Edward Witten

After re-examining a classic article by Hugh Everett, I will discuss the entropy of de Sitter space from the standpoint of an observer living in the space. This provides a nice example of the importance — for certain questions — of taking seriously the fact that in cosmology any observer is really part of the universe that is being observed.

A bstract: We examine the behavior of sequences of states in the large N limit of AdS/CFT duality in cases in which the bulk duals involve baby universes or black holes. Such sequences generally fail to converge as pure states. Under suitable conditions, such as diverging coarse-grained entropy, they can converge to mixed states for the large N algebra, as in the case of black holes. For Euclidean preparations that produce baby universes, the sequences do not converge, due to wormhole contributions, and so these states cannot admit large N limits. Nevertheless, appropriate averaging over N can lead to convergence to a mixed state. The associated algebras have nontrivial commutants, which can possibly be interpreted as operators in the baby universe.

By adapting previously known arguments concerning Ricci flow and the c-theorem, we give a direct proof that in a two-dimensional sigma-model with compact target space, scale invariance implies conformal invariance in perturbation theory. This argument, which applies to a general sigma-model constructed with a target space metric and B-field, is in accord with a more general proof in the literature that applies to arbitrary two-dimensional quantum field theories. Models with extended supersymmetry and a B-field are known to provide interesting test cases for the relation between scale invariance and conformal invariance in sigma-model perturbation theory. We give examples showing that in such models, the obstructions to conformal invariance suggested by general arguments can actually occur in models with target spaces that are not compact or complete. Thus compactness of the target space, or at least a suitable condition of completeness, is necessary as well as sufficient to ensure that scale invariance implies conformal invariance in models of this type.

In recent years, a surprisingly direct and simple rigorous understanding of quantum Liouville theory has developed. We aim here to make this material more accessible to physicists working on quantum field theory.

We investigate the differential geometry of the moduli space of instantons on S 3 × S 1 . Extending previous results, we show that a sigma-model with this target space can be expected to possess a large N = 4 superconformal symmetry, supporting speculations that this sigma-model may be dual to Type IIB superstring theory on AdS 3 × S 3 × S 3 × S 1 . The sigma-model is parametrized by three integers—the rank of the gauge group, the instanton number, and a ‘level’ (the integer coefficient of a topologically nontrivial B-field, analogous to a WZW level). These integers are expected to correspond to two five-brane charges and a one-brane charge. The sigma-model is weakly coupled when the level, conjecturally corresponding to one of the five-brane changes, becomes very large, keeping the other parameters fixed. The central charges of the large N = 4 algebra agree, at least semiclassically, with expectations from the duality.

We propose an algebra of operators along an observer’s worldline as a background-independent algebra in quantum gravity. In that context, it is natural to think of the Hartle-Hawking no boundary state as a universal state of maximum entropy, and to define entropy in terms of the relative entropy with this state. In the case that the only spacetimes considered correspond to de Sitter vacua with different values of the cosmological constant, this definition leads to sensible results.