New class of singular complex manifolds studied with degenerate theory.
problem Understanding singular complex manifolds.
method Developed degenerate Kodaira-Hodge theory for new class.
result New degenerate theory for singular complex manifolds.
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48 results for “class R theories”
Following the analogies between 3-dimensional topology and number theory, we study an idèlic form of class field theory for 3-manifolds. For a certain set of knots in a 3-manifold , we first present a local theory for each knot in , which is analogous to local class field theory, and then,…
The paper explores mapping class groups and their quantum field theory representations.
problem Understanding finite dimensional representations of mapping class groups.
method Survey of topological quantum field theory aspects.
result Discussion of finite dimensional representations in quantum field theory.
We construct characteristic classes of 4-manifold bundles using -Yang-Mills theory and Seiberg-Witten theory for families.
A version of smooth K-theory is constructed, which is adapted to the total Chern class instead of the Chern character (contrarily to previous theories). Some total Chern class morphism from this K-theory to Cheeger-Simons differential characters is constructed. This answers a question raised by U. Bunke.
We identify a large class R of three-dimensional N=2 superconformal field theories. This class includes the effective theories T_M of M5-branes wrapped on 3-manifolds M, discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern-Simons-matter theories w…
Study gauged supergravity, M5-branes, and class R theories, constraining supergravity coefficients and calculating partition functions.
problem Constraints and calculations in higher-derivative supergravity and related theories.
method Holography, Chern-Simons theory, 3d-3d correspondence, wrapped M5-branes.
result Constrained coefficients in supergravity Lagrangian, calculated partition functions for class R theories.
The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.
problem Function theory on Teichmüller space and dynamics of mapping class groups.
method Utilizes Thurston's theory and Sullivan's theory on discrete subgroups of hyperbolic space.
result Establishes connections between function theory, dynamics, and ergodic theory.
Garside groups and geometrymath.GR
Invites geometers to Garside theory for mapping class groups.
problem None explicitly stated, but related to geometric group theory.
method Garside theory applied to mapping class groups.
result No specific key result mentioned in the abstract.
We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.
We study a topological analogue of idèlic class field theory for 3-manifolds, in the spirit of arithmetic topology. We firstly introduce the notion of a very admissible link in a 3-manifold , which plays a role analogous to the set of primes of a number field. For such a pair , we intr…
Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.
problem Classifying theories with eight supercharges using mathematical tools.
method Assumes theories are given by genus g fibrations of Riemann surfaces, uses pseudo-periodic maps of negative type in mapping class group.
result Identifies dual graphs and 3d mirror quivers, unifies various SCFTs in combinatorial framework.
The central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum Teichmüller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is analogous to the result obtained in \cite{FS} for the Thompson groups.
In this note we give a simple, model-independent construction of Chern classes as natural transformations from differential complex K-theory to differential integral cohomology. We verify the expected behaviour of these Chern classes with respect to sums and suspension.
Study of IR phases in 3D class R theories linked to non-hyperbolic 3-manifolds.
problem Understanding IR phases of 3D class R theories associated with non-hyperbolic 3-manifolds.
method Analysis of IR phenomena through `exceptional' Dehn fillings and gauging of flavor symmetries.
result 3D class R theories associated with certain atoroidal non-hyperbolic 3-manifolds exhibit supersymmetry enhancement at low energy.
Let X --> B be a proper submersion with a Riemannian structure. Given a differential K-theory class on X, we define its analytic and topological indices as differential K-theory classes on B. We prove that the two indices are the same.
Study algebraic K-theory for specific groups of non-orientable surfaces.
problem Algebraic K-theory of group rings for specific non-orientable surface groups.
method Detailed analysis of group rings and algebraic K-theory.
result General formula for algebraic K-theory groups of mapping class groups of non-orientable surfaces.
Global EQG sums boundary states over manifold diffeomorphism classes.
problem Summing boundary states over manifold diffeomorphism classes.
method Formulated as classical statistical physics, weights determined by general principles.
result Hartle-Hawking state as a probability measure.
This paper contains the constructions of a real manifold version of relative K-theory, and of an extension of Karoubi's multiplicative K-theory suggested by U. Bunke (which I call ``free multiplicative K-theory'' in the sequel). Chern-Simons-Nadel type classes on relative K-theory are constructed, while it is proved th…
Develops noncommutative Cowen-Douglas theory for noncommuting operators.
problem Exploring noncommutative analogues of classical Cowen-Douglas theory.
method Defining noncommutative Cowen-Douglas class using matricial joint eigenvalues and showing equivalence classes are determined by associated noncommutative vector bundles.
result Unitary equivalence class of a tuple in the noncommutative Cowen-Douglas class is determined by the equivalence class of its associated noncommutative vector bundle.
Machine Learning has become very famous currently which assist in identifying the patterns from the raw data. Technological advancement has led to substantial improvement in Machine Learning which, thus helping to improve prediction. Current Machine Learning models are based on Classical Theory, which can be replaced b…
Topological twists for 4d N=2 theories depend on spacetime type, gerbe connections, and generalized spin-c structures.
problem Defining topologically twisted partition functions for 4d N=2 theories.
method Topological twisting for general 4d N=2 theories, introducing generalized spin-c structures.
result Topological partition functions depend on spacetime type, gerbe connections, and generalized spin-c structures.
The Chern classes of a K-theory class which is represented by a vector bundle with connection admit refinements to Cheeger-Simons classes in Deligne cohomology. In the present paper we consider similar refinements in the case where the classes in K-theory are represented by geometric families of Dirac operators. In low…
In this paper we give explicit formulas of differential characteristic classes of principal -bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and differential Euler class. Furthermore, we show that…
The study proves a generic multiplicity one theorem for -invariant minimal hypersurfaces.
problem Proving a generic multiplicity one theorem for -invariant minimal hypersurfaces.
method Equivariant min-max theory and analysis of -homology classes.
result Shows a generic multiplicity one theorem for -invariant minimal hypersurfaces.
Develops trace class operators and inverse Laplacian theory for infinite dimensions.
problem Understanding trace class operators and inverse Laplacian on infinite dimensional spaces.
method Presentation of trace class operators and construction of inverse Laplacian on closed manifolds.
result Original trace computations involving the inverse Laplacian on the torus.
Develops formal moduli theory for splitting complex supermanifolds.
problem Tackles the splitting problem of complex supermanifolds.
method Constructs a filtered dg Lie algebra to control splittings and transfers the theory to a minimal filtered -model.
result Recover classical obstruction classes as leading terms of Maurer-Cartan representatives and proves the existence of higher obstructions.
The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate local symmetries. The whole package provides a functional analytic framework to de…
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
Study approximates top Lyapunov exponents for surface mapping classes.
problem Approximating topological Lyapunov exponents for surface mapping classes.
method Periodic approximation and joint spectral radius extension.
result Top Lyapunov exponents can be approximated by periodic orbits.
The paper extends Euler class theory to measurable cocycles.
problem Understanding the structure of measurable cocycles and their cohomology.
method Constructing a parametrized Euler class in bounded cohomology and studying semicohomologous cocycles.
result The parametrized Euler class vanishes if and only if the cocycle can be lifted and admits an equivariant family of points.
In this paper, we introduce the notion of modular class of a Lie algebroid equipped with a Nambu structure satisfying some suitable hypothesis. We also introduce cohomology and homology theories for such Lie algebroids and prove that these theories are connected by a duality isomorphism when the modular class is nu…
Simplified Milnor-Schwarz lemma for geometric group theory.
problem Conditions for orbit maps to be quasi-isometries.
method Succinct treatment and applications to non-Archimedean groups.
result Sharpened results on mapping class groups and quasi-isometry classification.
We study gravity duals to a broad class of N=2 supersymmetric gauge theories defined on a general class of three-manifold geometries. The gravity backgrounds are based on Euclidean self-dual solutions to four-dimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions d…
The paper develops a statistical theory explaining overfitting in imbalanced classification.
problem Overfitting in high-dimensional imbalanced classification.
method Developed a statistical theory for support vector machines and logistic regression.
result Overfitting is more severe for the minority class due to truncation or skewing effects in high-dimensional data.
On Perrot's index cocyclesmath.DG
Simplified construction recovers Todd class using algebraic methods.
problem Recovering Todd class from a JLO-type cocycle.
method Algebraic framework, Dirac operator, new trace.
result Recovery of Todd class using novel methods.
Study finds modular classes help in proving Berezin volumes for supersymmetric theories.
problem Existence of Berezin volumes in supergeometric representation theory.
method Cohomological coherence criterion for modular classes of Q-manifolds.
result Established a method to prove the existence of invariant Berezin volumes.
Generative models characterized through learning theory.
problem Characterizing generative models using learning theory.
method Formalized Gold, Angluin, and Kleinberg's results; introduced uniform and non-uniform generation; characterized closure dimension.
result Incompatibility between generatability and predictability for certain hypothesis classes.
New algebraic theory classifies symplectic curves in complex projective space.
problem Classifying symplectic curves with specific singularities.
method Developed a novel algebraic theory of positive braids and conjugacy classes in the braid group.
result Established a complete classification of isotopy classes of degree three symplectic curves with -singularities.
Quantum map counts BPS states in special theories.
problem Computing protected spin characters in class S theories.
method Geometric approach from 5D supersymmetric Yang-Mills theory.
result Explicit computation of protected spin characters in various examples.
We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example,…
New proof links initial class bias to DNN trainability, challenging traditional understanding.
problem Understanding the initial class bias in DNNs and its impact on trainability.
method Theoretical proof linking initial class bias to mean field theories of DNNs.
result Efficient learning is connected to a network's prejudice towards a specific class, contradicting traditional understanding.
This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory.
Novel link classification connects quadratic forms and knot theory.
problem Classifying isotopy classes of links in 3D space.
method Established a correspondence between quadratic forms and isotopy classes of links.
result Class numbers of quadratic number fields measure link distinguishability.
Study virtual fundamental classes of derived manifolds, proving invariant vanishes.
problem Computing virtual fundamental classes for derived manifolds.
method Combining derived differential geometry and cosection localization.
result Stable pair invariants of hyperkähler fourfolds are zero.
Unified view of spectral networks linking geometry and gauge theory.
problem Understanding BPS states in gauge theories.
method Unified geometric and physical approaches, focusing on spectral networks.
result Spectral networks provide a framework for determining BPS spectra.
Study fractional structures on bundle gerbe modules using rational homotopy theory.
problem Understanding twisted Chern classes of torsion bundle gerbe modules.
method Sullivan's rational homotopy theory to realize twisted Chern classes at the level of classifying spaces.
result Introduction of fractional U-structures as a universal framework.
Researchers construct an index map for contact manifolds using K-theory.
problem Constructing an index for maximally hypoelliptic operators on contact manifolds.
method Using Higson's construction for symbol class in K-theory, they derive a series of maps whose induced map in K-theory is the Heisenberg Atiyah-Singer index map.
result Explicit construction of a series of maps leading to the Heisenberg Atiyah-Singer index map.