Abstract
We study the origin of the recently proposed effective theory of stress tensor exchanges based on reparametrization modes, that has been used to efficiently compute Virasoro identity blocks at large central charge. We first provide a derivation of the nonlinear Alekseev-Shatashvili action governing these reparametrization modes, and argue that it should be interpreted as the generating functional of stress tensor correlations on manifolds related to the plane by conformal transformations. In addition, we demonstrate that the rules previously prescribed with the reparametrization formalism for computing Virasoro identity blocks naturally emerge when evaluating Feynman diagrams associated with stress tensor exchanges between pairs of external primary operators. We make a few comments on the connection of these results to gravitational theories and holography.
Highlights
We demonstrate that the rules previously prescribed with the reparametrization formalism for computing Virasoro identity blocks naturally emerge when evaluating Feynman diagrams associated with stress tensor exchanges between pairs of external primary operators
We show that this nonlinear extension of the effective action (1.1) should still be viewed as a generating functional for stress tensor correlations on manifolds related to the plane by conformal transformations, with the derivative of the reparametrization mode acting as the corresponding source
After reviewing some of the basic properties of the Polyakov action in section 2, we provided a first principle derivation of the Alekseev-Shatashvili action as a nonlinear extension of the effective action (1.1) governing the reparametrization modes
Summary
The starting point for the construction of the effective action (1.1) in [2] was the generating functional for the connected stress tensor two-point function on the complex plane, W [δg]. The Polyakov action (2.3) is manifestly nonlocal in the background metric gij that acts as a source for the stress tensor It can be put in an alternative form through the introduction of an auxiliary variable φ solving φ = R,. Functional differentiation of the generating functional W [gij] yields connected stress tensor correlators on a background with fixed metric g0, Tij (x1) . Due to the Weyl anomaly, one could have feared that conformal transformations are not symmetries of the Polyakov action, and not true symmetries of the quantum theory. As mentioned below (2.9), the Liouville field shifts to the new value φ = −ω = ln ∂zΠ(z) , Such that, on the manifold obtained by conformal transformation from the complex plane, the vacuum energy (2.7) reduces to. We have recovered the well-known expression in terms of the Schwarzian derivative of Π
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