed a number of models for the PRE effect in unfolded proteins: (i) mutual diffusion of the two tethered spheres, (ii) mutual diffusion of the two tethered spheres subject to a harmonic potential, (iii) mutual diffusion of the two tethered spheres subject to a simulated mean-force potential (Smoluchowski equation); (iv) explicit-atom molecular dynamics simulation. The new models were used to predict the dependences of the PRE rates on the 1 H N residue number and static magnetic field strength; the results are appreciably different from the Gillespie-Shortle model. At the same time, the Gillespie-Shortle approach is expected to be generally adequate if the goal is to reconstruct the distance distributions between 1 H N spins and the paramagnetic center (provided that the characteristic correlation time is known with a reasonable accuracy). The theory has been tested by measuring the PRE rates in three spin-labeled mutants of the drkN SH3 domain in 2M guanidinium chloride. Two modifications introduced into the measurement scheme-using a reference compound to calibrate the signals from the two samples (oxidized and reduced) and using peak volumes instead of intensities to determine the PRE rates-lead to a substantial improvement in the quality of data. The PRE data from the denatured drkN SH3 are mostly consistent with the model of moderately expanded random-coil protein, although part of the data point toward a more compact structure (local hydrophobic cluster). At the same time, the radius of gyration reported by Choy et al. (J Mol Biol 2002;316:101) suggests that the protein is highly expanded. This seemingly contradictory evidence can be reconciled if one assumes that denatured drkN SH3 forms a conformational ensemble that is dominated by extended conformations, yet also contains compact (collapsed) species. Such behavior is apparently more complex than predicted by the model of a random-coil protein in good solvent/poor solvent.Additional Supporting Information may be found in the online version of this article.Abbreviations: drkN SH3, N-terminal SH3 domain of the Drosophila adapter protein drk; EDTA, ethylenediaminetetraacetic acid; ESR, electron spin resonance; FRET, fluorescence resonance energy transfer; MTSL, methanethiosulfonate spin label; NOE, nuclear Overhauser effect; NAG, N-acetyl-glycine.

We revisit the spectrum of pure quantum gravity in AdS 3. The computation of the torus partition function will-if computed using a gravitational path integral that includes only smooth saddle points-lead to a density of states which is not physically sensible, as it has a negative degeneracy of states for some energies and spins. We consider a minimal cure for this non-unitarity of the pure gravity partition function, which involves the inclusion of additional states below the black hole threshold. We propose a geometric interpretation for these extra states: they are conical defects with deficit angle 2π(1−1/N), where N is a positive integer. That only integer values of N should be included can be seen from a modular bootstrap argument, and leads us to propose a modest extension of the set of saddle-point configurations that contribute to the gravitational path integral: one should sum over orbifolds in addition to smooth manifolds. These orbifold states are below the black hole threshold and are regarded as massive particles in AdS, but they are not perturbative states: they are too heavy to form multi-particle bound states. We compute the one-loop determinant for gravitons in these orbifold backgrounds, which confirms that the orbifold states are Virasoro primaries. We compute the gravitational partition function including the sum over these orbifolds and find a finite, modular invariant result; this finiteness involves a delicate cancellation between the infinite tower of orbifold states and an infinite number of instantons associated with PSL(2, Z) images.

Abstract. We describe general constraints on the elliptic genus of a 2d supersymmetric conformal field theory which has a gravity dual with large radius in Planck units. We give examples of theories which do and do not satisfy the bounds we derive, by describing the elliptic genera of symmetric product orbifolds of K3, product manifolds, certain simple families of Calabi-Yau hypersurfaces, and symmetric products of the "Monster CFT". We discuss the distinction between theories with supergravity duals and those whose duals have strings at the scale set by the AdS curvature. Under natural assumptions, we attempt to quantify the fraction of (2,2) supersymmetric conformal theories which admit a weakly curved gravity description, at large central charge.