Three-Point Functions in ABJM and Bethe Ansatz

Abstract

We develop an integrability-based framework to compute structure constants of two subdeterminant operators and a single-trace non-BPS operator in ABJM theory in the planar limit. In this first paper, we study them at weak coupling using a relation to an integrable spin chain. We first develop a nested Bethe ansatz for an alternating SU(4) spin chain that describes single-trace operators made out of scalar fields. We then apply it to the computation of the structure constants and show that they are given by overlaps between a Bethe eigenstate and a matrix product state. We conjecture that the determinant operator corresponds to an integrable matrix product state and present a closed-form expression for the overlap, which resembles the so-called Gaudin determinant. We also provide evidence for the integrability of general sub-determinant operators. The techniques developed in this paper can be applied to other quantities in ABJM theory including three-point functions of single-trace operators.

On exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chains

“…which agrees with the convention in [9] if c = i. To get the corresponding K-matrix we have to solve the equation (5.28)…”

“…It was shown that these three-point functions can be calculated as an overlap between the finite volume multi-particle state and an integrable initial state. First these quantities were investigated in the AdS 5 /CF T 4 duality [7,8] and recently the method was generalized to the AdS 4 /CF T 3 duality [9].…”

“…Nevertheless, the exact overlaps can be guessed, since in the factorized formulas only the single-particle overlaps F ν (u ν j ) are the unknown quantities. To determine these quantities there exist two approaches which uses coordinate Bethe Ansatz [11,24,25,7,9] or Quantum Transfer Matrix [26,27,28,29] methods. However, we emphasize that these procedures are not suitable for proving the overlap formulas.…”

“…The ABJM theory in the scalar sector can be described as an alternating gl(4) spin chain where the quantum space is Λ ⊗ Λ ⊗L where Λ = (1, 1) = (1, 0, 0, 0) which is compatible with our untwisted monodromy matrix. In [9] the authors proposed an overlap for the two-site state…”

D-branes and orbit average

“…In the top-down construction of AdS/CFT based on string theory, operators with different conformal dimensions admit different holographic descriptions. For instance in the matrixlike large N limit, operators with O(1) conformal dimension 1 are dual to perturbative string states while operators with O(N ) and O(N 2 ) conformal dimensions correspond to D-branes and backreacted geometries including black holes. The best studied among them are operators dual to string states since they can be analyzed by various approaches such as supergravity, integrability, conformal bootstrap and perturbation theory.…”