Extended operators in conformal field theories and the quantum M2-brane.

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M2 branes in asymptotically AdS space can describe extended operators in the conformal field theory at the boundary via the AdS/CFT duality. We utilise this description to study extended operators at the semiclassical level.We study M2-branes in asymptotically AdS7 space, which describe surface operators in the (2,0) 6d CFT. By computing the divergence in their semiclassical partition function, we obtain the conformal anomaly for a large class of supersymmetric surface operators directly from their holographic description, matching indirect results for the anomaly in the literature.We aim to extend this to M2-branes dual to non-supersymmetric surface operators (where the M2-brane satisfy Neumann conditions) and ones with non-trivial R-symmetry breaking pattern.We also study M2-branes in AdS4, which describe vortex and Wilson loops in ABJM theory. We use this description to compute the expectation value of 1/2 and 1/3 BPS vortex loops beyond the classical limit. We are also working on computing correlation functions of operators on vortex loops using the holographic description. In particular, we focus on exactly marginal operators on the vortex loop. This involves computing Witten diagrams on the M2-brane involving these marginal operators at the boundary. We also use the bootstrap approach to study the correlation functions, which should confirm our holographic results.The tools we develop for the study of M2-brane partition functions' divergences can be applied to other branes in asymptotically AdS space, also representing extended operators in CFTs. In particular, we aim to find the near-boundary classical description of D3-branes representing surface operators in N=4 SYM, and use it to compute the classical anomaly for these surface operators.