Exact solution of the supersymmetric sinh-Gordon model with boundary

Abstract

The boundary supersymmetric sinh-Gordon model is an integrable quantum field theory in 1 + 1 dimensions with bulk N = 1 supersymmetry, whose bulk and boundary S matrices are not diagonal. We present an exact solution of this model. In particular, we derive an exact inversion identity and the corresponding thermodynamic Bethe Ansatz equations. We also compute the boundary entropy, and find a rich pattern of boundary roaming trajectories corresponding to c < 3/2 superconformal models.

The scaling supersymmetric Yang–Lee model with boundary

“…Indeed, the S matrix is that of the first SSG breather [40], [41] with α = 1/3. In particular, it coincides with the expression for the S matrix of the supersymmetric sinh-Gordon model given in [15] with B = −1/3.…”

“…The above expression for S SU SY essentially coincides with the one for the supersymmetric sinh-Gordon model given in [15] with B = − 1 3 , ε = +1, ϕ = φ + iπ 2 with φ real, and r = −ir. The only differences lie in the CDD factor F (θ ; φ) (which is absent from [15]) and the factor Y 1 : the expression given here is an analytic continuation of the one given in [15].…”

“…We now try to determine S(θ) for the boundary SYL model (4.1). By analogy with the bulk SYL model, as well as with the boundary YL model, we expect that the boundary S matrix of the boundary SYL model should be some reduction of that of the boundary supersymmetric sine-Gordon model [45], or equivalently, the boundary supersymmetric sinh-Gordon model [15]. We therefore consider 12) where the scalar factor S Y L (θ ; b) is given by (2.3), and S SU SY (θ ; φ) is given by…”

“…Furthermore, the CBC and the boundary perturbation must be compatible [10], [9]. By means of a "boundary TBA" [13] - [15], ratios of g factors of the boundary CFT can be computed from the bulk and boundary S matrices. (See also [16], [17].)…”

“…We then propose the boundary S matrix for the boundary SYL model. Our approach is to restrict the boundary S matrix of the boundary supersymmetric sinh-Gordon model [15], by imposing the various boundary bootstrap constraints [9]. We then show that the proposed boundary S matrix commutes with a supersymmetry-like charge.…”