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Register a new userOn the S-matrix of Liouville theory
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The S-matrix for each chiral sector of Liouville theory on a cylinder is computed from the loop expansion of correlation functions of a one-dimensional field theory on a circle with a non-local kinetic energy and an exponential potential. This action is the Legendre transform of the generating function of semiclassical scattering amplitudes. It is derived from the relation between asymptotic in- and out-fields. Its relevance for the quantum scattering process is demonstrated by comparing explicit loop diagrams computed from this action with other methods of computing the S-matrix, which are also developed.
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We consider the conformal higher spin (CHS) theory in d=4 that contains the s=1 Maxwell vector, s=2 Weyl graviton and their higher spin s=3,4,... counterparts with higher-derivative box^s kinetic terms. The interacting action for such theory can be found as the coefficient of the logarithmically divergent part in the induced action for sources coupled to higher spin currents in a free complex scalar field model. We explicitly determine some cubic and quartic interaction vertices in the CHS action from scalar loop integrals. We then compute the simplest tree-level 4-particle scattering amplitudes 11 -> 11, 22 -> 22 and 11 -> 22 and find that after summing up all the intermediate CHS exchanges they vanish. This generalises the vanishing of the scattering amplitude for external conformal scalars interacting via the exchange of all CHS fields found earlier in arXiv:1512.08896. This vanishing should generalise to all scattering amplitudes in the CHS theory and as in the conformal scalar scattering case should be a consequence of the underlying infinite dimensional higher spin symmetry that extends the standard conformal symmetry.
Using duality in optimization theory we formulate a dual approach to the S-matrix bootstrap that provides rigorous bounds to 2D QFT observables as a consequence of unitarity, crossing symmetry and analyticity of the scattering matrix. We then explain how to optimize such bounds numerically, and prove that they provide the same bounds obtained from the usual primal formulation of the S-matrix Bootstrap, at least once convergence is attained from both perspectives. These techniques are then applied to the study of a gapped system with two stable particles of different masses, which serves as a toy model for bootstrapping popular physical systems.
We investigate the structure of the quantum S-matrix for perturbative excitations of the Pohlmeyer reduced version of the AdS_5 x S^5 superstring following arXiv:0912.2958. The reduced theory is a fermionic extension of a gauged WZW model with an integrable potential. We use as an input the result of the one-loop perturbative scattering amplitude computation and an analogy with simpler reduced AdS_n x S^n theories with n=2,3. The n=2 theory is equivalent to the N=2 2-d supersymmetric sine-Gordon model for which the exact quantum S-matrix is known. In the n=3 case the one-loop perturbative S-matrix, improved by a contribution of a local counterterm, satisfies the group factorization property and the Yang-Baxter equation, and reveals the existence of a novel quantum-deformed 2-d supersymmetry which is not manifest in the action. The one-loop perturbative S-matrix of the reduced AdS_5 x S^5 theory has the group factorisation property but does not satisfy the Yang-Baxter equation suggesting some subtlety with the realisation of quantum integrability. As a possible resolution, we propose that the S-matrix of this theory may be identified with the quantum-deformed [psu(2|2)]^2 x R^2 symmetric R-matrix constructed in arXiv:1002.1097. We conjecture the exact all-order form of this S-matrix and discuss its possible relation to the perturbative S-matrix defined by the path integral. As in the AdS_3 x S^3 case the symmetry of the S-matrix may be interpreted as an extended quantum-deformed 2-d supersymmetry.
We consider the three-particle scattering S-matrix for the Landau-Lifshitz model by directly computing the set of the Feynman diagrams up to the second order. We show, following the analogous computations for the non-linear Schr{o}dinger model, that the three-particle S-matrix is factorizable in the first non-trivial order.
Similarly to the bosonic Liouville theory, the $mathcal{N}=2$ supersymmetric Liouville theory was conjectured to be equipped with the duality that exchanges the superpotential and the Kahler potential. The conjectured duality, however, seems to suffer from a mismatch of the preserved symmetries. More than fifteen years ago, when I was a student, my supervisor Tohru Eguchi gave a beautiful resolution of the puzzle when the supersymmetry is enhanced to $mathcal{N}=4$ based on his insight into the underlying geometric structure of the $A_1$ singularity. I will review his unpublished but insightful idea and present our attempts to extend it to more general cases.
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