# Moore-Seiberg construction The fact that one can obtain the same Riemann surface from different surgery procedures imposes consistency conditions on the conformal blocks. The Moore-Seiberg construction says that 4-point crossing on the sphere and 1-point modular covariance on the torus are sufficient to ensure these "sewing constraints" of the theory on arbitrary closed orientable Riemann surfaces. ## Refs - original works - [[1988#Moore, Seiberg]] - [[1989#Moore, Seiberg]]: main - reformulation - [[1998#Bakalov, Kirillov]]: fills gaps of Moore-Seiberg using a different approach - see also - [[1988#Sonoda]] - [[1989#Cardy]] - for [[0548 Boundary CFT|boundary CFTs]] - see that note - for [[0620 Non-orientable CFT|non-orientable CFTs]] - see that note ## Relation to TQFT The Moore-Seiberg data can be summarised in a basis-independent way as a [[0644 Modular tensor category|modular tensor category]]. ## Non-compact Riemann surfaces To define CFTs on Riemann surfaces with [[0548 Boundary CFT|boundaries]], four additional constraints need to be satisfied: 1. crossing symmetry of the boundary 4-point function on the disk - analogue of bulk crossing 2. two boundary operators and a single bulk operator on the disk - the boundary circle is partitioned into two intervals by the two operators insertions - two decompositions correspond to inserting the internal boundary operator on each of the two intervals 3. two bulk operators and a single boundary operator on the disk - turning both bulk operators into boundary ones first v.s. fusing the two bulk operators into an internal bulk operator first 4. boundary 2-point function on the cylinder - cutting along a bulk circle vs cutting into two boundary pair-of-pants According to [[2006#Fjelstad, Fuchs, Runkel, Schweigert]], the 4th condition is redundant if: - there is a unique closed-state vacuum; - the disk two-point function is non-degenerate - the sphere two-point function is non-degenerate