Abstract
We describe a p-dimensional conformal defect of a free Dirac fermion on a d-dimensional flat space as boundary conditions on a conformally equivalent space ℍp+1× mathbbm{S} d−p−1. We classify allowed boundary conditions and find that the Dirichlet type of boundary conditions always exists while the Neumann type of boundary condition exists only for a two-codimensional defect. For the two-codimensional defect, a double trace deformation triggers a renormalization group flow from the Neumann boundary condition to the Dirichlet boundary condition, and the free energy at UV fixed point is always larger than that at IR fixed point. This provides us with further support of a conjectured C-theorem in DCFT.
Highlights
For BCFTs with p = d − 1, a slight modification is needed since BCFTs is defined on a hemisphere HSd, and the boundary free energy is introduced as1 log D(p)
We will compute the renormalized free energy by evaluating the zeta function based on the method used in [62, 63]
We studied a free Dirac fermion on Hp+1 × Sq−1 as a DCFT
Summary
We summarise coordinate systems for a sphere Sd, a hemisphere HSd, a hyperbolic space Hd and Hp+1 ×Sq−1 and conformal maps among a flat space Rd and them. After a Weyl rescaling, the metric (2.4) reduces to a geometry Hp+1 × Sq−1 with radius R, ds2 = R2. The defect sits at the boundary of Hp+1. We classify allowed boundary conditions of a massless fermion on Hp+1 × Sq−1. For the product space Hp+1 × Sq−1, we consider a massless fermion from the beginning. We first decompose the fermionic field as ψ(z, x, θ) = ψHp+1 (z, x) ⊗ ψ ,Sq−1 (θ) ,. The mode with = 0 for q = 2 is allowed to have the boundary conditions corresponding to ∆−. The allowed boundary conditions for the massless fermion are classified as follows and are listed in table 1. The = 0 mode is allowed, resulting in a nontrivial boundary condition with ∆−=0 =. Two different boundary conditions are allowed, Dirichlet b.c.
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