Supersymmetry, homology with twisted coefficients and n-dimensional knots

Let $n$ be any natural number. Let $K$ be any $n$-dimensional knot in $S^{n+2}$. We define a supersymmetric quantum system for $K$ with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for $K$. Our Witten indexes are topological invariants for $n$-dimensional knots. Our Witten indexes are not zero in general. If $K$ is equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten indexes restrict the Alexander polynomials of $n$-knots. If one of our Witten indexes for an $n$-knot $K$ is nonzero, then one of the Alexander polynomials of $K$ is nontrivial. Our Witten indexes are connected with homology with twisted coefficients. Roughly speaking, our Witten indexes have path integral representation by using a usual manner of supersymmetric theory.Comment: 10pages, no figure

Dyson processes on the octonion algebra

We consider Brownian motion on symmetric matrices of octonions, and study the law of the spectrum. Due to the fact that the octonion algebra is nonassociative, the dimension of the matrices plays a special role. We provide two specific models on octonions, which give some indication of the relation between the multiplicity of eigenvalues and the exponent in the law of the spectrum

Exotic Differentiable Structures and General Relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as $S^7$, \R and $S^3\times {\bf R^1}.$ Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.Comment: 11 pages, LaTe

The anomaly line bundle of the self-dual field theory

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.Comment: 41 pages. v2: A few typos corrected. Version accepted for publication in CM

The Seven-sphere and its Kac-Moody Algebra

We investigate the seven-sphere as a group-like manifold and its extension to a Kac-Moody-like algebra. Covariance properties and tensorial composition of spinors under $S^7$ are defined. The relation to Malcev algebras is established. The consequences for octonionic projective spaces are examined. Current algebras are formulated and their anomalies are derived, and shown to be unique (even regarding numerical coefficients) up to redefinitions of the currents. Nilpotency of the BRST operator is consistent with one particular expression in the class of (field-dependent) anomalies. A Sugawara construction is given.Comment: 22 pages. Macropackages used: phyzzx, epsf. Three epsf figure files appende

Compactification, topology change and surgery theory

We study the process of compactification as a topology change. It is shown how the mediating spacetime topology, or cobordism, may be simplified through surgery. Within the causal Lorentzian approach to quantum gravity, it is shown that any topology change in dimensions $\geq 5$ may be achieved via a causally continuous cobordism. This extends the known result for 4 dimensions. Therefore, there is no selection rule for compactification at the level of causal continuity. Theorems from surgery theory and handle theory are seen to be very relevant for understanding topology change in higher dimensions. Compactification via parallelisable cobordisms is particularly amenable to study with these tools.Comment: 1+19 pages. LaTeX. 9 associated eps files. Discussion of disconnected case adde

Cosmological Creation of D-branes and anti-D-branes

We argue that the early universe may be described by an initial state of space-filling branes and anti-branes. At high temperature this system is stable. At low temperature tachyons appear and lead to a phase transition, dynamics, and the creation of D-branes. These branes are cosmologically produced in a generic fashion by the Kibble mechanism. From an entropic point of view, the formation of lower dimensional branes is preferred and $D3$ brane-worlds are exponentially more likely to form than higher dimensional branes. Virtually any brane configuration can be created from such phase transitions by adjusting the tachyon profile. A lower bound on the number defects produced is: one D-brane per Hubble volume.Comment: 30 pages, 5 eps figures; v2 more references added; v3 section 4 slightly improve