arXiv 1902.095672019-02hep-thcond-mat.str-el

边界上的三维阿贝尔规范理论

3d Abelian Gauge Theories at the Boundary

Lorenzo Di Pietro, Davide Gaiotto, Edoardo Lauria, Jingxiang Wu

备注：: 69 pages
DOI：: 10.1007/JHEP05(2019)091

摘要：

一个四维阿贝尔规范场可以耦合到一个具有$U(1)$对称性的三维CFT。这种耦合产生了一个连续的家庭边界共形场理论（BCFT）参数化的规范耦合$τ$在上半平面和选择的CFT的去耦限制$τ\to \infty$。在批量执行$SL(2,\mathbb{Z})$变换并进入新帧中的解耦极限时，在边界上发现不同的3d CFT，其通过Witten的$SL(2, \mathbb{Z})$动作与原始CFT相关[1]。特别地，实$τ$轴上的尖点对应于原始CFT的3d测量。我们研究了这种BCFT的一般性质。我们展示了如何表达散装的一个和两个点的功能，和半球自由能，在两个点的功能的边界电流和磁场。然后，我们考虑的情况下，其中的3D CFT是一个狄拉克费米子。由于3D对偶性，该BCFT通过批量$S$变换映射到自身，并且它还承认在边界上给出$O(2)$模型的解耦限制。我们计算边界算子的标度维数和半球自由能两个循环。使用$S$对偶改进的分析，我们外推的微扰结果，并找到良好的近似的观测值的$O(2)$模型。我们还考虑与其他理论的边界上的例子，如大- $N_f$狄拉克费米子-其中的外推到强耦合可以完全按顺序在$1/N_f$ -和一个自由的复杂标量。

英文摘要：

A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a $U(1)$ symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFT) parametrized by the gauge coupling $τ$ in the upper-half plane and by the choice of the CFT in the decoupling limit $τ\to \infty$. Upon performing an $SL(2,\mathbb{Z})$ transformation in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's $SL(2, \mathbb{Z})$ action [1]. In particular the cusps on the real $τ$ axis correspond to the 3d gauging of the original CFT. We study general properties of this BCFT. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. We then consider the case in which the 3d CFT is one Dirac fermion. Thanks to 3d dualities this BCFT is mapped to itself by a bulk $S$ transformation, and it also admits a decoupling limit which gives the $O(2)$ model on the boundary. We compute scaling dimensions of boundary operators and the hemisphere free-energy up to two loops. Using an $S$-duality improved ansatz, we extrapolate the perturbative results and find good approximations to the observables of the $O(2)$ model. We also consider examples with other theories on the boundary, such as large-$N_f$ Dirac fermions --for which the extrapolation to strong coupling can be done exactly order-by-order in $1/N_f$-- and a free complex scalar.

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