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We study surface operators in 3d Topological Field Theory and their relationswith 2d Rational Conformal Field Theory. We show that a surface operator givesrise to a consistent gluing of chiral and anti-chiral sectors in the 2d RCFT.The algebraic properties of the resulting 2d RCFT, such as the classificationof symmetry-preserving boundary conditions, are expressed in terms ofproperties of the surface operator. We show that to every surface operator onemay attach a Morita-equivalence class of symmetric Frobenius algebras in theribbon category of bulk line operators. This provides a simple interpretationof the results of Fuchs, Runkel and Schweigert on the construction of 2d RCFTsfrom Frobenius algebras. We also show that every topological boundary conditionin a 3d TFT gives rise to a commutative Frobenius algebra in the category ofbulk line operators. We illustrate these general considerations by studying indetail surface operators in abelian Chern-Simons theory.

Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory