Integrable open spin-chains in ${\mathrm{AdS}}_3/{\mathrm{CFT}}_2$ correspondences

We study integrable open boundary conditions for $\mathfrak{d}(2,1;\alpha {)}^2$ and $\mathfrak{p}\mathfrak{s}\mathfrak{u}(1,1|2{)}^2$ spin-chains. Magnon excitations of these open spin-chains are mapped to massive excitations of type-IIB open superstrings ending on D-branes in the ${\mathrm{AdS}}_3\times S^3\times S^3\times S^1$ and ${\mathrm{AdS}}_3\times S^3\times T^4$ supergravity geometries with pure R-R flux. We derive reflection matrix solutions of the boundary Yang-Baxter equation which intertwine representations of a variety of boundary coideal subalgebras of the bulk Hopf superalgebra. Many of these integrable boundaries are matched to D1- and D5-brane maximal giant gravitons.

Figure 1

Half of the D5-brane maximal giant graviton wrapping $S^1\times {S^3}^{\prime }\times S^1$ in ${\mathrm{AdS}}_3\times S^3\times {S^3}^{\prime }\times S^1$.

Figure 2

The D1-brane maximal giant graviton wraps a 1-cycle on a $S^1\times S^{1\prime }$ in ${\mathrm{AdS}}_3\times S^3\times S^{3\prime }\times S^1$.

Figure 3

The D1- and D5-brane maximal giant gravitons wrapping $S^1$ and $S^1\times T^4$ in ${\mathrm{AdS}}_3\times S^3\times T^4$.

Figure 4

Three-magnon scattering factorizes into a succession of two-magnon scattering events. The double red and blue line indicates the direct sum of left- and right-magnon states with scattering described by the complete $R$-matrix. The scattering of individual red or blue lines (left or right magnons) is described by partial $R$-matrices. This diagram gives both the Yang-Baxter equation for the complete $R$-matrix (treating double lines as a single composite line) and the Yang-Baxter equation for the partial $R$-matrices (choosing a red or blue line from each pair to give eight possibilities).

Figure 5

A two-magnon reflection off a singlet boundary factorizes into a succession of single-magnon reflections and two-magnon scattering events. The double red and blue line again indices the direct sum of left and right magnon states which scatter by complete $R$ and $K$-matrices.

Figure 6

A two-magnon reflection off a vector boundary factorizes into a succession of single-magnon reflections and two-magnon scattering events. The double red and blue line again indicates a direct sum of left and right magnons or boundary vector states.