TBA, NLO Lüscher correction, and double wrapping in twisted AdS/CFT

Abstract

The ground-state energy of integrably-twisted theories is analyzed in finite volume. We derive the leading and next-to-leading order (NLO) Lüscher-type corrections for large volumes of the vacuum energy for integrable theories with twisted boundary conditions and twisted S-matrix. We then derive the twisted thermodynamic Bethe ansatz (TBA) equations to describe exactly the ground state, from which we obtain an untwisted Y-system. The two approaches are compared by expanding the TBA equations to NLO, and exact agreement is found. We give explicit results for the O(4) model and for the three-parameter family of γ-deformed (non-supersymmetric) planar AdS/CFT model, where the ground-state energy can be nontrivial and can acquire finite-size corrections. The NLO corrections, which correspond to double-wrapping diagrams, are explicitly evaluated for the latter model at six loops.

A next-to-leading Lüscher formula

“…We shall observe, at least in a diagonal case, that the insertion of such defects modifies the ground-state TBA equations in the same way as resulting from the standard analytical continuation of the ground-state TBA [15,16,38]. More importantly for our purposes, the twist matrices involved in the NLO Lüscher-like formula for the twisted vacuum [37] will be replaced by the S-matrices describing the scattering between physical and virtual particles. This will give as a result the first proposal of NLO Lüscher-like formulas for the excited states' energies and rapidities, even in a non-diagonal integrable theory.…”

“…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…”

“…This will be done by using the experience with the calculation of double-wrapping corrections in the vacuum of the deformed O(4) σ-model and AdS/CF T [37], and by considering the physical excited states as momenta-dependent defects, which twist the boundary conditions of the vacuum. We shall observe, at least in a diagonal case, that the insertion of such defects modifies the ground-state TBA equations in the same way as resulting from the standard analytical continuation of the ground-state TBA [15,16,38].…”

“…Following the analysis of [37], we can easily write now the second term in the r.h.s. of (2.9) in integral form.…”

“…At the next-to leading order, like in [37], there is a diagonal contribution to the last term in (2.9), given by…”