Week 12.02.2023 – 18.02.2023

These lectures aim to provide a self-contained introduction to the modern conformal bootstrap method. The study of conformal field theory (CFT) will first be motivated and the “old” way of studying CFTs as endpoints of RG flows will be explained. The set of ideas necessary to understand the conformal bootstrap method will then be introduced, and both analytic and numerical implementations of the conformal bootstrap method will be discussed.

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In 1965, Birch and Swinnerton-Dyer formulated a conjecture on the Mordell-Weil rank $r$ of elliptic curves which also implies the convergence of the Nagao-Mestre sum. We show that if the Nagao-Mestre sum converges, then the limit equals $-r+1/2$, and study the connections to the Riemann hypothesis for E. We also relate this to Nagao's conjecture for elliptic curves. Furthermore, we discuss a generalization of the above results for the Selberg classes and hence (conjecturally) for larger classes of $L$-functions.

De Finetti promoted the importance of predictive models for observables as the basis for Bayesian inference. The assumption of exchangeability, implying aspects of symmetry in the predictive model, motivates the usual likelihood-prior construction and with it the traditional learning approach involving a prior to posterior update using Bayes’ rule. We discuss an alternative approach, treating Bayesian inference as a missing data problem for observables not yet obtained from the population needed to estimate a parameter precisely or make a decision correctly. This motivates the direct use of predictive models for inference, relaxing exchangeability to start modelling from the data in hand (with or without a prior). Martingales play a key role in the construction. This is joint work with Stephen Walker and Edwin Fong, based on the paper “Martingale Posteriors” to appear with discussion JRSS Series B.

We construct an N = 2 supersymmetric gauged quantum mechanics, by starting from the 3d Chern-Simons-matter theory holographically dual to massive Type IIA string theory on AdS_4 × S^6, and Kaluza-Klein reducing on S^2 with a background that is dual to the asymptotics of static dyonic BPS black holes in AdS4. The background involves a choice of gauge fluxes, that we fix via a saddle-point analysis of the 3d topologically twisted index at large N. The ground-state degeneracy of the effective quantum mechanics reproduces the entropy of BPS black holes, and we expect its low-lying spectrum to contain information about near-extremal horizons. Interestingly, the model has a large number of statistically distributed couplings, reminiscent of SYK models.