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This workshop will be devoted to conformal field theories (CFT), and in particular to the circle of ideas surrounding the conformal bootstrap program in three and four dimensions. The bootstrap has been fully successful for two-dimensional CFTs, but very little has been achieved in d >2. In view of recent advances, the time seems ripe to reconsider the higher-dimensional bootstrap.

The physical motivation for studying the four-dimensional theories is to learn more about Gauge theories such as QCD. Three-dimensional CFTs describe the critical behavior of condensed matter systems and holographically define quantum gravity in four dimensions.

In recent years, spectacular progress has been achieved towards the exact solution of some three- and four-dimensional CFTs, thanks to the AdS/CFT correspondence and to the application of integrability techniques. The best known examples are N=4 SYM in d=4 and the ABJM theory in d=3, for which the exact spectrum is largely understood. A complete solution of these theories will be a major breakthrough in theoretical physics. Given the spectrum, it is very natural to ask what are the constraints of crossing symmetry on higher-point functions.

The question can also be asked holographically, and we intend to discuss higher-point correlation functions in AdS/CFT. We also aim to discuss which properties of a three-dimensional CFT are crucial to define a consistent theory of quantum gravity.

An independent, but clearly related, line of research that has had some recent notable success is the study of general bounds in CFTs that follow from crossing symmetry and unitarity.

The goal of this workshop is to bring together people working in these different approaches. The main theme is the higher-dimensional bootstrap program which - we foresee - is coming back to life.

**Itzhak Bars**

**Freddy Cachazo**

**Miguel Costa**

**Francis Dolan**

**Mike Douglas**

**Liam Fitzpatrick**

**Nikolay Gromov**

**Simeon Hellerman**

**Diego Hofman**

**Jared Kaplan**

**Rob Myers**

**Hugh Osborn**

**Kyriakos Papadodimas**

**Joao Penedones**

**David Poland**

**Suvrat Raju**

**Leonardo Rastelli**

**Riccardo Rattazzi**

**Werner Ruehl**

**Slava Rychkov**

**Amit Sever**

**Emery Sokatchev**

**Balt van Rees**

**Alessandro Vichi**

**Pedro Vieira**

**Werner Ruehl,** Technische Universitat Kaiserslautern

**AdS/CFT correspondence: Applications and puzzles**

Though AdS/CFT correspondence can be considered to be proved in general, some puzzles remain that have to be understood. In conformal harmonic analysis of CFT representations always appear in dual pairs. In AdS-QFT this has been shown to occur in the same way only in some special examples. It is known that in a perturbative approach to any CFT the critical objects, such as field dimensions in particular their anomalous parts, can be extracted from a system of conformal bootstrap equations (Parisi-Symanzik equations). In the corresponding AdS-QFTs such bootstrap equations must also exist that determine the masses of the fields in terms of power series of the coupling constants. There cannot be a freedom of mass renormalization (to experimental values) as in QED. We propose some applications hitherto not investigated.