New geometry connects super loops to Chern character.
problem Understanding super parallel transport on quotient stacks.
method Super holonomy on loop stacks to construct Chern character.
result Global equivariant Chern character from super holonomy.
Non-trivial Clifford bundle from loop space tangent bundle.
problem Triviality obstruction of Clifford bundle on loop space.
method Constructing Clifford algebra bundle from loop space tangent bundle, showing non-triviality through Stiefel-Whitney and Pontrjagin classes.
result Clifford bundle is non-trivial, obstructed by manifold's Stiefel-Whitney and Pontrjagin classes.
This work interprets supergravity as a super Cartan geometry linking it to Yang-Mills theory.
problem Geometric interpretation of supergravity and its relation to Yang-Mills theory.
method Using enriched categories and super Cartan geometries, the authors link supergravity to Yang-Mills theory.
result Non-extended D=4 supergravity naturally arises in this framework.
The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects o…
We study the supersymmetric Wilson loop as introduced by Caron-Huot, which attaches to lightlike polygons certain edge and vertex operators, whose shape is determined by supersymmetry constraints. We state explicit formulas for the vertex operators to all orders in the Graßmann expansion, thus filling a gap in the lite…
Spinor bundle constructed on loop space for string manifolds.
problem Constructing spinor bundles for string manifolds.
method Using ideas from Stolz and Teichner, a spinor bundle and fusion product are defined for a string manifold.
result Existence of spinor bundle with fusion product on a manifold X if and only if X admits a string structure.
In this paper, we give a quantum interpretation of the Bismut-Chern character form (the loop space lifting of the Chern character form) as well as the Chern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by …
Paper tackles super-resolving labels for weakly labeled data.
problem Real-world data scarcity with expertly labeled data.
method Nested loop with KDE to super-resolve labels.
result KDE super-resolves labels more accurately than baselines.
There are two different notions of holonomy in supergeometry, the supergroup introduced by Galaev and our functorial approach motivated by super Wilson loops. Either theory comes with its own version of invariance of vectors and subspaces under holonomy. By our first main result, the Twofold Theorem, these definitions …
10D IIA Superspace is put on shell by imposing duality-symmetric Bianchi identities on super-flux densities.
problem Dimensional reduction of 11D supergravity to 10D IIA
method Cyclification of 11D supergravity
result Full 10D IIA supergravity is put on shell with duality-symmetric Bianchi identities.
The paper computes group factors and properties of Wilson loops in Chern-Simons theory.
problem Computing group factors and properties of Wilson loops in Chern-Simons theory.
method Developed a method for computing group factors of the perturbative series expansion of Wilson loops.
result Provided a combinatorial description of group factors with clear dependence on rank and representation.
Study super cluster algebras from super Plücker and Ptolemy relations.
problem Developing super cluster algebra structure in super Grassmannians.
method Analyzing super Plücker and Ptolemy relations, developing super cluster structure.
result New simple form of super Plücker relations for $\Gr_{r|1}(n|1)$.
Establishes higher T-duality for super M-branes.
problem Generalizing T-duality for super p-branes.
method Super L-infinity-algebraic T-duality for super WZW-terms.
result Spherical T-duality of super M5-branes.
Revises super unitary representations by relaxing Hilbert space definition.
problem Left-regular representations of super Lie groups are not super unitary.
method Weaken super Hilbert space definition and introduce a new metric.
result Left-regular representations of all super Lie groups are super unitary.
The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined W2-curvature tensor, computed curvature tensors and Ricci tensors, and studied curvature flatness. result Mixed Ricci-flat super twisted product semi-Riemannian manifolds can be expressed as super warped product manifolds.
Unified super-symmetry and higher fluxes using super-Lie-infinity algebras.
problem Unified extended super-symmetry and higher flux densities.
method Using super-Lie-infinity algebras and their extensions and cyclifications.
result Derivation of topological T-duality laws from super-Lie-infinity structure.
The paper studies super metrics and their local isometry groups.
problem Understanding the structure of super metrics and their isometry groups.
method Deriving canonical forms, computing covering groups, and applying Rogers' Theorem.
result For certain super metrics, the simply connected covering groups are super Lie groups with conventional super Lie algebras.
Researchers create holographic super-embeddings for M5 and M2 branes.
problem No concrete examples of super-embeddings for M5 and M2 branes existed.
method Constructed explicit holographic super-embeddings of probe M5 and M2 branes into their super-AdS backgrounds.
result Explicit holographic super-embeddings of M5 and M2 branes were successfully constructed.
Develops Riemannian geometry for noncommutative super surfaces.
problem No specific problem stated; focuses on mathematical development.
method Introduces metric and connections on noncommutative super surfaces, showing compatibility and zero torsion under certain conditions.
result Noncommutative super surfaces have a well-defined Riemannian geometry with properties analogous to classical Riemannian geometry.
Defines super stable maps and proves quotient superorbifolds for genus zero.
problem Defines stable supercurves and super stable maps of genus zero.
method Uses labeled trees and slice theorem for super Lie groups.
result Proves moduli space of stable supercurves and super stable maps are quotient superorbifolds.
Defines semi-symmetric metric connection on super warped products.
problem Computing curvature and Ricci tensors on super warped products.
method Introduced semi-symmetric metric connection and conditions for Einstein spaces.
result Conditions for super warped product spaces to be Einstein with semi-symmetric metric connection.
Authors derive McShane identity for super tori.
problem Deriving McShane identity for super tori.
method Super Teichmüller theory and supergeometry.
result Asymptotic growth rate of length spectra established.
Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short intro…
New theorem disproves Angle Defect for super triangles.
problem Angle Defect Theorem for N=1 super hyperbolic geometry.
method Action of OSp(1|2) on real super Minkowski space and brute-force computation.
result Disproves Angle Defect Theorem and provides novel additive function.
Constructs the moduli space of super J-holomorphic curves.
problem Defines and constructs the moduli space of super J-holomorphic curves.
method Uses component fields of a super differential equation and a transversality argument.
result Constructs the moduli space of super J-holomorphic curves as a smooth subsupermanifold.
The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of no…
The paper studies special warped products with a specific connection on super Riemannian manifolds.
problem Investigating curvature and Ricci tensors on super warped product spaces with a semi-symmetric non-metric connection.
method Defined a semi-symmetric non-metric connection, computed curvature and Ricci tensors, and introduced and analyzed two types of super warped product spaces.
result Conditions for two super warped product spaces with a semi-symmetric non-metric connection to be Einstein spaces are provided.
Super tau-covers extend bihamiltonian hierarchies' symmetries.
problem Extending symmetries of bihamiltonian hierarchies.
method Constructing super tau-covers for bihamiltonian integrable hierarchies.
result Symmetries of bihamiltonian hierarchies extended to super tau-covers.
A super Lie group is a group whose operations are G∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of …
Defines a super analog of Plücker embedding for Grassmannian.
problem Generalizing Plücker embedding to super vector spaces with mixed parity.
method Combines super vector space and its parity-reversion to construct embedding.
result Proves the embedding is an isomorphism to a weighted projective space.
We show how to construct an N=1 superconformal vertex algebra (SCVA) from any Riemannian manifold. When the Riemannian manifold has special holonomy groups, we discuss the extended supersymmetry. When the manifold is complex or Kähler, we also generalize the construction to obtain N=2 SCVA's. We study the BRST cohomolo…
Unified solution to M-theory problems using super-exceptional geometry.
problem M-theory's Lagrangian construction, gauge field origin, and supersymmetric enhancement.
method Combining homotopy theory, supergeometry, and super-homotopy theory.
result Unified description of M-theory Lagrangians and emergence of gauge fields.
The paper develops a theory of C∞-superrings and their superschemes.
problem Developing a theory for C∞-superrings and superschemes. method Proving an equivalence between categories of fair affine C∞-superschemes and fair C∞-superrings. result A key equivalence between fair affine C∞-superschemes and fair C∞-superrings. Scheme resolves super-brane topology via equivariant structures.
problem Topological issues in super-brane dynamics.
method Geometric scheme for Green-Schwarz cocycles, equivariant structures.
result Equivariant structures resolve super-brane topology.
Characterizes super-replication prices in a financial market model.
problem Characterizing prices in a financial market model.
method Characterizes prices as the supremum of mono-prior super-replication prices through extreme priors and martingale measures.
result Super-replication prices are the supremum of mono-prior super-replication prices.
Defines super projective modules and explores their properties.
problem Exploring the geometric-algebraic link in super geometry.
method Defined and explored super projective modules over supersmooth functions.
result Module of vector fields over a supersphere is a super projective module.
This article provides a brief discussion of the functional of super Riemann surfaces from the point of view of classical (i.e. not "super-) differential geometry. The discussion is based on symmetry considerations and aims to clarify the "borderline" between classical and super differential geometry with respect to the…
Matrix formulas for super Teichmüller spaces generalize previous work and yield super λ-lengths.
problem Calculating super λ-lengths on bordered surfaces with marked points.
method Using holonomy matrices of elements in the supergroup OSp(1|2) to compute super λ-lengths in decorated super Teichmüller spaces.
result Matrix formulas for arcs on bordered surfaces yield super λ-lengths in Penner-Zeitlin's decorated super Teichmüller space.
Extends Wigner's representation to study super hyperbolic geometry.
problem Understanding geometry in super hyperbolic three-space.
method Extended Wigner's representation of the Lorentz group to OSp_C(1|2) and applied to Minkowski (3,1|4)-dimensional super space.
result Proof of divergence of the volume of a typical ideal tetrahedron in super hyperbolic three-space.
Extends differential geometry concepts to manifolds with super tangent bundles.
problem No specific problem stated; extending differential geometry to super tangent bundles.
method Introduces super tangent bundle and extends differential geometry concepts.
result Basic notions of differential geometry extended to manifolds with super tangent bundles.
Paper constructs super integrable systems on color Lie algebra.
problem Super integrable systems on color Lie algebra.
method Using non-isospectral problems with matrices from color Lie algebra sp1(6), constructing (1+1)- and (2+1)-dimensional systems. result Super integrable systems and their Hamiltonian structures constructed on color Lie algebra sp1(6). Note on instabilities in super-time-stepping methods for Heston model.
problem Instabilities in super-time-stepping methods applied to Heston model.
method Exploration of explicit super-time-stepping schemes (RK-Chebyshev, RK-Legendre) for Heston model.
result Relevance of stability remarks beyond super-time-stepping schemes.
Super-convergence allows neural nets to train faster with large learning rates.
problem Training neural networks too slowly.
method Training with large learning rates and one learning rate cycle.
result Neural networks can be trained an order of magnitude faster.
The super or Z_2-graded Schouten-Nijenhuis bracket is introduced. Using it, new generalized super-Poisson structures are found which are given in terms of certain graded-skew-symmetric contravariant tensors Λof even order. The corresponding super `Jacobi identities' are expressed by stating that these tensors have zero…
The hidden M-algebra is integrated into a super-Lie group, allowing for compactification of extra dimensions.
problem Integrating the hidden M-algebra into a super-Lie group to model super-exceptional spacetimes.
method Left-invariant extension of the decomposed M-theory 3-form, providing a computer-checked re-derivation and streamlined conception of super-Lie groups.
result Lattice subgroups of the hidden M-group allow toroidal compactification of hidden dimensions, akin to topological T-duality.
We review a construction of a new class of algebraic curves, called super-A-polynomials, and their quantum generalizations. The super-A-polynomial is a two-parameter deformation of the A-polynomial known from knot theory or Chern-Simons theory with SL(2,C) gauge group. The two parameters of the super-A-polynomial encod…
Extends super-replication theorem with dynamic strategies and transaction costs.
problem Dynamic super-replication under proportional transaction costs.
method Generalizes admissible strategies and defines a well-defined super-replication price process.
result Well-defined super-replication price process in dynamic setting.
Study on super-Toda system on Riemann surfaces, analyzing solutions with four types of bubbling.
problem Analyzing solutions to the super-Toda system on Riemann surfaces.
method Blow-up analysis, energy gap results for spinor parts of solutions.
result Showed energy identities for spinor parts of a blow-up sequence of solutions.