arXiv subjects
Publications and source records attributed to D. Boulatov.
We show that the spectral dimension d_s of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c <= 1. The same arguments provide a simple proof of the known result d_s= 4/3 for branched polymers.
The matrix model with a Bethe-tree embedding space (coinciding at large $N$ with the Kazakov-Migdal ``induced QCD'' model \cite{KM}) is investigated. We further elaborate the Riemann-Hilbert approach of \rf{Mig1} assuming certain holomorphic properties of the solution. The critical scaling (an edge singularity of the density) is found to be $γ_{str} = -\frac{1}π \arcos D$, for $|D|<1$, and $γ_{str} = -\frac{1}π \arcos \frac{D}{2D-1}$, for $D>1$. Explicit solutions are constructed at $D=\frac{1}{2}$ and $D=\infty$.
The 2D lattice gauge theory with a quantum gauge group $SL_q(2)$ is considered. When $q=e^{i\frac{2π}{k+2}}$, its weak coupling partition function coincides with the one of the G/G coset model ({\em i.e.} equals the Verlinde numbers). However, despite such a remarkable coincidence, these models are not equivalent but, in some certain sense, dual to each other.
A model is proposed which generates all oriented $3d$ simplicial complexes weighted with an invariant associated with a topological lattice gauge theory. When the gauge group is $SU_q(2)$, $q^n=1,$ it is the Turaev-Viro invariant and the model may be regarded as a non-perturbative definition of $3d$ simplicial quantum gravity. If one takes a finite abelian group $G$, the corresponding invariant gives the rank of the first cohomology group of a complex \nolinebreak $C$: $I_G(C) = rank(H^1(C,G))$, which means a topological expansion in the Betti number $b^1$. In general, it is a theory of the Dijkgraaf-Witten type, $i.e.$ determined completely by the fundamental group of a manifold.