Investigates geometric aspects of double field theory and its membrane sigma-model formulation.
problem Capturing geometric and non-geometric flux backgrounds in DFT.
method Determines a splitting and projection of the Courant algebroid, constructs a membrane sigma-model, and analyzes gauge invariance.
result Unified description of geometric and non-geometric flux backgrounds in DFT.
We review the AKSZ construction as applied to the topological open membranes and Poisson sigma models. We describe a generalization to open topological p-branes and Nambu-Poisson sigma models.
This thesis reviews Leibniz algebroids and generalized geometry for string theory.
problem Describing string and membrane backgrounds using generalized geometry.
method Review of Leibniz algebroids, generalized geometry, and Nambu-Poisson structures.
result Foundation laid for describing string and membrane backgrounds in generalized geometry.
Study topological A/B-models using double field theory and generalized geometry.
problem Tackles topological A/B-models within a double field theory framework.
method AKSZ-type BV constructions, Courant sigma-model, generalized complex structure.
result Introduces S-duality in membrane sigma-model based on generalized complex structure.
Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solut…
Paper formulates governing equations for membrane O surfaces.
problem Formulating equations for membrane O surfaces.
method Formulated governing equations for membrane O surfaces of the 1st and 2nd kind.
result Membrane O surfaces are a subclass of Demoulin's Ω surfaces.
New model considers spontaneous curvature for lipid bilayer membranes, including discontinuities.
problem Modeling discontinuities at interfaces in lipid bilayer membranes.
method Introduced a family of energies for smooth surfaces and phase fields, derived a sharp interface limit.
result Theoretical result extends classical model by assigning bending energy to tangential discontinuities.
Study connects hyperbolic geometry to membrane shapes.
problem Understanding the shapes of biological membranes.
method Relates geometry of hyperbolic space to Helfrich model.
result Establishes a connection between membrane shapes and hyperbolic geometry.
We define a topological quantum membrane theory on a seven dimensional manifold of G2 holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is CY3×S1 quantum amplitudes of non-local observables …
New multi-spiral approach improves packaging of thick membranes.
problem Deploying thick membranes on curved surfaces efficiently.
method Multi-spiral folding approach with prismatic folding lines.
result Improved deployment performance of thick membranes on curved surfaces.
New theory shows how membranes can break symmetry.
problem Understanding symmetry breaking in membranes with boundaries.
method Applied bifurcation theory and reduced membrane equation.
result Existence of symmetry breaking bifurcation in membrane solutions.
This review reports some theoretical results on the Geometry of membranes. The governing equations to describe equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are derived from the variation of free energies of these structures. Some analytic solutions to these e…
Study the stability of membranes using Helfrich energy and second variation formula.
problem Stability of membranes under various conditions.
method Developed and applied a second variation formula for the Helfrich energy for a class of surfaces.
result Studied the second variation of the area functional for a specific example.
Proves C1,1 regularity for multiple membrane solutions.
problem Stationary C1,α solutions to the multiple membrane problem. method Uses C1,1-regularity estimate. result Proves C1,1 regularity for stationary solutions. Theory and methods for particle dynamics in curved lipid membranes.
problem Understanding particle behavior in curved lipid membranes.
method Developed theory and computational methods for hydrodynamic coupling.
result Membrane curvature and topology affect particle mobility.
We construct a gauge fixed action for topological membranes on G2-manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. Moreover on M×S1 the theory naturally reduces…
The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed…
Periodic surfaces have a limited number of bending modes, equal to their membrane modes.
problem Understanding the limitations of bending modes in periodic surfaces.
method Analyzing deformation modes of periodic, piecewise smooth, simply connected surfaces.
result Effective membrane modes and bending modes are orthogonal, limiting the total number of modes to 3.
This review reports some key results in theoretical investigations on configurations of lipid membranes and presents several challenges in this field which involve (i) exact solutions to the shape equation of lipid vesicles; (ii) exact solutions to the governing equations of open lipid membranes; (iii) neck condition o…
A surface functional theory for p-dimensional extended objects, the p-branes, was proposed in previous papers. The field equations for toroidal p-branes was exactly solved in d=p+2 dimensions, yielding equally spaced mass-squared spectrum with massless states. In this paper, we obtain the asymptotic distribution of m…
Transforming cylindrical packings into bicontinuous surfaces.
problem Understanding the early development of bicontinuous structures in plant plastids.
method Geometric modeling and computational simulations of cylinder packings.
result Specific cylinder packings with cubic symmetry transform into TPMS.
Bayesian modeling predicts hydroxide ion conductivity in polymer membranes.
problem Quantitative relationship between hydrophilic domain size and hydroxide ion conductivity in polymer membranes is unknown.
method Bayesian sparse modeling applied to copolymer composition data.
result Composition-derived features are identified as critical for predicting hydroxide ion conductivity.
We construct membrane homology groups $\h(M)$ associated with each compact connected oriented smooth manifold, and show that $\h(M)$ is matrix graded algebra.
Proves existence and regularity of spherical minimizers for lipid membrane energy.
problem Existence and regularity of minimizers for the Canham-Helfrich energy.
method Establishes lower semicontinuity and proves existence through weak convergence of immersions.
result Proves existence and regularity of minimizers for the Canham-Helfrich energy on spheres.
Study of symplectic manifolds degenerating into singular spaces.
problem Understanding degenerations of symplectic manifolds into singular spaces.
method Topological framework focusing on Lagrangian manifolds and holomorphic membranes.
result Degenerations into singular toric varieties yield exotic Lagrangian tori.
Study derives the limit of lipid bilayer membranes, proving their smooth transition across interfaces.
problem Understanding the smooth transition of lipid bilayer membranes across interfaces.
method Rigorous derivation of the Γ-limit for rotationally symmetric two-phase bilayer membranes. result Limit membranes are C1 across interfaces and can consist of multiple topological spheres. Reviews uses of nonlinear sigma models in various systems.
problem Understanding nonlinear sigma models in different physical contexts.
method General discussion and focus on geometrical interpretations.
result Connection between sigma models and various geometries.
Study finds minimizers for complex membrane models without symmetry assumptions.
problem Minimizing the Canham-Helfrich functional in multiple phases for heterogeneous biological membranes.
method Reformulated as oriented curvature varifolds, proving existence without symmetry assumptions.
result Existence of minimizers for single- and multiphase models under constraints.
Generalizes momentum sections to higher-dimensional gauged sigma models.
problem Understanding momentum sections in Hamiltonian mechanics and sigma models.
method Introduces a generalization of momentum sections on pre-multisymplectic manifolds.
result Shows a connection between constrained Hamiltonian systems and gauged sigma models.
Generates tubular and membranous shapes using curvature functionals.
problem Difficult analysis of tubular and membranous shapes.
method Modeling as curvature optimization problem, phase-field formulation, GPU algorithm.
result Wide continuum of shape textures discovered.
Proves Payne conjecture for buckling and membrane eigenvalues.
problem Proving Payne conjecture for buckling and membrane eigenvalues.
method Analytical proof for buckling and membrane eigenvalues.
result Proves Payne conjecture for n-dimensional case (n≥2). Generalizes Hamiltonian structures to Dirac structures for new mechanics models.
problem Extending Hamiltonian structures to non-symplectic manifolds.
method Introduces Hamiltonian Lie algebroids and momentum sections over Dirac structures.
result Constructs new sigma models based on these structures.
We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the membrane case.
For a membrane in the plane the multiplicity of the k-th eigenvalue is known to be not greater than 2k−1. Here we prove that it is actually not greater than 2k−3, for k≥3.
Study connects supermanifold Laplacian to harmonic superfunctions via sigma model.
problem Understanding harmonic superfunctions on supermanifolds.
method Calculus of variations applied to a supersymmetric sigma model.
result Relates Laplacian to harmonic superfunctions on supermanifolds.
Sigma models linked to Gross-Neveu models via quiver varieties.
problem Understanding the relationship between sigma models and Gross-Neveu models.
method Exploring the mathematical correspondence between sigma models and Gross-Neveu models, including their geometric and trigonometric/elliptic deformations.
result Sigma models are mathematically equivalent to Gross-Neveu models under certain conditions.
This paper mainly aims to establish the well-posedness on time interval [0,ε−21T] of the classical initial problem for the bosonic membrane in the light cone gauge. Here ε is the small parameter measures the nonlinear effects. In geometric, the bosonic membrane are timelike submanifo…
This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional t…
Constructs zero-curvature representations for sigma-models.
problem Equations of motion for sigma-models with complex homogeneous spaces.
method Constructs zero-curvature representations and shows gauge-equivalence in symmetric cases.
result Zero-curvature representations for sigma-models in non-symmetric cases.
The Poisson--Weil sigma model, worked out by us recently, stems from gauging a Hamiltonian Lie group symmetry of the target space of the Poisson sigma model. Upon gauge fixing of the BV master action, it yields interesting topological field theories such as the 2--dimensional Donaldson-Witten topological gauge theory a…
Summarizes geometric connections between sigma models and Gross-Neveu models.
problem Understanding geometric connections between sigma models and Gross-Neveu models.
method Geometric facts and connections to nilpotent orbits, Springer resolutions, and quiver varieties.
result Sheds light on the general setup of the correspondence.
Grassmannian sigma models extend Gross-Neveu model formulations.
problem Understanding sigma models on Grassmannian targets.
method Chiral Gross-Neveu model formulations for orthogonal and symplectic Grassmannians.
result One-loop β-functions proportional to dual Coxeter numbers. Study shows weak solutions' regularity for a sigma model with coarse gravitino.
problem Analyzing the regularity of solutions to a nonlinear sigma model with limited gravitino regularity.
method Examined solutions in Lp space with p>4 and derived precise regularity results. result Precise regularity results depend on the value of p. Study sigma-models on flag manifolds with curvature zero and relate to chiral models.
problem Understanding sigma-models on flag manifolds with zero curvature.
method Use zero-curvature representation and nilpotent orbits theory.
result Established relation between flag manifold sigma-models and principal chiral models.
New sigma models compute graviton scattering amplitudes from quaternionic geometry.
problem Computing graviton scattering amplitudes from quaternionic geometry.
method Introducing new twistor sigma models that encode finite non-linear perturbations of flat structures.
result Provides a first-principles derivation of Hodges' formula for MHV graviton amplitudes.
We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…
Study on smooth solutions of a nonlinear sigma model with gravitino fields.
problem Analyzing smooth solutions of a modified nonlinear sigma model.
method Geometric setup, Euler-Lagrange equations, Rivière's regularity theory, Riesz potential theory.
result Smoothness of weak solutions proven using advanced mathematical theories.
One has believed that low energy effective theories of the Higgs branch of gauged linear sigma models correspond to supersymmetric nonlinear sigma models, which have been already investigated by many works. In this paper we discuss a explicit derivation of supersymmetric nonlinear sigma models from gauged linear sigma …