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Quantum spectral curve for the \(\eta\)-deformed \(\mathrm{AdS}_{5} \times S^{5}\) superstring. (English) Zbl 1375.81204

Summary: The spectral problem for the \(\mathrm{AdS}_{5} \times S^{5}\) superstring and its dual planar maximally supersymmetric Yang-Mills theory can be efficiently solved through a set of functional equations known as the quantum spectral curve. We discuss how the same concepts apply to the \(\eta\)-deformed \(\mathrm{AdS}_{5} \times S^{5}\) superstring, an integrable deformation of the \(\mathrm{AdS}_{5} \times S^{5}\) superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the \(\mathrm{AdS}_{5} \times S^{5}\) superstring, like the relation between the XXZ and XXX spin chains, or the sausage and the \(S^2\) sigma models for instance. We derive the quantum spectral curve for the \(\eta\)-deformed string by reformulating the corresponding ground-state thermodynamic Bethe ansatz equations as an analytic \(Y\) system, and map this to an analytic \(T\) system which upon suitable gauge fixing leads to a P{\(\mu\)} system - the quantum spectral curve. We then discuss constraints on the asymptotics of this system to single out particular excited states. At the spectral level the \(\eta\)-deformed string and its quantum spectral curve interpolate between the \(\mathrm{AdS}_{5} \times S^{5}\) superstring and a superstring on “mirror” \(\mathrm{AdS}_{5} \times S^{5}\), reflecting a more general relationship between the spectral and thermodynamic data of the \(\eta\)-deformed string. In particular, the spectral problem of the mirror \(\mathrm{AdS}_{5} \times S^{5}\) string, and the thermodynamics of the undeformed \(\mathrm{AdS}_{5} \times S^{5}\) string, are described by a second rational limit of our trigonometric quantum spectral curve, distinct from the regular undeformed limit.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
81T60 Supersymmetric field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
14H81 Relationships between algebraic curves and physics
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B23 Exactly solvable models; Bethe ansatz
82B30 Statistical thermodynamics
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