Free energy and defect C-theorem in free scalar theory

Abstract

We describe conformal defects of p dimensions in a free scalar theory on a d-dimensional flat space as boundary conditions on the conformally flat space ℍp+1× 𝕊d−p−1. We classify two types of boundary conditions, Dirichlet type and Neumann type, on the boundary of the subspace ℍp+1 which correspond to the types of conformal defects in the free scalar theory. We find Dirichlet boundary conditions always exist while Neumann boundary conditions are allowed only for defects of lower codimensions. Our results match with a recent classification of the non-monodromy defects, showing Neumann boundary conditions are associated with non-trivial defects. We check this observation by calculating the difference of the free energies on ℍp+1× 𝕊d−p−1 between Dirichlet and Neumann boundary conditions. We also examine the defect RG flows from Neumann to Dirichlet boundary conditions and provide more support for a conjectured C-theorem in defect CFTs.

Free energy and defect C-theorem in free fermion

“…For DCFT with p < d − 1, our conjecture however has passed only several checks in field theory [12,18,21,[30][31][32] and holography [28,29,33]. Recently, we provide further evidence of our conjecture in a simple model, a conformally coupled scalar field [34] (See also related works [35,36]). Instead of putting the conformally coupled scalar field on the sphere, we put it on H p+1 ×S q−1 , where q = d−p is a codimension of the defect, and impose boundary conditions at the boundary of H p+1 .…”

“…In this paper, we extend our analysis [34] to a free fermion to provide a further check of our conjecture. To compare with a scalar field, a defect free energy of a free fermion in higher dimensions has been studied only in BCFT [36,37] and DCFT with a twocodimensional defect in the context of entanglement entropy [49,50].…”

“…as in the scalar case [34,35] and the holographic case [35]. It is expected that the holographic result should be the same as the free fermion result because the anomaly coefficient does not depend on the strength of a coupling constant and the holography can be applied if the number of fermions is large.…”

“…where we used the awkward suffix for α n,d+1 and β n,d+1 to use the same notation in the scalar case [34]. Since α 0,d+1 = 0, we can omit the n = 0 term in the summation for even d whenever we want.…”

“…For even q, the combination of the Hurwitz zeta functions can be simplified using a formula ) in [34] for the derivation. Unfortunately, it is difficult to simplify the equations anymore, so we perform the integral (4.61) numerically.…”