From Defects to Boundaries

Abstract

In this paper we describe how relativistic field theories containing defects are equivalent to a class of boundary field theories. As a consequence previously derived results for boundaries can be directly applied to defects, these results include reduction formulas, the Coleman–Thun mechanism and Cutcosky rules. For integrable theories the defect crossing unitarity equation can be derived and defect operator found. For a generic purely transmitting impurity we use the boundary bootstrap method to obtain solutions of the defect Yang–Baxter equation. The groundstate energy on the strip with defects is also calculated.

Asymptotic Bethe Ansatz on the GKP vacuum as a defect spin chain: Scattering, particles and minimal area Wilson loops

“…The extra term 2D(u m ) in the exponent is interpreted as the effect of two purely transmitting (i.e. without reflection [46]) defects related to the tips of the GKP string. The rational factor in the right hand side of (2.8) takes into account the internal degrees of freedom: solving (2.7) one obtains u q,k in terms of u m : plugging this result in the rational term in (2.8) one obtains, together with the products over the various S(u m , u m ′ ), the phase change due to the scattering between an excitation with rapidity u m and the other excitations with rapidity u m ′ .…”

“…We wish also to notice the possibility of interpreting the free boson c = 1 2D CFT (Coulomb gas) correlation function formulae by means of this one. Now, we can make explicit the gaussian measure in (11.45) as a kinetic term so to read W (g) hex as a quantum mechanics partition function for the field X g (u) 46) where the action S (g) [X g ], directly expressed in terms of the hyperbolic rapidity θ, has the form…”

A next-to-leading Lüscher formula

“…If we consider the physical particles as defects introduced in the compactified space of a cylinder, then we have that the usual defect transmission phase [39,40] is given by the S-matrix, which describes the scattering of a probe particle against N excitations, that is, in a relativistic case…”

“…Upon a double Wick rotation, the defect line, which defines, in our case, the asymptotic Bethe equations for the physical theory, 2 becomes a defect operator [37,39,40] (see figure 1), which modifies the expression of the mirror 3 partition function: 5) where, introducing the mirror theory rapidityθ = θ+i/2, the defect operator, for a diagonal theory with single species particles, is given by 4 6) with A, A † being the Zamolodchikov-Faddeev annihilation and creation operators, respectively, in the mirror theory. In particular, the n-particle matrix element of the mirror partition function can be calculated as…”

Finite volume form factors in the presence of integrable defects

“…The T + transmission factor is parametrized such that for its physical domain of rapidities (θ < 0) its argument is always positive. Unitarity and crossing symmetry relates these two amplitudes as [2]:…”