Global Aspects of T-Duality, Gauged Sigma Models and T-Folds

The gauged sigma-model argument that string backgrounds related by T-dual give equivalent quantum theories is revisited, taking careful account of global considerations. The topological obstructions to gauging sigma-models give rise to obstructions to T-duality, but these are milder than those for gauging: it is possible to T-dualise a large class of sigma-models that cannot be gauged. For backgrounds that are torus fibrations, it is expected that T-duality can be applied fibrewise in the general case in which there are no globally-defined Killing vector fields, so that there is no isometry symmetry that can be gauged; the derivation of T-duality is extended to this case. The T-duality transformations are presented in terms of globally-defined quantities. The generalisation to non-geometric string backgrounds is discussed, the conditions for the T-dual background to be geometric found and the topology of T-folds analysed.Comment: Minor corrections and addition

Random matrix theory within superstatistics

We propose a generalization of the random matrix theory following the basic prescription of the recently suggested concept of superstatistics. Spectral characteristics of systems with mixed regular-chaotic dynamics are expressed as weighted averages of the corresponding quantities in the standard theory assuming that the mean level spacing itself is a stochastic variable. We illustrate the method by calculating the level density, the nearest-neighbor-spacing distributions and the two-level correlation functions for system in transition from order to chaos. The calculated spacing distribution fits the resonance statistics of random binary networks obtained in a recent numerical experiment.Comment: 20 pages, 6 figure

Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles

We apply some methods of homology and K-theory to special classes of branes wrapping homologically nontrivial cycles. We treat the classification of four-geometries in terms of compact stabilizers (by analogy with Thurston's classification of three-geometries) and derive the K-amenability of Lie groups associated with locally symmetric spaces listed in this case. More complicated examples of T-duality and topology change from fluxes are also considered. We analyse D-branes and fluxes in type II string theory on ${\mathbb C}P^3\times \Sigma_g \times {\mathbb T}^2$ with torsion $H-$flux and demonstrate in details the conjectured T-duality to ${\mathbb R}P^7\times X^3$ with no flux. In the simple case of $X^3 = {\mathbb T}^3$, T-dualizing the circles reduces to duality between ${\mathbb C}P^3\times {\mathbb T}^2 \times {\mathbb T}^2$ with $H-$flux and ${\mathbb R}P^7\times {\mathbb T}^3$ with no flux.Comment: 27 pages, tex file, no figure

A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on B\"ar and Pf\"affle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on the space of piecewise geodesic paths which is the pullback by a natural section of Mathai and Quillen's Thom form of a bundle over this space. In the case of closed paths, the bundle is the tangent space to the space of geodesic paths, and the integral of this form passes in the limit to the supertrace of the heat kernel.Comment: 14 pages, LaTeX, no fig

The Loop Group of E8 and Targets for Spacetime

The dimensional reduction of the E8 gauge theory in eleven dimensions leads to a loop bundle in ten dimensional type IA string theory. We show that the restriction to the Neveu-Schwarz sector leads naturally to a sigma model with target space E8 with the ten-dimensional spacetime as the source. The corresponding bundle has a structure group the group of based loops, whose classifying space we study. We explore some consequences of this proposal such as possible Lagrangians and existence of flat connections.Comment: 17 pages, main section improved, change in title, reference and acknowledgement adde

Type I D-branes in an H-flux and twisted KO-theory

Witten has argued that charges of Type I D-branes in the presence of an H-flux, take values in twisted KO-theory. We begin with the study of real bundle gerbes and their holonomy. We then introduce the notion of real bundle gerbe KO-theory which we establish is a geometric realization of twisted KO-theory. We examine the relation with twisted K-theory, the Chern character and provide some examples. We conclude with some open problems.Comment: 23 pages, Latex2e, 2 new references adde

Tunneling in Fractional Quantum Mechanics

We study the tunneling through delta and double delta potentials in fractional quantum mechanics. After solving the fractional Schr\"odinger equation for these potentials, we calculate the corresponding reflection and transmission coefficients. These coefficients have a very interesting behaviour. In particular, we can have zero energy tunneling when the order of the Riesz fractional derivative is different from 2. For both potentials, the zero energy limit of the transmission coefficient is given by $\mathcal{T}_0 = \cos^2{\pi/\alpha}$, where $\alpha$ is the order of the derivative ($1 < \alpha \leq 2$).Comment: 21 pages, 3 figures. Revised version; accepted for publication in Journal of Physics A: Mathematical and Theoretica

Ground state properties and dynamics of the bilayer t-J model

We present an exact diagonalization study of bilayer clusters of t-J model. Our results indicate a crossover between two markedly different regimes which occurs when the ratio J_perp/J between inter-layer and intra-layer exchange constants increases: for small J_perp/J the data suggest the development of 3D antiferromagnetic correlations without appreciable degradation of the intra-layer spin order and the d_(x2-y2) hole pairs within the planes persist. For larger values of J_perp/J local singlets along the inter-layer bonds dominate, leading to an almost complete suppression of the intra-layer spin correlation and the breaking of the intra-layer pairs. The ground state with two holes in this regime has s-like symmetry. The data suggest that the crossover may occur for values of J_perp/J as small as 0.2. We present data for static spin correlations, spin gap, and electron momentum distribution and spectral function of the `inter-layer RVB state' realized for large J_perp/J. The latter deviates from the single layer ground state, making it an implausible candidate for modelling high-temperature superconductors.Comment: Revtex-file, 6 PRB pages, figures appended as uu-encoded postscript. Hardcopies of figures (or the entire manuscript) can be obtained by e-mailing to: [email protected]

Twisted topological structures related to M-branes

Studying the M-branes leads us naturally to new structures that we call Membrane-, Membrane^c-, String^K(Z,3)- and Fivebrane^K(Z,4)-structures, which we show can also have twisted counterparts. We study some of their basic properties, highlight analogies with structures associated with lower levels of the Whitehead tower of the orthogonal group, and demonstrate the relations to M-branes.Comment: 17 pages, title changed on referee's request, minor changes to improve presentation, typos correcte

Nonassociative strict deformation quantization of C*-algebras and nonassociative torus bundles

In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with $H$-flux. We also show that the Octonions fit nicely into our theory.Comment: 15 pages, latex2e, exposition improved, to appear in LM