1210.2398 (Jonathan Maltz)

Gauge Invariant Computable Quantities In Timelike Liouville Theory    [PDF]

Jonathan Maltz

Timelike Liouville theory admits the sphere $\mathbb{S}^{2}$ as a real saddle point, about which quantum fluctuations can occur. An issue that occurs when computing the expectation values of standard classical quantities, like the distance between points in this fluctuating geometry, is that even after fixing the system to conformal gauge by imposing $g_{\mu\nu} = e^{2\b\phi}\tilde{g}_{\mu\nu}$, where $\phi$ is the Liouville field and $\tilde{g}_{\mu\nu}$ is a reference metric which gives a coordinate system for the geometry, a $SL_{2}(\mathbb{C})$ gauge symmetry of the reference coordinates still remains. Under this symmetry the coordinates and Liouville field transform in such a way as to leave the physical manifold invariant. Meaning that until the gauge is fixed, the position of two points on the reference sphere does not uniquely determine two points on the physical manifold. In this paper it is shown through perturbative analysis that after fixing to conformal gauge, the remaining zero mode due to $SL_{2}(\mathbb{C})$ coordinate transformations of the reference sphere can be dealt with but using standard Fadeev-Popov methods employing the gauge condition that the dipole of the coordinate system is a fixed vector, and then integrating over all values of this dipole. A Green's function is obtained and used to perturbatively compute the expectation value of the length of a geodesic on the $\mathbb{S}^{2}$ under fluctuations of a Semi-classical Timelike Liouville field. When computed to second order in the Timelike Liouville coupling $\b$, it is shown that this quantity is well defined and doesn't suffer from any power law or logarithmic divergences as a naive power counting argument might suggest.

View original: http://arxiv.org/abs/1210.2398