A Model of Three-Dimensional Lattice Gravity

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.Comment: 20 page

Conformal transformations and the SLE partition function martingale

We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick's theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half plane. This is then applied to stochastic Loewner evolutions (SLE). We give a rigorous derivation of the equations, obtained previously by the authors, that connect the stochastic Loewner equation to the representation theory of the Virasoro algebra. We give a new proof that this construction enumerates all polynomial SLE martingales. When one of the hulls removed from the upper half plane is the SLE hull, we show that the partition function is a famous local martingale known to probabilists, thereby unravelling its CFT origin.Comment: 41 pages, 4 figure

Conformal Field Theories of Stochastic Loewner Evolutions

Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLE processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLE zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems for generalized crossing probabilities whose coefficients are multipoint CFT correlation functions. It provides a direct link between conformal correlation functions and probabilities of stopping time events in SLE evolutions. We point out a relation between SLE processes and two dimensional gravity and conjecture a reconstruction procedure of conformal field theories from SLE data.Comment: 38 pages, 3 figures, to appear in Commun. Math. Phy

Bootstrapping Multi-Parton Loop Amplitudes in QCD

We present a new method for computing complete one-loop amplitudes, including their rational parts, in non-supersymmetric gauge theory. This method merges the unitarity method with on-shell recursion relations. It systematizes a unitarity-factorization bootstrap approach previously applied by the authors to the one-loop amplitudes required for next-to-leading order QCD corrections to the processes e^+e^- -> Z,\gamma^* -> 4 jets and pp -> W + 2 jets. We illustrate the method by reproducing the one-loop color-ordered five-gluon helicity amplitudes in QCD that interfere with the tree amplitude, namely A_{5;1}(1^-,2^-,3^+,4^+,5^+) and A_{5;1}(1^-,2^+,3^-,4^+,5^+). Then we describe the construction of the six- and seven-gluon amplitudes with two adjacent negative-helicity gluons, A_{6;1}(1^-,2^-,3^+,4^+,5^+,6^+) and A_{7;1}(1^-,2^-,3^+,4^+,5^+,6^+,7^+), which uses the previously-computed logarithmic parts of the amplitudes as input. We present a compact expression for the six-gluon amplitude. No loop integrals are required to obtain the rational parts.Comment: 43 pages, 8 figures, RevTeX, v2-v4 clarifications and minor correction

Numerical Study of Finite Size Scaling for First Order Phase Transitions

I present results of simulations of the q=10 and q=20 2-d Potts models in the transition region. The asymptotic finite size behavior sets in only for extremely large lattices. We learn from this simulation that finite size scaling cannot be used to decide that a transition is first order.Comment: Talk presented at the Workshop on Dynamics of First Order Transitions, HLRZ, Forschungszentrum J\"ulich,Germany, June 1-3, 1992, 7 pages, 2 PostScript Figures (typing mistakes in figure captions corrected

Can conventional forces really explain the anomalous acceleration of Pioneer 10/11 ?

A conventional explanation of the correlation between the Pioneer 10/11 anomalous acceleration and spin-rate change is given. First, the rotational Doppler shift analysis is improved. Finally, a relation between the radio beam reaction force and the spin-rate change is established. Computations are found in good agreement with observational data. The relevance of our result to the main Pioneer 10/11 anomalous acceleration is emphasized. Our analysis leads us to conclude that the latter may not be merely artificial.Comment: 9 pages, no figur

Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales

A statistical mechanics argument relating partition functions to martingales is used to get a condition under which random geometric processes can describe interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs to satisfy this condition leads to some natural processes, which we study in this note. We give examples of such multiple SLEs and discuss how a choice of conformal block is related to geometric configuration of the interfaces and what is the physical meaning of mixed conformal blocks. We illustrate the general ideas on concrete computations, with applications to percolation and the Ising model.Comment: 40 pages, 6 figures. V2: well, it looks better with the addresse

Null-vectors in Integrable Field Theory

The form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine-Gordon model. This space has to be isomorphic to that of the corresponding minimal model of conformal field theory. We describe the subspaces which correspond to the Verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion (Neveu-Schwarz or Ramond depending on the particular primary field). The description of null-vectors relies on the relation between form factors and deformed hyper-elliptic integrals. The null-vectors correspond to the deformed exact forms and to the deformed Riemann bilinear identity. In the operator language, the null-vectors are created by the action of two operators \CQ (linear in the fermion) and \CC (quadratic in the fermion). We show that by factorizing out the null-vectors one gets the space of operators with the correct character. In the classical limit, using the operators \CQ and \CC we obtain a new, very compact, description of the KdV hierarchy. We also discuss a beautiful relation with the method of Whitham.Comment: 36 pages, Late