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Title:
Introduction to the Conformal Bootstrap for Mathematicians
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Abstract:
Conformal field theories are a rich class of quantum field theories in general number of dimensions which can be defined and studied fully rigorously. In fact, the definition of a conformal field theory fits in one line: It is a unitary representation V of the conformal group G (G=the universal cover of SO(2,n)), which is a direct sum of irreducible lowest-weight modules, and which admits a certain kind of G-invariant multiplication. In my talk, I will explain this definition and some of its consequences. In particular, I will review that the definition implies stringent bounds on the weights of the irreducible components in V. These bounds are typically obtained using linear and/or semi-definite programming. The bounds are often sharp -- saturated by physical conformal field theories. In certain special cases, the sharpness of the bounds can be proved using magic functions for sphere packing in 8 and 24 dimensions. The ultimate goal of the conformal bootstrap is to classify conformal field theories starting from the above definition. The most tantalizing finding is that the bounds reveal the existence of many non-trivial solutions of the axioms, yet no explicit description of any non-trivial solution is presently known.
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