Beyond E11

We study the non-linear realisation of E11 originally proposed by West with particular emphasis on the issue of linearised gauge invariance. Our analysis shows even at low levels that the conjectured equations can only be invariant under local gauge transformations if a certain section condition that has appeared in a different context in the E11 literature is satisfied. This section condition also generalises the one known from exceptional field theory. Even with the section condition, the E11 duality equation for gravity is known to miss the trace component of the spin connection. We propose an extended scheme based on an infinite-dimensional Lie superalgebra, called the tensor hierarchy algebra, that incorporates the section condition and resolves the above issue. The tensor hierarchy algebra defines a generalised differential complex, which provides a systematic description of gauge invariance and Bianchi identities. It furthermore provides an E11 representation for the field strengths, for which we define a twisted first order self-duality equation underlying the dynamics.Comment: 97 pages. v2: Minor changes, references added. Published versio

On the extendedness of eigenstates in a hierarchical lattice: a critical view

We take a critical view at the basic definition of extended single particle states in a non-translationally invariant system. For this, we present the case of a hierarchical lattice and incorporate long range interactions that are also distributed in a hierarchical fashion. We show that it is possible to explicitly construct eigenstates with constant amplitudes (normalized to unity) at every lattice point for special values of the electron- energy. However, the end-to-end transmission, corresponding to the above energy of the electron in such a hierarchical system depends strongly on a special correlation between the numerical values of the parameters of the Hamiltonian. Keeping the energy and the distribution of the amplitudes invariant, one can transform the lattice from conducting to insulating simply by tunning the numerical values of the long range interaction. The values of these interactions themselves display a fractal character.Comment: 19 pages, 5 figure

Boolean Operations, Joins, and the Extended Low Hierarchy

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL_2, yet their join is in EL_2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL_ and prove that EL_2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL_2 lower bounds for certain notions generalizing Selman's P-selectivity, which may be regarded as an interesting result in its own right.Comment: 12 page

Cooperativity Beyond Caging: Generalized Mode Coupling Theory

The validity of mode coupling theory (MCT) is restricted by an uncontrolled factorization approximation of density correlations. The factorization can be delayed and ultimately avoided, however, by explicitly including higher order correlations. We explore this approach within a microscopically motivated schematic model. Analytic tractability allows us to discuss in great detail the impact of factorization at arbitrary order, including the limit of avoided factorization. Our results indicate a coherent picture for the capabilities as well as limitations of MCT. Moreover, including higher order correlations systematically defers the transition and ultimately restores ergodicity. Power-law divergence of the relaxation time is then replaced by continuous but exponential growth.Comment: 4 pages, 2 figure

Solving the Hierarchy Problem without Supersymmetry or Extra Dimensions: An Alternative Approach

In this paper, we propose a possible new approach towards solving the gauge hierarchy problem without supersymmetry and without extra spacetime dimensions. This approach relies on the finiteness of string theory and the conjectured stability of certain non-supersymmetric string vacua. One crucial ingredient in this approach is the idea of ``misaligned supersymmetry'', which explains how string theories may be finite even without exhibiting spacetime supersymmetry. This approach towards solving the gauge hierarchy problem is therefore complementary to recent proposals involving both large and small extra spacetime dimensions. This approach may also give a new perspective towards simultaneously solving the cosmological constant problem.Comment: 33 pages, LaTeX, 3 figure

Strange eigenstates and anomalous transport in a Koch fractal with hierarchical interaction

Stationary states of non-interacting electrons on a Koch fractal are investigated within a tight binding approach. It is observed that if a hierarchically long range hopping is allowed, a suitable correlation between the parameters defining the Hamiltonian leads to spectacular changes in the transport properties of finite, but arbitrarily large fractals. Topologically identical structures, that are found to support the same distribution of the amplitudes of eigenstates, are conducting in some cases and insulating in the others, depending on the choice of the hierarchy parameter. The values of the hierarchical parameter themselves display a self-similar, fractal character.Comment: 6 pages, 5 figure

Gardner's deformations as generators of new integrable systems

We re-address the problem of construction of new infinite-dimensional completely integrable systems on the basis of known ones, and we reveal a working mechanism for such transitions. By splitting the problem's solution in two steps, we explain how the classical technique of Gardner's deformations facilitates -- in a regular way -- making the first, nontrivial move, in the course of which the drafts of new systems are created (often, of hydrodynamic type). The other step then amounts to higher differential order extensions of symbols in the intermediate hierarchies (e.g., by using the techniques of Dubrovin et al. [1,2] and Ferapontov et al. [3,4]).Comment: Accepted to Proc. Int. workshop 'Physics and Mathematics of Nonlinear Phenomena' (June 22-29, 2013; Gallipoli (LE), Italy), 6 page