New theory captures framing anomaly in gauge theory.
problem Capturing framing anomaly in gauge theory.
method Constructs a relative Crane-Yetter theory from non-semisimple data.
result Establishes invertibility property for the theory.
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48 results for “Crane-Yetter theory”
Proves Witten-Reshetikhin-Turaev 3-TQFT as a boundary condition of Crane-Yetter 4-TQFT.
problem Proving a boundary condition for Crane-Yetter 4-TQFT.
method Extending ideas of Crane-Yetter and Jordan, proving Crane-Yetter 4-TQFT and its non-semisimple version are once-extended TQFTs, defining a boundary condition.
result Reconstructs Witten-Reshetikhin-Turaev 3-TQFT and its non-semisimple versions using Crane-Yetter 4-TQFT.
We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. In the case of the three-dimensional invariant, we prove a duality formula relating …
Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds.They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. I…
A general state-sum construction of 2-dimensional topological quantum field theories with defectsmath.QA
We derive the general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2-D TQFTs without defects (cf. Lauda \& Pfeiffer \cite{LP}). From the extended Pachner moves (Crane \& …
Semisimple 4D field theories can't distinguish smooth 4-manifolds.
problem Detecting exotic smooth structures in 4-manifolds.
method Proving field theories lead to stable invariants, distinguishing only homeomorphic and homotopy equivalent manifolds.
result Semisimple 4D field theories can't distinguish homotopy equivalent 4-manifolds.
New (3+1) TQFTs created from non-semisimple categories.
problem Creating TQFTs from non-semisimple ribbon categories.
method Using skein theory and admissible skein modules, defining TQFTs with specific algebraic conditions.
result Explicit realization of a TQFT based on the cobordism hypothesis.
A universal state summath.QA
Defines a universal state sum construction for various TQFTs.
problem No specific problem stated; universal construction for TQFTs.
method Defines a universal state sum construction using n-categories with specific conditions.
result Produces state sums from n-categories and handle decompositions of n+1-manifolds.
We construct a state-sum type invariant of smooth closed oriented -manifolds out of a -crossed braided spherical fusion category (-BSFC) for a finite group. The construction can be extended to obtain a -dimensional topological quantum field theory (TQFT). The invariant of -manifolds generalizes s…
This paper develops 4-manifold invariants using Hopf algebras.
problem Creating 4-manifold invariants from Hopf algebras.
method Using Hopf triplets and trisection diagrams, the authors construct 4-manifold invariants.
result Every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds.
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.
Survey of Floer theories and their connections.
problem None explicitly stated; focuses on surveying theories.
method None explicitly stated; focuses on surveying theories.
result None explicitly stated; focuses on surveying theories.
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
The paper defines strong emergence in field theories and proves it exists between certain theories.
problem Defining and proving the existence of strong emergence phenomena between field theories.
method Formal definition and sufficient conditions for emergence, proving existence in Euclidean background.
result Strong emergence exists between certain parameterized Lagrangian field theories.
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.
Unified Higgs bundle vacua from M-theory on Spin(7) spaces.
problem Unifying Higgs bundle vacua from different string compactifications.
method Developed formalism for M-theory on local Spin(7) spaces and constructed explicit solutions.
result Unified 3D effective field theory from 4D M- and F-theory vacua.
Researchers find new -conifolds in -theory with potential field theory duals.
problem Exploring the field theory interpretation of -theory -conifolds.
method Constructing -holonomy orbifolds from circle bundles over Calabi-Yau cones.
result Many UV perturbative gauge theories have an infrared dual described by smooth -holonomy backgrounds in -theory.
We survey three different ways in which K-theory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted K-theory, and we illustrate with some finite models. Part 2 is a review of pfaffians of Dirac operators, anomalies, a…
Distributivity in algebraic structures appeared in many contexts such as in quasigroup theory, semigroup theory and algebraic knot theory. In this paper we give a survey of distributivity in quasigroup theory and in quandle theory.
Survey on algebraic K- and L-theory conjecture.
problem Algebraic K- and L-theory of groups rings.
method Not specified in the abstract, likely involves algebraic and geometric approaches.
result Applications to algebra, geometry, group theory, and topology.
Study pin manifolds using Clifford linear Dirac operator and KO-theory.
problem Index theory on Pin manifolds.
method Clifford linear Dirac operator and differential KO-theory.
result Systematic treatment of index theory on Pin manifolds.
Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes rema…
In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one term and two term (rack) homology theori…
Quantum field theory uses Lorentzian bordisms to describe time evolution.
problem Describing the time evolution of quantum field theories.
method Defines a functorial field theory on Lorentzian bordism pseudo-category.
result Lorentzian bordisms naturally arise in algebraic quantum field theory.
This thesis proposes a global geometric formulation of Extended Field Theories.
problem Global understanding of Extended Field Theories remains an open problem.
method Introducing an atlas for the principal infinity-bundle, unifying metric and higher gauge field.
result Global abelian T-duality and Poisson-Lie T-duality are automatically recovered.
This is the revised version of the second 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. Some proofs have been improved and …
The paper quantizes hybrid topological-holomorphic field theories on .
problem Quantizing hybrid topological-holomorphic field theories rigorously.
method Constructing perturbative, one-loop quantizations on .
result The one-loop obstruction to quantization vanishes when .
3D dual field theories for Virasoro minimal models constructed using Seifert fiber spaces.
problem Constructing 3D dual field theories for Virasoro minimal models.
method 3D-3D correspondence and Seifert fiber spaces.
result 3D dual field theories constructed for Virasoro minimal models.
We propose a new partially topological theory in three dimensions which couples Chern-Simons theory to matter. The 3-manifolds needed for this construction admit transverse holomorphic foliation (THF). The theory depends only on the choice of such a structure, but not on a choice of metric and in this sense, it is topo…
We show that Chern-Simons gauge theory with appropriate cutoffs is equivalent, term by term in perturbation theory, to a Fermionic theory with a nonlocal interaction term. When an additional cutoff is placed on the Fermi fields, this Fermionic theory gives rise to a convergent perturbation expansion. This leads us to c…
From Euclidean field theory to hyperkähler Floer theory via regularized polysymplectic geometrymath.SG
Abstract: Linking field theory to Floer theory via regularization.
problem Finding periodic solutions of Hamilton's equation.
method Regularization scheme for polysymplectic formalism linking Euclidean field theory to hyperkähler Floer theory.
result Proved a cuplength estimate.
Unified treatment of gauge theories and Yang-Mills theory duality.
problem Unified treatment of gauge theories and Yang-Mills theory duality.
method Cohomological localization techniques and Atiyah-Singer index theorem.
result Unified framework and simplified derivations of localization formulas.
New theory connects string theory to swampland distance conjecture.
problem Connecting string theory to swampland distance conjecture.
method Deformations of the heterotic superpotential, treating separately for large fluxes or large distances, integrating out fields to obtain a new field theory.
result New holomorphic theory defined, connects to swampland distance conjecture.
Identifies all perturbative vacua in bosonic string theory.
problem Identifying all perturbative vacua in bosonic string theory.
method Completely identified perturbative vacua through string fluctuations.
result Derivation of path-integrals up to any order from fluctuations.
In this short note we show how Dubrovin's integrable hierarchies, defined using the Gromov-Witten theory of a closed symplectic manifold, generalizes to Hamiltonian Floer theory. In particular, we show how the required generalization of the PSS isomorphism, relating Gromov-Witten theory and Hamiltonian Floer theory, ca…
Introduction to gauge theorymath.DG
Lecture notes on gauge theory for manifold invariants.
problem Constructing invariants of manifolds using gauge theory.
method Gauge-theoretic approach to manifold invariants.
result Introduction to Seiberg-Witten gauge theory.
Paper introduces a new geometric homology theory and applies it to Gromov-Witten theory.
problem Developing a new homology theory for orbifolds with corners.
method Using stratification and triangulation theories of Lie groupoids and their orbit spaces, extending to Lie groupoids with corners.
result Proposes and proves the geometric homology theory (GHT), a flexible generalization of singular homology.
New geometric approach realizes 5D bulk theories with 4D edge modes.
problem Realizing novel higher-dimensional junctions of theories coupled to localized edge modes.
method M-theory on singular, asymptotically conical G2-holonomy orbifolds.
result Geometric approach shows how bulk generalized symmetries are inherited in the boundary system.
Paper reinterprets marginal productivity theory using vectorial products, challenging traditional ethical interpretations.
problem Challenges traditional ethical interpretations of marginal productivity theory.
method Formulates marginal productivity theory using vectorial marginal products, contrasting with traditional scalar approach.
result Vectorial marginal products conflict with traditional distributive shares picture of property.
Category theory generalizes finite type invariants using diagrams systems.
problem Generalizing finite type invariants using category theory.
method Relating generating sets for generalized finite type theories with diagrams systems.
result Demonstrates the correspondence between finite type theories and diagrams systems.
We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.
New theory challenges traditional machine learning assumptions.
problem Traditional machine learning theories are critiqued.
method A new theory is proposed and discussed.
result Learning true probabilities is not equivalent to other learning goals.
Abelian Chern-Simons theorymath.DG
We give a construction of the abelian Chern-Simons gauge theory from the point of view of a 2+1 dimensional topological quantum field theory. The definition of the quantum theory relies on geometric quantization ideas which have been previously explored in connection to the nonabelian Chern-Simons theory [JW,ADW]. We f…
There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field theory combined.
Quantum theory improves counting overlapping clusters.
problem Counting overlapping clusters in machine learning.
method Applied quantum theory using path integral technique.
result Quantum theory provides a robust statistical method for counting clusters.
The paper proves UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
problem Proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
method Rigorously proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories on .
result Proves vanishing anomalies for hybrid topological-holomorphic field theories, allowing for the definition of a factorization algebra structure for quantum observables.
Note on advancements in nonlinear elliptic equations' regularity theory.
problem Nonlinear elliptic equations and their regularity.
method De Giorgi-Nash-Moser theory, Krylov-Safonov theory, Evans-Safonov theory.
result Contributions to Hilbert's 19th problem and fully nonlinear equations.
The abstract discusses open problems in isoparametric theory.
problem Open problems in isoparametric theory.
method Survey and propose open problems.
result Discussion of open problems in isoparametric theory.