from Part IV - Anomalies and Spacetimes
Published online by Cambridge University Press: 05 September 2026
The d=2 Weyl anomaly finds another important application in the context of fundamental strings. Fundamental strings start from two-dimensional quantum field theories whose dynamical degrees of freedom are the embedding maps of the string worldsheet into spacetime. The usual formulation, known as the Polyakov string, involves a worldsheet metric, the integration over which should bring us back to the Nambu-Goto type. The auxiliary nature of the worldsheet metric, combined with how the d=2 metric can be brought into a conformally flat form, demands that the d=2 quantum field theory thus defined over the worldsheet should be invariant under the d=2 renormalization flow. A surprising consequence is that this alone forces the spacetime manifold, in which fundamental strings are moving about, to be Ricci flat in the simplest model and, more generally, to be a solution to an Einstein-like equation. We derive this central fact in the spacetime presence of the metric, the Kalb-Ramond antisymmetric tensor field, and a dilaton field, which form a universal multiplet that appears in all string models, which implies that a version of quantum gravity must reside inside any string theory model.
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