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We consider correlation functions of operators and the operator product expansion in two-dimensional quantum gravity. First we introduce correlation functions with geodesic distances between operators kept fixed. Next by making two of the operators closer, we examine if there exists an analog of the operator product expansion in ordinary field theories. Our results suggest that the operator product expansion holds in quantum gravity as well, though special care should be taken regarding the physical meaning of fixing geodesic distances on a fluctuating geometry.
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We compute the most general embedding space two-point function in arbitrary Lorentz representations in the context of the recently introduced formalism in arXiv:1905.00036 and arXiv:1905.00434. This work provides a first explicit application of this approach and furnishes a number of checks of the formalism. We project the general embedding space two-point function to position space and find a form consistent with conformal covariance. Several concrete examples are worked out in detail. We also derive constraints on the OPE coefficient matrices appearing in the two-point function, which allow us to impose unitarity conditions on the two-point function coefficients for operators in any Lorentz representations.
We revisit the computation of instanton effects to various correlation functions in ${cal N}=4$ SYM and clarify a controversy existing in the literature regarding their consistency with the OPE and conformal symmetry. To check these properties, we examine the conformal partial wave decomposition of four-point correlators involving combinations of half-BPS and Konishi operators and isolate the contribution from the conformal primary scalar operators of twist four. We demonstrate that the leading instanton correction to this contribution is indeed consistent with conformal symmetry and compute the corresponding corrections to the OPE coefficients and the scaling dimensions of such twist-four operators. Our analysis justifies the regularization procedure used to compute ultraviolet divergent instanton contribution to correlation functions involving unprotected operators.
This work was carried out in 1985. It was published in Russian in Yad. Fiz. 44, 498 (1986) [English translation Sov. J. Nucl. Phys. 44, 321 (1986)]. None of these publications are available on-line. Submitting this paper to ArXiv will make it accessible. *** A simple method is developed that makes it possible to determine the $k$-loop coefficient of the $beta$-function if the operator product expansion for certain polarization operators in the $(k -1)$ loop is known. The calculation of the two-loop coefficient of the Gell-Mann-Low function becomes trivial -- it reduces to a few algebraic operations on already known expressions. As examples, spinor, scalar, and supersymmetric electrodynamics are considered. Although the respective results for $beta^{(2)}$ are known in the literature, both the method of calculation and certain points pertaining to the construction of the operator product expansion are new.
Local terms in the Operator Product Expansion in Superconformal Theories with extended supersymmetry are identified. Assuming a factorized structure for these terms their contributions are discussed.
We propose a new type of gauge in two-dimensional quantum gravity. We investigate pure gravity in this gauge, and find that the system reduces to quantum mechanics of loop length $l$. Furthermore, we rederive the $c!=!0$ string field theory which was discovered recently. In particular, the pregeometric form of the Hamiltonian is naturally reproduced.
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