For theories with mixed topological, holomorphic, and ordinary spacetime dimensions, OPE coefficients are proposed to be sheaf cohomology classes, with singular derived coefficients appearing under explicit dimension-counting conditions.
Higher Operations in Perturbation Theory
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abstract
We discuss the role of formal deformation theory in quantum field theories and present various ``higher operations'' which control their deformations, (generalized) OPEs, and anomalies. Particular attention is paid to holomorphic-topological theories where we systematically describe and regularize the Feynman diagrams which compute these higher operations in free and perturbative scenarios, including examples with defects. We prove geometrically that the resulting higher operations satisfy expected ``quadratic axioms,'' which can be interpreted physically as a form of Wess-Zumino consistency condition for BRST symmetry. We discuss a higher-dimensional analogue of Kontsevich's formality theorem, which proves the absence of perturbative corrections in TQFTs with two or more topological directions. We discuss at some length the relation of our work to the theory of factorization algebras and provide an introduction to the subject for physicists.
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On coefficients of operator product expansions for quantum field theories with ordinary, holomorphic, and topological spacetime dimensions
For theories with mixed topological, holomorphic, and ordinary spacetime dimensions, OPE coefficients are proposed to be sheaf cohomology classes, with singular derived coefficients appearing under explicit dimension-counting conditions.