The Partition Function for Topological Field Theories

We use a Hodge decomposition and its generalization to non-abelian flat vector bundles to calculate the partition function for abelian and non- abelian BF theories in $n$ dimensions. This enables us to provide a simple proof that the partition function is related to the Ray-Singer torsion defined on flat vector bundles for all odd-dimensional manifolds, and is equal to unity for even dimensions.Comment: 23 pages, plain-TeX fil

Thermodynamics and Statistical Mechanics of Induced Liouville Gravity

In this paper we describe a Liouville gravity which is induced by a set of quantum fields (constituents) and represents a two-dimensional analog of Sakharov's induced gravity. The important feature of the considered theory is the presence of massless constituents which are responsible for the appearance of the induced Liouville field. The role of the massive constituents is only to induce the cosmological constant. We consider the instanton solutions of the Euclidean Liouville gravity with negative and zero cosmological constants, some instantons being interpreted as two-dimensional anti-de Sitter $AdS_2$ black holes. We study thermodynamics of all the solutions and conclude that their entropy is completely determined by the statistical-mechanical entropy of the massless constituents. This shows, in particular, that the constituents of the induced gravity are the true degrees of freedom of $AdS_2$ black holes. Special attention is also paid to the induced Liouville gravity with zero cosmological constant on a torus. We demonstrate the equivalence of its thermodynamics to the thermodynamics of BTZ black holes and comment on computations of the BTZ black hole entropy.Comment: 22 pages, latex, no figure