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.
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.
We prove that semisimple 4-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth 4-manifolds and homotopy equivalent simply connected closed oriented smooth 4-manifolds. We show that all currently known 4-dimensiona…
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.
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.
Extends field theory foundations to infinitesimal spaces, simplifying complex concepts.
problem Develop rigorous foundations for field theory, especially for infinitesimal spaces.
method Formulates local Lagrangian field theory in a new category of thickened smooth sets.
result Establishes a firm foundation for field theory, including tangent bundles and perturbative considerations.
The paper quantizes hybrid topological-holomorphic field theories on RmimesCn.
problem Quantizing hybrid topological-holomorphic field theories rigorously.
method Constructing perturbative, one-loop quantizations on RmimesCn. result The one-loop obstruction to quantization vanishes when m≥1. The paper reviews a correspondence between Double Field Theory and bundle gerbes.
problem Exploring a geometric interpretation of Double Field Theory.
method Interpreting Double Field Theory as a field theory on the total space of bundle gerbes.
result Double Field Theory can be seen as a higher geometric field theory.
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.
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 Rd′imesCd. result Proves vanishing anomalies for hybrid topological-holomorphic field theories, allowing for the definition of a factorization algebra structure for quantum observables.
Paper introduces Floer theory for field theories, proving periodic solutions for particle-field systems.
problem Defining Hamiltonian Floer theory for covariant field theories, especially those with degenerate action functionals.
method Regularization procedure to handle degeneracy, leading to Floer curves that converge to periodic solutions.
result Existence of Floer curves and space-time periodic solutions for coupled particle-field systems.
Geometrically describes Jacobi equations for field theories with dissipation.
problem Describing field theories with dissipation geometrically.
method Prolongation of the Lagrangian on a k-cosymplectic formulation to describe Jacobi equations and a modified Lagrangian for variational formulation.
result Variational formulation of field theories with dissipation.
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive struct…
Derives localization formulas in Batalin-Vilkovisky formalism.
problem Localization in Batalin-Vilkovisky formalism.
method Equivariant localization formulas in Batalin-Vilkovisky formalism.
result Derives localization formulas in Batalin-Vilkovisky formalism.
The paper axiomatizes strong emergence in parameterized field theories and proves existence theorems.
problem Formalizing and proving existence of strong emergence in parameterized field theories.
method Axiomatization and proof of existence theorems for strong emergence between Lagrangian field theories.
result Existence of strong emergence phenomena between parameterized Lagrangian field theories.
Machine learning explores symmetries in field theory and algebra.
problem Understanding symmetries in field theory and algebra.
method Using neural networks to analyze conformal field theory and Lie algebra representation theory.
result Recent advances in machine learning have uncovered new symmetries.
Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.
problem Analyzes non-conservative field theories, especially dissipative systems.
method Introduces evolution k-contact k-vector fields and develops two Hamilton-Jacobi theories.
result Recover ordinary contact Hamilton-Jacobi theory as k=1, and enlarges application range.
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 consider the hypothesis that the C-field 4-flux and 7-flux forms in M-theory are in the image of the non-abelian Chern character map from the non-abelian generalized cohomology theory called J-twisted Cohomotopy theory. We prove for M2-brane backgrounds in M-theory on 8-manifolds that such charge quantization of the…
New geometric framework for non-conservative field theories with time-dependent terms.
problem Describing non-conservative field theories with explicit space-time dependence.
method Combining k-cosymplectic and k-contact formulations to develop Hamiltonian and Lagrangian formalisms.
result Illustrated with the nonlinear damped wave equation, demonstrating the new formalism's applicability.
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 K of knots in a 3-manifold M, we first present a local theory for each knot in K, which is analogous to local class field theory, and then,…
Reduces field theories using Poisson-Poincaré method.
problem Reduction of field theories using Poisson-Poincaré method.
method Poisson-Poincaré reduction for field theories.
result Reduction procedure for field theories.
Develops a new geometric framework for non-conservative field theories.
problem Non-conservative field theories in classical physics.
method Multisymplectic and contact geometries, variational field equations, jet bundle description.
result Introduces variational field equations in multicontact manifolds.
Non-linear image reconstruction and signal analysis deal with complex inverse problems. To tackle such problems in a systematic way, I present information field theory (IFT) as a means of Bayesian, data based inference on spatially distributed signal fields. IFT is a statistical field theory, which permits the construc…
Statistical field theory aids in understanding deep learning complexities.
problem Complexity and lack of theoretical understanding in deep learning.
method Statistical field theory as a theoretical framework.
result Field theory provides insights into generalization, bias, and feature learning.
Analyzes Poisson structures on solution spaces of Hamiltonian field theories.
problem Defining Poisson bracket structures on solution spaces of first order Hamiltonian field theories.
method Examines mechanical point systems and field theories without gauge symmetries, introduces symplectic structures; for gauge theory, free electrodynamics, a pre-symplectic tensor is used to induce a Poisson structure.
result Existence of Poisson structures on solution spaces of Hamiltonian field theories, including free electrodynamics.
Defines new canonical lifts for field theories, analyzing Klein-Gordon, Polyakov string, and Einstein-Cartan gravity.
problem Analyzing natural Noether symmetries and conserved quantities in field theories.
method Defining canonical lifts to study field theories and applying Noether's theorem.
result New geometrical interpretation of Virasoro constraint in string theory.
In the paper [4] is presented a theory which unifies the gravitation theory and the mechanical effects, which is different from the Riemannian theories like GTR. Moreover it is built in the style of the electomagnetic field theory. This paper is a continuation of [4] such that the complex variant of that theory yields …
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…
We describe the elements of a novel structural approach to classical field theory, inspired by recent developments in perturbative algebraic quantum field theory. This approach is local and focuses mainly on the observables over field configurations, given by certain spaces of functionals which are studied here in dept…
Develops Marsden-Meyer-Weinstein reduction for k-contact field theories.
problem None explicitly stated, but related to field theories and contact geometry.
method Marsden-Meyer-Weinstein reduction techniques applied to k-contact field theories. result Clarifies and corrects previous contact reduction literature.
Constructs maps from field theories to complexified K-theory and elliptic cohomology.
problem Identifying geometric models for Chern characters in supersymmetric field theories.
method Higher-dimensional generalization of Fei Han's method, involving super moduli spaces and derived geometry.
result Provides evidence for the Stolz--Teichner program and geometric models for Chern characters.
Finite presentations for skein algebras linked to gauge field theory.
problem Understanding finite presentations for skein algebras and their relationship to gauge field theory.
method Provided finite presentations and deduced properties of stated skein algebras.
result Stated skein algebras are Koszul and isomorphic to quantum moduli algebras in gauge field theory.
Develops a new framework for anomaly description in quantum field theories.
problem Anomalies in quantum field theories.
method Extended functorial field theories and symmetric monoidal bicategories.
result Explicit construction of anomaly 2-cocycles and 't Hooft anomalies.
Topological conformal field theories are defined using only basic results from the theory of quasiconformal mappings.
We extend the topological field theory (``itsy bitsy topological field theory"') of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on …
In a previous paper, we have shown that the geometry of double field theory has a natural interpretation on flat para-Kähler manifolds. In this paper, we show that the same geometric constructions can be made on any para-Hermitian manifold. The field is interpreted as a compatible (pseudo-)Riemannian metric. The tangen…
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.
Given a family of Dirac operators with vanishing spectral flow we construct a thin-invariant rank-one field theory in the sense of Turner and Willerton arXiv:math.AT/0201116. Our construction of the field theory generalizes the one of the index gerbe by Lott, arXiv:math.DG/0106177, and it also complements the relation …
A quantum field theory generalization, Baaquie, of the Heath, Jarrow, and Morton (HJM) term structure model parsimoniously describes the evolution of imperfectly correlated forward rates. Field theory also offers powerful computational tools to compute path integrals which naturally arise from all forward rate models. …
Simplified neural network EFTs reveal a single critical condition.
problem Understanding neuron statistics in neural networks at initialization.
method Diagrammatic approach to effective field theories (EFTs).
result A single condition governs criticality of all neuron preactivations.
Derives stress-energy identities in Liouville theory on compact surfaces.
problem Stress-energy tensor correlation functions on compact Riemann surfaces.
method Varying correlation functions with respect to background metric, treating different types of variations separately.
result Stress-energy correlation functions expressed as differential operators acting on primary field correlation functions.
Study General Relativity using field theories and Poisson brackets.
problem Defining a Poisson bracket structure on solution spaces of field theories.
method Applying Poisson bracket structure to first order Hamiltonian field theories, focusing on General Relativity as a gauge theory.
result Established a Poisson bracket structure for General Relativity.
TKFT models computation via smooth vector fields, simulating functions in a single dynamical step.
problem Modeling computation in a single step.
method Established Topological Kleene Field Theory (TKFT) as a new model of computation.
result Any computable function can be simulated in a single go of a dynamical system.
Renormalization in neural networks linked to quantum field theory.
problem Implementing renormalization in neural networks.
method Mapping neural networks to quantum field theory, applying renormalization techniques.
result Changing weight standard deviation corresponds to a renormalization flow.
In a recent formulation of a quantum field theory of forward rates, the volatility of the forward rates was taken to be deterministic. The field theory of the forward rates is generalized to the case of stochastic volatility. Two cases are analyzed, firstly when volatility is taken to be a function of the forward rates…
I briefly review my proposal about how to extend the geometric Hamilton-Jacobi theory to higher derivative field theories on fiber bundles.
Paper studies flows of spinor fields with flux for unified theories.
problem Existence of covariantly constant spinors in unified theories.
method Introduces parabolic flows of spinor fields to find stationary points.
result Establishes short-time existence and smoothing estimates for spinor flows.