arXiv research

We explore homotopies in quantum field theory formalism.

Classifies two-dimensional extended homotopy field theories with aspherical targets.

Characterizes a specific type of Courant algebroid with a Calabi-Yau structure.

Given a discrete group G and a spherical G-fusion category whose neutral component has invertible dimension, we use the state-sum method to construct a 3-dimensional Homotopy Quantum Field Theory (HQFT) with target the Eilenberg-MacLane space K(G,1).

3D HQFTs constructed using graded monoidal categories.

This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of 'quantum invariants' of manifolds inspired by ideas originating from physics.…

A 3-dimensional homotopy quantum field theory (HQFT) can be described as a TQFT for surfaces and 3-cobordisms endowed with homotopy classes of maps into a given space. For a group $π$, we introduce a notion of a modular crossed $π$-category and show that such a category gives rise to a 3-dimensional HQFT with target sp…

We review the relation between homotopy algebras of conformal field theory and geometric structures arising in sigma models. In particular we formulate conformal invariance conditions, which in the quasi-classical limit are Einstein equations with extra fields, as generalized Maurer-Cartan equations.

In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinement…

Defines discrete differential geometry concepts in homotopy type theory.

Semisimple 4D field theories can't distinguish smooth 4-manifolds.

Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.

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 …

Introduces a new geometric framework for field theories.

Develops homotopies for Lagrangian field theory using advanced algebraic structures.

We study a variation of Turaev's homotopy quantum field theories using 2-categories of surfaces. We define the homotopy surface 2-category of a space $X$ and define an $\cS_X$-structure to be a monoidal 2-functor from this to the 2-category of idempotent-complete additive $k$-linear categories. We initiate the study of…

Study homotopy sheaves on categories and their presheaves, proving descent properties.

New examples of manifolds that are homotopy but not simple homotopy equivalent.

Graph potentials link to topological QFTs, with computational methods.

Higher gauge theory via differential nonabelian cohomology

We define a model for the homology of manifolds and use it to describe the intersection product on the homology of compact oriented manifolds and to define homological quantum field theories which generalizes the notions of string topology introduced by Chas and Sullivan and homotopy quantum field theories introduced b…

Study proves existence of multiple geodesics in a specific metric space.

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a…

Study of profinite quandles with constructions and characterizations.

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations -- such as the so-called thin homotopy -- and the resulting interp…

Motivated by the study of the interrelation between functorial and algebraic quantum field theory, we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of the associated representations in cohomol- ogy. Furthermore, we observe that the …

Introduces quadratic linking degree in algebraic geometry.

Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.

Study the topology of stable vector fields and Lyapunov functions on R^n.

New mathematical framework connects M-theory charges to stable homotopy groups.

In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining technique…

Unified classification of equivariant principal bundles using higher homotopy theory.

The paper explores how fields in higher dimensions are quantized.

A key open problem in M-theory is the mechanism of "gauge enhancement", which supposedly makes M-branes exhibit the nonabelian gauge degrees of freedom that are seen perturbatively in the limit of 10d string theory. In fact, since only the twisted K-theory classes represented by nonabelian Chan-Paton gauge fields on D-…

We formalize higher dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for s…

Starting from a higher Courant bracket associated to exceptional generalized geometry, we provide a systematic derivation of all types of fluxes and their Bianchi identities for four-dimensional compactifications of M-theory. We show that these fluxes may be understood as generalized Wess-Zumino terms in certain topolo…

Homotopy momentum map extends Noether's theorem in general relativity.

We consider maps into Riemannian manifolds of non-positive curvature and start developing a systematic PDE theory. We control the Sobolev $H^{2,2}$-norm of such a map in terms of its energy, the $L^2$-norm of its tension field and a topological term depending on the homotopy class. We also solve a Dirchlet problem with…

Characterizes Alexandrov spaces with Cohen-Macaulay actions.

Link concordance equals homotopy for high-dimensional spheres.

We introduce and study algebraic structures underlying 2-dimensional Homotopy Quantum Field Theories (HQFTs) with arbitrary target spaces. These algebraic structures are formalized in the notion of a twisted Frobenius algebra. Our work generalizes results of Brightwell, Turner, and the second author on 2-dimensional HQ…

New method compares geometric and standard cup products.

By studying spaces of flow graphs in a closed oriented manifold, we construct operations on its cohomology, parametrized by the homology of the moduli spaces of compact Riemann surfaces with boundary marked points. We show that the operations satisfy the gluing axiom of an open homological conformal field theory. This …

Overview of 3D TQFTs and 3-manifold invariants.

V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generaliz…

Combines techniques to remove tameness condition in Morse-Smale flows.

This is a survey of motivations, constructions and applications of higher prequantum geometry. In section 1 we highlight the open problem of prequantizing local field theory in a local and gauge invariant way, and we survey how a solution to this problem exists in higher differential geometry. In section 2 we survey ex…

Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.