The continuum limit of sl(N/K) integrable super spin chains

Abstract

I discuss in this paper the continuum limit of integrable spin chains based on the superalgebras sl(N/K). The general conclusion is that, with the full ``supersymmetry'', none of these models is relativistic. When the supersymmetry is broken by the generator of the sub u(1), Gross Neveu models of various types are obtained. For instance, in the case of sl(N/K) with a typical fermionic representation on every site, the continuum limit is the GN model with N colors and K flavors. In the case of sl(N/1) and atypical representations of spin j, a close cousin of the GN model with N colors, j flavors and flavor anisotropy is obtained. The Dynkin parameter associated with the fermionic root, while providing solutions to the Yang Baxter equation with a continuous parameter, thus does not give rise to any new physics in the field theory limit. These features are generalized to the case where an impurity is embedded in the system

The BaxterQ-operator for the gradedSL(2|1) spin chain

“…The construction of Q−operators can be extended to spin chains with the SL(2|N ) sym-metry [40,41,42]. These models have recently attracted attention in light of the gauge/string duality, most notably for the maximally supersymmetric N = 4 gauge theory.…”

Integrability and the AdS/CFT correspondence

“…The Bethe Ansatz techniques for integrable spin chains with simple Lie algebras as the symmetry algebras have been developed in [29], [30] and subsequently generalized to the case of superalgebras [31]. This allowed Beisert and Staudacher [4] to write down the one-loop Bethe equations of the planar N = 4 SYM theory.…”

Bethe ansatz equations for general orbifolds of 𝒩 = 4 SYM

“…6 As a consequence of the orbifold projection, some of the physical operators (49) vanish identically. Using equation (50) to commute γ(g) past all the fields in the operator shows that a necessary condition for non-vanishing operators is that the total single trace operator must be invariant under the simultaneous action of R(g) B A on all the spins. However, while necessary, this is not sufficient.…”

“…8 The reason is that the single site operators W A are all selected via the orbifold projection (28)- (29) to transform according to (50) under the finite group Γ. The space of orbifold operators is spanned by the orbit basis, defined by…”

“…Introduction of the twist field causes appearance of an extra vertical cut in the dual graph (dotted lines). Should this cut be located in the interaction region it can be shifted away using the interchange relation (50). Representation matrix R(g) being diagonalized as in (55), this shift results in a mere phase factor σ A (g)σ B (g) −1 .…”