Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

Journal of High Energy Physics, Sep 2015

We give a derivation of quantum spectral curve (QSC) — a finite set of Riemann-Hilbert equations for exact spectrum of planar \( \mathcal{N}=4 \) SYM theory proposed in our recent paper Phys. Rev. Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system — a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.

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Quantum spectral curve for arbitrary state/operator in AdS5/CFT4

Received: February spectral curve for arbitrary state/operator in A 1 2 3 4 7 8 9 S 1 2 4 7 8 9 /CFT 1 2 4 7 8 9 Nikolay Gromov 1 2 4 6 7 8 9 Vladimir Kazakov 1 2 3 4 5 7 8 9 S´ebastien Leurent 0 1 2 4 7 8 9 Dmytro Voling 1 2 4 7 8 9 h 1 2 4 7 8 9 The Strand 1 2 4 7 8 9 London WC 1 2 4 7 8 9 R 1 2 4 7 8 9 LS 1 2 4 7 8 9 U.K. 1 2 4 7 8 9 St.Petersburg INP 1 2 4 7 8 9 Open Access 1 2 4 7 8 9 c The Authors. 1 2 4 7 8 9 0 Institut de Math ́ematiques de Bourgogne, UMR 5584 du CNRS 1 Place Jussieu , 75005 Paris , France 2 24 , rue Lhomond 75005 Paris , France 3 Universit ́e Paris-VI 4 Gatchina , 188 300, St.Petersburg , Russia 5 School of Natural Sciences, Institute for Advanced Study 6 Mathematics Department, King's College London 7 College Green , Dublin 2 , Ireland 8 Universit ́e de Bourgogne , 9 avenue Alain Savary, 21000 DIJON , France 9 Princeton , NJ08540 , U.S.A We give a derivation of quantum spectral curve (QSC) - a finite set of Riemann-Hilbert equations for exact spectrum of planar N = 4 SYM theory proposed in our recent paper Phys. Rev. Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system - a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits. AdS-CFT Correspondence; Integrable Field Theories 1 Introduction Notations and conventions Spectral parameter and Riemann sheets Multi-indices and sum conventions Quantum spectral curve from analytic Y-system 3.1 Inspiration from TBA TBA equations as a set of functional equations Generalization and extension Asymptotics at large u Quantum spectral curve as an analytic Q-system Q-system — General algebraic description Definition of Q-system and QQ-relations A complete basis for parameterization of all Q-functions Q-system and AdS/CFT spectral problem P and Q as Q-functions A different point of view: from analytic Q-system to QSC Complex conjugation and reality Particular case of the Left-Right symmetric states Exact Bethe equations Large volume limit Asymptotic solution of the QSC equations Finding Qα|β Exploring the results su(2|2) Q-functions explicitly QQ-relations and dualities Asymptotic Bethe Ansatz equations Equation for the energy and cyclicity condition Numbers of roots and the conserved global charges Quasi-classical approximation Constraining pre-exponents Quasi-classical limit of the discontinuity relations Quasi-classical limit of T-functions and characters of monodromy matrix Conclusions and discussion A Mathematica code to check derivations A.1 Details of derivation for section 2 Details of the relations between the Q-, T- and Y-systems B.1.1 Statements Proof of uniqueness of the distinguished gauge A.2 QQ-relations B.4 Conclusions Wronskian parameterization Analyticity of Qs in the mirror basis Deeper look into analytic structure Fundamental Q-system Unitarity and global charges C.1 Representation theory for psu(2, 2|4) C.2 Unitarity constraints from analyticity of QSC Kac-Dynkin-Vogan diagrams Long multiplets (typical representations) Short multiplets (atypical representations) Unimodularity and projectivity Quantization of charges Main unitarity constraint The discovery and exploration of integrability in the planar AdS/CFT correspondence has a long and largely successful history [1]. In the two most advanced examples of 4the planar spectrum of anomalous dimensions of some simple but non-protected single trace limits and also numerically, with sufficiently high precision [3–5], by means of an explicit but immensely complicated Thermodynamic Bethe Ansatz (TBA) formalism [6–8]. One should admit that the complexity of the TBA-like equations appeared to be in a stark contradiction with the elegant integrability concept for the spectrum of these beautiful maximally super-symmetric gauge theories. Fortunately, the situation was not hopeless as some signs of hidden simplicity started to emerge here and there. The system of integral nonlinear TBA equations was known to have a reformulation in terms of a simple and universal Y-system [2, 6] supplied by a relatively simple analytic data [9]. Furthermore, the Y-system, an infinite system of nonlinear functional equations is equivalent to the integrable Hirota bilinear equation (T-system) [2] which by itself was known to be integrable. The integrability of the latter would imply that it can be rewritten in terms of a finite number of Q-functions of the spectral parameter — (...truncated)


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Nikolay Gromov, Vladimir Kazakov, Sébastien Leurent, Dmytro Volin. Quantum spectral curve for arbitrary state/operator in AdS5/CFT4, Journal of High Energy Physics, 2015, pp. 187, Volume 2015, Issue 9, DOI: 10.1007/JHEP09(2015)187