Operator expansions, layer susceptibility and two-point functions in BCFT

Abstract

We show that in boundary CFTs, there exists a one-to-one correspondence between the boundary operator expansion of the two-point correlation function and a power series expansion of the layer susceptibility. This general property allows the direct identification of the boundary spectrum and expansion coefficients from the layer susceptibility and opens a new way for efficient calculations of two-point correlators in BCFTs. To show how it works we derive an explicit expression for the correlation function 〈ϕiϕi〉 of the O(N) model at the extraordinary transition in 4 − ϵ dimensional semi-infinite space to order O(ϵ). The bulk operator product expansion of the two-point function gives access to the spectrum of the bulk CFT. In our example, we obtain the averaged anomalous dimensions of scalar composite operators of the O(N) model to order O(ϵ2). These agree with the known results both in ϵ and large-N expansions.

Fusion of conformal defects in four dimensions

“…The defects we consider breaks the O(N )-symmetry of the bulk scalars. This means that the RG flow might be similar to that in the extraordinary phase transition near a boundary [23,24]. 9 We can also proceed to study fusion of higher dimensional defects.…”

Superconformal boundaries in 4 − ϵ dimensions

“…The importance of ζ was noted in [57], where they observed that the contribution of a single conformal block in the boundary expansion is proportional to ζ∆ − d−1 2 . This allows one to extract the boundary CFT data directly from the susceptibility without the need to reexpress everything in terms of the correlation function F (ξ).…”

“…Crucially, this integral transform is invertible and one can recover the two-point function in terms of the susceptibility. This idea has been recently used to compute the one-loop twopoint function of the order parameter in the extraordinary phase transition of the O(N ) model [56,57]. One can also apply it to the O(N ) model in the large-N limit, see [58] for the three-dimensional case with a φ 6 potential.…”

“…We will evaluate this integral with a trick that has been used in the literature in similar situations [56,57,59]. The idea is to split the integration region between z > z and z < z .…”

On analytic bootstrap for interface and boundary CFT

“…Bootstrap techniques for BCFT and ICFT were studied in [1]. See [4][5][6][7][8][9][10][11] for recent works on BCFTs.…”