Calculating endomorphism valued cohomology: Singlet spectrum in superstring models

We present a computational strategy based on exact sequences for determining the dimension of endomorphism valued cohomology groups for complete intersections in complex projective space. This cohomology group gives rise to part of the gauge singlet spectrum in superstring compactifications. We establish the underlying justification for the known deformation theoretical algorithm, and by comparison with the exact sequence method, indicate its limitations.Work supported in part by NSF contract PHY-82-09011Work supported in part by an NSF Postdoctoral Fellowship and DOE contract DE-FG02-88ER25065Work supported in part by NSF contract PHY-82-15249     

The chiral determinant and the eta invariant

For { _y },y, a one parameter family of invertible Weyl operators of possibly non-zero index acting on spinors over an even dimensional compact manifoldX, we express the phase of the chiral determinant det _– in terms of the invariant of a Dirac operator acting on spinors over ×X.Supported in part by NSF Grant No. PHY-82-15249Supported in part by NSF Grant PHY 8605978 and the Robert A. Welch Foundation     

New gravitational tests of early universe cosmology

Gravitational waves and lenses were among the earliest predictions of general relativity. I demonstrate here how both these phenomena can, in conjunction with newly discovered astrophysical objects, be used to test fundamental aspects of early universe cosmology, including (a) scenarios for galaxy formation, and (b) nonadiabatic expansion before and after nucleosynthesis.Research supported in part by the N.S.F. under Grant No. PHY-82-15249.     

Lattice systems with a continuous symmetry

We prove that the expansion in powers of the temperatureT of the correlation functions and the free energy of the plane rotator model on ad-dimensional lattice is asymptotic to all orders inT. The leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series. We also prove the inverse power decay of the pair correlation at low temperatures ford=3.Supported by NSF Grant No. MCS 78-01885Supported by NSF Grant No. PHY 78-15920Supported by NSF Grant No. DMR 73-04355Supported by NSF Grant No. PHY-7825390 A01On leave from: Institut de Physique Théorique, Université de Louvain, BelgiumAlso: Department of Physics     

Dark matter and inflation

I demonstrate that dark matter consisting of any type or types of stable weakly interacting elementary particle is incompatible with the minimal predictions of inflation, based on present observation of galaxy clustering, and assuming galaxies are good tracers of mass in the universe. If we wish to resolve this problem by particle physics alone, we seem to be driven to the possibility that the initial dark matter was unstable.Research supported in part by the National Science Foundation under Grant No. PHY-82-15249 and also by the NSERC (Canada).This essay received the first award from the Gravity Research Foundation for the year 1984-Ed.     

Global anomalies on orbifolds

We consider global anomalies for heterotic string theory formulated on orbifolds. The vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies. For abelian orbifolds level matching implies these cohomology conditions, so suffices for the absence of anomalies. For nonabelian orbifolds level matching does not suffice, and there are additional constraints. We give some examples to illustrate these new constraints.The first author is partially supported by an NSF Postdoctoral Research Fellowship. The second author is supported in part by the NSF contract no. PHY 82-15249, and in part by a fellowship from the Harvard Society of Fellows     

An intermediate phase with slow decay of correlations in one dimensional 1/|x−y|2 percolation,Ising and Potts models

We rigorously establish the existence of an intermediate ordered phase in one-dimensional 1/|x–y|² percolation, Ising and Potts models. The Ising model truncated two-point function has a power law decay exponent which ranges from its low (and high) temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary. Similar results are obtained for percolation and Potts models.Alfred P. Sloan Research Fellow. Research supported in part by NSF Grants No. PHY-8706420 and PHY-8645122Research supported in part by NSF Grant No. DMS-8514834 and AFOSR Contract F49620-86-C0130 at the Arizona Center for Math. Sciences     

Details of the non-unitarity proof for highest weight representations of the Virasoro algebra

We give an exposition of the details of the proof that all highest weight representations of the Virasoro algebra forc<1 which are not in the discrete series are non-unitary.This work was supported in part by DOE grant DE-FG02-84ER-45144, NSF grant PHY-8451285 and the Sloan Foundation     

Initial Data Engineering

We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.Partially supported by a Polish Research Committee grant 2 P03B 073 24Partially supported by the NSF under Grants PHY-0099373 and PHY-0354659Partially supported by the NSF under Grant DMS-0305048 and the UW Royalty Research Fund     

A note on the Ising model in high dimensions

We consider thed-dimensional Ising model with a nearest neighbor ferromagnetic interactionJ(d)=1/4d. We show that asd the+phase (and the — phase) approaches a product measure with density given by the mean field approximation. In particular the spontaneous magnetization converges to its mean field value. A similar result holds for the unique Gibbs measure of the system subject to an external fieldh0.Part of this work was done while this author was visiting Rutgers University, supported by NSF grant DMR-86-12369 and Princeton University, support by NSF grant PHY-85-15288-A01Partially supported by a NSF grant to Cornell UniversityPartially supported by NSF grant DMR 86-12369Supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell and by a NSF grant to Cornell University. This author was supported by the NSF grant DMR-86-12369 while visiting Rutgers University (when this work was started). On leave from São Paulo University     

Preferred shapes and modes of internal motion in a four-boson system

A four-body quantum-mechanical system is studied and through suitable manipulations of its wave functions, dominant shapes and modes of internal motion are ascribed to a number of its low-lying 0⁺ states.Research supported in part by NSF Grant No. PHY-8712229 and Grant No. PHY-8945627     

Supertori are algebraic curves

Super Riemann surfaces of genus 1, with arbitrary spin structures, are shown to be the sets of zeroes of certain polynomial equations in projective superspace. We conjecture that the same is true for arbitrary genus. Properties of superelliptic functions and super theta functions are discussed. The boundary of the genus 1 super moduli space is determined.Research partially supported by the DOE (DE-AC02-82-ER-40073) and NSF (PHY-85-21588)     

On the diffusion of a fast molecule

We consider the motion of a heavy particle in interaction with an infinite ideal gas of slow atoms. We prove that the velocity of the heavy particle is, in a suitable limit, modeled by a deterministic process. We also treat the process of rescaled velocity fluctuations around a certain deterministic motion and show that this is appropriately modeled by a nonhomogeneous diffusion process.Supported in part by NSF grants PHY-8201708 and DMR81-14726.Supported in part by NSF grant PHY-8003298 and the Seton Hall University, Research and Faculty Development Council.     

Preferred shapes and modes of internal motion in a five-boson system

A five-body system is studied using harmonic-oscillator basis functions and through suitably-defined shape-density functions which are based on its wave functions, we are able to ascribe dominant shapes and modes of internal motion to a number of its low-lying 0⁺ states.Research supported in part by NSF Grant No. PHY-8712229 and Grant No. PHY-8945627     

Hyperscaling inequalities for percolation

A set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved. One of the consequences of the inequalities is the lower boundd _C6 for the upper critical dimension. The proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as Ising ferromagnets.Supported by NSF Grant PHY-85-15288-A01     

Perturbation theory and non-renormalizable scalar fields

We study how to set up systematic summation rules that could permit us to interpret the divergent expressions arising in the perturbation theory of :P(): _d when one does not allow any renormalization besides the usual coupling constants, mass and wave function renormalizations.Supported in part by NSF grant PHY-8342570. Address after March 1985: Theoretische Physik, ETH-Hönggerberg, CH-8093 Zürich, SwitzerlandSupported in part by NSF grant MCS-8108814 (A03)     

Invariant states of a thermally conducting barrier

For an infinite one-dimensional system representing a thermally conducting barrier and two semi-infinite reservoirs which it separates, we prove the existence of a unique stationary probability distribution, to which essentially any initial distribution converges for large times.Research partially supported by NSF grant No. PHY-8003298.Research partially supported by NSF grant No. PHY-8201708.     

On the curvature of piecewise flat spaces

We consider analogs of the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge; considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.Supported in part by NSF MCS-810-2758-A-02Supported in part by Deutsche Forschungsgemeinschaft and NSF PHY-81-09110-A-01On leave of absence from Freie Universität Berlin     

Topology of monopole catalysis

We give the necessary mathematical conditions for strong monopole catalysis.Researach supported in part by the National Science Foundation under Grant No PHY 82-15249.     

Asymptotic freedom for quantum chromodynamics with top yukawa and Higgs interactions

Asymptotic freedom of QCD is extended to the enlarged system including the top Yukawa and Higgs interactions. Necessary and sufficient conditions for asymptotic freedom are given. By expansion with respect to powers of, all couplings are determined which are compatible with asymptotic freedom. It is found that the Higgs coupling is a function of the top coupling and that both couplings have upper bounds which correspond to the nontrivial case of reduction. The ultraviolet behavior of the coupling is controlled in all orders of the expansion.Work supported in part by the NSF: PHY-91-23780.