Title: Consistency of 2d Conformal field theory with boundary and Swiss-cheese operad

Abstract:
 Conformal field theory can be formulated algebraically using the
 associativity and commutativity (a.k.a. bootstrap equation) of operator
 product expansions (OPEs). However, the algebraic structure of
 conformal field theory differs from ordinary algebra in that it has
 convergence regions where the products are well-defined.
 In this talk, we will show that, in the case of 2d conformal field
 theories with/without boundary, the combinatorial structure of the
 convergence region of OPEs can be described based on the fundamental
 groupoid of the Swiss-Cheese/E2 operad (PaPB/PaB operad).

 Mathematically, we will show that
 (1) The PaPB/PaB operad acts on the representation category of a vertex
 operator algebra imposing a certain finiteness.
 (2) The bootstrap equations imposed on conformal field
 theories with/without boundary in physics are generators of the
 PaPB/PaB operad.
 (3) From this, we can prove the consistency of conformal field theory
 under the finiteness of (1).
 If time permits, we will also explain some explicit constructions of
 solutions of the bootstrap equations for boundary conformal field
 theory (D branes).