Regularizing the Functional Integral in 2d-Quantum Gravity

Abstract

We show that the usual conjectures made to understand Polyakov’s quantization of 2D-gravity in the functional integral approach arise from an explicit calculation using heat kernel regularization. The connection of these results with string theory is briefly commented upon.

Non-critical strings at high energy

“…The functional measures Dĝx Dĝψ and Dĝφ are built from the L 2 norms on the functions with respect to the fiducial metricĝ (see [8], [31]). …”

Двухмерная Гравитация, Теория Лиувилля И Некритические Струны

“…Следующим шагом является преобразование интеграла (9). Для этого мы воспользуемся известной классической теоремой Гаусса, согласно которой произвольную метрику g.r^ на двухмерной сфере всегда можно преобразовать к произвольно заданному виду (с то пологическим ограничением (8)) преобразованиями (И) одновремен но с конформными преобразованиями:…”

“…Имеются лишь различного рода аргументы в пользу -его справедливо сти, основной из которых -одинаковые трансформационные свойства левой и правой частей (34) при конфо(рмных преобразованиях [9]. Предписание (34) позволяет представить производящий функцио нал некритической струнной теории в виде…”

Duality and the Renormalization Group

“…Simply using the requirements that duality and the RG commute as motions in the parameter space of the sigma model, we have thus been able to determine the two-loop beta function to be 12) for an entirely generic metric background, again without any Feynman diagram calculations. Because we used an extremely restrictive class of backgrounds, it was not possible to determine the value of λ (the correct value is λ = 1 2 ).…”

“…But, in fact, the above has not been entirely in vain: we know that, in the scheme in which the EA is given by (4.2) Recently, a set of corrected duality transformations has been found [11] that keep invariant the two-loop EA in a particular scheme. In that scheme, it is claimed that the matrix G ij connecting the equations of motion and the Weyl anomaly coefficients is purely numerical, that is, it contains no spacetime derivative operators acting onβ i [10], [12]. That is certainly a crucial advantage if one is interested in studying a c-theorem for generalized sigma models.…”