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Register a new userLiouville Theory, AdS$_2$ String, and Three-Point Functions
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Shota Komatsu
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This is a write-up of the lectures given in Young Researchers Integrability School 2017. The main goal is to explain the connection between the ODE/IM correspondence and the classical integrability of strings in AdS. As a warm up, we first discuss the classical three-point function of the Liouville theory. The starting point is the well-known fact that the classical solutions to the Liouville equation can be constructed by solving a Schrodinger-like differential equation. We then convert it into a set of functional equations using a method similar to the ODE/IM correspondence. The classical three-point functions can be computed directly from these functional equations, and the result matches with the classical limit of the celebrated DOZZ formula. We then discuss the semi-classical three-point function of strings in AdS2 and show that one can apply a similar idea by making use of the classical integrability of the string sigma model on AdS2. The result is given in terms of the massive generalization of Gamma functions, which show up also in string theory on pp-wave backgrounds and the twistorial generalization of topological string.
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Starting from the known expression for the three-point correlation functions for Liouville exponentials with generic real coefficients at we can prove the Liouville equation of motion at the level of three-point functions. Based on the analytical structure of the correlation functions we discuss a possible mass shell condition for excitations of noncritical strings and make some observations concerning correlators of Liouville fields.
The possibility of extending the Liouville Conformal Field Theory from values of the central charge $c geq 25$ to $c leq 1$ has been debated for many years in condensed matter physics as well as in string theory. It was only recently proven that such an extension -- involving a real spectrum of critical exponents as well as an analytic continuation of the DOZZ formula for three-point couplings -- does give rise to a consistent theory. We show in this Letter that this theory can be interpreted in terms of microscopic loop models. We introduce in particular a family of geometrical operators, and, using an efficient algorithm to compute three-point functions from the lattice, we show that their operator algebra corresponds exactly to that of vertex operators $V_{hat{alpha}}$ in $c leq 1$ Liouville. We interpret geometrically the limit $hat{alpha} to 0$ of $V_{hat{alpha}}$ and explain why it is not the identity operator (despite having conformal weight $Delta=0$).
We compare calculations of the three-point correlation functions of BMN operators at the one-loop (next-to-leading) order in the scalar SU(2) sector from the integrability expression recently suggested by Gromov and Vieira, and from the string field theory expression based on the effective interaction vertex by Dobashi and Yoneya. A disagreement is found between the form-factors of the correlation functions in the one-loop contributions. The order-of-limits problem is suggested as a possible explanation of this discrepancy.
We consider mixed three-point correlation functions of the supercurrent and flavour current in three-dimensional $1 leq mathcal{N} leq 4$ superconformal field theories. Our method is based on the decomposition of the relevant tensors into irreducible components to guarantee that all possible tensor structures are systematically taken into account. We show that only parity even structures appear in the correlation functions. In addition to the previous results obtained in arXiv:1503.04961, it follows that supersymmetry forbids parity odd structures in three-point functions involving the supercurrent and flavour current multiplets.
We make an ansatz for the Mellin representation of the four-point amplitude of half-BPS operators of arbitrary charges at order $lambda^{-frac{5}{2}}$ in an expansion around the supergravity limit. Crossing symmetry and a set of constraints on the form of the spectrum uniquely fix the amplitude and double-trace anomalous dimensions at this order. The results exhibit a number of natural patterns which suggest that the bootstrap approach outlined here will extend to higher orders in a simple way.
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