Three approaches towards Floer homology of cotangent bundles

Consider the cotangent bundle of a closed Riemannian manifold and an almost complex structure close to the one induced by the Riemannian metric. For Hamiltonians which grow for instance quadratically in the fibers outside of a compact set, one can define Floer homology and show that it is naturally isomorphic to singular homology of the free loop space. We review the three isomorphisms constructed by Viterbo (1996), Salamon-Weber (2003) and Abbondandolo-Schwarz (2004). The theory is illustrated by calculating Morse and Floer homology in case of the euclidean n-torus. Applications include existence of noncontractible periodic orbits of compactly supported Hamiltonians on open unit disc cotangent bundles which are sufficiently large over the zero section.Comment: 30 pages, 6 figures. To appear in J. Symplectic Geom. (Stare Jablonki conference issue

Realization of the Three-dimensional Quantum Euclidean Space by Differential Operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by $q$-deformation. Simultaneously, angular momentum is deformed to $so_q(3)$, it acts on the $q$-Euclidean space that becomes a $so_q(3)$-module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on $C^{\infty}$ functions on $\mathbb{R}^3$. On a factorspace of $C^{\infty}(\mathbb{R}^3)$ a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a $q$-lattice.Comment: 13 pages, late

Brane Tensions and Coupling Constants from within M-Theory

Reviewing the cancellation of local anomalies of M-theory on R^10 x S^1/Z_2 the Yang-Mills coupling constant on the boundaries is rederived. The result is lambda^2 = 2^(1/3) (2 pi) (4 pi kappa^2)^(2/3) corresponding to eta = lambda^6/kappa^4 = 256 pi^5 in the `upstairs' units used by Horava and Witten and differs from their calculation. It is shown that these values are compatible with the standard membrane and fivebrane tensions derived from the M-theory bulk action. In view of these results it is argued that the natural units for M-theory on R^10 x S^1/Z_2 are the `downstairs' units where the brane tensions take their standard form and the Yang-Mills coupling constant is lambda^2 = 4 pi (4 pi kappa^2)^(2/3).Comment: 11 pages, no figures, Latex2e, amsmath, amsfonts, typo in abstract correcte