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

渐近AdS空间中的M2膜可以通过AdS/CFT对偶性在边界处描述共形场论中的扩展算子。我们利用这一描述在半经典水平上研究了扩展算子。我们研究了渐近AdS7空间中的m2 -膜，它描述了（2,0）6d CFT中的表面算子。通过计算它们的半经典配分函数的散度，我们直接从它们的全息描述中得到了一类超对称曲面算子的共形异常，与文献中关于异常的间接结果相匹配。我们的目标是将其扩展到m2膜对偶到非超对称表面算子（其中m2膜满足诺伊曼条件）和具有非平凡r -对称破缺模式的算子。我们还研究了AdS4中的m2膜，它描述了ABJM理论中的涡旋和威尔逊环。我们利用这一描述计算了超出经典极限的1/2和1/3 BPS涡旋回路的期望值。我们也在利用全息描述计算涡旋环上算子的相关函数。特别地，我们关注的是涡环上的精确边缘算子。这涉及到计算m2膜上的Witten图，涉及到边界上的这些边缘算子。我们还使用自举法研究了相关函数，这应该证实了我们的全息结果。我们开发的用于研究m2 -膜配分函数散度的工具可以应用于渐近AdS空间中的其他膜，也表示cft中的扩展算子。特别地，我们的目标是找到N=4 SYM中表示表面算子的三维膜的近边界经典描述，并用它来计算这些表面算子的经典异常。

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.