Four Dimensional Quantum Yang-Mills Theory and Mass Gap

A quantization procedure for the Yang-Mills equations for the Minkowski space $\mathbf{R}^{1,3}$ is carried out in such a way that field maps satisfying Wightman axioms of Constructive Quantum Field Theory can be obtained. Moreover, by removing the ultra violet cut off, the spectrum of the corresponding QCD Hamilton operator is proven to be positive and bounded away from zero, except for the case of the vacuum state, which has vanishing energy level. The whole construction is gauge invariant. The particles corresponding to all solution fields are bosons. As expected from QED, if the coupling constant converges to zero, then so does the mass gap. The results are proved first for the model with the bare coupling constant, and then for a model with a running coupling constant by means of renormalization.Comment: With respect to the preceding version of this paper, the gauge invariance of the construction has been proved and the construction of the probability measure making the Hamiltonian QCD selfadjoint has been rewritten with more clarit

Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the 'instantaneous arbitrage capability' generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the (NFLVR) condition if and only if $0$ is in the discrete spectrum of the connection Laplacian on the cash flow bundle or of the Dirac Laplacian of the twisted cash flow bundle with the exterior algebra bundle. We apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles for complete arbitrage free markets to markets not satisfying the (NFLVR). Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler characteristic of the asset nominal space is a topological obstruction to the the (NFLVR) condition, and, by means of the Bochner-Weitzenb\"ock formula, the non vanishing of the homology group of the cash flow bundle is revealed to be a topological obstruction to (NFLVR), too. Asset bubbles are defined, classified and decomposed for markets allowing arbitrage.Comment: arXiv admin note: substantial text overlap with arXiv:1406.6805, arXiv:0910.167

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact oriented K\"ahler manifold is a special case of a Dirac operator. The Green function for the Dirac Laplacian over a manifold with boundary allows to express the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash-Moser generalized inverse function theorem we prove the existence of complex submanifolds of a compact oriented variety satisfying globally a certain partial differential equation under a certain injectivity assumption. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for non singular projective algebraic varieties.Comment: Definition section has been amende