Researchers analyze sign choices for O-planes in orientifolds, proving stabilisation and deriving topological constraints.
problem Assigning sign choices to O-planes in orientifolds of type II string theory.
method Investigating sequences of invariant p-gerbes and coboundary maps to derive sign choices and topological constraints.
result Sign choice homomorphisms stabilise with the dimension of the orientifold and topological constraints on sign configurations are derived.
We study topological open string amplitudes on orientifolds without fixed planes. We determine the contributions of the untwisted and twisted sectors as well as the BPS structure of the amplitudes. We illustrate our general results in various examples involving D-branes in toric orientifolds. We perform the computation…
New supersymmetric vacua found on special geometric spaces.
problem Finding new supersymmetric vacua in type II supergravities.
method Using flux vacua on four-dimensional Minkowski times six-dimensional solvmanifolds, with localized orientifold planes and D-branes.
result Discovered new supersymmetric vacua not T-dual to torus vacua.
Construct M-Theory lifts of type IIA orientifolds.
problem Lift type IIA orientifolds to M-Theory.
method Construct M-Theory on twisted connected sum G2 manifolds.
result Two building blocks correspond to open and closed string sectors.
New framework solves string theory's RR-field tadpole cancellation problems.
problem Precise global nature of RR-field tadpole cancellation conditions in string theory.
method Formulated M-theory C-field on flat M-orientifolds using equivariant cohomotopy.
result Equivariant cohomotopy implies anomaly cancellation conditions for M-branes and D-branes.
We give a precise and concise formulation of the orientifold construction in Type II superstring theory. Our results include anomaly cancellation on the worldsheet and a spacetime computation of the background Ramond-Ramond charge.
The simplest orientifolds of the WZW models are obtained by gauging a Z_2 symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion g \mapsto (ζg)^{-1}, where ζis an element of the center of G. It reverses the …
We study refined topological string theory in the presence of orientifolds by counting second-quantized BPS states in M-theory. This leads us to propose a new integrality condition for both refined and unrefined topological strings when orientifolds are present. We define the SO(2N) refined Chern-Simons theory which co…
Real bundle gerbes classify geometric cycles in twisted KR-homology.
problem Classifying geometric cycles in twisted KR-homology.
method Real bundle gerbes and their modules as building blocks.
result Geometric cycles generate a real-oriented generalised homology theory.
In superstring theory spin structures are present on both the 2-dimensional worldsheet and 10-dimensional spacetime. We present a new proposal for the B-field in superstring theory and demonstrate its interaction with worldsheet spin structures. Our formulation generalizes to orientifolds, where various twistings appea…
We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs (L,H) with L a flat line bundle and H∈H3(M,L) a degree 3 cla…
New tools identify potential counter-examples to string theory conjectures.
problem Classical de Sitter solutions with specific conditions.
method Developed new tools and constraints to explore parameter space.
result Identified corners of parameter space where counter-examples could be found.
Review of non-abelian gerbes and their applications in string theory.
problem Anomaly cancellation and geometric description of T-duals in string theory.
method Systematic construction of non-abelian gerbes via descent.
result Extension of non-abelian gerbes to orientifold sigma models and T-duals.
We construct new examples of torsional heterotic backgrounds using duality with orientifold flux compactifications. We explain how duality provides a perturbative solution to the type I/heterotic string Bianchi identity. The choice of connection used in the Bianchi identity plays an important role in the construction. …
We propose a new, precise integrality conjecture for the colored Kauffman polynomial of knots and links inspired by large N dualities and the structure of topological string theory on orientifolds. According to this conjecture, the natural knot invariant in an unoriented theory involves both the colored Kauffman polyno…
Unified classification of equivariant principal bundles using higher homotopy theory.
problem Unified classification of equivariant principal bundles.
method Smooth Oka principle, singular-cohesive homotopy theory, internally describing principal bundles.
result Unified classification results for equivariant principal bundles.
Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to CP2. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…
Study bifurcation in plane-to-plane germs with specific properties.
problem Understanding the structure of bifurcations in a specific class of geometric objects.
method Explicit description of the bifurcation diagram of the topologically A-versal unfolding.
result Explicit description of the bifurcation diagram for a specific class of plane-to-plane germs.
We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.
The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.
problem Characterizing semiaffine stable planes.
method Analyzing properties of lines and points in stable planes.
result Semiaffine stable planes are either affine, projective, or punctured projective planes.
Extends results on automorphism groups of flat Minkowski planes to toroidal circle planes.
problem Understanding automorphism groups of toroidal circle planes.
method Extending Schenkel's results to toroidal circle planes.
result Automorphism groups of toroidal circle planes have dimensions at most 6 and can have dimensions at least 4 or kernels of dimension 3.
Stable planes are locally isomorphic to classical projective planes.
problem Characterizing stable planes that are locally isomorphic to classical projective planes.
method Analyzing properties of stable planes and comparing them to classical projective planes over specific fields.
result Simply connected stable planes with connected lines are isomorphic to open subplanes of classical projective planes.
Only vertical planes are asymptotic to other planes in 3D space.
problem Characterizing asymptotic planes in 3D space.
method Proof of uniqueness for complete translators with finite topology.
result Vertical planes are the only asymptotic planes in 3D space.
Study of elementary planes in Apollonian orbifold with unusual equidistribution.
problem Equidistribution failure in Apollonian orbifold.
method Complete list of elementary planes, boundary data analysis.
result Uniform boundedness of elementary plane areas and closed union.
Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…
Criteria for sharksfin and deltoid singularities from plane to plane, with applications.
problem Identifying and understanding singularities in plane-to-plane mappings.
method Providing criteria and geometric meanings for singularities.
result Geometric meanings and criteria for sharksfin and deltoid singularities.
Crooked planes in 3D Minkowski space can be foliated.
problem Understanding foliations between crooked planes in 3D Minkowski space.
method Showed that any two disjoint crooked planes are leaves of a crooked foliation.
result Answered a question about foliations between crooked planes in 3D Minkowski space.
We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…
New manifolds with many hyperbolic planes, not homogeneous.
problem Constructing non-homogeneous manifolds with many hyperbolic planes.
method Constructing manifolds with geodesics in hyperbolic planes.
result Non-homogeneous manifolds with many hyperbolic planes exist.
Researchers found parallel mean curvature tori in complex projective and hyperbolic planes.
problem Finding tori with parallel mean curvature vectors.
method Explicit determination of tori in complex projective and hyperbolic planes.
result Explicit determination of tori with parallel mean curvature vectors in both complex projective and hyperbolic planes.
For a pair of points in a smooth locally convex surface in 3-space, its mid-plane is the plane containing its mid-point and the intersection line of the corresponding pair of tangent planes. In this paper we show that the limit of mid-planes when one point tends to the other along a direction is the Transon plane of th…
The study explores conformal planes with finite areas.
problem Geometry of conformal planes with finite areas.
method Analyzes several questions about conformal planes.
result Exploration of conformal planes with finite areas.
Calculates first exit times for Brownian motion in Euclidean and hyperbolic planes.
problem Computing expected first exit times for Brownian motion.
method Analytical computation for Brownian motion in Euclidean and hyperbolic planes.
result Results in expected first exit times for specified domains.
We construct a compact nonpositively curved squared 2-complex whose universal cover contains a flat plane that is not the limit of periodic flat planes.
This thesis explores bifoliated planes and their group actions from Anosov flows.
problem Understanding bifoliated planes and their group actions from Anosov flows.
method Analyzes bifoliated planes associated with Anosov flows and describes their properties.
result Shows left-orderability of groups acting on bifoliated planes and describes bifoliated planes from specific families of Anosov flows.
We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…
The paper extends curve curvature types from normed to gauge planes.
problem Extending curve curvature classification to gauge planes.
method Using gauge analogue of Birkhoff orthogonality and differential geometry.
result Four curvature types (Minkowski, normal, circular, arc-length) are identified.
Explains groups made from two complex hyperbolic plane movements.
problem None explicitly stated; focuses on explanation.
method Expository article.
result Provides insight into groups generated by two isometries.
Two proofs show that removing a loop from a plane circuit splits the plane.
problem Proving the Weak Jordan Theorem about plane circuits.
method Detailed presentation of Thomassen's and Filippov's proofs.
result The complement of any loop in a plane circuit is disconnected.
Study fake projective planes using recent results to show bicanonical map is always an embedding and construct an exceptional collection.
problem Analyzing Keum's fake projective planes and their geometric properties.
method Apply recent results from Galkin et al. [GKMS15] to study fake projective planes.
result The bicanonical map of Keum's fake projective planes is always an embedding.
New findings on plane waves in 3D spacetimes, showing non-unimodular elliptic plane waves are unique.
problem Classifying Lorentz homogeneous spaces of dimension 3, focusing on plane waves.
method Revisiting and relaxing usual completeness assumptions, characterizing homogeneous plane waves.
result Non-unimodular elliptic plane waves are unique and non-extendable, geodesically complete only if symmetric.
The paper classifies singularities of plane congruences and affine distance functions.
problem Classifying singularities of plane congruences and affine distance functions.
method Classification through 2-parameter plane congruences in \(\mathbb{R^4}\) and affine normal plane congruences.
result Generic singularities of plane congruences and affine distance functions are classified.
Computed distortion coefficients for the α-Grushin plane.
problem Analyzing the distortion coefficients of the α-Grushin plane.
method Using generalised trigonometric functions and synthetic curvature conditions.
result Estimates for distortion coefficients and a curvature condition conjecture.
Classifies toroidal circle planes with 3D automorphism groups.
problem Classifying toroidal circle planes with specific automorphism groups.
method Using almost simple Lie groups and their isomorphism to PSL(2, R), a framework for classification is described.
result Three-dimensional connected automorphism groups are isomorphic to PSL(2, R).
The paper studies stable mappings of plane curves using distance-squared functions.
problem Stability of mappings of plane curves.
method Investigation of compositions of plane curves and generic distance-squared mappings.
result Stable mappings of plane curves are explored.
Study compact plane waves, showing they are essentially standard.
problem Understanding the topology and dynamics of compact plane waves.
method Analyzing quotients of homogeneous plane waves by discrete subgroups.
result Compact quotients of homogeneous plane waves are essentially standard.
Proves Ptolemaean Inequality and Theorem in complex hyperbolic plane.
problem None explicitly stated, but related to complex hyperbolic geometry.
method Proof of Ptolemaean Inequality and Theorem in complex hyperbolic plane with Cygan metric.
result Proves Ptolemaean Inequality and Ptolemaeus' Theorem in the closure of complex hyperbolic plane.
New classification of 16D planes with specific automorphism groups.
problem Classifying 16-dimensional projective planes with certain automorphism groups.
method Detailed analysis and classification of 16-dimensional planes with specific automorphism properties.
result Classification of 16-dimensional planes with a group of dimension at least 35, excluding planes fixed by exactly one flag.