Combining decision procedures for the reals

We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which "local" decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let Tadd[Q] be the first-order theory of the real numbers in the language of ordered groups, with negation, a constant 1, and function symbols for multiplication by rational constants. Let Tmult[Q] be the analogous theory for the multiplicative structure, and let T[Q] be the union of the two. We show that although T[Q] is undecidable, the universal fragment of T[Q] is decidable. We also show that terms of T[Q]can fruitfully be put in a normal form. We prove analogous results for theories in which Q is replaced, more generally, by suitable subfields F of the reals. Finally, we consider practical methods of establishing quantifier-free validities that approximate our (impractical) decidability results.Comment: Will appear in Logical Methods in Computer Scienc

Critical solutions in topologically gauged N=8 CFTs in three dimensions

In this paper we discuss some special (critical) background solutions that arise in topological gauged ${\mathcal N}=8$ three-dimensional CFTs with SO(N) gauge group. These solutions solve the TMG equations (containing the parameters $\mu$ and $l$) for a certain set of values of $\mu l$ obtained by varying the number of scalar fields with a VEV. Apart from Minkowski, chiral round $AdS_3$ and null-warped $AdS_3$ (or Schr\"odinger(z=2)) we identify also a more exotic solution recently found in $TMG$ by Ertl, Grumiller and Johansson. We also discuss the spectrum, symmetry breaking pattern and the supermultiplet structure in the various backgrounds and argue that some properties are due to their common origin in a conformal phase. Some of the scalar fields, including all higgsed ones, turn out to satisfy three-dimensional singleton field equations. Finally, we note that topologically gauged ${\mathcal N}=6$ ABJ(M) theories have a similar, but more restricted, set of background solutions.Comment: 34 pages, v2: minor corrections, note about a new solution added in final section, v3: two footnotes adde

Chiral Fermions on the Lattice through Gauge Fixing -- Perturbation Theory

We study the gauge-fixing approach to the construction of lattice chiral gauge theories in one-loop weak-coupling perturbation theory. We show how infrared properties of the gauge degrees of freedom determine the nature of the continuous phase transition at which we take the continuum limit. The fermion self-energy and the vacuum polarization are calculated, and confirm that, in the abelian case, this approach can be used to put chiral gauge theories on the lattice in four dimensions. We comment on the generalization to the nonabelian case.Comment: 31 pages, 5 figures, two refs. adde

The Conserved Charges and Integrability of the Conformal Affine Toda Models

We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.Comment: 18 pages, LaTeX, (one appendix and one reference added, small changes in introduction and conclusions, eqs.(5.14) and (5.19) improved, final version to appear in Int. J. Modern Phys. A