Folkert Müller-Hoissen

Spontaneous compactification with quadratic and cubic curvature terms

Abstract Certain curvature squared terms, which supplement the Einstein action in the low energy limit of string theories, allow for spontaneous compactification to R (1,3) × S n , n ⩾4. A cosmological constant in N = 4 + n dimensions is necessary. This is no longer the case if we add certain cubic curvature terms to the lagrangian. Solutions in N ⩾ 10 dimensions are obtained. The field equations considered contain at most second derivatives of the metric.

Noncommutative differential calculus and lattice gauge theory

The authors study consistent deformations of the classical differential calculus on algebras of functions (and, more generally, commutative algebras) such that differentials and functions satisfy nontrivial commutation relations. For a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice. These differential calculi generate a lattice on a space continuum. The whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus. In this framework one just has to express the Lagrangian for the continuum theory in terms of differential forms. This expression then also makes sense for the deformed differential calculus. There is a natural integral associated with the latter. Integration of the Lagrangian over a space continuum then produces the correct lattice action for a large class of theories. This is explicitly shown for the scalar field action and the action for SU(m) gauge theory.

Discrete differential calculus: Graphs, topologies, and gauge theory

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘‘reduction’’ of the ‘‘universal differential algebra’’ and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a ‘‘Hasse diagram’’ determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two‐point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an ‘‘internal’’ discrete space (a la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, a ‘‘symmetric lattice’’ is also studied which (in a certain continuum limit) turns out to be related to a ‘‘noncommutative differential calculus’’ on manifolds.

Discrete Riemannian geometry

Within a framework of noncommutative geometry, we develop an analog of (pseudo-) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs (the vertices of the latter are given by the elements of the finite set). Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (nonlocal) tensor product over the algebra of functions, as considered previously by several authors. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined in the same way as in continuum Riemannian geometry. In particular, in the case of the universal differential calculus on...

Quantum mechanics on a lattice and q-deformations

Abstract Based on previous work on a relation between noncommutative differential calculus and lattice structures it is shown that q-calculus arises as an exponentiated form of lattice calculus. In particular, quantum mechanics on a lattice can also be described as “q-deformed quantum mechanics”.

Bi-differential calculi and integrable models

The existence of an infinite set of conserved currents in completely integrable classical models, including chiral and Toda models as well as the KP and self-dual Yang-Mills equations, is traced back to a simple construction of an infinite chain of closed (respectively, covariantly constant) 1-forms in a (gauged) bi-differential calculus. The latter consists of a differential algebra on which two differential maps act. In a gauged bi-differential calculus these maps are extended to flat covariant derivatives.

Dimensionally continued Euler forms: Kaluza-Klein cosmology and dimensional reduction

The most general gravity Lagrangian in more than four dimensions is considered which leads to field equations with at most second derivatives of the metric. It consists of a series of dimensionally continued Euler forms and allows spontaneous compactification. The field equations are elaborated for the usual Kaluza-Klein cosmology ansatz and solved in the special case where the extra dimensions form a sphere with constant radius. The dimensional reduction of the theory to four dimensions is discussed as well.

From continuum to lattice theory via deformation of the differential calculus

Abstract It is shown that a lattice (gauge) theory can be obtained from a continuum theory via deformation of the standard differential calculus in such a way that functions and differentials no longer commute. Any lagrangian for a continuum theory which can be expressed in terms of differential forms is also defined for the deformed differential calculus. Using an integral which is naturally associated with the deformed differential calculus, one obtains an action for a lattice theory.

From Chern-Simons to Gauss-Bonnet

Abstract The relation between SO(1, 2 p − 1) Chern-Simons forms and SO(1, 2 p − 2) Gauss-Bonnet forms in 2 p − 1 dimensions is discussed in detail based on recent results by Chamseddine. This approach singles out a special linear combination of Gauss-Bonnet terms as a lagrangian for gravity in odd dimensions. We show that in eleven dimensions this theory naturally admits spontaneous compactification over the four-dimensional Minkowski space and in some sense even distinguishes four dimensions. Similar results are obtained in any odd dimension. We also find a certain degeneracy of the field equations which might cause a problem for such a “topological gauge theory of gravity”.

Gravity actions, boundary terms and second-order field equations

Abstract The Lovelock action is a natural generalization of the Einstein-Hilbert gravity action in higher (more than four) dimensions. Certain boundary terms have to be added to this action because of various reasons which we briefly review. Explicit expressions of these boundary terms can already be found in Cherns famous paper [6] on the generalization of the Gauss-Bonnet theorem. Dimensional reduction of the Lovelock action leads to a generalized Einstein-Yang-Mills (+ scalar field or σ-models) action in lower dimensions. Since the field equations derived from the Lovelock action are of second order only, the same holds for the dimensionally reduced action. The necessary boundary terms for the latter action are then obtained by dimensional reduction of the boundary terms in the Lovelock action. We consider reduction from n to n −1 dimensions in some detail and obtain generalizations of Horndeskis non-minimally coupled Einstein-Maxwell action with second-order field equations in more than four dimensions.