Abstract
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 → ∞. Upon performing an SL(2, 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, 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-Nf Dirac fermions — for which the extrapolation to strong coupling can be done exactly order-by-order in 1/Nf — and a free complex scalar.
Highlights
The objective of this paper is to study conformal invariant boundary conditions for free Abelian gauge theory in four-dimensions
A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a U(1) symmetry living on a boundary
A striking property of these boundary conformal field theories (BCFT) is that they are typically well-defined on some open patch in the space of the four-dimensional gauge coupling
Summary
The objective of this paper is to study conformal invariant boundary conditions for free Abelian gauge theory in four-dimensions. [19,20,21,22,23,24,25] for recent results and the review [26]) We will apply this strategy to a very nice boundary condition for a U (1) gauge theory, which is conjecturally associated to a free Dirac fermion at two distinct cusps and to the O(2) model at two other cusps [27,28,29,30,31]. Applying the general strategy to compute the hemisphere partition function to this case, and taking the limit λ → ∞, we obtain the 1/Nf correction to the sphere partition function of large-Nf QED3 Another example with a Z2 duality acting on τ is conjecturally obtained in the case where the theory on the boundary is a free complex scalar, or equivalently the U(1) GrossNeveu model [34, 35]. We show how to obtain QED3 with 2 fermionic flavors starting with this setup, using the extended electric-magnetic duality group Sp(4, Z) that acts on the two bulk gauge fields
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