Supersymmetry and Fredholm modules over quantized spaces

The purpose of this paper is to apply the framework of non-commutative differential geometry to quantum deformations of a class of Kähler manifolds. For the examples of the Cartan domains of type I and flat space, we construct Fredholm modules over the quantized manifolds using the supercharges which arise in the quantization of supersymmetric generalizations of the manifolds. We compute an explicit formula for the Chern character on generators of the Toeplitz ^*-algebra.Supported in part by the National Science Foundation under grant DMS-9206936Supported in part by the Department of Energy under grant DE-FG02-88ER25065Supported in part by the Consiglio Nazionale delle Ricerche (CNR)     

Superspace formulation of the Chern character of a theta-summable Fredholm module

We apply the concepts of superanalysis to present an intrinsically supersymmetric formulation of the Chern character in entire cyclic cohomology. We show that the cocycle condition is closely related to the invariance under supertranslations. Using the formalism of superfields, we find a path integral representation of the index of the generalized Dirac operator.Supported in part by the Department of Energy under grant DE-FG02-88ER25065     

QuantumK-theory

We construct a cocycle on an infinite dimensional generalization of ap-summable Fredholm module. Our framework is related to Connes' cyclic cohomology and is motivated by our work on index theory on infinite dimensional manifolds. Thep-summability condition is characteristic of dimensionO(p). We replace this assumption by the requirement that there exists an underlying heat kernel which is trace class. Then we use the heat kernel to regularize states in dimension-independent fashion. Our cocycle may be interpreted as an infinite dimensional Chern character.Supported in part by the National Foundation under Grant DMS/PHY 86-45122     

Asymptotically commuting families of operators

We study families of symmetric operators (Q _n) with domains given by the range of self-adjoint contraction semigroups (e ^−tHn ). Assuming the asymptotic commutativity, lim _n [Q _n, e^−tHn]=0, and certain other estimates, we establish the existence and properties of a limiting self-adjoint operatorQ=lim _n Q _n. We apply these results to the study of an elementary supersymmetry algebra. Supported in part by the National Science Foundation under Grant DMS/PHY 88-16214.     

Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q ². In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ ₊=P _? QP ₊, whereP _±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q ₊) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q ₊)=n?1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom.     

Linear time domain model of the acoustic potential field

A new time domain formulation of the acoustic wave is developed to avoid approximating assumptions of the linearized scalar wave equation that limit its validity to low Mach particle velocity modeling or to a smooth potential field in a stationary medium. The proposed model offers precision of the moving frame while retaining the form of the widely used linearized scalar wave equation although with respect to modified coordinates. It is applicable to field calculations involving transient waves with unlimited particle velocity, propagating in inhomogenous fluids or in those with time varying density. The model is based on the exact flux continuity equation and the equation of motion, both using the moving reference frame. The resulting closed-form free space scalar wave equation employing total derivatives is converted back to the partial differential form by using modified independent variables. The modified variables are related to the common coordinates of space and time following integral expressions involving transient particle velocity representing wave radiated by each point of a stationary source. Consequently, transient field produced by complex surface velocity sources can be calculated following existing surface integrals of the radiation theory although using modified coordinates. The use of the proposed model is presented in a numerical simulation of a transient velocity source vibrating at selected magnitudes, leading to the determination of the propagating pressure and velocity wave at any point.     

A priori estimates forN=2 Wess-Zumino models on a cylinder

We establish bounds uniform in the ultraviolet cutoff (i.e., in the number of degrees of freedom) for a family of two-dimensional Wess-Zumino models. These estimates are useful in proving existence of the models, as well as in investigating their properties. For example, we require these estimates for the analysis of the supercharge and of the Hamiltonian. These are the fundamental a priori estimates for elliptic regularity in infinite dimensions.Supported in part by the National Science Foundation under Grant DMS/PHY 86-45122