Evaluation of the operatorial Q-system for non-compact super spin chains

Abstract

We present an approach to evaluate the full operatorial Q-system of all $\mathfrak{u}(p,q|r+s)$-invariant spin chains with representations of Jordan-Schwinger type. In particular, this includes the super spin chain of planar $\mathcal{N}=4$ super Yang-Mills theory at one loop in the presence of a diagonal twist. Our method is based on the oscillator construction of Q-operators. The Q-operators are built as traces over Lax operators which are degenerate solutions of the Yang-Baxter equation. For non-compact representations these Lax operators may contain multiple infinite sums that conceal the form of the resulting functions. We determine these infinite sums and calculate the matrix elements of the lowest level Q-operators. Transforming the Lax operators corresponding to the Q-operators into a representation involving only finite sums allows us to take the supertrace and to obtain the explicit form of the Q-operators in terms of finite matrices for a given magnon sector. Imposing the functional relations, we then bootstrap the other Q-operators from those of the lowest level. We exemplify this approach for non-compact spin $-s$ spin chains and apply it to $\mathcal{N}=4$ SYM at the one-loop level using the BMN vacuum as an example.

arXiv comment31 pages, v2: minor changes, version published in JHEP, v3: typo fixed

References in corpus (38)

• Review of AdS/CFT Integrability: An Overview
• Deforming field theories with $U(1)\times U(1)$ global symmetry and their gravity duals
• Exactly Marginal Operators and Duality in Four Dimensional N=1 Supersymmetric Gauge Theory
• Integrable Structure of Conformal Field Theory, Quantum KdV Theory and Thermodynamic Bethe Ansatz
• Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation
• High energy QCD as a completely integrable model
• Lax Pair for Strings in Lunin-Maldacena Background
• Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relation
• Quantum spectral curve for AdS_5/CFT_4
• Beauty and the Twist: The Bethe Ansatz for Twisted N=4 SYM
• New integrable non-gauge 4D QFTs from strongly deformed planar N=4 SYM
• Quantum spectral curve for arbitrary state/operator in AdS${}_5$/CFT${}_4$
• One-point Functions in Defect CFT and Integrability
• Hidden Grassmann Structure in the XXZ Model II: Creation Operators
• One-point Functions in AdS/dCFT from Matrix Product States
• Hidden Grassmann structure in the XXZ model
• Baxter's Relations and Spectra of Quantum Integrable Models
• Baxter Q-Operators and Representations of Yangians
• Baxter's Q-operators for supersymmetric spin chains
• T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve
• Asymptotic representations and Drinfeld rational fractions
• Quasiclassical QCD Pomeron
• NNLO BFKL Pomeron eigenvalue in N=4 SYM
• Quantum spectral curve as a tool for a perturbative quantum field theory
• Solutions of the T-system and Baxter equations for supersymmetric spin chains
• Quantum Group Representations and Baxter Equation
• Baxter's Q-operators and operatorial Backlund flow for quantum (super)-spin chains
• Quantization of Integrable Systems and a 2d/4d Duality
• Oscillator Construction of su(n|m) Q-Operators
• Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M|N)
• Discrete Hirota's equation in quantum integrable models
• A Generalized Q-operator for U_q(\hat(sl_2)) Vertex Models
• Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansätze and the R-Matrix Formalism
• Oscillator versus prefundamental representations
• Q-operators for the open Heisenberg spin chain
• A Q-operator for the twisted XXX model
• Noncompact sl(N) spin chains: BGG-resolution, Q-operators and alternating sum representation for finite dimensional transfer matrices
• Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz

Cited by in corpus (6)

• QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains
• Non-compact quantum spin chains as integrable stochastic particle processes
• New Compact Construction of Eigenstates for Supersymmetric Spin Chains
• The non-compact XXZ spin chain as stochastic particle process
• The fate of the Konishi multiplet in the β-deformed Quantum Spectral Curve
• Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin