We prove that when anomalies are canceled, the compact space partition functions can be expressed through holomorphic blocks associated to R2xT2. We also study 4d N = 1 theories on T2 fibrations over S2. This fits the interpretation of the 3d theories as codimension two defects. Moreover, particular degenerate correlators are mapped to S3 and S2xS1 partition functions of 3d N = 2 theories. We derive the 3-point functions for two different q-deformed CFTs, and we show that non-chiral correlators can be identified with S5 and S4xS1 partition functions of certain 5d N = 1 theories. In this work we define a class of q-deformed CFT correlators, where chiral blocks are controlled by the q-Virasoro algebra and are identified with R4xS1 instanton partition functions. The perturbative integrand is identified with the product of CFT 3-point functions, while each copy of the non-perturbative instanton partition function is identified with conformal blocks of the Virasoro algebra. Remarkably, the AGT correspondence identifies the S4 partition function of class S theories with Liouville CFT correlators. An example of this phenomenon is given by the S4 partition function of 4d N = 2 theories, which can be written as an integral of two copies of the R4 Nekrasov partition function. Partition functions on different compact spaces can be obtained by fusing the holomorphic blocks with pairings reflecting the geometric decomposition of the backgrounds.
The compact space partition functions of the considered theories can be decomposed into products of holomorphic blocks which are identified with partition functions on elementary geometries. In this thesis we study partition functions of supersymmetric gauge theories on compact backgrounds in various dimensions, with particular focus on infinite dimensional symmetry algebras encoded in these observables.
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