Area law proven for 2D Yang-Mills in generalized axials.
problem Proving area law for 2D Yang-Mills theory in generalized axials.
method Homotopy invariance of iterated integrals, complex gauge-fixing, integration cycle deformation.
result Wilson loop expectation values obey area law up to second order.
This paper compares exact 2D Yang-Mills theory with perturbative approaches.
problem Investigating the relationship between exact and perturbative approaches in 2D Yang-Mills theory.
method Mathematical rigorous formulation of perturbative quantization and comparison of Wilson loop expectation values.
result Exact lattice Wilson loop expectations on S2 agree with perturbative expectations in holomorphic gauge for simple closed curves to all orders. New 2D TQFT connects Chern-Simons theory on circle bundles to gauge theory.
problem Localization of Chern-Simons theory on circle bundles.
method Caloron correspondence and symplectic structure.
result Wilson loops on circle fibers described naturally.
We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…
5D gauge theories are dual to 3D and 2D models via Floer homologies.
problem Exploring dualities in 5D gauge theories and their 3D and 2D counterparts.
method Using Landau-Ginzburg models and Floer homologies, the paper establishes dualities between different gauge theories and their associated homologies.
result Dual A∞-categories of Floer homologies are derived, proving mirror symmetry and Langlands duality. Research constructs 2D Dijkgraaf-Witten theory with defects.
problem Modeling 2D Dijkgraaf-Witten theory with defects.
method State-sum model with triangulation and flag-like requirements, internal degrees of freedom for defects.
result Generalizations of 2-cocycles for defect-free theory, examples from group characters.
2d GLSM connects Berry connections to Coulomb branch via difference equations.
problem Connecting Berry connections to Coulomb branch via difference equations.
method Boundary 2d GLSM, 3d A-twisted gauge theory, spectral data of monopoles.
result Coulomb branch algebra actions derived from 2d GLSM.
We study 4d superconformal indices for a large class of N=1 superconformal quiver gauge theories realized combinatorially as a bipartite graph or a set of "zig-zag paths" on a two-dimensional torus T^2. An exchange of loops, which we call a "double Yang-Baxter move", gives the Seiberg duality of the gauge theory, and t…
Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.
problem Understanding symmetries and anomalies in 2D Yang-Mills theory.
method Combining continuum methods, topological defects, and higher gauge theory.
result Unified description of higher and lower form gauge fields, identifying spontaneous symmetry breaking.
Novel A∞-categories derived from gauge theories for manifold homologies.
problem Categorifying manifold homologies via gauge theories.
method 3d and 8d gauged Landau-Ginzburg models, higher A∞-categories. result Derived novel A∞-categories for various manifold homologies. The paper explores gauge theory invariants and their duals via topological-holomorphic twist.
problem Understanding gauge theory invariants and their duals in 4d and 2d.
method Topological-holomorphic twist of N=4 supersymmetric gauge theory.
result Derived novel topological and holomorphic invariants and their Langlands duals.
Study of surface defects in gauge theories leads to duality and separation of variables.
problem Understanding surface observables and their transitions in gauge theories.
method Utilized Fourier transformations and spectral problems to derive dualities and separation of variables.
result Exact duality between spectral problems of spin chains and Gaudin models.
Novel gauge-theoretic Floer homologies defined from 5d N=2 theory, linking 4, 3, and 2-manifolds.
problem Defining and linking novel Floer homologies for different manifold dimensions.
method Physics of a topologically-twisted 5d N=2 gauge theory, Vafa-Witten, Hitchin, and BF configurations.
result Derived novel gauge-theoretic and symplectic Floer homologies, and Atiyah-Floer correspondences.
Physical proof of Floer homology and Verlinde formula using N=2 gauge theory.
problem Proving mathematical conjectures in Floer homology and quantum cohomology.
method Topologically-twisted N=2 gauge theory on a four-manifold with boundary.
result Physical proofs of various Floer homology and quantum cohomology results.
Study connections on Lie and Courant algebroids, defining basic curvature and Atiyah cocycle.
problem Understanding connections on Lie and Courant algebroids and their compatibility.
method Revisit and define basic curvature for Lie algebroids, introduce basic curvature for Courant algebroids, and use Atiyah cocycle for gauge theory.
result Basic curvature tensor for Courant algebroids and its relation to the Atiyah cocycle.
T-duality rules for 2D (2,1) supersymmetric models clarified.
problem Clarify T-duality in (2,1) supersymmetric sigma-models.
method Gauging of sigma-models in (2,1) superspace, finding complexified duality transformations.
result Complexified duality transformations are equivalent to Buscher transformations and diffeomorphisms.
Reformulates compatibility condition for generalized metrics in string theory.
problem Compatibility condition between generalized metrics and Dirac structures.
method Formulated using two connections in Dirac-geometric terms.
result Transverse generalized metric needed for string theory dynamics.
Lectures detail field theory dynamics and exact WKB analysis.
problem Understanding quantum field theories with four supercharges.
method Combining WKB analysis with 2D quantum field theories.
result Partition function characterizes non-perturbative dynamics.
We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-ty…
Novel gauge-theoretic Floer homologies defined for 7, 6, and 5-manifolds.
problem Defining novel gauge-theoretic Floer homologies for different dimensions.
method Topologically-twisted 8d N=1 gauge theory on Spin(7)-manifolds. result Derivation of novel Floer homologies and A∞-categories. We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra g leads naturally to the appearance of the "generalized tangent bundle" TM≡TM⊕T∗M by means of composite fields. Gauge transformations of the composite fields foll…
Study of 2d gauged linear sigma models to derive difference equations and spectral data.
problem Understanding monopole solutions and their spectral data in 2d gauged models.
method Analyzing ground states and cohomology of supercharges to derive difference modules and equations.
result Derived novel difference equations for brane amplitudes and hemisphere partition functions.
This is the second in a series of papers discussing in the framework of gerbe theory canonical and geometric aspects of the 2d nonlinear sigma model in the presence of conformal defects in the worldsheet. Employing the formal tools worked out in the first paper of the series, 1101.1126 [hep-th], a thorough analysis of …
New K-theory approach classifies anyonic topological phases in 2D semimetals.
problem Classifying interacting topological phases remains open.
method TED K-theory of configuration spaces of points in the Brillouin torus.
result Classifies su(2)-anyonic topological order in 2D semimetals.
New method defines Wilson surface observables on arbitrary 2D surfaces.
problem Defining Wilson surface observables on arbitrary surfaces.
method Using equivariant cohomology and Poisson σ-models.
result Wilson surface observables are nontrivial for non-simply connected groups.
The algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl(2,R) [semidirect product] R^2 Lie algebra. We present a simple calculus for calculations in its universal enveloping algebra. As an application, we derive generating functions for t…
Improved lattice field theory simulations with local-Autoregressive Conditional Normalizing Flow.
problem Efficiently sampling lattice field theories with computational challenges.
method Integrates locality into autoregressive conditional normalizing flows.
result Autocorrelation times improved by orders of magnitude for φ4 theory on a 2D lattice. We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface Σ that separates two subsystems of quantum strongly coupled N=4 SU(N) superconformal gauge theory. We extend this result and calculate en…
We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riema…
We find canonical gauges for higher gauge theories in 2- and 3-gauge theories.
problem Finding good gauges for connections in higher gauge theories.
method Defined Coulomb gauges for 2- and 3-connections in strict 2- and 3-gauge theories.
result Coulomb gauges are essentially unique and linear in critical dimensions.
We study complex Chern-Simons theory on a Seifert manifold M3 by embedding it into string theory. We show that complex Chern-Simons theory on M3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the d…
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.
New discretization method for gauge theories preserves gauge invariance rigorously.
problem Discretization of continuum gauge theories with polynomial curvature.
method Holonomy-based discretization preserving gauge invariance.
result Gauge-fixing is fully rigorous for discretized Yang-Mills theories.
We propose a general notion of algebraic gauge theory obtained via extracting the main properties of classical gauge theory. Building on a recent work on transferring curved A∞-structures we show that, under certain technical conditions, algebraic gauge theories can be transferred along chain contractions. Sp…
Gauge theory for families helps compare 4-manifold groups.
problem Comparing diffeomorphism and homeomorphism groups of 4-manifolds.
method Gauge theory applied to families of 4-manifolds.
result Improved understanding of group comparisons in 4-manifolds.
We consider dimensional reduction of gauge theories with arbitrary gauge group in a formalism based on equivariant principal bundles. For the classical gauge groups we clarify the relations between equivariant principal bundles and quiver bundles, and show that the reduced quiver gauge theories are all generically buil…
Survey of gauge theory for families of 4-manifolds.
problem Understanding gauge theory for families of 4-manifolds.
method Developed gauge theory for diffeomorphism groups of 4-manifolds.
result New applications of gauge theory to diffeomorphism groups.
Perturbs gauge theories on manifolds with boundary.
problem Quantum gauge theories on manifolds with boundary.
method Cohomological symplectic (BV-BFV) formalism for a general perturbative quantization scheme.
result Explicit examples of abelian BF theory and its perturbations.
Higher gauge theory via differential nonabelian cohomology
problem Global infrared completion of higher gauge fields
method Maxwell-type higher gauge fields
result Electromagnetic flux quantization
Survey on advanced gauge theory concepts.
problem Understanding higher gauge theory structures.
method Introduction to higher structures and connections on higher principal bundles.
result Summarized applications and principles of higher gauge theories.
Defines 2-holonomy for surface knots in higher gauge theory.
problem Analyzing holonomy invariants in higher gauge theory.
method Defines and analyzes 2-holonomy for surface knots.
result Determines covariance properties of 2-holonomy.
Develops a new approach to describe gauge theories with background fields using presymplectic structures.
problem Describing gauge theories with background fields using presymplectic structures.
method Extension of the presymplectic BV-AKSZ approach to include background fields.
result Gauge theories with background fields correspond to presymplectic gauge PDEs over gauge PDEs describing background fields.
In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra v and its automorphism 2-group Aut(v). Gauge transformat…
Develops a new sampling method for gauge theories.
problem Sampling from SU(N) gauge theories. method Gauge-equivariant flows for SU(N) variables. result Constructs a class of flows respecting matrix conjugation symmetry.
Defines mathematical Coulomb branches for 3D gauge theories.
problem Mathematical definition of Coulomb branches in 3D gauge theories.
method Provisional mathematical definition based on existing work.
result Provides a new framework for understanding gauge theories.
We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which include new 3d N=2 theories T[M_3] associated with rational homology spheres and new results for Vafa-Witten partition fun…
It is argued that the enlargement of the gauge group found in non-commutative gauge theory is more fundamentally thought of as a consequence of the non-locality of the construction and that it was already encountered in an earlier discussion of a non-local gauge theory.
We study the relation between the space of representation classes of the fundamental group of a Riemann surface and gauge theory on trivalent graphs. We construct a partial gauge fixing in the latter gauge theory. As an application we get a proof of a conjecture of Florentino.