Effective Quantum Field Theory for the Thermodynamical Bethe Ansatz
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Springer
Received: December 28, Revised: December 30, Accepted: January 25, Published: February 6,
2019 2019 2020 2020
Ivan Kostov Institut de Physique Th´eorique, DSM, CEA, URA2306 CNRS, Saclay, F-91191 Gif-sur-Yvette, France
E-mail: [email protected] Abstract: We construct an effective Quantum Field Theory for the wrapping effects in 1+1 dimensional models of factorised scattering. The recently developed graph-theoretical approach to TBA gives the perturbative desctiption of this QFT. For the sake of simplicity we limit ourselves to scattering matrices for a single neutral particle and no bound state poles, such as the sinh-Gordon one. On the other hand, in view of applications to AdS/CFT, we do not assume that the scattering matrix is of difference type. The effective QFT involves both bosonic and fermionic fields and possesses a symmetry which makes it one-loop exact. The corresponding path integral localises to a critical point determined by the TBA equation. Keywords: Bethe Ansatz, Integrable Field Theories, Topological Field Theories ArXiv ePrint: 1911.07343
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP02(2020)043
JHEP02(2020)043
Effective Quantum Field Theory for the Thermodynamical Bethe Ansatz
Contents 1 Introduction
1
2 Degrees of freedom of the effective QFT 2.1 Space and time wrapping operators 2.2 Operator form of the Bethe-Yang equations 2.3 Free-field realisation
3 3 6 7 8 8 9 11 12
4 Path integral and localisation
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5 Example: the Sinh-Gordon model
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6 Concluding remarks
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A Conventions about the scattering in physical and mirror kinematics
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B Operator representation of the Gaudin determinant
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1
Introduction
In 1+1 dimensional field-theoretical models with factorised scattering, the momenta and the energies of the particles forming a multi-particle excitation are not changed by the Hamiltonian evolution. This property makes possible to use the notion of a particle with given rapidity even after the theory is confined on a circle with asymptotically large circumference R. Asymptotically large signifies much larger than the correlation length, so that the exponential effects can be neglected. The asymptotic energy spectrum is computed by solving the Bethe-Yang equations which determine the allowed values of the momenta compatible with the periodic boundary conditions. The leading order exponential corrections to the energy of the ground state, known as wrapping corrections, come from virtual particles wrapping the space circle. To compute the energy spectrum at finite volume R, one should evaluate the sum over all possible wrapping processes. An efficient and elegant way to do that, suggested by Alexey Zamolodchikov [1–3], is to compactify also the time circle at asymptotically large distance L and formulate the problem in the mirror channel where the space and the time are interchanged.
–1–
JHEP02(2020)043
3 Partition function on the torus 3.1 The partition function as a sum over on-shell states 3.2 Rewriting the sum as a co
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