Abstract
Extended objects such as line or surface operators, interfaces or boundaries play an important role in conformal field theory. Here we propose a systematic approach to the relevant conformal blocks which are argued to coincide with the wave functions of an integrable multi-particle Calogero-Sutherland problem. This generalizes a recent observation in [1] and makes extensive mathematical results from the modern theory of multi-variable hypergeometric functions available for studies of conformal defects. Applications range from several new relations with scalar four-point blocks to a Euclidean inversion formula for defect correlators.
Highlights
System of sewing constraints [2]
Applications range from several new relations with scalar four-point blocks to a Euclidean inversion formula for defect correlators
While initially the two-dimensional bootstrap started from the crossing symmetry of bulk four-point functions to gradually bootstrap correlators involving extended objects, better strategies were adopted later which depart from some of the sewing constraints involving extended objects
Summary
Before we begin discussing our new Calogero-Sutherland approach to defect blocks, we want to summarize the main results that are present in the existing conformal field theory literature. The setup that has received most attention involves two bulk fields in the presence of a p-dimensional defect. For such correlators, the conformal blocks are known at least as series expansions [15, 16] or, more explicitly, through relations with scalar fourpoint blocks which exist for some special cases, see subsection 2.3. Results on conformal blocks in the more generic setup when none of the defects is point-like are scarce, see [28], where the number of independent cross-ratios was counted and. This subsection contains a parametrization of defect crossratios in terms of new geometric variables that will turn out to be well adapted to our Calogero-Sutherland models later on
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