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)