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

ADS空间中的M2膜可以通过ADS/CFT对偶来描述共形场理论中边界上的扩展算符。利用这种刻画，我们在半经典水平上研究了推广的算子。我们研究了渐近ADS7空间中的M2-膜，它们描述了(2，0)6维CFT中的曲面算子。通过计算半经典配分函数的散度，我们直接从其全息描述中得到了一大类超对称曲面算子的共形反常，与文献中关于这种反常的间接结果相吻合.我们的目标是将这一结果推广到对偶到非超对称曲面算子(M2膜满足Neumann条件)和具有非平凡R对称破缺模式的M2膜.我们还研究了AdS4中描述涡旋和威尔逊环的M2膜.我们使用这种描述来计算超出经典极限的1/2和1/3 bps涡环的期望值。我们还使用全息描述来计算涡旋环上算子的关联函数。特别地，我们关注涡旋环上的精确边缘算子。这涉及到在M2膜上计算涉及边界上的这些边缘算子的Witten图。我们还使用Bootstrap方法研究了相关函数，这应该证实了我们的全息结果。我们开发的研究M2膜配分函数发散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=4SYM中表示曲面算子的D3-膜的近边界经典描述，并用它来计算这些曲面算子的经典反常。

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.