# Poster presented by Elmar Bittner

Poster Session

Topic: Gravity and Random Surfaces

### Two-dimensional quantum gravity with matter fields

Abstract: We consider two versions of quantum Regge calculus. The Standard Regge Calculus where the quadratic link lengths $q$ of the simplicial manifold vary continuously and the Discrete Regge Model where they are restricted to two possible values. These manifolds are coupled with $Z_2$ spin. We simulated the partition function \hbox{$Z=\sum_{s} \int {\cal D} \mu(q) \exp( -I(q) - K E(q,s))$}, where \hbox{$I(q)= \lambda \sum_i A_i$} is the gravitational action with the cosmological constant $\lambda$, and hbox{$E(q,s)=\frac{1}{2} \sum_{\langle ij \rangle} A_{ij} \frac{(s_i - s_j)^2}{q_{ij}}$} is the energy of the Ising spins $s_i$, $s_i = \pm 1$, being located at the vertices $i$ of the lattice. $A_i$ and $A_{ij}$ are barycentric areas associated with the vertices $i$ and the edges $\langle ij \rangle$, respectively. We use the same path-integral measure ${\cal D} \mu(q)$ as in the pure gravity simulations, which is chosen to render the Discrete Regge Model particularly simple. By applying reweighting techniques we determined the maxima of $C$ and $\chi$, and extracted the critical exponents. The Standard Regge Calculus and the Discrete Regge Model produce the same critical exponents of the Ising transition which agree with the Onsager exponents for regular static lattices. The KPZ exponents are definitely excluded. The gravitational action does not affect the critical exponents of the Ising phase transition.

Massimo Campostrini,   campo@mailbox.difi.unipi.it