TcCFT · Tools to Carve out Conformal Field Theories

Tools to Carve out Conformal Field Theories

The Bootstrap philosophy employs mathematical consistency (e.g. symmetries and quantum mechanics) to map out the space of quantum theories - including strongly coupled models - and make sharp predictions for their physical observables. While the Bootstrap idea dates back to the early 1940s with its roots in the S-matrix approach to the strong nuclear force, in recent years it has experienced a wave of renewed interest and successes owing to the development of new computational tools and physical insights. A notable development has been the application of the Bootstrap to Conformal Field Theories (CFTs) in d>2 dimensions, known as the Conformal Bootstrap. The constraint of Conformal Symmetry makes CFTs easier to study than generic QFTs since, given a spectrum and operator product expansion (OPE) coefficients, it specifies all observables. The OPE coeffients can be determined by solving the crossing equation (fig 1), which enforces associativity of the OPE - known as crossing symmetry. The ubiquity and universality of Conformal Field Theories in physical systems means that successes of the Conformal Bootstrap have potentially far-reaching impact in various branches of theoretical physics. Indeed, CFTs describe a wealth of physical phenomena such as: Critical phenomena, the String Theory world sheet and, via gauge-gravity duality, UV complete theories of gravity, and more, for which there are few quantitative methods available to study their properties – in particular non-perturbatively. The most celebrated results of the Conformal Bootstrap programme to date have been borne out of the development of powerful numerical machinery based on convex optimisation, which provide bounds on OPE data consistent with crossing symmetry. In contrast, the fruits of efforts to tackle the Conformal Bootstrap problem analytically are taking longer to mature. The last couple of years however have seen pivotal progress, to which this project has contributed directly. Most significantly, powerful inversion techniques have been developed which give a recipe to extract OPE data from a given CFT correlator. This provides a long-awaited CFT analogue of the celebrated S-matrix inversion formula of Froissart and Gribov. Inversion techniques also give rise to the decomposition of individual operator exchanges under crossing, known as crossing kernels, which provide basic building blocks needed to solve the crossing equation analytically. Ultimately the development of analytic tools to solve non-perturbatively a QFT is of utmost importance to push forward our understanding of quantum phenomena in general, phase transitions in condensed matter and via gauge-gravity duality also the emergent nature of quantum gravity which are among the most fundamental problems in nature and for which even a small step forward would constitute an important advance for the whole society.

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