On three-point functions in the AdS4/CFT3 correspondence

We develop an integrability-based framework to compute structure constants of two sub-determinant 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.

“…17 See [110] for results in the SU(2)×SU(2) sector. 18 It would also be useful to revisit the analysis in [77] in order to understand the difference between N = 4 SYM and ABJM.…”

Section: A Partially Contracted Giant Gravitonmentioning

confidence: 99%

Three-point functions in ABJM and Bethe Ansatz

Yang

Jiang

Komatsu

et al. 2022

“…34 For early examples, see [71][72][73][74][75]. 35 The factors of p cancel between the coefficient of (C.2) and the sum in the trace.…”

Section: C1 Truncations and Flowsmentioning

confidence: 99%

Boundary terms and three-point functions: an AdS/CFT puzzle resolved

Freedman

Pilch

Pufu

et al. 2017

N = 8 superconformal field theories, such as the ABJM theory at ChernSimons level k = 1 or 2, contain 35 scalar operators O IJ with ∆ = 1 in the 35 v representation of SO(8). The 3-point correlation function of these operators is non-vanishing, and indeed can be calculated non-perturbatively in the field theory. But its AdS 4 gravity dual, obtained from gauged N = 8 supergravity, has no cubic A 3 couplings in its Lagrangian, where A IJ is the bulk dual of O IJ . So conventional Witten diagrams cannot furnish the field theory result. We show that the extension of bulk supersymmetry to the AdS 4 boundary requires the introduction of a finite A 3 counterterm that does provide a perfect match to the 3-point correlator. Boundary supersymmetry also requires infinite counterterms which agree with the method of holographic renormalization. The generating functional of correlation functions of the ∆ = 1 operators is the Legendre transform of the on-shell action, and the supersymmetry properties of this functional play a significant role in our treatment.

“…Given the close similarity we have found between the counting formulae for flavoured and unflavoured quivers, we expect that the results on orthogonal bases, correlators and TFT2 [1] will also have simple generalizations from unflavoured quivers to flavoured quivers. An interesting avenue would be to explore the links between this permutation group TFT2 approach to correlators with methods based on integrable models (see e.g [41,42]).…”

Section: Discussionmentioning

confidence: 99%

Quivers, words and fundamentals

Mattioli

Ramgoolam

2015

A systematic study of holomorphic gauge invariant operators in general N = 1 quiver gauge theories, with unitary gauge groups and bifundamental matter fields, was recently presented in [1]. For large ranks a simple counting formula in terms of an infinite product was given. We extend this study to quiver gauge theories with fundamental matter fields, deriving an infinite product form for the refined counting in these cases. The infinite products are found to be obtained from substitutions in a simple building block expressed in terms of the weighted adjacency matrix of the quiver. In the case without fundamentals, it is a determinant which itself is found to have a counting interpretation in terms of words formed from partially commuting letters associated with simple closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. For finite ranks of the unitary gauge groups, the refined counting is given in terms of expressions involving Littlewood-Richardson coefficients.