Lower bound for energies of harmonic tangent unit-vector fields on convex polyhedra

We derive a lower bound for energies of harmonic maps of convex polyhedra in $\R^3$ to the unit sphere $S^2,$ with tangent boundary conditions on the faces. We also establish that $C^\infty$ maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy.Comment: Acknowledgment added, typos removed, minor correction

Fluctuation, time-correlation function and geometric Phase

We establish a fluctuation-correlation theorem by relating the quantum fluctuations in the generator of the parameter change to the time integral of the quantum correlation function between the projection operator and force operator of the ``fast'' system. By taking a cue from linear response theory we relate the quantum fluctuation in the generator to the generalised susceptibility. Relation between the open-path geometric phase, diagonal elements of the quantum metric tensor and the force-force correlation function is provided and the classical limit of the fluctuation-correlation theorem is also discussed.Comment: Latex, 12 pages, no figures, submitted to J. Phys. A: Math & Ge

Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals

We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of Γ-development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing Q-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy

Energies of S^2-valued harmonic maps on polyhedra with tangent boundary conditions

A unit-vector field n:P \to S^2 on a convex polyhedron P \subset R^3 satisfies tangent boundary conditions if, on each face of P, n takes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices of P. We consider fields which are continuous elsewhere. We derive a lower bound E^-_P(h) for the infimum Dirichlet energy E^inf_P(h) for such tangent unit-vector fields of arbitrary homotopy type h. E^-_P(h) is expressed as a weighted sum of minimal connections, one for each sector of a natural partition of S^2 induced by P. For P a rectangular prism, we derive an upper bound for E^inf_P(h) whose ratio to the lower bound may be bounded independently of h. The problem is motivated by models of nematic liquid crystals in polyhedral geometries. Our results improve and extend several previous results.Comment: 42 pages, 2 figure

Quantum pumping and dissipation: from closed to open systems

Current can be pumped through a closed system by changing parameters (or fields) in time. The Kubo formula allows to distinguish between dissipative and non-dissipative contributions to the current. We obtain a Green function expression and an $S$ matrix formula for the associated terms in the generalized conductance matrix: the "geometric magnetism" term that corresponds to adiabatic transport; and the "Fermi golden rule" term which is responsible to the irreversible absorption of energy. We explain the subtle limit of an infinite system, and demonstrate the consistency with the formulas by Landauer and Buttiker, Pretre and Thomas. We also discuss the generalization of the fluctuation-dissipation relation, and the implications of the Onsager reciprocity.Comment: 4 page paper, 1 figure (published version) + 2 page appendi

Time-Dependent Warping, Fluxes, and NCYM

We describe the supergravity solutions dual to D6-branes with both time-dependent and time-independent B-fields. These backgrounds generalize the Taub-NUT metric in two key ways: they have asymmetric warp factors and background fluxes. In the time-dependent case, the warping takes a novel form. Kaluza-Klein reduction in these backgrounds is unusual, and we explore some of the new features. In particular, we describe how a localized gauge-field emerges with an analogue of the open string metric and coupling. We also describe a gravitational analogue of the Seiberg-Witten map. This provides a framework in supergravity both for studying non-commutative gauge theories, and for constructing novel warped backgrounds.Comment: 32 pages, LaTeX, references adde

Maslov indices and monodromy

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the resulting restrictions on the monodromy matrix are derived

Non-commutative gauge theory on D-branes in Melvin Universes

Non-commutative gauge theory with a non-constant non-commutativity parameter can be formulated as a decoupling limit of open strings ending on D3-branes wrapping a Melvin universe. We construct the action explicitly and discuss various physical features of this theory. The decoupled field theory is not supersymmetric. Nonetheless, the Coulomb branch appears to remain flat at least in the large N and large 't Hooft coupling limit. We also find the analogue of Prasad-Sommerfield monopoles whose size scales with the non-commutativity parameter and is therefore position dependent.Comment: 15 pages, 1 figure, reference adde

On Supergravity Solutions of Branes in Melvin Universes

We study supergravity solutions of type II branes wrapping a Melvin universe. These solutions provide the gravity description of non-commutative field theories with non-constant non-commutative parameter. Typically these theories are non-supersymmetric, though they exhibit some feature of their corresponding supersymmetric theories. An interesting feature of these non-commutative theories is that there is a critical length in the theory in which for distances larger than this length the effects of non-commutativity become important and for smaller distances these effects are negligible. Therefore we would expect to see this kind of non-commutativity in large distances which might be relevant in cosmology. We also study M5-brane wrapping on 11-dimensional Melvin universe and its descendant theories upon compactifying on a circle.Comment: 25 pages, latex file; v2: typos corrected, Refs. adde

Supersymmteric Null-like Holographic Cosmologies

We construct a new class of 1/4-BPS time dependent domain-wall solutions with null-like metric and dilaton in type II supergravities, which admit a null-like big bang singularity. Based on the domain-wall/QFT correspondence, these solutions are dual to 1/4-supersymmetric quantum field theories living on a boundary cosmological background with time dependent coupling constant and UV cutoff. In particular we evaluate the holographic $c$ function for the 2-dimensional dual field theory living on the corresponding null-like cosmology. We find that this $c$ function runs in accordance with the $c$-theorem as the boundary universe evolves, this means that the number of degrees of freedom is divergent at big bang and suggests the possible resolution of big bang singularity.Comment: 26 pages;v2 references adde