Abstract
The mathfrak{psu}left(2,left.2right|4right) integrable super spin chain underlying the AdS/CFT correspondence has integrable boundary states which describe set-ups where k D3-branes get dissolved in a probe D5-brane. Overlaps between Bethe eigenstates and these boundary states encode the one-point functions of conformal operators and are expressed in terms of the superdeterminant of the Gaudin matrix that in turn depends on the Dynkin diagram of the symmetry algebra. The different possible Dynkin diagrams of super Lie algebras are related via fermionic dualities and we determine how overlap formulae transform under these dualities. As an application we show how to consistently move between overlap formulae obtained for k = 1 from different Dynkin diagrams.
Highlights
All known overlap formulae contain as a key ingredient the Gaudin matrix [17] of the Bethe eigenstate, or more precisely an object which can be expressed as the superdeterminant of the Gaudin matrix [5]
Dynkin diagrams and Cartan matrices for super Lie algebras are not unique [19] but related via a set of fermionic dualities [20], and this immediately raises a question in relation to the newly derived overlap formulae for integrable super spin chains, namely: how do these formulae transform under fermionic dualities? This question is the main focus of our work
Recent results on integrable one-point functions in domain wall versions of N = 4 SYM [5, 6] have triggered a need to know how overlap formulae for super spin chains can be translated between different gradings of the underlying super Lie algebra
Summary
The Bethe-ansatz spectrum of an integrable spin chain with a rational R-matrix is neatly encoded in the group-theory data. Expectation values of local operators induced by the D-brane are naturally given by an overlap between the boundary state and on-shell Bethe eigenstates. This description proved highly efficient in computations of correlation functions in the presence of domain walls [1, 2] or of very large determinant operators [24, 25]. Where the bracket D3D5| denotes a boundary state in the psu(2, 2|4) spin chain It can be explicitly constructed in perturbation theory by evaluating Feynman diagrams in the presence of the defect [5, 27,28,29]. Understanding how overlaps transform under fermionic duality is the main goal of this paper
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